Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~jsy1001/thesis.pdf Äàòà èçìåíåíèÿ: Tue Jul 20 19:18:57 1999 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:58:44 2012 Êîäèðîâêà: ISO8859-5 Ïîèñêîâûå ñëîâà: planetary alignment

Infrared Imaging with COAST
John Stephen Young

St John's College, Cambridge and Cavendish Astrophysics

A dissertation submitted for the degree of Doctor of Philosophy in the University of Cambridge 26 March 1999



iii

Preface
This dissertation describes work carried out in the Astrophysics Group of the Department of Physics, University of Cambridge, between October 1995 and March 1999. Except where explicit reference is made to the work of others, this dissertation is the result of my own work, and includes nothing which is the outcome of work done in collaboration. No part of this dissertation has been submitted for a degree, diploma, or other qualification at any University. This dissertation does not exceed 60,000 words in length.



v

Acknowledgements
Many people say that this is the only page of a PhD. thesis worth reading. I hope that is not the case here. This is, however, the only page not written in the passive voice, and the only one which might make you smile. Above all, I would like to thank my supervisor, Professor John Baldwin, for always being available to give advice and encouragement, and for assisting with many hours of alignment and even more hours of observing. The shortbread was much appreciated! Many thanks are also due for his reading of this thesis. It has been a pleasure to work with all of the members of the COAST team. None of the work described in this thesis would have been possible without the NICMOS camera built by Martin Beckett. I would like to thank him for taking the time to explain it to me. I am also grateful to Craig Mackay for revealing some of the undocumented features of CCD controllers. I would like to thank Chris Haniff for many useful discussions, and for reading Chapter 8. Peter Warner was always willing to hack the Nord software to accommodate the infrared system, and participated in most of the observing with COAST. David Burns wrote the original versions of most of the analysis software. Donald Wilson contributed his expertise to the artificial star and mirror drives used in the IR beam combiner. Thanks to Richard Wilson for help with the Betelgeuse stuff. Peter Lawson also participated in the WHT run which yielded the Betelgeuse data. Amanda George and Debbie Pearson took the photographs of COAST. All of the above, plus John Rogers, Roger Boysen, and David St-Jacques contributed to the very enjoyable COAST meetings on Friday afternoons. Of those outside Cambridge, I would like to thank Professor Michael Scholz for supplying model Mira CLVs, and Dr. Joseph Lehar for modifying his versions of the Caltech VLBI model-fitting software. My office mates deserve a mention for putting up with me. Keith and Mat should go first, for surviving all 3 1 years of me (and the canteen food). Klaus, Youri, Firouzeh, Dave, Marcel and 2 Anja all helped to brighten things up too. Special thanks are due to my long-time housemates Matt, Emma (honorary housemate), Tim, Steve, and Anna. I also want to thank Dave, Pip and Fred for keeping me in touch with the real world. I owe the largest debt of gratitude to my family. Thanks to Mum, Dad and Suzanne for love and support from afar, and to my grandparents for the same from slightly closer.



To my family



The Cambridge Optical Aperture Synthesis Telescope


x


Contents

1

Introduction and Historical Perspective 1.1 1.2 1.3 1.4 High resolution imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of optical stellar interferometry . . . . . . . . . . . . . . . . . . . . . . History of infrared stellar interferometry . . . . . . . . . . . . . . . . . . . . . . This work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5

2

Infrared Detectors for Interferometry 2.1 2.2 Detector principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.3 Quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Read noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Readout time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dark current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Well capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Array size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 9 9 9 9 10 10 10 10 11 11 11

NICMOS camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 2.3.3 2.3.4 NICMOS device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dewar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CCD controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


xii 2.3.5 2.4

CONTENTS Host computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 12 15 15 15 16 16 16 17 17 18 23 23 24 28 28 30 31 33 36 37 37 42 42 45 46 46

Pixel readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Optical Systems 3.1 Components of an optical/infrared interferometer . . . . . . . . . . . . . . . . . 3.1.1 3.1.2 3.1.3 3.1.4 3.2 Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acquisition/autoguiding . . . . . . . . . . . . . . . . . . . . . . . . . . Path compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Possible correlator designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 3.2.2 All-on-one v. pair-wise beam combination . . . . . . . . . . . . . . . . . Image plane v. pupil plane combination . . . . . . . . . . . . . . . . . .

3.3

Alignment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 General problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared-specific problems . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Alignment solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 Measuring the pupil positions . . . . . . . . . . . . . . . . . . . . . . . Reference directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Focusing the artificial star . . . . . . . . . . . . . . . . . . . . . . . . . Scheme for fine adjustment of the beam directions . . . . . . . . . . . . Results and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6 Optical throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fringe visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Path stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


CONTENTS 4 NICMOS3 Infrared Camera 4.1 4.2 Sampling requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Readout of the NICMOS array . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 4.2.2 4.3 4.4 NICMOS features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extrinsic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii 47 47 49 51 53 55 59

Future improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Stellar Observations, Data Reduction and Analysis 5.1 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 5.2.2 5.2.3 5.3 5.4 5.5 5.6 5.7 Telescope alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finding fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recording fringe data . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 62 62 63 67 70 71 72 75 76 78

Visibility amplitude estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure phase estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capella observations and data reduction . . . . . . . . . . . . . . . . . . . . . . Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Effective temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

The Wavelength-dependent Morphology of Betelgeuse 6.1 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations and data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 6.2.2 6.2.3 Measurements with COAST at 1.3 Åm . . . . . . . . . . . . . . . . . . . Measurements with COAST at 905 nm . . . . . . . . . . . . . . . . . . Measurements with the WHT . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 83 84 84


xiv 6.3

CONTENTS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 6.3.2 6.3.3 6.3.4 6.4 Fourier data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Apparent sizes and limb-darkening . . . . . . . . . . . . . . . . . . . . 85 85 90 90 97

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.4.1 6.4.2 Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Infrared limb-darkening . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Cyclic Variations in the Angular Diameter of Cygni 7.1

109

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.1.1 7.1.2 Mira variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Diameter changes of Miras . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.2

Observations and data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2.1 7.2.2 Measurements with COAST . . . . . . . . . . . . . . . . . . . . . . . . 110 Measurements with the WHT . . . . . . . . . . . . . . . . . . . . . . . 111

7.3 7.4 7.5

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 Mira Variables 8.1 8.2 8.3

121

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Observations and data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3.1 8.3.2 8.3.3 Simple models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Photospheric diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 128


CONTENTS 8.3.4 8.3.5 8.3.6 8.4

xv Variation with phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Effective temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Linear radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.4.1 8.4.2 8.4.3 8.4.4 Asymmetries and limb-darkening . . . . . . . . . . . . . . . . . . . . . 135 Previously-published angular diameters . . . . . . . . . . . . . . . . . . 137 Effective temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Linear radii and pulsation modes . . . . . . . . . . . . . . . . . . . . . . 138

8.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9

Conclusions 9.1

143

The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145


xvi

CONTENTS


List of Figures
2.1 2.2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Circuit for calculation of reset noise . . . . . . . . . . . . . . . . . . . . . . . . DCS circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Image plane beam combiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of a temporal fringe pattern . . . . . . . . . . . . . . . . . . . . . . . . Pupil plane beam combiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power spectrum of three-baseline data . . . . . . . . . . . . . . . . . . . . . . . Layout of the optical components inside the COAST building . . . . . . . . . . . Beam-splitter unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photograph of the IR beam combiner . . . . . . . . . . . . . . . . . . . . . . . . Effect of pupil shift with defocused artificial star . . . . . . . . . . . . . . . . . . Setup for focusing artificial star . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 14 19 20 20 22 25 26 27 29 32 34 35 36 43 44 52 53 54 56

3.10 Centroid coordinate plotted against beam angle . . . . . . . . . . . . . . . . . . 3.11 Angular displacements due to a thin prism . . . . . . . . . . . . . . . . . . . . . 3.12 Non-parallel-sided beam-splitter and compensating plates . . . . . . . . . . . . . 3.13 Power spectra of internal fringes and stellar fringes . . . . . . . . . . . . . . . . 3.14 Transmission curve of J band filter and telluric absorption features . . . . . . . . 4.1 4.2 4.3 4.4 Real-time code in "observe" . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power spectrum of three-baseline data on Capella . . . . . . . . . . . . . . . . . Data stream from fast readout cycle . . . . . . . . . . . . . . . . . . . . . . . . Noise features due to pick-up at specific frequencies . . . . . . . . . . . . . . . .


xviii 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

LIST OF FIGURES Residual noise features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of signal against read number for successive reads with only occasional resets Stellar fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Histogram of visibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative average visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-summed power spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the integration method for visibility amplitude estimation . . . . . uv coverage for Capella observations . . . . . . . . . . . . . . . . . . . . . . . . Image of Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-component model for Capella . . . . . . . . . . . . . . . . . . . . . . . . . Visibility curve and closure phases for Betelgeuse at 1.3 Åm . . . . . . . . . . . . Visibility curves and closure phases for Betelgeuse at 905 nm . . . . . . . . . . . Visibility curve for Betelgeuse at 700 nm . . . . . . . . . . . . . . . . . . . . . . Closure phases for Betelgeuse at 700 nm . . . . . . . . . . . . . . . . . . . . . . COAST infrared image of Betelgeuse . . . . . . . . . . . . . . . . . . . . . . . WHT 700 nm image of Betelgeuse . . . . . . . . . . . . . . . . . . . . . . . . . Visibility functions and intensity profiles for Hestroffer limb-darkened models . . 57 58 63 65 66 68 69 71 74 77 86 88 89 89 92 93 94

Brightness distributions for the best-fit models for Betelgeuse at 700 nm and 905 nm 98 Best-fit two-parameter limb-darkened disk model for Betelgeuse at 1.3 Åm . . . . 101

6.10 Blackbody hotspot model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.11 Synthetic TiO spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.1 7.2 7.3 7.4 8.1 Visibility curve for Cyg at 905 nm . . . . . . . . . . . . . . . . . . . . . . . . 114 Gaussian plus one unresolved feature model for Cyg . . . . . . . . . . . . . . 115 Variation of diameter with pulsation phase for Cyg . . . . . . . . . . . . . . . 117 Light curve for Cyg at 905 nm . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Visibility curves and closure phases for T Cep and Cyg . . . . . . . . . . . . . 125


LIST OF FIGURES 8.2 8.3 8.4 8.5

xix

Visibility curves for R Cas and o Cet . . . . . . . . . . . . . . . . . . . . . . . . 125 J band centre-to-limb intensity profiles for Mira models . . . . . . . . . . . . . . 131 Stellar radius plotted against pulsation period . . . . . . . . . . . . . . . . . . . 139 Stellar radius plotted against pulsation period for Miras from van Belle et al. (1996) 141


xx

LIST OF FIGURES


List of Tables
3.1 3.2 3.3 3.4 5.1 5.2 5.3 6.1 6.2 6.3 7.1 7.2 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Reflectivity of mirror coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . Transmission of optical components . . . . . . . . . . . . . . . . . . . . . . . . Predicted IR throughput of COAST . . . . . . . . . . . . . . . . . . . . . . . . Quantum efficiency of NICMOS device . . . . . . . . . . . . . . . . . . . . . . Two-component model for Capella . . . . . . . . . . . . . . . . . . . . . . . . . Predicted and observed separation and position angle for Capella . . . . . . . . . Synthetic V 39 40 41 41 76 76 78 82 95 96



J colours for G- and K-type giant stars . . . . . . . . . . . . . . .

Log of Betelgeuse observations . . . . . . . . . . . . . . . . . . . . . . . . . . . Best-fit disk plus one unresolved feature models for 905 nm 97/11/21 COAST data Best-fit disk plus one unresolved feature models for 700 nm WHT data . . . . . .

Log of Cyg observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Apparent angular sizes for Cyg . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Log of Mira observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 One-parameter models for Miras . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Adopted uniform disk diameters for calibrator stars . . . . . . . . . . . . . . . . 126 Best-fit disk plus one unresolved feature models . . . . . . . . . . . . . . . . . . 129 Best-fit disk elliptical disk models . . . . . . . . . . . . . . . . . . . . . . . . . 129 Fundamental properties of Mira models . . . . . . . . . . . . . . . . . . . . . . 131 Diameter scaling factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133


xxii 8.8 8.9

LIST OF TABLES Photospheric angular diameters for Miras . . . . . . . . . . . . . . . . . . . . . 134 Mean photospheric angular diameters for Miras . . . . . . . . . . . . . . . . . . 134

8.10 Effective temperatures for Miras . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.11 Linear radii of Miras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136


Chapter 1

High Resolution Infrared Imaging: Introduction and Historical Perspective
1.1 High resolution imaging
escopes is limited by turand with flawless optical by diffraction from the fiby 1 22 D,where

The detail which can be seen by conventional optical and infrared tel bulence in the Earth's atmosphere. In the absence of an atmosphere, components, the imaging performance of a telescope would be limited nite primary mirror. The diffraction limit for a circular aperture is given

is the light wavelength and D is the aperture size, i.e. the limit is inversely proportional to the size of the aperture in wavelengths. Thus the angular resolution of a diffraction-limited 10 m telescope in the visible part of the spectrum ( 500 nm) would be 12 milliseconds of arc. The diffraction-limited resolution of a 10 m telescope is insufficient for many astrophysical problems. Binary systems in which the two constituent stars are close enough to interact would appear as single images, and the disks of all but the nearest, most evolved single stars would remain unresolved. Higher angular resolution is essential for understanding the early and late stages of stellar evolution, and for imaging the inner regions of active galactic nuclei. However, telescopes with large one-piece mirrors are hugely expensive and time-consuming to build. It is difficult to imagine telescopes with mirrors much more than 10 m in diameter being constructed within the next ten years. Radio astronomers ran into this problem much earlier. Because radio wavelengths are one hundred thousand times longer than optical wavelengths, the diffraction-limited resolution of the world's largest radio antennas is still one minute of arc. The solution is to combine the signals from physically separated radio dishes. The van Cittert-Zernicke (VCZ) theorem states that the degree of coherence between the parts of a wavefront received at two separate points from a distant source is the Fourier transform of the brightness distribution of the source. The VCZ theorem tells us that it is not necessary for all parts of a telescope mirror to be present at once. Thus it is possible to use an array of small telescopes to synthesise a much larger telescope. The synthesised telescope


2

CHAPTER 1. INTRODUCTION AND HISTORICAL PERSPECTIVE

will have the angular resolution of a single-mirror telescope the size of the largest separation in the array. This technique, called aperture synthesis, was pioneered at the Mullard Radio Astronomy Observatory (MRAO) in the 1950s and 1960s. The superposition of the parts of a wavefront received at two separated telescopes produces an interference pattern. Hence a telescope which works in this way is called an interferometer. The contrast, or visibility, of the interference pattern (or fringe pattern) is directly related to the degree of coherence of the two wavefronts, and has both an amplitude and a phase. The interference pattern produced by a pair of telescopes is analogous to the pattern of light and dark fringes produced in a Young's slit experiment. A two-element interferometer is sufficient to measure the angular size of an object, but more telescopes are needed for imaging through the atmosphere. The individual telescopes making up an interferometer cannot be arbitrarily large, as all groundbased optical and infrared telescopes (not just the elements of an interferometer) suffer from image degradation caused by the atmosphere, which can drastically reduce the contrast of interference fringes. The image distortions are caused by both spatial and temporal variations of refractive index in the turbulent air above the telescope, called seeing. Hence the atmosphere acts as a constantly changing lens which distorts the image. The refractive index variations cause variations in optical path, and hence distortions of the phase of an incoming light wavefront (e.g. from a star) across an aperture. The phase variations may be characterised by a spatial scale and a timescale. The spatial scale is a length r0 (Fried 1966), equal to the diameter of a circular aperture over which the RMS variation in the phase of a light wavefront is one radian. Thus r0 is roughly the size of the largest telescope which is unaffected by the atmosphere. At visible wavelengths this size is only about 10 cm. The timescale of the refractive index variations can be derived by assuming that an unchanging screen of turbulence is blown across the telescope by the wind. This yields a characteristic time t0 in which the RMS phase variation, relative to that of the undistorted wavefront, is one radian. At visible wavelengths, t0 is typically 5 ms at good sites. This is the longest exposure time which will "freeze" the seeing. If a short exposure (i.e. t0 ) image is made with a telescope whose primary mirror is larger than a few times r0 , the resulting image is a speckle pattern, consisting of an image of the target convolved with a random pattern of bright "speckles". To obtain diffraction-limited resolution for long-exposure images, the optical train of the telescope must include Adaptive Optics (AO), i.e. a system to measure the wavefront distortions introduced by the atmosphere, plus a deformable mirror to correct the wavefronts. The wavefront must be measured and corrected before it has changed significantly, i.e. in a time less than t0 . The number of degrees of freedom needed for the deformable mirror, and hence the number of measurements which must be made in each time interval, increases with the size of the telescope aperture. In order to measure the wavefront distortions when the target is faint, an AO system must observe a nearby bright point source through the same column of atmosphere as the target of interest. There may be no real stars sufficiently close to a particular target to act as references, in which case a guide "star" can be generated artificially using powerful lasers. Even without laser guide stars, AO systems are complicated and expensive. Nevertheless, during the last few years a number of systems have come into routine use at observatories world-wide.


1.2. HISTORY OF OPTICAL STELLAR INTERFEROMETRY

3

The simplest form of AO system is tip-tilt correction, which only corrects the mean tilt of the wavefront across the telescope aperture. This is equivalent to taking out the mean image motion. Tip-tilt correction is particularly useful for optical interferometry, since it allows the aperture size to be increased to 3r0 without seriously reducing the visibility of the interference fringes. Even if the wavefront distortions across all of the telescopes making up an optical interferometer were completely corrected, the visibility phases would contain no information about the astronomical target. This is because the atmosphere randomly perturbs the paths of light rays from the source to the different telescopes in the array, i.e. the atmosphere adds a random phase to the signal from each antenna. However, the sum of the phases around a triangle of three antennas remains unaffected by the atmosphere. This quantity, called the closure phase (Jennison 1958), is vital to provide the phase information needed to reconstruct images. Atmospheric turbulence places strict constraints on optical interferometers. The individual telescopes must not be larger than a few times r0 , so that the images from the telescopes, which are superposed to form a fringe pattern, do not break up into a number of speckles. Measurements of the contrast of the fringe pattern must be made in a time t0 , otherwise the pattern would become blurred. The coherence length must also be longer than the atmospherically-induced optical path fluctuations (otherwise there would not always be fringes), thus a narrow optical bandwidth must be used. All of these constraints restrict optical aperture synthesis to bright targets. Use of the standard statistical theory of atmospheric turbulence (Kolmogorov 1941) leads to the conclusion that r0 , and hence t0 (which is proportional to r0 divided by the wind speed), increase with wavelength as the 6/5 power (Fried 1966). Also the optical path variations caused by the atmosphere are a smaller number of wavelengths in the infrared, and so a wider fractional bandwidth can be used. The combination of these effects is the potential for the signal-to-noise in a fringe measurement to increase as 4 6 , if the optimum telescope size and fractional bandwidth are employed. For this reason, it is preferable to operate an interferometer at infrared, rather than optical, wavelengths. However, early experiments used visible light, as suitable infrared detectors were not available.

1 .2

History of optical stellar interferometry

The use of interferometry to obtain high angular resolution at optical wavelengths was in fact proposed by Fizeau in 1868, decades before interferometry was used in the radio. The first experiment was performed by Stephan in 1874, who failed to resolve any stars with an interferometer Ä constructed by masking all of a large telescope but two small apertures, spaced 50-65 cm apart. Stephan correctly concluded that all stars had angular diameters much smaller than 0.16 seconds Ä of arc. The first astronomical angular diameter measurements were made by Michelson, who measured the diameters of Jupiter's moons in 1891, and the diameter of the red supergiant star Betelgeuse in 1921. The latter measurement required the use of two mirrors at either end of a 20 foot beam mounted on the 100 inch Hooker telescope. These measurements were made by varying the spacing of the two apertures until the observer determined that the interference fringes (watched by eye) had disappeared. In practice the fringes often disappeared because of optical


4

CHAPTER 1. INTRODUCTION AND HISTORICAL PERSPECTIVE

misalignments and bending of the telescope structure, hence the experiment was very difficult to perform. Similar problems affected the dedicated 50 foot interferometer which was subsequently constructed (Pease 1931). Further development in this field was stalled until the 1970s by the lack of detectors suitable for fringe measurement. The entire history of aperture synthesis at optical and infrared wavelengths has been closely linked to the development of required technology in the areas of opto-mechanics, opto-electronics, and computing. The relevant theory and image reconstruction techniques were developed in the 1950s and 1960s for radio interferometry. The early 1980s saw the introduction of infrared detectors which were sufficiently sensitive to detect the small numbers of photons collected by stellar interferometers in the permitted integration times. Because of the potential for better signal-to-noise in the IR compared with visible wavelengths, several of the two-element stellar interferometers built subsequently were designed to operate in the infrared.

1.3

History of infrared stellar interferometry

All of the infrared interferometric telescopes described here had one feature in common. None were designed to combine the light from more than two telescopes simultaneously, and hence they could not measure closure phases or make images. Nevertheless all of the telescopes listed below have been successful at making stellar diameter measurements, which have led to significant astrophysical results. The instruments have also contributed to the understanding of the practice of interferometry at IR wavelengths. `` Ä The first-generation of infrared interferometers included the Interferometre a 2 Telescopes (I2T, Ä Di Benedetto 1985) at the Observatoire de Calern in Provence, and the University of Wyoming's Infrared Michelson Array (IRMA, Benson 1991). Both operated at a wavelength of 2.2 Åm and used single-element InSb detectors (the I2T could also operate in the visible). The Infrared Optical Telescope Array (IOTA, Dyck et al. 1995), with larger telescopes located on a better site in Arizona, is the successor to IRMA, run by a consortium of US and French institutions. IOTA is now capable of operating at wavelengths between 1.0 and 2.4 Åm, thanks to the recent incorporation of a NICMOS3 detector. The most recent stellar interferometer to commence operations is the Palomar Testbed Interferometer (Colavita et al. 1995). This interferometer operates at 2.2 Åm, and was built by the Jet Propulsion Laboratory to experiment with the new technique of dual-star astrometry. Two stars are observed simultaneously in order to accurately measure their relative positions. This method will be used on the forthcoming Keck interferometer to search for planets orbiting nearby stars, by attempting to detect the orbital motion of the parent star. The interferometers mentioned above all detect near-IR photons incoherently, after the light beams from different telescopes have been interfered to form a fringe pattern. Several other experiments have used heterodyne detection techniques (as used by sub-millimetre and radio telescopes) at wavelengths of 10 Åm. Two early experiments (Johnson et al. 1974; Assus et al. 1979) were


1.4. THIS WORK

5

forerunners of the successful Infrared Spatial Interferometer (ISI, Danchi et al. 1988). Heterodyne detection is generally noisier than incoherent detection at shorter wavelengths, and this thesis is only concerned with the 1-3 Åm wavelength range. Because of bright limiting magnitudes, all of the targets observed by optical and near-infrared interferometers thus far have been stars, particularly cool, late-type stars (which are large enough to be resolved on baselines 20 m). The peaks of the (approximately) black-body spectra of stars with effective temperatures 3000 K are at wavelengths near 1 Åm, and so IR interferometers can operate in the band which maximises the detected flux. More importantly, the infrared region (above 1 Åm) is relatively free of spectral lines, so an appropriate band may be used to image the "true" surface of the star. A potential problem with this application is that cool stars are enveloped by dust, but fortunately the optical depth of a 1000 K dust shell is minimised in the near-infrared. At shorter wavelengths there is significant scattering by the dust grains, whereas at wavelengths longer than about 3 Åm, thermal emission from the dust begins to dominate. Hence observations at 5-10 Åm can be used to determine the spatial distribution of the circumstellar dust. Dust also obscures young stellar objects (YSOs) in nearby star-forming regions. Many YSOs should be bright enough to be imaged by future aperture synthesis arrays. The cores of active galactic nuclei are yet more ambitious targets, but the best chance is at IR wavelengths: AGN are typically several magnitudes brighter in the infrared than in the visible. For example, the core of NGC 1068, one of the brightest AGN, has a K band (2.2 Åm) magnitude of 8.0 (McCarthy et al. 1982). The astronomical targets accessible to IR interferometers are discussed in more detail by Dyck and Kibblewhite (1986).

1.4

This work

The first optical/IR aperture synthesis image from separated telescopes was made at visible wavelengths, in 1995 (Baldwin et al. 1996), using the Cambridge Optical Aperture Synthesis Telescope (COAST). COAST is the first optical interferometer specifically designed to produce images, by measuring closure phase as well as visibility amplitude. COAST was designed to work at visible wavelengths. The aim of this project was to add the capability of imaging at near-infrared wavelengths, following on from the work of Beckett (1995). The project was made possible by the arrival of new detector technology, discussed in Chapter 2. This work describes the problems encountered in modifying COAST to operate as an imaging telescope in the near-infrared, and the solutions employed in overcoming these problems. Many of the solutions have more general applications in interferometry. The effectiveness of the solutions is clearly demonstrated by the number of astronomical results obtained with the working COAST IR system. These results, described in Chapters 5-8, include high-quality images of Capella and Betelgeuse, and a large number of diameter measurements of Mira variables. The astrophysical implications of these results, combined with high angular resolution observations at visible wavelengths, are also discussed.


6

CHAPTER 1. INTRODUCTION AND HISTORICAL PERSPECTIVE

Chapter 2 is a brief review of the principles of operation of modern infrared array detectors, required in order to follow the discussion in later chapters. Chapter 3 describes the optical systems of a typical stellar interferometer, with particular emphasis on beam-combining schemes which could be used at infrared wavelengths. The relative merits of the different schemes when a noisy detector is used are discussed. The pupil plane beam combiner chosen for IR operation at COAST is described, as are the fundamental problems in aligning such a system. Possible solutions to these problems are discussed, including the methods finally adopted at COAST. These procedures have been used at COAST to accurately align the four-way IR beam combiner and maintain its alignment throughout the observing season. Chapter 4 discusses the readout mode used for the NICMOS3 camera at COAST to sample pupil plane fringes at up to 2.5 kHz. The functionality of the software used to control the NICMOS array and store fringe data is described. Chapter 5 describes the procedures used when observing with COAST in the infrared, illustrated by observations of the binary star Capella. I discuss the methods for extracting visibility amplitudes and closure phases from the raw data, and subsequently using them to reconstruct an image of the astronomical target. Chapter 6 is concerned with contemporaneous high resolution imaging of the M-type supergiant star Betelgeuse in a number of wavebands from 0.7-1.3 Åm, performed with COAST, and with the William Herschel Telescope by the technique of non-redundant masking. The implications for the possible origin of the "hotspots" frequently detected on Betelgeuse are discussed. Chapter 7 presents the results of a programme to monitor the angular diameter of the Mira variable star Cygni. This programme was carried out with COAST and the William Herschel Telescope, over a period of 17 months. Chapter 8 describes a programme to make near-continuum (1.3 Åm) diameter measurements of a sample of Mira variables with COAST, and the implications of the measurements for the effective temperatures, physical diameters and pulsation modes of the sample stars. Chapter 9 is a summary of the most important ideas from each chapter. Developments planned for the IR instrumentation at COAST are discussed, with suggestions for future astronomical programmes.


Chapter 2

Infrared Detectors for Interferometry
Infrared interferometry is limited by detector performance, so I shall continue by describing the principles of operation of infrared detectors, and how these principles impinge upon the design and performance of real detectors. I will discuss the importance of each aspect of detector performance for stellar interferometry, identifying the limits imposed by the detector on how faint or complex a source can be imaged by aperture synthesis. Finally, I will describe the mechanical and electronic components which make up the NICMOS3 camera system used at COAST, and examine the processes involved in first addressing a pixel of the NICMOS array, then reading the signal stored at that pixel.

2 .1

Detector principles

All IR detectors suitable for use in interferometers operating at wavelengths between one and five microns are based on semiconductors, and detect individual photons directly. An incoming photon with energy greater than the bandgap energy is absorbed within the semiconductor material, exciting a bound electron into the conduction band. The remaining net positive charge behaves as a positively-charged particle, called a hole. Thus a electron-hole pair is created, which can carry a measurable electric current (a photocurrent). Infrared detectors must have low bandgap energies, as photons with wavelengths longer than c hc Ebandgap are not detected. This cut-off wavelength (which is temperature dependent) is about 1.05 Åm for silicon, and so more exotic materials must be used for most of the infrared region. Popular choices include InGaAs, which is sensitive up to 1.7 Åm, InSb (c 5 4 Åm) and HgCdTe, whose composition can be tuned to give a cut-off wavelength in the range 2.4-4.8 Åm (Joyce 1992). The simplest form of semiconductor detector is a photoconductor. An applied electric field causes a photocurrent when incident photons generate electron-hole pairs within the semiconductor. The photocurrent is sensed by measuring the voltage across a series resistor. However, this voltage is sensitive to changes in the external resistance or in the intrinsic resistance of the semiconductor. Instead, most high-performance detectors use the photovoltaic effect: a diode junction is created


8

CHAPTER 2. INFRARED DETECTORS FOR INTERFEROMETRY

within the semiconductor and biased to create an electric field across the junction. The photocurrent discharges the diode capacitor, and so the voltage across the diode at the end of an integration gives the accumulated signal. Arrays of detectors are desirable for efficient direct imaging, but are necessarily much more complex than single pixel detectors. A possibility is to use small arrays of self-contained detectors, each with its own readout circuitry, but these are expensive and the pixels cannot be closely packed. Shared addressing and readout circuitry must be manufactured in silicon, as the technology to make complex integrated circuits in other semiconductors is still immature. Thus highperformance infrared arrays employ hybrid designs, where a detector layer is bonded to a silicon layer containing the readout circuits. A hybrid device may use a Charge-Coupled Device (CCD)1 as a readout circuit. The CCD is used to transfer the charge from the pixel of interest to a shared readout circuit. Alternatively the device may be a direct-readout array, in which each pixel has its own readout circuit in the silicon layer. A set of switches called a multiplexor allows any individual readout circuit to be connected to the output of the device. Both types of hybrid array are difficult to manufacture, as differential thermal expansion between the two layers can cause the array to break apart. Partly as a result of this, such IR arrays are very expensive.

2.2

Performance

As discussed in the previous chapter, stellar interferometers operate in a photon-starved regime. The fringe patterns must be measured in integration times shorter than t0 (which is typically tens of milliseconds in the near-infrared). These constraints determine which of the limitations of IR detectors are important for interferometry.

2.2.1

Quantum efficiency

The Quantum Efficiency (QE) of a detector is the fraction of incident photons (with appropriate wavelengths) which are detected. The QE is the product of the probabilities that: 1. A photon incident on the front surface of the detector will reach the photon-sensitive semiconductor layer. This depends upon the design of the device. 2. A photon will generate an electron-hole pair within the semiconductor. This depends on the composition and temperature of the semiconductor. 3. An electron-hole pair will be detected by the readout circuitry. This depends on the design of the diode junction and readout circuit.
In this context, CCD refers to a readout circuit which operates by charge transfer, which should not be confused with visible wavelength detectors, usually referred to as CCDs, which have the photon-sensitive region and the (chargecoupled) readout circuit combined in a single silicon integrated circuit.
1


2.2. PERFORMANCE

9

Because only a few photons may be incident on the detector during each integration time, the QE of the detector is crucially important. The quantum efficiency and read noise of the detector limit the faintest target for which fringes can be detected by an interferometer. Modern direct-readout arrays typically have quantum efficiencies in the range 20-80%.

2.2.2

Read noise

This is the component of the noise on the signal from a single pixel which is independent of the signal level. Read noise arises from the process by which the charge stored on the diode capacitor is converted to the final signal. If the readout procedure is optimised, the read noise is usually limited by electronic noise in the amplifier(s). The smallest signal which can be detected goes as the square of the read noise.

2.2.3

Readout time

Long readout times are usually needed to take out the effects of high frequency changes in bias voltages etc., and so the time taken to read out a pixel with the minimum read noise can be tens or hundreds of microseconds. If measuring a fringe pattern requires a large number of pixel reads, the total readout time can approach or exceed t0 , leading to blurring of the fringe pattern.

2.2.4

Linearity

Infrared detectors which use the photovoltaic effect are non-linear, i.e. the output signal is not proportional to the number of photons detected. As arriving photons discharge the diode capacitor, the bias across the junction, and hence its capacitance, decreases. Thus the detector is non-linear. This non-linearity is important for all applications of IR detectors, but the relationship between detected photons and output signal is monotonic, and so can be calibrated.

2.2.5

Dark current

Dark current is the signal received which does not originate from the target of interest. It has two components: electron-hole pairs are generated thermally within the semiconductor material, as well as by arriving photons. To minimise the resulting signal, IR detectors are normally operated at low temperatures. There is a further contribution to the dark signal from background radiation incident on the detector. In the infrared, at wavelengths longer than 2 Åm, most of the background is thermal radiation from objects at room temperature within the field of view of the detector. The intrinsic dark current is almost always negligibly small for the short exposures used in interferometry, but the thermal background signal can be a problem if the detector sees large areas at room temperature.


10

CHAPTER 2. INFRARED DETECTORS FOR INTERFEROMETRY

2.2.6

Well capacity

This is the signal needed to saturate the detector, i.e. to discharge the diode capacitor. The well capacity is determined by the bias applied to the diode junction, and is of little importance to interferometric applications in the near-infrared. At longer wavelengths, thermal background radiation can saturate array detectors in the time it takes to read them out.

2.2.7

Array size

Larger format arrays increase the speed at which astronomical surveys can be carried out. Infrared

arrays with 1024 ? 1024 pixels are now available, and 2048 square arrays are promised in the very near future. However, interferometers can work satisfactorily with single element detectors or with small (128 square) arrays, depending upon the fringe measurement scheme chosen.

2.3

NICMOS camera

This thesis is concerned with the operation of COAST in the infrared, using the IR camera system built by Martin Beckett. I will now describe the components of this system. The processes of addressing and reading a pixel of the array are explained in the following section. For more details the reader is referred to Beckett (1995).

2.3.1

NICMOS device

The camera uses a NICMOS3 detector chip, manufactured by Rockwell International. The device was designed for use in the Hubble Space Telescope, hence its acronym, which stands for NearInfrared Camera and Multi-Object Spectrograph. It is a hybrid direct-readout array with 256 ? 256 40 Åm square pixels, although the array is in fact operated as four 128 square quadrants. The detector material is HgCdTe, with a cut-off wavelength of 2.4 Åm. The quantum efficiency of the NICMOS3 is 50% for wavelengths from 1.0 Åm to the cut-off wavelength. The device in use at COAST is an engineering grade chip, which has a large number of inoperative pixels. Each pixel has a detector diode within the HgCdTe layer, which is electrically connected to its own unit cell in the silicon layer by indium bonds. The unit cell circuit contains a Field-Effect Transistor (FET) with very high gate resistance, so that the voltage across the diode at the end of an integration can be read without any of the charge leaking away. This property of non-destructive reads can be used to reduce the effective read noise, by making multiple reads of the same signal level. The silicon layer also contains the multiplexor, a matrix of switches which can connect any pixel in the array to an output amplifier, which provides the final signal from the detector chip.


2.3. NICMOS CAMERA

11

2.3.2

Dewar

The NICMOS device is mounted in an standard Oxford Instruments liquid-nitrogen-cooled dewar, which has two functions:

? ?

Cool the NICMOS device itself, to minimise the intrinsic dark current. Ensure that the photo-sensitive parts of the device see only cold surfaces in addition to the desired input light beam, to minimise the thermal background signal.

The second requirement includes cooling of the interference filter used to select the spectral band for observing.

2.3.3

Interface box

Cables link the NICMOS device inside the dewar to an external electronic circuit, designed to interface the chip to a standard controller (Astromed 3200) intended for visible-wavelength CCD cameras. The interface box has the following functions:

? ? ?

Select a particular quadrant of the NICMOS device, as the CCD controller can only handle one at a time. In conjunction with the CCD controller, implement Double-Correlated Sampling (DCS) to reduce the read noise. The operation of DCS is described in the next section. Incorporate a source follower based upon an n-channel FET, for compatibility with the controller.

2.3.4

CCD controller

Operating a direct-readout array is much simpler than controlling a CCD, as only very simple analogue circuitry is required -- the minimum is just that needed to digitise the final signal. The controller must also generate digital signals to control pixel addressing and readout within the IR device. The controller has two components, a Digital Drive Electronics (DDE) unit, controlled by a simple RISC (Reduced Instruction Set Computer) microprocessor on an expansion card inside a Personal Computer (PC). The two components communicate via a 16-bit RS-422 bus.

2.3.5

Host computer

The IBM-compatible personal computer's purpose is to load and run programs on the RISC processor, in order to operate the NICMOS array. The PC provides a simple user interface, and


12

CHAPTER 2. INFRARED DETECTORS FOR INTERFEROMETRY

R

C

VN

Figure 2.1: Circuit for calculation of reset noise. See text. is responsible for storing image data or for transferring it to a Norsk Data mini-computer by a custom parallel link, in real-time (i.e. while the camera is being operated).

2.4

Pixel readout

A pixel is addressed by clocking two shift registers in the multiplexor, using digital signals from the CCD controller. It is not possible to address a particular pixel without first skipping the preceding rows, then the preceding columns on the desired row. It is the controller in the COAST camera system which limits the rate at which rows and columns can be skipped. To reset a pixel, it is connected to a bias voltage (supplied by the CCD controller) through the multiplexor, thus charging up the diode capacitor. During an integration, photons are incident on the chip, and the resulting photoelectrons discharge the capacitor. At the end of the integration, the voltage remaining across the capacitor must be measured in order to read the final signal level. When the reset switch is released after the reset there is an uncertainty of

ä

kT C in the charge on

the capacitor, where k is Boltzmann's constant, T is the temperature of the NICMOS device, and C is the capacitance of the diode junction. This uncertainty leads to kT C, or reset, noise. Reset noise is caused by thermal motions of charge carriers in a circuit with finite resistance. Consider the circuit in Figure 2.1 (this argument is taken from Rieke 1994). C and R are equal to the capacitance and resistance of the diode junction respectively. This circuit has one degree of freedom, the noise voltage VN , which has an associated thermal energy of 1 kT when in thermodynamic equilibrium. 2 The energy stored in the capacitor is 1 CV 2 , hence 2 1 ? 2? C VN 2 The charge on the capacitor is given by Q
?

1 kT 2

(2.1)

CV , so Q
2 N

?

kT C

(2.2)

In the NICMOS device the reset noise is about 100 electrons RMS, which would be the major noise contribution if not corrected. Fortunately, the correction is easily made, by measuring the reset


2.4. PIXEL READOUT

13

level immediately after the reset, before the integration begins. The drawbacks of this technique are that twice as much time is spent reading each pixel, and that the effective read noise for the ä differential measurement is a factor of 2 greater than that for a single read. The controller contains dedicated electronics for removing reset noise when it is used with a CCD camera. This technique of Double-Correlated Sampling (DCS) cannot be used to remove reset noise with a direct readout array, but is nevertheless useful for reducing the noise from other sources which is associated with any single read of a pixel. The DCS circuits are shown in Figure 2.2. The interface box incorporates a potential divider, driven from the same bias voltage supplied to the NICMOS device, to provide a reference voltage. The interface also contains a low noise FET switch which can connect either the NICMOS or the reference to the controller. The reference voltage, or "dummy pixel" is set to approximately the level of a real pixel immediately after a reset. The controller contains positive and negative unity gain amplifiers, an integrator, and an analogue-to-digital converter (ADC). During the first half of the DCS cycle, the reference is connected, the positive amplifier is selected, and the signal is integrated for a fixed time (5-50 Ås). For the second half of the cycle, the signal from the NICMOS is integrated for the same time, but through the negative amplifier. Finally the voltage at the integrator output is digitised. Thus a differential measurement of the real and dummy pixels is made (this differential measurement should not be confused with that used to remove reset noise, which involves the entire DCS procedure twice, once straight after a reset, then again after the desired integration time). The final signal is in fact proportional to dummy minus real, to ensure that the result increases with decreasing pixel charge (recall that arriving photons discharge the diode capacitor). The integration procedure averages out any high frequency noise, and since the reference is supplied by the bias voltage used for the real pixels, slow drifts in the bias supply are cancelled out by the differential measurement. Using the DCS procedure with 20 Ås integration times, and measuring the reset level to remove kT C noise, the COAST NICMOS system has a typical read noise of 23 electrons (Beckett 1995).


14

CHAPTER 2. INFRARED DETECTORS FOR INTERFEROMETRY

NICMOS
+V

Interface

Controller

x(+1) Dummy

ADC
NICMOS Output NICMOS x(-1)

0V

Figure 2.2: DCS circuitry. The diagram is from Beckett (1995).


Chapter 3

Optical Systems
This chapter describes the optical components of COAST, especially those specific to infrared operation. The difficulties associated with aligning these many components are discussed. An alignment procedure which has been used successfully at COAST is presented. After alignment by this method, the beam combiner can produce fringe patterns with very high visibilities. Various aspects of the performance of the beam combiner are discussed, including optical throughput, visibility losses, and closure phase errors.

3.1

Components of an optical/infrared interferometer

I shall now briefly describe the optical components of a generic visible/infrared interferometer and their functions, illustrated by the example of COAST. For more information on the design of COAST, see Baldwin et al. (1994).

3.1.1

Telescopes

The purpose of the telescopes is to collect light from the target of interest, and produce smalldiameter light beams which can be steered into the other components of the interferometer. The part of the telescope mirror which is actually used must be 3r0 in diameter, so that the resulting image is a single speckle. Since June 1998, COAST has had five telescopes, which can be arranged in many different configurations, on foundations along the arms of a Y. The maximum baseline which will fit on the site is 100 metres. Each telescope is a fixed 40 cm Cassegrain design, illuminated by a steerable flat mirror which tracks the astronomical target. The telescope produces a 25 mm diameter collimated light beam, which is steered into an optics laboratory through alloy pipes.


16

CHAPTER 3. OPTICAL SYSTEMS

3.1.2

Acquisition/autoguiding

The field of view of the telescopes, looking through the long light pipes from inside the optics building, is just a few seconds of arc. The size of this field is comparable to the pointing accuracy of the telescopes, and so each telescope incorporates a wide-field video camera for acquisition purposes. For aperture diameters comparable to or larger than r0 , the fringe contrast will be drastically reduced by wavefront distortions caused by the atmosphere above the telescopes, unless the mean tilt of the wavefront is removed (Tango and Twiss 1980). For this purpose, each telescope also incorporates a piezo-actuated fast steering mirror, driven by error signals from a CCD camera operating at frame rates of up to 250 Hz, which is located inside the optics building. This CCD system is also used for acquisition.

3.1.3

Path compensation

To form interference fringes in white light, the light beams which are combined must have travelled equal paths from the source. In a stellar interferometer, the relative paths for the beams from the different telescopes change continuously as the target moves across the sky. The interferometer must incorporate a path compensation system to take out these variations. At COAST, path compensation is performed by reflecting the beams from movable mirrors mounted on trolleys, which run on precisely-aligned rail tracks. The position of the mirrors on each trolley are continuously monitored by a laser interferometer, and kept at the desired position by a servo-controlled loudspeaker voice coil. The trolley is driven along the track to keep the voice coil displacement small.

3.1.4

Correlator

The correlator consists of a beam combiner and optical or infrared detector(s). The beam combiner is an optical system designed to form interference pattern(s) which can be measured by the chosen detector system. The correlator must be designed to allow the visibility amplitudes and phases on all of the interferometer baselines to be measured, with the capability of measuring the phase on at least three baselines simultaneously, in order to measure a closure phase. All of the components of the interferometer, except the correlator, will serve for observations at both visible and near-infrared wavelengths, providing that the effective telescope aperture is reduced for operation at the shorter wavelengths. However, an interferometer optimised for a narrower spectral region would probably use different coatings on its optical surfaces to those employed in a general-purpose design. COAST has two beam combiners, one which operates at visible wavelengths (650-950 nm), and an alternative combiner, optimised for infrared wavelengths, which was added later. The IR beam


3.2. POSSIBLE CORRELATOR DESIGNS combiner is the subject of the remainder of this chapter.

17

Selection of one of the two beam combiners is achieved by means of four dichroic mirrors mounted on a kinematic slide. These can be slid into position to intercept the light beams from the four telescopes, after path compensation, in order to reflect infrared light into the IR beam combiner. A fraction of the visible wavelength light is transmitted through the dichroic mirrors. The dichroics may be removed so that all wavelengths propagate in this direction. These light beams meet partially-aluminised glass plates which reflect the central portion of the beams into the visible beam combiner (recall that only a fraction of the telescope aperture may be used at optical wavelengths). The outer parts of the pupil are directed onto the autoguider CCD camera.

3.2
3.2.1

Possible correlator designs
All-on-one v. pair-wise beam combination

Consider a beam combiner which has M input light beams from M telescopes. Two extreme possibilities for the design of the beam combiner are as follows. The M beams may all be mixed

together to form a single fringe pattern, containing M ÄM 1Å 2 sets of fringes. Each set must have a unique spatial or temporal frequency, so that the fringes on different baselines may be separated. This is all-on-one beam-combination. Alternatively, M ÄM 1Å 2 separate fringe patterns may be formed, each from a single pair of beams. These are not the only choices of beam combiner design: hybrid schemes, such as that used at the Navy Prototype Optical Interferometer (Mozurkewich 1994) can be envisaged, where more than two but fewer than M beams go into each separate fringe pattern. In the case where all M ÄM 1Å 2 baselines are measured simultaneously, Buscher (1988) showed that, with a noiseless detector, the all-on-one scheme gives the best signal-to-noise for visibility amplitude measurement at all light levels. Is this also true with a noisy detector? The signal-tonoise ratio for visibility measurement from a generic fringe pattern is given by SNR V 2N 2N 3 V
2 2Ç

ä

N

2Ç

2n2 4

(3.1)

where V is the fringe visibility, N is the number of photons detected in the pattern, n is the number of detector reads made and is the noise for each read (Nightingale 1991). If M beams are combined to form a single pattern, the visibility measured on one of the baselines is inversely proportional to the number of beams, so V V0 M , and the number of photons in the pattern is proportional to the number of beams i.e. N MN0 . If the beams are combined pair-wise to form M ÄM 1Å 2 fringe patterns, each made up of two beams, the visibility in each pattern will be V0 2. Each beam must be split M 1 ways, so N 2N0 ÄM 1Å. At high light levels, Equation 3.1 simplifies to SNR V N 2. Substituting the above expressions for N and V we find that the SNR for all-on-one beam combination is SNRa V0 N0 ä 2M
ä

ä

(3.2)


18 whereas that for pair-wise combination is SNRp

CHAPTER 3. OPTICAL SYSTEMS

V0 N0 ä 2 M 1 V ä N2 2n2
2 2

ä

(3.3)

In the low light-level extreme, when Equation 3.1 simplifies to SNR (3.4)

the results for all-on-one and pair-wise beam combination are SNRa and SNRp
Ä

V ä0

2 N0 2na 2

(3.5)

M



2 V02 N0 ä2 1Å2 2n p

(3.6)

If we assume that the highest frequency fringes in each pattern have four samples per fringe, n
a

2M ÄM



1Å and n

p

4. Hence in the low light-level limit, SNRa SNRp 2 ÄM



1Å M

(3.7)

So the all-one-one scheme is better for visibility amplitude measurement with a noisy detector at all light levels. The advantage of the all-on-one scheme is greatest at low photon rates. In this limit the read noise imposes a large penalty for splitting each beam. As pointed out by Buscher, and by Mozurkewich, the all-on-one scheme also avoids the two principal disadvantages of the pair-wise scheme. Firstly, a pair-wise beam combiner is inevitably very large, with many optical components. Secondly, a small change in the optical path in one sub-combiner (which combines one pair of beams) of the pair-wise design will add a systematic error to the closure phases involving that baseline. Calibration observations or laser metrology are needed to remove the bias. On the other hand, the all-on-one design is prone to cross-talk between the baselines, which can limit the accuracy of the closure phases. Also, if fringes are encoded temporally (see next section), the detector must be read out very fast to measure more than a few baselines at once. COAST was designed to measure a maximum of six baselines simultaneously (from four telescopes). An all-on-one design is clearly better for this application.

3.2.2

Image plane v. pupil plane combination

A beam combiner may produce a fringe pattern in the image plane, or in the pupil plane. These terms can be explained as follows:

Image plane The beams are simply imaged together e.g. by a lens, to form a spatial fringe pattern on an array detector (Figure 3.1). The resulting image will be an Airy pattern crossed by fringes. A unique


3.2. POSSIBLE CORRELATOR DESIGNS

19

2D

1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000

3D

D

1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 11111111111111 00000000000000 Detector 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 1111111111111111111111111111 0000000000000000000000000000 11111111111111 00000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000 1111111111111111111111111111 0000000000000000000000000000

Figure 3.1: Simple scheme for making an image plane fringe pattern. Four pathcompensated beams are arranged in a line with non-redundant spacings, and focused onto an array detector by a lens. spatial frequency for each baseline can be obtained by arranging the input beams in a line with nonredundant spacings, as in Figure 3.1, or by using a non-redundant two-dimensional arrangement.

Pupil plane Two beams are superposed by matching their positions and directions. The optical path of one beam is changed, in order to scan through the white-light fringe position. The resulting fringe pattern is the intensity of a combined beam plotted against time (Figure 3.2). More beams can be superposed in the same way, and if their paths are scanned at suitable rates, the extra sets of fringes will have different temporal frequencies. In practice, beam-splitters (partially reflective mirrors) are used to superpose the light beams. A schematic of a pupil plane beam combiner is shown in Figure 3.3. This beam combiner accepts four input beams, one from each of four telescopes, and gives four output beams. Each output beam contains equal amounts of light from the four telescopes. The system combines beams in pairs: beam 1 is combined with beam 2 at one beam-splitter, beam 3 is mixed with beam 4 at a second beam-splitter, then the mixture of beams 1 and 2 is combined with the mixture of beams 3 and 4 at each of two further beam-splitters. This scheme is readily extended to larger numbers of beams.

Fundamental limits Buscher (1988) showed that the image plane and pupil plane schemes are equivalent for signal-tonoise calculations, if the only noise sources are atmospheric phase perturbations and photon noise, and if the temporal fringe pattern is scanned very fast. This led to the conclusion that the choice of scheme depended solely upon which was easier to implement in practice. The results of Buscher lead to a different conclusion if the effect of read noise is considered. In this case extra reads carry a significant noise penalty, and so limit the faintest source which can be observed.


20

CHAPTER 3. OPTICAL SYSTEMS

Figure 3.2: Example of a temporal fringe pattern. The upper plot shows fringes from the COAST IR system, obtained using an internal light source. The horizontal axis is time in milliseconds, and the vertical axis is proportional to the number of photons detected in each one millisecond integration. In the time interval plotted, the path delay (shown in the lower graph with the same time axis) was swept through the white-light fringe position four times. As the path delay exceeds the coherence length of the light, the fringes disappear.

1+2

3+4

1+2

3+4

mirror

beam-splitter 1 2 3 4

Figure 3.3: Pupil plane beam combiner. The combiner takes in four input beams (labelled 1-4), at the bottom of the diagram, and produces four output beams at the top. Each output beam contains equal amounts of the four input beams. The small rectangles are mirrors with 50% reflectivity.


3.2. POSSIBLE CORRELATOR DESIGNS A minimum of 10M ÄM

21

from M beams (i.e. with M ÄM 1Å 2 baselines). If the minimum beam separation is twice the beam diameter (the use of smaller separations will result in aliasing), there will be five of the lowest frequency fringes across the central lobe of the Airy pattern. The other baselines are assumed to be at integer multiples of this spatial frequency. The finest fringes have four pixels per fringe (coarser sampling will cause significant visibility loss). A further assumption is that the image can be compressed into a single row of pixels. If the detector is a CCD camera, the compression may be done by on-chip binning: the charge from a sub-array (in this case a line of pixels) is combined before the readout process, so that the read noise is only superimposed on the total signal. If on-chip binning is not possible, the compression can be done with a cylindrical lens. The longest permitted exposure time between readouts is approximately equal to t0 . In the case of the temporal fringe pattern described above, there must be at least 4 ? M ÄM 1Å 2 samples during each period of the slowest fringes. To give the total number of detector reads this number must be multiplied by the number of output beams ( M ) from the pupil plane beam combiner. There is a further problem: the atmosphere continually changes the relative paths of the different beams, and to first order this corresponds to a change in fringe frequency. To avoid significant leakage of fringe power from one baseline to the baselines closest in frequency, the time interval taken to scan through one lowest-frequency fringe should be much shorter than t0 (Buscher 1988). This result does not apply to an image plane fringe pattern, where temporal path variations lead to changes in fringe phase, but not fringe frequency (the sets of fringes are shifted relative to each other, but the fringe spacings do not change). The pupil plane fringe frequencies must also be well above the highest scintillation frequency. Scintillation is the intensity fluctuations caused by the temporal refractive index variations in the atmosphere (stars "twinkle" when observed with the naked eye because of scintillation). The effects of leakage and scintillation are illustrated by the power spectrum of some pupil plane fringes from COAST, shown in Figure 3.4.



1Å pixel reads are needed to sample an image plane fringe pattern formed

If the pupil plane fringe period is taken to be t0 5, compared with an integration time of t0 for the image plane fringe pattern, there will be M times more detector reads (M outputs ?

2M ÄM 1Å reads ? 5 10M 2 ÄM 1Å, compared with 10M ÄM 1Å) every t0 if beam combination is done in the pupil plane, causing a significant read noise penalty.

Practical problems Designing an image plane system which produces the optimum fringe pattern over a wide range of possible operating wavelengths is very difficult (Beckett 1995). The fundamental problem is that the size of the image on the detector is proportional to the wavelength. Hence the minimum number of pixel reads can only be achieved at one particular wavelength. Also, if the detector is a direct-readout array, on-chip binning cannot be used to compress the fringe pattern in the direction parallel to the fringes, and so this must be done with optics. It is likely that at least several rows of pixels would have to be read out. Together, these effects mean that at least four


22

CHAPTER 3. OPTICAL SYSTEMS

Figure 3.4: Power spectrum of a three-baseline temporal fringe pattern. The data was taken with COAST at visible wavelengths on a real star. The spectrum contains fringe peaks at 220, 440 and 660 Hz, and a further peak due to scintillation below 100 Hz, all superposed on a "white" background of photon noise. The natural width of each fringe peak is inversely proportional to the time taken to sweep through the fringe packet. The number of fringes in the packet is the same in each case (set by the fractional bandwidth, 5% for these data) and so higher frequency fringe peaks are wider. The fringe peaks are further broadened by optical path variations due to the atmosphere, as described in the text. times more pixels than the theoretical minimum must be read after each exposure, which cancels out the factor calculated above if four beams are to be combined at once. Nevertheless, an image plane system is likely to have lower light losses than a pupil plane beam combiner, as the number of optical components is smaller. It may also be easier to align, although experience might teach us otherwise!

Conclusion A four-way pupil plane beam combiner was chosen for use in the infrared at COAST. Four pixels of the NICMOS3 array are used as single element detectors, to measure the intensities of the four output beams. This choice was motivated by the lack of on-chip binning in the NICMOS device, and the consequent difficulty of building an optimised, achromatic image plane combiner. The existing visible-wavelength beam combiner was a pupil plane combiner, so the possibility of using existing control and data analysis software, together with the value placed on previous experience with this design, swayed the decision in favour of a pupil plane beam combiner.


3.3. ALIGNMENT PROBLEMS Fringe tracking

23

It is interesting to consider which system would be better for visibility and closure phase measurement in a fringe-tracking interferometer i.e. one in which the optical path variations introduced by the atmosphere are measured and actively corrected, using a servo system. In the design considered here, some light is split off after path compensation and used to measure and correct the residual optical path variations caused by the atmosphere (the design of the beam combiner used in the fringe tracker itself is beyond the scope of this work). The fringe tracker could be used for visibility and closure phase measurement, but if science observations are needed in a different waveband from the optimal fringe tracking band, the system outlined here would give better fringe tracking performance. The fringe pattern is effectively frozen in time while the fringe tracker is operating. An image plane system may use exposures of several minutes between readouts (the limit is set by the effect of earth rotation on the measured visibilities and closure phases). In the pupil plane case, the problem of fringe power leakage between baselines goes away. However, arbitrary scan times still cannot be used, because of the effect of scintillation. The beams from any two telescopes scintillate independently, which leads to a time-varying reduction in the fringe visibility. Both image plane and pupil plane beam combination systems are affected. The effect is small however: if the intensity of the two beams differs by a factor of 3, the instantaneous visibility is reduced by 13%. The average visibility is only slightly reduced. Scintillation causes a further problem if a pupil plane beam combiner is used. The time-variation of the detected intensity caused by scanning through the fringe pattern must be separated from that due to scintillation in individual image plane case, because the major fringe pattern, which does not affect plane fringes are separated from the beams. This aspect of scintillation is not a problem in the effect of scintillation is to change the intensity of the whole the inferred fringe visibility. Without fringe tracking, pupil scintillation by scanning much faster than the scintillation

timescale. Fringe tracking opens up the possibility of scanning much slower (probably in discrete steps) than the intensity variations due to scintillation, so that they are averaged out. In a fringe-tracking interferometer, the pupil plane and image plane systems are nearly equivalent, even if the detector has significant read noise. The splitting up of the input light in a pupil plane beam combiner must be offset against the difficulty of building an optimised image plane combiner.

3 .3
3.3.1

Alignment problems
General problems

A pupil plane beam combiner must precisely superpose the input beams from the telescopes. For high visibility fringes the beam angles must be matched to 4 divided by the pupil diameter. At


24

CHAPTER 3. OPTICAL SYSTEMS

COAST in the infrared, this angle is 2.7 seconds of arc (for a wavelength of 1.3 Åm and a 25 mm pupil). The pupil positions must be matched to a few per-cent of the pupil diameter. In the case of a four-way pupil plane beam combiner, these requirements lead to 32 variables: the directions and positions (in a plane perpendicular to the beam direction) of four beams (beams 1-4 in Figure 3.3) at each of two independent outputs (each of the upper two beam-splitters in Figure 3.3 produces two combined beams which are not independent of each other). The number of degrees of freedom in the combiner depends upon its precise design, but is always much greater than the number of variables -- the system is complicated and under-constrained. The layout of the optical components within the COAST building is shown in Figure 3.5. The IR beam combiner (Figure 3.7 is a photograph of the beam combiner) is on its own optical table at the top right of the diagram. The visible beam combiner is on the right side of the left-hand table. Both designs have the same logic as the beam combiner in Figure 3.3, but the simple layout has been sheared to reduce the angles of incidence on the beam-splitters. The IR beam combiner has 64 degrees of freedom (not counting those associated with the mirrors which direct the combined beams onto the detector): the position on the table and angles about two axes of a total of 16 mirrors and beam-splitters. There are 48 degrees of freedom in the visible beam combiner. Despite this complexity, the visible beam combiner can be aligned using measurements of the beam positions within the combiner and of the beam directions on exit. The infrared beam combiner is very similar to the visible-wavelength combiner, but despite this similarity, Martin Beckett found that the infrared combiner is much more difficult to align than its visible counterpart (Beckett 1995).

3.3.2

Infrared-specific problems

The beam combiner contains beam-splitter units (Figure 3.6) consisting of a glass plate with a 50% reflective dielectric multi-layer coating on one face. This beam-splitting plate is used with a compensating plate of the same thickness, which ensures that all routes through the system pass through the same thickness of glass. The three surfaces which are not intended to be reflective are anti-reflection coated. A 10 degree angle of incidence is used so that the two principal planes of polarisation behave in the same way. Aligning the IR combiner is difficult because it is impossible to make dielectric multi-layer coatings which are 50% reflective from visible wavelengths out to the cut-off wavelength of the NICMOS detector at 2.4 Åm. It is similarly impossible to make anti-reflection coatings which work over such a wide band. Thus all four faces of a beam-splitter unit optimised for the infrared can have comparable reflectivity at visible wavelengths. The beam-splitter and compensating plates used in the IR beam combiner were manufactured by Comar Instruments. The manufacturer's curves show that the beam-splitting surfaces have

50 ? 6% reflectivity over the wavelength range 1.02-2.32 Åm, and that the anti-reflection coated surfaces are 99 ? 1% transmissive from 1.04-2.35 Åm. However, the reflectivities of the two types of surface vary by 50% at wavelengths below 1 Åm, and change very rapidly with wavelength.


3.3. ALIGNMENT PROBLEMS

IR camera

Acquisition /autoguiding

Visible beamcombining
Beam-splitters APD detectors

Multi-channel Spectrometer
CCD

Infrared beam-combining

Visible/IR selection: removable dichroics

Incoming beams after path compensation

Incoming beams from telescopes (150mm above)

Mirrors reflect central part of pupil

Path compensation
Metrology system

Figure 3.5: Layout of the optical components inside the COAST building

25


26
Beam-splitter

CHAPTER 3. OPTICAL SYSTEMS
Compensator

1+2

1+2

1

2

50% reflective surface

Figure 3.6: Beam-splitter unit. The unit combines two input beams (1 and 2), to give two combined beams (labelled 1 Ç 2), each of which contains equal fractions of 1 and 2. The 50% reflective dielectric multi-layer coating is on the right-hand surface of the beam-splitter plate. The angles of incidence of beams 1 and 2 on this surface are 10 degrees, to avoid polarisation effects. Note the sideways shifts of the beams, which must be taken into account when aligning the beam combiner. A beam combiner which contains four such beam-splitter units cannot be aligned using visible light, as it is very easy to make the mistake of co-aligning beams which have been reflected from the wrong faces. Such a beam combiner must be aligned in the infrared. This requirement makes the alignment more difficult. Most importantly, it becomes very difficult to measure the position of the pupil at many positions within the beam combiner. The pupil position of a particular input beam at any point within the combiner may be measured as follows. An illuminated target is placed at the desired point, and imaged through a theodolite which is pointed in a fixed direction and focused on the target. The theodolite is in front of the beam combiner, at the point where the beam from the relevant telescope enters the optics building. This procedure can only be used for the infrared combiner if the imaging is performed in the infrared, which would require a dedicated IR camera. This was judged to be prohibitively expensive in 1995, so we elected to perform the alignment using only the NICMOS3 camera intended for fringe measurement. A single reference source is used for alignment of the IR combiner. This "artificial star" consists of an illuminated pinhole behind a (reasonably) achromatic lens, and injects a collimated beam of white light into the beam combiner at one of the outputs (kinematic feed mirrors allow a choice of two outputs for this purpose). The visible beam combiner incorporates a similar artificial star. In both cases, the beam combiner works in reverse to produce light at each of the inputs. These beams travel through the path-compensation system to the point where the light beams from the telescopes would enter the building, where flat mirrors are placed to reflect the beams back into the combiner (i.e. the system is auto-collimated), where they are recombined and then focused onto the detector.


3.3. ALIGNMENT PROBLEMS

Figure 3.7: Photograph of the infrared beam combiner at COAST. The beam-splitter units are in the foreground. Some of the mirrors within the beam combiner can be seen behind the beam-splitters. The blue-looking plates in the background, at the end of the tubular trolley rails, are the infrared-selecting dichroics. 27


28

CHAPTER 3. OPTICAL SYSTEMS

If three of the four input beams are blocked, the position of the image from the remaining beam gives the direction of that beam1 . However, the double pass through the beam combiner makes the logic of the alignment procedure more complicated. Also, the path length from the reference source to the detector is inevitably long (20 metres or more). In an auto-collimated optical system, the overall pupil shift through the system is zero, as the outgoing and incoming light rays are coincident. However, the artificial star is not necessarily auto-collimated if seen at a different beam combiner output from the one where it was originally injected. Hence for some routes through the system, the long optical path may lead to a significant pupil shift (a 10 arc-sec misalignment will cause a 1 mm pupil shift over a 20 m path), which will be confused with a beam tilt unless the the artificial star is well focused at the wavelength of observation. To see how a pupil shift p can cause a displacement of the image on the detector, examine the diagram in Figure 3.8. The dimensions have been wildly distorted to illustrate the effect. The pupil is defined at the camera, as would be the case when observing a real star. x s p u by similar triangles, and hence the shift is given by x where total o as the sp D

vÌ (which is negative) gives the position of the virtual image of the pinhole, and D is the ptical path from the artificial star lens to the camera lens. As vÌ tends to minus infinity (i.e. beam becomes better collimated), the image shift tends to zero.



vÌ

(3.8)

After focusing the artificial star at visible wavelengths, I measured vÌ 13 9m at 1 3 Åm by the technique outlined in Section 3.4.3. This amount of defocus would give a displacement on the chip of 4.2 Åm for a 1 mm pupil shift (assuming D 20 m), equivalent to an angle shift of 6 arc-sec. Clearly the lens is not sufficiently achromatic over this wavelength range, and so the artificial star must be focused in the infrared.

3 .4
3.4.1

Alignment solutions
Measuring the pupil positions

The requirement on the accuracy of the final pupil positions is not as stringent as that on the beam angles. Both the visible and IR combiners use the same mounts for mirrors and beam-splitters. The angles of the components in these mounts have proved to be stable to a second of arc over long timescales. Since tilts of up to 10 arc-sec do not lead to significant pupil shifts over the path from one end of the beam combiner to the other, the pupil positions do not drift significantly. Hence the solution adopted for making the pupil positions correct is to kinematically replace the
In the infrared beam combiner, the NICMOS detector can be used to measure the image position. The avalanche photo-diodes used in the visible system are single element detectors, and so the visible beam combiner incorporates a video camera for this purpose.
1


3.4. ALIGNMENT SOLUTIONS

detector Artificial star Camera

-v' x p

pinhole D u

s

Figure 3.8: Effect of pupil shift with defocused artificial star. A small pupil is defined at the camera lens.

29


30

CHAPTER 3. OPTICAL SYSTEMS

beam-splitter and compensating plates with a set optimised for visible wavelengths. The beam combiner can then be aligned using visible light, with the procedures developed for the visible beam combiner. The pupil position at any point in the system can be measured by placing a target there and looking at the target through a theodolite focused on that point. However, the beam-splitter and compensating plates cannot be made perfectly parallel-sided. The scatter in the wedge angles of our plates is three seconds of arc. There are eight plates in the beamcombiner, and so exchanging the visible plates for the infrared ones introduces misalignments greater than the required angular accuracy of 2.7 arc-sec, although the pupil positions are not changed significantly for a single pass through the beam combiner. We measure the wedge angles and choose IR replacement plates and their orientations in order to minimise the changes, but significant misalignments will inevitably remain.

3.4.2

Reference directions

Both the visible and infrared beam combiners must superpose the same beams from the telescopes, therefore the alignment of the two systems must be matched. However, it is difficult to tie the alignment of the two beam combiners together, because the "visible" artificial star can't be seen with the infrared camera2 , and the "infrared" artificial star is not very useful when observed in the visible, due to multiple reflections from each beam-splitter unit in the IR combiner at visible wavelengths. This is one reason for swapping the beam-splitters optimised for the infrared for ones which work at visible wavelengths: both artificial stars can then be seen with a theodolite located at the entrance to the optics building. The angles of the visible artificial star, as seen from the theodolite bench, are used as a secondary reference (the primary reference is a target which can be mounted at fixed positions on the trolley rails). The auto-collimating flats are always adjusted so that the four beams from the visible artificial star are co-aligned at one particular output of the visible beam combiner. We assume that the angle of the visible artificial star off of each of the flats is the same as the angle of a beam from a real star. Hence we make the angle at which the IR artificial star hits each of the flats (from both artificial star feed mirrors) match that of the visible artificial star. This is obviously the same as co-aligning four correct input beams at each of the two beam combiner outputs where the feed mirrors are located. This automatically co-aligns the beams at the other two outputs. The system is now aligned for visible light. When the beam-splitters and compensating plates are changed, the artificial star no longer hits each of the flats at the correct angle. An adjustment scheme which uses angles measured with the infrared camera must be applied to fix this. The size of the adjustments can be minimised by replacing each plate by one with a similar wedge angle. A beam-splitter plate in a particular orientation may be replaced by another plate in one of the two orientations which have the reflective face towards the compensating plate (see Figure 3.6). There
It must be transmitted through the infrared-selecting dichroic mirrors (labelled in Figure 3.5), which are at least 90% reflective for infrared light.
2


3.4. ALIGNMENT SOLUTIONS

31

are four possible orientations for a replacement compensating plate. We have four visible beam-

splitter/compensating plate pairs and five infrared pairs. There are thus 5C4 4!Ä2 ? 4Å4 491520 ways to replace the visible beam-splitters and compensating plates with plates which work in the infrared. The most important criterion for choosing the optimum combination is to minimise the angular displacement on the way out from the artificial star to the auto-collimating flats of the beam which is unchanged by the subsequent adjustment scheme. Then after the adjustment scheme is applied, the artificial star will come off the flats at almost the same angles as a real star enters the building. Hence the result after subsequently passing through the beam combiner is the same for both. The wedge angles of all of the plates were measured by inserting each one into a beam from the "visible" artificial star and measuring the angular displacement of the beam with the theodolite. The changes on swapping from visible to IR plates were calculated for each of the 491520 combinations (with the help of a computer!). The combination selected was that which minimised the tilt of the beam which would be unchanged by the adjustment scheme (the minimum displacement was 0 ? 6 arc-sec in both coordinates), while keeping the other displacements as small as possible (the optimum solution gave displacements in the range 7-12 arc-seconds).

3.4.3

Focusing the artificial star

After changing to the infrared beam-splitter and compensating plates, the "infrared" artificial star must be focused at IR wavelengths. To measure the defocus, we can use the fact that a pupil shift causes an image shift on the detector if the artificial star is not focused. A mask is placed over the camera lens, to define two small apertures a distance y apart (see Figure 3.9). The position of the conjugate of the pinhole is related to the artificial star defocus (measured by vÌ )by 1 v D vÌ 1



1 f

(3.9)

where f is the focal length of the camera lens, and the other symbols are defined in Figure 3.9. The principle of similar triangles can be used to obtain 1 y v 2 between v and the image separation x is v sy y
1 2Ä

y



xÅ s, hence the relationship



x

(3.10)

In general, the camera will not be focused on infinity, hence s is unknown. However, if the optical path D from star to camera is altered, the change in the image separation will be entirely due to the star defocus. Suppose that the image separation is measured for two path lengths D1 and D2 , giving separations x1 and x2 . The counterpart of Equation 3.9 for this differential measurement is 1 v1 Following from Equation 3.10, 1 v1



1 v2

1 D
1

1 vÌ D2 x2

vÌ

(3.11)



1 v2


sy

x1

(3.12)


CHAPTER 3. OPTICAL SYSTEMS

Artificial star

Camera

detector

-v'

y pinhole

x

s D u
Figure 3.9: Setup for focusing the artificial star. Two separated pupils are defined at the camera lens.

v

32


3.4. ALIGNMENT SOLUTIONS

33

Equating the right-hand sides of Equations 3.11 and 3.12, and making the approximation Äx2 x1
Å ÄsyÅ Äx2





x1

ÅÄ

fyÅ gives 1 D
1

D 1 2
vÌ D
2 2Å

vÌ

x2


fy

x1

(3.13)

Solving for vÌ yields the result vÌ D
1Ç

D

2

?

æ
ÄD1 Ç



4 D1 D 2

2Ç x

2



fy x 1 ÄD 2



D

1Å

(3.14)

This tells us how far the artificial star is from being focused, in terms of the measured image separations x1 and x2 , the distances D1 and D2 , and the known focal length f . The position of a large (i.e. two pixels in diameter) image of the pinhole can be measured to 0.05 pixel (2 Åm) by centroiding. Putting values D1 20 m, D2 50 m, f 143 mm and y 20 mm into Equation 3.14 gives vÌ 207 m, which corresponds to a difference from the ideal pinholelens distance of 0.1%. With this defocus, a 1 mm pupil shift would give an image shift equivalent to a 0.9 arc-sec error in the beam direction, much better than the desired accuracy of 2.7 arc-sec. The sign of x indicates on which side of the conjugate of the pinhole the detector plane lies. The sign of x1 x2 indicates whether the beam from the artificial star is converging or diverging, and hence whether the positive or negative solution of Equation 3.14 is appropriate. In practice an iterative process is used to collimate the artificial star. The image separation x should be measured for two values of the optical path D which differ as much as possible. The direction in which to move the artificial star lens relative to the pinhole should then be inferred from the sign of x1 x2 , and an adjustment made. The procedure should be repeated until x1 x2 ,at which point the artificial star will be focused. The camera lens can then be moved until x focus the camera. This procedure has been used successfully at COAST. 0, in order to

3.4.4

Scheme for fine adjustment of the beam directions

The relative angles of the four beams (each corresponding to a different telescope) which make up each combined beam must now be made equal. This is done by means of a fine adjustment scheme. To measure the relative angles, each input beam in turn is allowed to enter the beamcombiner (the others being blocked), and the position of the image at the pixel which corresponds to the desired output beam is measured. The optics feeding the detector are designed to give a coarse pixel scale, so that all the light from a beam falls into a single pixel, to minimise the read noise and readout time when recording fringes. The image positions must be measured to a small fraction of a pixel, so a 5 mm diameter aperture is introduced, which diffracts the light over several pixels. The centroid of the large image, calculated over a 3 ? 3 pixel region, can be measured to 0.05 pixel, which gives the angle of the beam to three seconds of arc. The nine-pixel cell has two practical advantages over a quad cell: when the beam is centred in the cell, it falls at the centre rather than the corner of a pixel,


34

CHAPTER 3. OPTICAL SYSTEMS

1 0.9 0.8 Centroid coordinate 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 Beam displacement /arc-sec 50 60

Figure 3.10: This plot shows the linear relationship between centroid coordinate and beam angle. See text for details. and the central pixel can be saturated without affecting the inferred beam angle. The readout time for such a small sub-array is insignificant, so the coordinates of a beam may be updated while an adjustment is made. Simulations show that a 5 mm aperture size gives a linear relationship between image displacement and centroid coordinate for displacements up to 0.7 pixel from the centre of the nine-pixel cell (Figure 3.10). The beam combiner combines four beams by mixing two beams at each of four beam-splitters (beam 1 is combined with beam 3 at one beam-splitter, beam 2 is combined with beam 4 at another, and then the mixture of beams 1 and 3 is combined with the mixture of beams 2 and 4 at each of two further beam-splitters -- see Figure 3.3 for an illustration), and can be divided up into four sub-units, each of which consists of a beam-splitter unit with one or more of the input beams relayed off a mirror. Each sub-unit combines two beams. A misalignment in the form of a small tilt of any component in the sub-unit can be corrected by tilting a relay mirror on either of the input beams, to equalise their angles of incidence on the beam-splitter. There is no problem with the double pass through the system: the same adjustment works for both a real star (on a single pass) and for an artificial star which either has or has not passed out through the sub-unit before going in through it. A further tilt of the same mirror can also compensate for both the beam-splitter and compensating plates not being parallel-sided. To see the effect of non-parallel plates, first consider Figure 3.11. The tilt of a beam transmitted through the thin prism is equal to for beams incident on either side of the prism (although the sense of the tilt is reversed). The top part of the figure illustrates this. Reverse the ray direction to see that the angular displacement is the same in both cases. If the inside face of the prism is


3.4. ALIGNMENT SOLUTIONS

35



2

Figure 3.11: Angular displacements on transmission through a thin prism (top) and on reflection from its inner face (bottom). See text for explanation. reflective, the shift on reflection from this face is twice that for transmission through the prism. Consider the images of the prism and incident beam in the reflective face, as shown by the dashed lines in the lower part of Figure 3.11. The tilt is clearly the same as if the beam had been transmitted through a prism of twice the wedge angle. Near to normal incidence, the tilt is proportional to the wedge angle, hence the tilt is 2. Thick parallel-sided pieces of glass can be added to these thin prisms, to transform them into the beam-splitter and compensating plates. This will only shift the beams sideways, without changing their directions. Now examine the beam-splitter/compensating plate combination in Figure 3.12. If only the beamsplitter (top) is not parallel-sided, the tilts from Figure 3.11 imply that the two beams may be co-aligned at both outputs by tilting one input beam through an angle , as indicated. Finally, add the non-parallel compensating plate (bottom). By symmetry, the resulting tilts are the same multiples of the shift due to a single transmission as for the beam-splitter. Hence the beams can be co-aligned again by a further tilt of one of the inputs. The fine adjustment scheme is a sequence which corrects the alignment of the four sub-units in turn, by tilting one component in each. Each such adjustment co-aligns a pair of input beams at one or more of the outputs of the beam combiner. The four sub-units must be corrected in a particular order, which depends on which mirrors we choose to adjust. Otherwise an adjustment may misalign a sub-unit which has just been corrected. A 20 arc-second tilt causes a maximum pupil shift of 0.5 mm (a small fraction of the 25 mm pupil diameter) at the detector on a single pass through the beam combiner. Similar adjustment schemes can be devised for pupil plane beam combiners which combine more than four beams by combining pairs in sub-units. It should always be possible to find a suitable adjustment scheme.


36

CHAPTER 3. OPTICAL SYSTEMS

Beam-splitter

This superposes beams at both outputs

Compensator

Figure 3.12: Effect of non-parallel-sided beam-splitter and compensating plates. The solid line represents one input beam, and the dashed line represents the other. Dotted lines show the ray directions if the plates were parallel-sided. Note that the sideways shifts of the rays have been omitted.

3.4.5

Results and summary

The following is a summary of the alignment procedure used for the COAST infrared system:

1. Use beam-splitters/compensating plates with coatings optimised for visible wavelengths, in kinematic mounts 2. Focus artificial star at visible wavelengths 3. Align in visible, get pupil positions correct 4. Swap plates for IR versions. Minimise effect of different wedge angles 5. Focus artificial star at IR wavelengths 6. Apply fine adjustment scheme

? ? ?

Use relative angles of input beams measured at detector Treat beam-combiner as several sub-units, each of which combines two beams Tilt of single component fixes each sub-unit


3.5. PERFORMANCE

37

The relative angles of the beams at the outputs of the beam combiner are usually stable to six seconds of arc over timescales of a week, and so the only realignment which needs to be done regularly is the fine adjustment scheme, which takes a few minutes. The entire alignment procedure can be performed in a day. The procedure is much easier to carry out than to justify in detail! The relative elevations of the four beams can change by up to 10 arc-sec when the IR-selecting dichroic mirrors are slid out of position then back in again, probably due to twisting of the plate on which the mirrors are mounted. This can be quickly corrected with the fine adjustment scheme, but is nevertheless a problem when visible and IR observations are interleaved on the same night, when the resulting misalignments lead to slightly reduced fringe visibilities. Despite this problem, the alignment of the COAST infrared system can be maintained over long periods using the techniques described above. This has enabled infrared observations to be made whenever the weather conditions have been suitable.

3.5

Performance

The acceptable levels of wavefront degradation and path stability within the optical systems of COAST are set by the atmosphere. As long as the wavefront distortions and path fluctuations within the instrument are much smaller than those caused by the atmosphere above the telescope, the precision of visibility and closure phase measurements will not be compromised. Although almost ideal performance can be achieved in these respects, the optical throughput of all stellar interferometers is very low. This prevents faint objects from being observed.

3.5.1

Optical throughput

An optical/infrared interferometer inevitably has a large number of optical surfaces, which, equally inevitably, cannot be made perfectly reflective or transmissive over a wide band. The limiting magnitude of the interferometer depends critically on the overall throughput of the system, and so the number of surfaces must be minimised and their coatings carefully chosen, in order to achieve the best performance. The design of COAST attempts to minimise the number of reflections from telescope to detector. Thus at visible wavelengths, COAST can generally observe fainter stars than its competitors. At the time of writing, fringes have been detected on a (red) star with V Ç7 6 at 905 nm. However, the infrared performance has been compromised in several respects, because IR capability was added after COAST had been constructed. I now present a calculation of the throughput of COAST in the infrared J , H and K bands. This calculation is similar to that performed by Beckett (1995), but draws on an additional three years of experience with the infrared system. Over this period, different coatings for the IR-selecting dichroics and for the mirrors in the beam combiner have been introduced.


38 Mirror coatings

CHAPTER 3. OPTICAL SYSTEMS

Metal mirror surfaces must be used to give high reflectivity over the broad bands needed at COAST. The three metals commonly used for mirrors are aluminium, silver, and gold. All are usually evaporated onto a glass substrate (possibly on top of a layer of chromium, to improve adhesion) in high vacuum (Hass 1965). Aluminium is the only one with high reflectivity from the ultraviolet to the visible, and has the further advantage that it forms a thin oxide coating which protects the surface and inhibits further corrosion. Silver coatings have higher reflectivity than aluminium coatings at visible wavelengths and in the infrared, but the metal corrodes rapidly without a chemically-inert overcoat. Gold coatings do not corrode, but they are mechanically fragile, and so also require a protective overcoat. The reflectivity of a gold film is comparable to that of silver in the infrared and no worse than aluminium in the red, but decreases dramatically at shorter wavelengths. Conditions at COAST are conducive to corrosion of metal mirror surfaces. The air both inside the telescope enclosures and the optics building is very humid, despite the presence of dehumidifiers, which run continuously except when observing. The atmosphere is also polluted by dust from a nearby cement factory. Both aluminium and silver mirrors have been used extensively at COAST. Aluminium coatings with a thin magnesium fluoride overcoat have proved extremely durable, even on the telescopes. Silver mirrors, despite a protective MgF2 overcoat, become visibly dull and yellow after two years in the COAST building. This is probably due to reaction with atmospheric sulphur. Because of these corrosion problems, all but four of the 14 silver mirrors in the IR beam combiner have been replaced by aluminium ones. Gold mirrors would give better IR performance, but would make the initial alignment in visible light more difficult. Silver mirrors remain in use in the pathcompensation trolleys. Normal-incidence reflectivities for freshly deposited aluminium, silver, and gold films are given in Table 3.1 (Melles-Griot 1990). The protective overcoatings on the mirrors used at COAST are sufficiently thin to have almost no effect on the infrared reflectivity. All of the beam combiner mirrors are used near to normal incidence, but the angles of incidence are larger for the small flat mirrors on the telescopes and for the trolley mirrors. The reflectivities given in Table 3.1 are not necessarily representative of the actual mirrors in use at COAST, both because of the effect of corrosion, and because the reflectivity of a metal film is highly dependent upon the conditions in which it was deposited (Hass 1965). The values given in different manufacturers' catalogues can differ by several per-cent. Burns (1997) measured the reflectivity of the mirrors in the path compensation trolleys and the visible beam combiner at COAST, in a number of visible wavebands. The silver mirrors in the beam combiner had 94% reflectivity at 800 nm, whereas the silver mirrors in the path compensation trolleys were much worse -- the typical combined reflectivity for the pair of mirrors in one trolley was 66%. Burns explained this as the effect of dust settling on the lower mirror in the


3.5. PERFORMANCE Table 3.1: Normal-incidence reflectivity of freshly deposited mirror coatings, from Melles-Griot (1990) Material Aluminium Silver Gold Reflectivity J H K 0.92 0.95 0.97 0.95 0.98 0.98 0.96 0.98 0.98

39

trolley, and hence both corroding the mirror surface and causing light loss by scattering. Even uncorroded mirrors can have poor reflectivity: some aluminium mirrors from COAST, immediately after being recoated, were found to be only about 85% reflective. The aluminium tip-tilt mirrors have a proprietary overcoat which enhances their reflectivity in the visible. The infrared reflectivity is unknown, but is assumed to be 0.90 in the calculations below.

Dichroic mirrors As discussed by Beckett (1995), infrared selection at COAST requires reflection of long wavelengths and transmission of short wavelengths, whereas the opposite arrangement would be preferred. Two different sets of dichroic mirrors have been used. Glass plates with thin gold coatings give 90% reflectivity above 1 Åm, but only 30% transmission in the visible, which reduces the amount of light available for autoguiding. With these dichroics, the autoguider sets the limiting magnitude of COAST for IR observations of very red stars. An alternative set, designed for observations in the 1.0-1.6 Åm range and using dielectric multi-layer coatings, have almost 100% reflectivity over this narrower infrared band, and 97% transmission in the range 600-900 nm.

Transmissive components The number of transmissive components in a stellar interferometer should be kept to the bare minimum, as there will be a 4% loss at each air-glass interface. This loss can be reduced with a suitable anti-reflection coating, but it is impossible to design anti-reflection coatings which work over very wide bands, so some surfaces may be better uncoated. The glass windows where the light pipes from the telescopes meet the COAST building are a case in point. These windows currently have an anti-reflection coating designed for visible wavelengths which reduces the overall IR transmission to 0.65-0.80 (see Table 3.2), compared with 0.88 for a bare glass plate. The IR beam-splitting and compensating plates have anti-reflection coatings optimised for the infrared, and so the loss at each interface is 1% for all wavelengths from 1.0 to 2.4 Åm. The camera lens and dewar window are not anti-reflection coated, but could probably be given similar coatings to the beam-splitter plates.


40

CHAPTER 3. OPTICAL SYSTEMS Table 3.2: Transmission of optical components Component Lab window Lens Dewar window Filter Transmission J H K 0.70 0.85 0.85 0.65 0.65 0.85 0.85 0.85 0.80 0.85 0.85 0.70

The transmissions of the H and K filters in Table 3.2 are those of new filters obtained from OCLI. The value for the J band relates to the filter used at COAST up to the time of writing. A replacement has been acquired from OCLI which has a better in-band transmission of 0.75. These values include surface reflections.

Predicted throughput The components met by the beam from one telescope as it propagates through the optical system to the NICMOS detector are as follows: Outside: Path compensation: IR selection: Beam combiner: Camera: 3 aluminium telescope mirrors, 1 enhanced aluminium mirror, 1 aluminium mirror, lab window 2 silver mirrors 1 dichroic mirror 4 aluminium mirrors (assumes beam-splitter units are loss-less) 2 aluminium mirrors, lens, window, filter

The overall throughput, given in Table 3.3, is the product of the reflection or transmission coefficients for the individual components. Note that the transmitted power is split four ways by the beam combiner. The predicted J band throughput is almost a factor of two worse than that given by Beckett (1995), mostly because of the change to aluminium mirrors in the beam combiner. If the camera lens and dewar window were anti-reflection coated, and the mirror coatings in the beam-combiner were changed to gold, the predicted J band throughput would improve by 71%. Significant throughput gains can clearly be made.

Measurements No direct measurements of the overall throughput of COAST in the infrared have been made, due to the lack of a calibrated light source. Measurements of the IR reflectivity or transmission of individual components in situ, as carried out by Burns (1997) in the visible, would require a more compact infrared detector than the NICMOS system.


3.5. PERFORMANCE Table 3.3: Predicted infrared throughput of COAST Components Reflections (w. gold dichroics) Reflections (w. dielectric dichroics) Transmissions Total (w. gold dichroics) Total (w. dielectric dichroics) Throughput J H K 0.32 0.35 0.33 0.11 0.12 0.47 - 0.40 0.19 - 0.52 - 0.40 0.21 -

41

Table 3.4: Adopted quantum efficiency of the NICMOS device NICMOS QE J 0.44 H 0.48 K 0.60

An estimate of the overall IR throughput can be made by comparing the detected flux from a star with that predicted from the calculations above. The energy flux at the top of the Earth's atmosphere from a star with an apparent magnitude of zero, at a wavelength of 1.3 Åm, is 3 18 ? 10 9 W m2 Åm (Wamsteker 1981), which implies a photon flux of 2 0 ? 1010 photon s m2 Åm. Absorption by water vapour in the atmosphere limits the effective bandwidth of the COAST IR system with the J band filter to 0.15 Åm (see next section). The continuum transmission of the

atmosphere is unknown, and thus assumed to be 100%. With an effective telescope aperture of 240 mm, the expected photon rate at the detector, for one beam combiner output, is therefore 2 0 ? 1010 ? ? 0 1202 ? 0 15 ? 0 12 ? 1 ? 0 92 3 7 ? 106 photon s (the factor 0.92 is for diffraction 4 by the aperture stop in front of the detector). The fraction of these photons which will be detected is determined by the quantum efficiency of the NICMOS device. The QE of our device has never been measured, so I have adopted values based on measurements of a small sample of NICMOS3 detectors (Rieke et al. 1993). These values are given in Table 3.4. We would expect to detect 1 6 ? 106 photon s at one beam combiner output, from a 0 magnitude star in the J band with a 240 mm aperture. The most recent measurements of Vega (which is defined to be 0 magnitude in all photometric bands) were made on 4 August 1998. The detected photon rate was measured for three different telescopes in each of three target pixels, corresponding to three different outputs of the beam combiner. The measured signal rates were between 3 9 ? 105 photon sand 9 2 ? 105 photon s. There is likely to have been significant extinction, as the observations were made at airmass 1.5 from a site at sea level. Hence these results suggest that the expected throughput has been achieved on some routes through the system. The throughput of the other routes can probably be improved by identifying and replacing problem components.


42

CHAPTER 3. OPTICAL SYSTEMS

3.5.2

Fringe visibility

Internal fringes between two beams can be obtained using the artificial star in auto-collimation. The visibility of the fringes will be reduced from unity, due to

? ? ? ?

Wavefront errors caused by imperfect optical surfaces Misalignments Intensity differences between the two interfered beams The use of finite integration times in sampling the temporal fringe pattern

A misalignment will only affect the visibility if the difference in the directions of the interfered beams is larger than 4D,where D is the pupil size, or if the positions of the two beams differ by more than a few per-cent of D. A small pupil size may be defined using an aperture immediately in front of the camera lens. If this pupil is sufficiently small there will be no visibility losses due to misalignments. Measurements with a 5 mm pupil at 1 3 Åm typically give visibilities in the range 0.80-0.95,

after removing the known visibility reduction due to the finite integration time. If a 10 mm pupil is used, the visibilities fall to 0.60-0.90. A 50% difference in the intensities of the two beams will only cause a 2% reduction in fringe visibility, and misalignments must still be negligible for this pupil size, so most of the reduction must be due to the cumulative effect of optical surface errors along the double pass through the system. At longer wavelengths, the size of the surface errors is a smaller fraction of the wavelength, so the visibility loss will be smaller. Astronomical observations involve only a single pass through the system, in which case the visibility reductions within COAST are much smaller than those caused by the atmosphere. The visibility values typically measured on stars are discussed in Chapter 5.

3.5.3

Spectral response

A strange feature was noticed in the spectra of internal fringe data, which remained unexplained for some time. The spectra of data from real stars are somewhat different. Figure 3.13 shows the power spectra of some internal fringe data (top) and of stellar fringes (bottom). Both data sets were taken through the broad ( 0 2) J band filter, with the same path delay modulation. In the bottom plot, there is little smearing of the fringe peak due to atmospheric path variations, because a high fringe frequency was chosen. The signal-to-noise is also very high. Consequently the spectral response of the infrared system in the two situations can be compared. In internal data, a visibility minimum is always seen at the frequency which corresponds to a wavelength of 1350 nm. The visibility always increases again at lower frequencies (i.e. longer wavelengths). However, the height of the secondary peak is not a constant fraction of the main


3.5. PERFORMANCE

43

Figure 3.13: Power spectra of internal fringes (top) and stellar fringes (bottom), illustrating the effect of absorption by water vapour on the spectral response of the COAST IR system with the J band filter. The lower horizontal axis is temporal frequency, and the upper horizontal axis is the wavelength which would give fringes at that frequency. See text for discussion.


44

CHAPTER 3. OPTICAL SYSTEMS

0.8 0.7 0.6 Transmission (%) 0.5 0.4 0.3 0.2 0.1 0 1100

1150

1200

1250 1300 1350 Wavelength (nm)

1400

1450

1500

Figure 3.14: Transmission curve of J band filter (top) and the corresponding part of the solar spectrum, as observed from the Earth's surface (Kurucz et al. 1984, bottom). The large feature in the spectrum is entirely due to absorption by water vapour in the Earth's atmosphere. The horizontal axes of both plots are wavelength in nm.


3.5. PERFORMANCE

45

peak height. A visibility minimum occurs at the same wavelength in stellar data. The secondary peak at longer wavelengths is not always seen above the photon noise. When present, it is much smaller than the peak in the internal data. The shape of the fringe peak should be equal to the transmission curve of the J band filter (shown in the top part of Figure 3.14), if everything else in the system has a flat spectral response across the filter passband. In fact, there is significant absorption by water vapour, both in the atmosphere above the telescopes, and within the COAST building. The spectrum of the relevant absorption feature is shown in the bottom part of Figure 3.14. The short-wavelength edge of the feature is at 1350 nm, and maximum absorption is at 1370 nm. The feature extends up to 1500 nm. This feature has precisely the shape needed to explain the power spectra, provided there is significant absorption inside the optics building, and much more absorption when observing a star through the Earth's atmosphere. It may seem surprising that absorption by water vapour can be detected inside the building, but the air paths needed to obtain fringes with the artificial star are approximately 30 m, and the relative humidity is usually at least 40%. For astronomical observations, the atmospheric absorption effectively cuts off the J band filter at 1350 nm. This has important consequences. The fractional bandwidth at 1.3 Åm is limited to 12%, seriously reducing the detected signal. The asymmetric bandpass can introduce systematic errors in the closure phases, whose magnitude is discussed below. Finally, the effective bandpass must be known in order to interpret COAST data. However, a narrower band should make this interpretation easier.

3.5.4

Closure phases

The effective bandpass of the IR system for internal fringes, with the J band filter in place, is highly asymmetric. If fringes are recorded on three baselines simultaneously (the three sets of fringes should have different temporal frequencies), and the three fringe envelopes do not overlap perfectly in time, the resulting closure phase will be biased from its true value (which is zero for an unresolved target such as the artificial star). Closure phases as large as ten degrees have been measured, corresponding to constant offsets equal to half the width of the fringe envelope. Good overlap gives values within one or two degrees of zero, as expected. This level of systematic error is very serious, but such large errors should not be expected, and indeed are not observed, in closure phase measurements on real stars. The atmosphere continuously changes the relative positions of the three fringe envelopes, and so there will only be a systematic error in the closure phase if the average positions of the three fringe packets differ significantly. The effective bandpass for stellar observations is much less asymmetric than that for internal measurements (compare the upper and lower plots in Figure 3.13), so the overlap requirements are less stringent. In practice, it is possible to centre the fringes well enough to make closure phase measurements on a star which are repeatable to 5Ö . The results presented in Chapters 5 and 6 clearly demonstrate


46

CHAPTER 3. OPTICAL SYSTEMS

that there is no systematic bias in a reasonably-sized data set. The topic of astronomical closure phase measurements is revisited in Chapter 5.

3.5.5

Path stability

Short-term path fluctuations within the optics building can be measured by following the phase of internal fringes. The path difference between two beams can change by 0.65 Åm in 30 seconds. Most of this variation is on timescales longer than 5 s. In any case, it is much smaller than the amplitude of the low frequency path changes introduced by the atmosphere, which is 25 Åm for observations on a 20 m baseline (using a result from Tango and Twiss 1980, assuming that r 20 cm at 1 3 Åm ).
0

Long-term path stability is very good. Only two paths within the beam combiner must be matched to obtain fringes at all outputs with the same external path delays. The difference between these two paths usually drifts by 5 Åm in a month, and so adjustments are infrequent. The external path delays required to obtain fringes with the artificial star change by about 40 Åm over the same period. These path variations are unimportant compared with those caused by temperature changes in the differential air paths in the pipes out to the telescopes.

3.5.6

Conclusions

The visibility losses and path variations within the aligned infrared beam combiner are much smaller than those caused by the Earth's atmosphere. The throughput of the beam combiner at 1.3 Åm is roughly that expected from the coatings used, but could probably be improved by the use of IR-optimised anti-reflection coatings and gold-coated mirrors. Observations through a standard J band filter suffer from problems with water vapour absorption, but these could be overcome by using a narrower filter.


Chapter 4

NICMOS3 Infrared Camera
This chapter returns to the subject of the NICMOS3 camera system described in Chapter 2. The readout mode used at COAST for sampling pupil plane fringes, and the software used to control the array are described. Some unexpected features of the NICMOS array which are relevant to operation in this mode are discussed. Methods which could be used to increase the maximum frame rate and decrease the effective read noise are also discussed.

4 .1

Sampling requirements

The principles of the COAST infrared correlator were described in Section 3.2.2. To recap, the correlator consists of a pupil plane beam combiner which, in conjunction with the path compensation trolleys, generates a temporal fringe pattern at each of four outputs of the beam combiner. Each fringe pattern is sampled by a sequence of reads of a single pixel of the NICMOS detector array. The temporal fringe pattern contains N sets of fringes, from N baselines. For simplicity, assume that the positions of the path compensation trolleys are modulated so that the fringe frequencies are 1 Tscan 2 Tscan N Tscan (i.e. Tscan is the period of the low-frequency fringes). To Nyquist sample the fastest fringes, the intensity of the combined beam must be measured at a rate of at least 2N Tscan. In order to determine the maximum value of Tscan , we must estimate the minimum value of the spacing of the fringe peaks in frequency space. Each fringe peak has a natural width which is equal to the reciprocal of the time taken to sweep through the fringe envelope. The number of fringes in the envelope is equal to 2 , where is the fractional bandwidth of the optical filter used. Hence the number of fringes is the same for all baselines. By definition, the time taken to sweep through one fringe is the reciprocal of the fringe frequency, hence the natural width of the peak in frequency space is proportional to its mean frequency. Thus, if the fractional bandwidth is too great, the high frequency peaks will run into each other. The largest permissible number of equally-spaced baselines is . This limit may be important at IR wavelengths, where wide fractional bandwidths can be used. For


48

CHAPTER 4. NICMOS3 INFRARED CAMERA

example, a 20% fractional bandwidth allows a maximum of five simultaneous baselines. The use of unequal spacings cannot improve matters significantly. The fringe peaks are further broadened by atmospheric turbulence. To first order, an atmospherically-induced change in the path difference between two beams while scanning through the fringes is equivalent to a change in fringe frequency. The magnitude of this effect was calculated by Buscher (1988). The fraction of the fringe power leaked from one baseline to adjacent baselines depends only on t0 and Tscan , but the amount of leakage which can be tolerated depends upon the relative visibilities on adjacent baselines. The leaked power is proportional to the square of the visibility, so Tscan 0 4t0 is sufficiently fast if the visibilities are comparable, but even scanning as quickly as Tscan t0 10 will lead to a change of a few per-cent in the smallest visibility if the visibilities on adjacent baselines differ by a factor of ten. A further problem is scintillation. Atmospheric refractive index variations cause intensity fluctuations in the beams from the telescopes. These lead to fluctuations in the (uncorrelated) background in the fringe pattern, as well as to variations in the instantaneous fringe visibility. The fluctuations in the background signal can be removed by subtracting the fringe patterns from the two outputs of the combining beam-splitter unit, but this only works if both patterns are sampled at the same times. This is the case with the APDs at COAST, but not with the NICMOS detector. Only one pixel within each quadrant of the NICMOS array may be read at a time. Consequently, the fringes must be scanned rapidly enough for the lowest frequency fringe peak in the spectrum of the raw data to be above the highest frequency at which there is significant scintillation (compared with the "white" background of photon and read noise). At COAST, this frequency is 100-150 Hz for still air conditions, i.e. somewhat smaller than the spacing of the peaks suggested in the previous paragraph. In order to measure a closure phase on a target which is unresolved on one baseline in the triangle but reasonably resolved on the other two, a scan time Tscan t0 10 should be sufficient. With three simultaneous baselines, the sampling interval required is t t0 60. In the shortest-wavelength infrared band centred on 1 3 Åm, t0 is typically 20 ms, hence it is necessary to sample the fringes at approximately 3 kHz. If it were possible to sample faster than this, what would be the optimum sampling rate with a noisy detector? The signal-to-noise ratio for a single sweep through the fringe envelope is given by Equation 3.1 (the expression applies to both temporal and spatial fringe patterns). If the integration time used is an appreciable fraction of the fringe period, this will lead to a reduction in the measured visibility V , compared to the value V0 which would be measured with very fine sampling (Buscher 1988): V V0 tint sincÄtint t
Å

V0

tint 1 t



Ä

tint 6

Å2

(4.1)

Here tint is the integration time, which in general will be slightly shorter than the sampling interval t ,and is the fringe frequency. With two samples per fringe (i.e. tint t T 2), V V0 0 64, four samples per fringe gives V V0 0 90, but increasing the number of samples per fringe to eight gives only a small gain: V V0 0 97 in this case. At very low light levels, when Equation 3.1


4.2. READOUT OF THE NICMOS ARRAY

49

simplifies to Equation 3.4, the optimum number of samples per fringe is about three. Even at high light levels, when the read noise is totally insignificant, the gains from having more than four samples per fringe are minimal.

4.2

Readout of the NICMOS array

I will now consider the sequence of operations on the NICMOS array needed to sample the temporal fringe pattern. The processes of addressing, resetting and reading a pixel were treated in Section 2.4. The precise timing of these operations determines whether sampling rates of can be achieved in practice. 3kHz

The operations on the NICMOS device are controlled by the RISC microprocessor incorporated in the CCD controller system. All of the array functions are mapped to an address space. The microprocessor is very simple, and only supports memory access and simple loops. RISC programs must contain less than 256 instructions. The four output beams from the beam combiner are focused onto four separate pixels of the NICMOS array, which are approximately at the corners of a square. However, each target pixel must be in a unique column, as it was discovered that resetting one pixel affects all others in that column. This effect is contrary to the published specification, but has been seen by other workers (e.g. Rieke et al. 1993). Rockwell have acknowledged that it is a feature of production devices (MillanGabet, private communication). As the NICMOS multiplexor does not allow addressing of a chosen pixel, all the other pixels in the array must be skipped without being read. The spacings of the target pixels, which are set by the optics used to feed the camera (described in detail by Beckett 1995), are about 90 pixels horizontally and vertically. The operations performed on the array are as follows (this sequence is exactly that presented by Beckett): all pixels up to the first target pixel are skipped, then the first target pixel is read (read 1), then reset, then immediately read again (read 2). More pixels are then skipped until the second target pixel is reached, then the same read, reset, read sequence is performed on this pixel. The same operations follow for the third and fourth pixels. The entire sequence is repeated after a short pause to allow each pixel the desired total integration time. Note that each of the four target pixels continues to accumulate signal while the others are being read out. In one cycle, the dead time for each target pixel (when it is not accumulating signal) is just the time taken to reset the pixel (23 Ås). The read immediately after each reset is needed in order to subtract the reset level, which would otherwise be the dominant noise source ( 100 electrons). The signal accumulated by a particular pixel in one integration is given by (read 1) minus (read 2 from the previous cycle). As described in Section 2.4, each read of a pixel makes use of the double-correlated sampling (DCS) circuitry in the CCD controller, to reduce noise. The resulting read noise falls with increasing DCS integration time until it reaches a threshold value. The minimum read noise of 16 electrons (this is an average over one quadrant of the array) is obtained using DCS integration times of 20 Ås. Each pixel read requires two DCS integrations, so this time plus a small overhead


50

CHAPTER 4. NICMOS3 INFRARED CAMERA

gives the final pixel read time of 45 Ås. Since two complete reads are needed for each sample, and the noise on the two reads is uncorrelated, the overall read noise is

ä

2

?

16

23 electrons.

The multiplexor can be driven at the nominal 6 MHz clock speed of the RISC microprocessor (when I took over the NICMOS camera, the RISC code was written to drive the multiplexor much more slowly than this). The column shift register in the NICMOS chip is double-edge clocked (i.e. a column is skipped on both edges of the clock signal) and so only two RISC clock cycles are needed to skip an adjacent pair of columns. The drawback of double-edged clocking is that different RISC code is needed to read pixels in even and odd columns. Only three clock cycles are required to skip a complete row of pixels. When reading all four pixels, 660 RISC clock cycles are spent skipping over unused rows and columns (with the usual locations of the target pixels on the chip), compared with 2660 clock cycles to read and reset the target pixels. If just the top two pixels are read out, these numbers become 241 and 1330 clock cycles respectively. The time required to skip over rows and columns of the NICMOS device is thus insignificant compared with the time taken to read out the target pixels. No significant gains in frame rate can be made by changing the feed optics to put the four target pixels closer together on the chip. A small number of additional clock cycles (20 when reading two pixels, 27 when reading four) are required to set up the array at the start of each sequence, to begin clocking along the rows containing the target pixels, and to perform an extra signal digitisation at the end of the program (this is needed to transfer the result of the previous digitisation into the RISC data memory). Thus the total numbers of clock cycles needed to run the two-pixel and four-pixel RISC programs are 1591 and 3347 respectively. The final overhead is the time needed to transfer the five or nine words of data (two reads per pixel plus one extra) from the RISC memory to the memory of the host PC. The transfer time is 4.2 Ås per 16-bit word. The total run times for the two-pixel and four-pixel sequences are therefore 0.29 ms and 0.60 ms respectively. The sequence of operations listed above is performed in a tight loop, the timing of which is tied to reference signals at 1 Hz and 5 kHz. This synchronisation of the IR camera with the reference signals was planned but not actually implemented by Beckett. The reference signals are also used by the path compensation trolleys and avalanche photodiode detectors at COAST, and thus detector reads and trolley sweeps are kept synchronised. The use of a 5 kHz reference signal restricts the permitted frame times t to integer multiples of 0.2 ms, i.e. frame rates of 5 kHz, 2.5 kHz, 1.67 kHz, 1.25 kHz, 1 kHz etc. A software program on the host PC, called "observe", performs high-level control of the NICMOS device when recording fringes. To start the process of recording data, this program downloads the appropriate code into the RISC program memory, waits for a pulse from the 1 Hz reference signal, then waits for a negative-going edge of the 5 kHz reference, at which point it signals the RISC processor to start running its program. While the RISC program is running on its own processor, the PC calculates the signal level from the last completed integration of each pixel, by subtracting


4.2. READOUT OF THE NICMOS ARRAY

51

the reset level measured in the previous cycle. The PC waits for the RISC program to end, then reads the resulting data back into its own memory. The PC now regularly checks the value of a counter triggered off the negative edge of the reference signal, until it notices that the next negative edge of the reference signal will indicate the end of the cycle. The reference is then read continuously, and when the edge is detected, the RISC program is restarted and the sequence repeats. The edge can be detected with a precision of 2 Ås, so the shortest sampling period is accurate to 0.5%. All data storage at COAST is handled by the Norsk Data computer (Nord). To give the potential for active correction of the position of the fringe envelope within the trolley sweep, data is transferred from the IR PC to the Nord while data taking continues. This real-time data transfer takes place over a custom 16-bit parallel link, with all handshaking done in software. The handshaking operations are carried out while the PC is waiting for the RISC program to finish, and for lower frame rates, while waiting for the 5 kHz counter near the end of each cycle. In order to keep to the desired cycle time, "observe" uses a buffer, implemented in software, which fills up in those cycles when the Nord is busy saving some data to disk. The timing within the real-time segment of "observe" is illustrated in Figure 4.1, for the cases of reading four pixels at 1 kHz (top) and four pixels at 1.25 kHz (bottom). This sequence can run continuously for long periods (the longest test run was 200 seconds). The software can also run in a mode where data is not sent to the Nord, but instead is stored in the PC's memory. The sequence runs for a set time (up to 26 seconds at the fastest frame rate), after which the data is displayed, and the process repeated. A similar scheme is used for real-time data transfer from the avalanche photodiode detectors to the Nord, although here the buffer is a dedicated electronic system. Fringe envelope tracking has already been implemented for the APDs, and only modifications to the Nord software are needed to extend it to the infrared system. The fastest sampling rates achieved by the method just described are 1.25 kHz when reading four pixels, and 2.5 kHz when reading only the top two pixels in the square. The latter rate is just fast enough to make closure phase measurements using 270 Hz spacings between the fringe peaks, with the fastest fringes sampled at three samples per fringe. The summed power spectrum (i.e. the sum of many power spectra, each of a single sweep through the fringe envelope) of some data taken on a star with 2.5 kHz sampling is shown in Figure 4.2. The value of Tscan chosen for these data (3.7 ms) is equal to 0 2t0 if t0 20 ms (a typical value at 1 3 Åm). Therefore some leakage of fringe power between baselines would be expected in the J band, but not in the H and K bands.

4.2.1

NICMOS features

Figure 4.3 shows the signal in two of the four target pixels, for the first four seconds of the readout loop described above. The minimal amount of light falling on the chip would have been undetectable in the 1 ms integration time. Two unexpected features can be seen in this plot. The first effect is the strange response of the pixels in the first two seconds. The effect remains unexplained, but is easily circumvented by ignoring the first few seconds of each run when analysing the data.


CHAPTER 4. NICMOS3 INFRARED CAMERA

counter

1

2

3

4

5

5kHz clock signal
calculate pixel values, add to buffer buffer->Nord wait for RISC read data buffer->Nord from RISC wait for edge

PC

1kHz sampling RISC
run RISC program wait for PC

PC

calculate pixel values, add to buffer

buffer->Nord

wait for RISC

read data wait for edge from RISC

1.25kHz sampling RISC
run RISC program wait for PC

Figure 4.1: This figure illustrates the timing of operations within the real-time segment of "observe", when running the four-pixel RISC program at two frame rates: 1 kHz (top) and 1.25 kHz (bottom).

52


4.2. READOUT OF THE NICMOS ARRAY

53

Figure 4.2: Summed power spectrum of three-baseline COAST IR data on Aurigae (Capella), taken through the J band filter. The spectrum contains fringe peaks at 270, 540 and 810 Hz, and a further peak due to scintillation below 150 Hz. The data were sampled at 2.5 kHz. On the night these data were taken, coherence times of 16 ms were measured in this waveband. The second feature is that the signal per integration (i.e. the difference between the final read and the read of the reset level), although constant after the first two seconds, is non-zero. In contrast, repeated reads at regular intervals, without resets, give a small ( 1 dn/ms) positive signal when the camera shutter is closed. Rieke et al. (1993) found evidence for a decaying reset level, using a different readout mode for a NICMOS3 device. The offsets seen with the COAST camera may be due to the same effect. The offset varies between pixels, and depends upon the frame rate. If not taken into account, it will bias the measured visibility. However, for a given pixel and frame rate, the offset is constant to within four electrons over 12 hours, hence it need only be measured occasionally. The offset is removed at the analysis stage by adding a constant value (different for each pixel) to every sample.

4.2.2

Extrinsic noise

The infrared camera system is prone to picking up electronic noise from external sources, but only at specific frequencies. This is because the readout electronics act as a very high gain amplifier for signals at harmonics of the pixel rate (which is the characteristic frequency of the signal at the output amplifier on the NICMOS device). The same effect is seen in visible CCD systems. For the readout scheme discussed here, there are a number of such characteristic frequencies, as there are a number of different time intervals between pixel reads. This interference gives rise to very narrow features in the power spectra of the data, which can affect any data analysis performed in the Fourier domain. A bad example of this effect is shown in the top part of Figure 4.4. COAST incorporates a lot of


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CHAPTER 4. NICMOS3 INFRARED CAMERA

Figure 4.3: Plot of the signal (in data numbers) in two of the four pixels for the first four seconds of the readout loop described in Section 4.2. The lights in the building were switched off and the light beams from the telescopes were blocked, so the photon flux would have been undetectable in the 1 ms integration time.


4.3. FUTURE IMPROVEMENTS

55

electronic equipment, so there are many sources of interference. It was discovered that the worst interference originated from a number of (rather old) Pericom computer terminals. The pick-up had been exacerbated by the presence of an earth loop, formed between the dewar and the interface box, both of which were earthed at different points on the metal optic table. After isolating the interface box from the table, the number and amplitude of the noise features was reduced (compare the lower plot in Figure 4.4 with the upper one). Further improvements were achieved by replacing the Pericom located nearest to the infrared camera, and by lengthening the reset pulse in the readout cycle (which must change one or more of the high-gain frequencies to a less harmful value). The residual noise features, with all the equipment in the building switched on, are shown in Figure 4.5. There are features at 100, 200, 300, and 400 Hz, probably originating from the mains power supply. We tried filtering the supply to the CCD controller, with no discernible effect. A further feature is visible at 270 Hz. The largest features above 100 Hz (below this the features are masked by scintillation) only have amplitudes twice that of the white noise level. The features are very narrow, so the total power due to extrinsic noise is completely insignificant compared with any detectable amount of fringe power.

4 .3

Future improvements

Increasing the frame rate without increasing the read noise requires a new approach. If the resets can be eliminated, the reads to measure the reset level can also be removed, giving a factor of two increase in speed. However, the loop can only be sustained until the pixels saturate. There are also potential problems with the inherent non-linear response of the NICMOS device. To test the feasibility of this scheme, the following sequence of operations was performed on the (illuminated) array: 1. Clock to pixel of interest 2. Reset pixel 3. Read pixel N times, at intervals t 4. Repeat from step 2 Figure 4.6 is a plot of signal against read number for the case N 300, t 0 2 ms. The response of the pixel is highly non-linear at the start of the first cycle (compare this with Figure 4.3), but in subsequent cycles it is approximately linear over almost the full 16-bit range of the analogueto-digital converter. The gain was set to 2.2 electrons per data number, so the response is linear up to 120 000 electrons. Higher gain settings may increase the usable dynamic range, as the well capacity is 250 000 electrons for the current 0.5 V bias setting (Beckett 1995).


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CHAPTER 4. NICMOS3 INFRARED CAMERA

Figure 4.4: Fourier transforms of two 60 second data files, taken before (top) and after (bottom) removing an earth loop, showing narrow features due to pick-up at specific frequencies.


4.3. FUTURE IMPROVEMENTS

57

Figure 4.5: Amplitude spectrum of a 20 s dark file recorded under observing conditions (top) and the summed power spectrum of the same file (bottom), illustrating the small size of the residual noise features, after taking the steps described in the text.


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70000 60000 50000 Signal (dn) 40000 30000 20000 10000 0 0 100 200 300 400 500 600 Read number 700 800 900

Figure 4.6: Plot of signal against read number for successive reads at 0.2 ms intervals, with a reset after every 300 reads. The detector was illuminated during this experiment. The obvious way of using this scheme at COAST is to synchronise the resets with the start of the path compensation trolley sweeps. If a pixel saturates during a sweep, the PC software must detect this, run a RISC program to perform a reset, then reload the original program. The PC must also interpolate the data point(s) which are missed while this process takes place. The scheme will clearly be much simpler to implement if the pixels do not saturate during a sweep. Currently it is necessary to use sweeps of 0.25 s duration, as the next shortest permitted duration (0.125 s) of a sweep at the required speed does not cover the full range of path variation introduced by the atmosphere (shorter sweeps can be used when fringe envelope tracking is implemented). If t 0 4 ms, the well capacity of 250 000 electrons then implies that the maximum signal per integration is 100 electrons. Bright stars give much more signal than this. To permit observations of targets with a range of IR fluxes, a selection of removable neutral density filters could be used. However, these filters will emit significant thermal radiation in the K band, unless they are installed inside the dewar. Instead of neutral density filters, shorter sweeps could be used for targets which are bright enough for the position of the fringe packet within the sweep to be determined. If the fringe envelope is not wholly within the sweep, the data can be discarded. This method is used with a NICMOS detector at the IOTA interferometer (Millan-Gabet et al. 1999). The IOTA system also uses the non-destructive readout property of the NICMOS device to reduce the effective read noise, by performing multiple reads of the signal accumulated by each pixel. The IOTA NICMOS camera can perform a single read in just 21 Ås, with 12 electron read noise (although this rises to 17e in the readout mode used for recording fringes) (Millan-Gabet et al. 1999). The COAST system requires a 45 Ås readout time to achieve comparable (16e) read noise.


4.4. SUMMARY

59

If a readout time of 20 Ås is used, the read noise is 22e (see Figure 3.11 in Beckett 1995). The superior performance of the IOTA system may be due to the use of purpose-built electronics, including an amplifier located inside the dewar, rather than a CCD controller. The readout time needed to attain the minimum read noise for the COAST system precludes the use of multiple reads unless

? ?

Only one baseline is measured at a time, so that slow frame rates can be used. Fringes are recorded at only one or two of the four beam combiner outputs.

Hence the technique will only be useful for improving the limiting magnitude for stellar diameter measurements, i.e. when closure phases are not required.

4.4

Summary

The COAST NICMOS3 camera system is used in a mode where two or four pixels are read at a high frame rate, which is used for sampling a temporal fringe pattern. The performance of the array in this mode can be summarised as follows:

? ? ? ?

The effective read noise is 23 electrons per sample. The maximum frame rate is 1.25 kHz when reading four pixels, and 2.5 kHz when reading two pixels. Thedead timeis just 23 Åsper sample. Data is transferred in real-time to the Norsk Data computer.

Both the beam combiner and detector work well enough to measure visibility amplitudes and closure phases on stars. The next chapter describes how these measurements are made, and the process of reconstructing an image from the data. The subsequent chapters discuss astronomical results obtained using the working COAST infrared system.


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Chapter 5

Stellar Observations, Data Reduction and Analysis
5 .1 Introduction

The previous two chapters have described the infrared beam combiner and the NICMOS camera, the two components of the infrared correlator system for COAST. The performance of the correlator with internal light sources has been discussed. However, COAST was intended for observations of stars -- that is the subject of this chapter. Usually, the aim of a night's observing is to make enough accurate visibility amplitude and closure phase measurements, evenly distributed in the Fourier plane, to reconstruct an image of the astronomical target. We must decide on an observing procedure, which takes into account the effects of the atmosphere on visibility and closure phase measurements, and the limitations of the available image reconstruction techniques. Should the visibility amplitudes on all baselines be measured at once, or one baseline at a time? How much observing time should be used for each data point? Theoretical predictions of atmospheric effects on the measurement of visibility amplitudes and closure phases were used in the design of the correlator. To what extent is the expected performance achieved when observing through the real atmosphere? The reduction and analysis of the fringe data are also addressed in this chapter. The procedures used for extracting visibility amplitudes and closure phases from the temporal fringe patterns recorded by the IR correlator are described. Two methods of retrieving information about the sky brightness distribution from the visibilities and closure phases are discussed: image reconstruction, and fitting simple models for the brightness distribution. The discussion in this chapter is illustrated by results from COAST observations of the binary star Capella ( Aurigae), made on 25 October 1997. These results are compared with published data in the final section. The astrophysical implications of the measurements are discussed.


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5.2
5.2.1

Observations
Telescope alignment

The alignment procedure discussed in Chapter 3 includes all components inside the COAST optics building, but excludes the telescopes. The beam from each telescope has been reflected from five mirrors by the time it enters the building, so the direction and position of the beam are likely to be incorrect unless adjusted. Large variations in the beam directions ( 20 arc-seconds) are caused by thermal expansion or contraction of the piezo-electric actuators for the tip-tilt mirrors. The alignment of the other components is fairly stable, hence they need only occasional adjustment. Any changes in the mean angle of a beam are taken out by the autoguider, but the reference directions used by the autoguider must match those used to align the components inside the building, if the beam combiners are to superpose the beams of starlight correctly. These reference directions can be adjusted by tilting the mirrors which steer (part of) the light beams from the telescopes onto the autoguider CCD. Before commencing infrared observations on a particular night, the autoguider reference directions are set to match the alignment of the IR beam combiner, as follows. A bright star is acquired, and the autoguider is switched into fast-guiding mode. In this mode the image on the CCD from each telescope is kept in the centre of a quad-cell, by a servo system. The beam direction which corresponds to this image position is adjusted by tilting the autoguider feed mirror for that beam. A small (5 mm diameter) aperture is placed in front of the infrared camera, to spread the star image over several pixels on the NICMOS detector. The position of the large image in a nine-pixel cell is measured by centroiding, in the same way that the artificial star image position is measured during the internal alignment procedure (see Section 3.4.4). The positions of the images from all four telescopes are measured, using one of the beam combiner outputs. If necessary, the autoguider mirrors are adjusted (once the angles of the piezo-actuated tip-tilt mirrors have been corrected, the residual adjustments are typically five seconds of arc) to make the image positions on the IR chip identical, and hence co-align the four beams (if the beam combiner has been aligned internally, the beams from the four telescopes will then be co-aligned at all four outputs).

5.2.2

Finding fringes

After adjusting the autoguider, the path delays needed to obtain stellar fringes must be determined. The microcomputer which controls the path compensation trolleys calculates the geometrical delay, given the coordinates of the target star and the baseline geometry. The calculated delay is typically in error by 500 Åm, due to uncertainties in the above parameters. The slowest trolley sweeps used for IR data have amplitudes of 87 Åm in path delay, so the observers must search for the fringe envelope in the region of the calculated position. Searching for infrared fringes is performed one baseline at a time, by blocking two of the beams from the telescopes with remotelycontrolled shutters on the beam-combining table, and scanning the relative path delay for the two remaining beams. The IR camera is run in "local" mode, i.e. data is taken without being transferred to the Nord, and the raw data is displayed at intervals of one or two seconds (0.7 seconds of


5.2. OBSERVATIONS

63

Figure 5.1: Stellar fringes at 1 3 Åm, on the red giant star Bootis. The horizontal axis is time in ms, and the vertical axis is signal in data numbers per ms (the gain of the system is 2.2 detected photons per dn). The full 40 cm telescope apertures were used. Boo is slightly resolved on this baseline -- the mean visibility amplitude is 0.25. Note that the trolley sweeps used when recording these data were somewhat shorter than those normally used for visibility amplitude measurements. data is plotted in Figure 5.1). The target star must be sufficiently bright for fringes to be visible in the raw data, above the photon and read noise. In good seeing conditions, this is possible for stars with J 2 0. The search procedure need only be carried out when starting observations of a new star, as the offsets for positions up to 10 degrees apart on the sky do not differ by more than 100 Åm. Hence changes in the delay offsets can be tracked as the target moves across the sky.

5.2.3

Recording fringe data

All-on-one beam combiners were chosen for COAST because they give the best signal-to-noise for simultaneous measurement of all baselines. However, it may not be possible to measure all baselines era is 2.5 data (the shown in at once without introducing systematic errors. The fastest sampling rate for the IR camkHz. Recall the effects of the atmosphere on the power spectra of pupil plane fringe summed power spectrum of some infrared data recorded using a 2.5 kHz frame rate is Figure 4.2 on page 53). Atmospherically-induced path changes while scanning through

the fringes broaden the fringe peaks in the power spectrum, hence the outer parts of adjacent peaks overlap. This leakage between baselines is usually insignificant if we are only interested in the power at the central frequency, as is the case when estimating a closure phase from the data. However, the total power under each fringe peak, above the noise background, must be used as an estimator for the visibility amplitude. If three-baseline data is used for this purpose, we may include some power from an adjacent baseline, or subtract an incorrect background level. The techniques used for extracting visibilities and closure phases from the data are discussed in more detail in Sections 5.3 and 5.4. When observing in the infrared at COAST, three-baseline data, recorded with one of the four beams from the telescopes blocked, is only used for closure phase estimation, at least at 1 3 Åm


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CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

(there will be less leakage at longer wavelengths). Separate single-baseline files, recorded with two beams blocked, are used for visibility estimation. For these files, the frame rate can be reduced, to increase the signal-to-noise at low light levels.

Aperture sizes and integration times The full 40 cm telescope apertures can be used at a wavelength of 1.3 Åm, as they are almost always smaller than 3r0 . However, the following effects conspire to reduce the optimum aperture size:

? ?

The 50 cm siderostat mirrors do not fully illuminate the 40 cm primary mirrors of the telescopes for targets more than 37Ö above the southern horizon. The path compensation trolleys use pairs of mirrors at right-angles to each other to reflect the light beams from the telescopes, hence errors in the directions of the trolley rails affect the beam directions. The autoguider corrects for these beam tilts, but the restoring tilts are made out at the telescopes, hence the beam positions vary by the sidereal motion of the star. 1 mm as the trolleys track

?

The choice of the four target pixels on the IR detector is restricted -- they must be in four different, even-numbered columns. This restriction on the directions of the combined beams can lead to their positions at the camera being wrong by 2 mm, as there are not enough degrees of freedom in the camera feed optics to adjust the position and direction of all four combined beams independently. There is evidence for vignetting of the outer few millimetres of some of the beams, either within the camera or by its feed optics.

The use of smaller apertures also makes the visibilities less susceptible to variations in seeing conditions (see below). Aperture sizes of 240 mm, defined by a 15 mm stop at the camera lens, are now generally used for 1.3 Åm observations at COAST. The maximum sampling rate of 2.5 kHz must be used for closure phase data when a wide bandpass is chosen, to separate the three baselines. For single-baseline data, the optimum rate is that which gives approximately four samples per fringe for the lowest possible fringe frequency (which is that which keeps the fringe peak above the scintillation). A 1 kHz frame rate satisfies these criteria under typical seeing conditions at COAST.

Visibilities Visibility amplitudes measured on unresolved stars are much lower than those obtained using internal light sources, because the interfered wavefronts have been distorted by atmospheric turbulence. The point-source visibility is very sensitive to the seeing conditions when 2-3r0 apertures are used with tip-tilt correction. At COAST, unresolved stars give visibilities in the range 0.35- 0.55 (after removing the known visibility reduction due to the finite integration time) for observations at 1.3 Åm using 240 mm apertures. If the only visibility loss is due to atmospheric turbulence,


5.2. OBSERVATIONS

65

Figure 5.2: Histogram of stellar visibilities, each from two successive sweeps within a 75 s data stream. The target (Capella), although resolved, is very bright, so photon noise has a negligible effect: the signal-to-noise for each visibility measurement is approximately 200:1. the expected value for 2r0 apertures with instantaneous tip-tilt correction is 0.65 (Tango and Twiss 1980; Buscher 1988). With no tip-tilt the theoretical visibility is 0.30. The COAST autoguider uses integration times between 2 and 15 ms. The overall update time includes a further 5 ms to read out four quad-cells on the CCD. Simulations of COAST by Scott (1998) suggest that the visibilities measured in the IR have the expected values, given the delay in making the tip-tilt corrections. The point-source visibility varies on two different timescales. The visibility fluctuates by 25%

of its value between successive scans through the fringe envelope (this effect can be seen in Figure 5.1). The effect is illustrated more quantitatively by Figure 5.2, which shows a histogram of visibilities from successive sweep periods in a 75 s recording. The distribution of visibilities is approximately Gaussian, with standard deviation 0.05 (28% of the mean). The timescale for the visibility variations appears to be short enough for them to be treated statistically. If the underlying distribution does not change significantly on short timescales, it should be possible to measure the visibility to 2% by averaging 156 scans through the fringe envelope (i.e. 39 s of data). The assumption of stationary statistics seems to hold in practice, as data streams longer than 20 s give approximately the same visibility, except in poor seeing conditions. Figure 5.3 shows the cumulative average visibility for the data file used to make Figure 5.2. Here the visibility has converged after 14 seconds. The visibility estimated from 40 s of data can vary by 20-30% during the course of a night, as the seeing conditions change. There may also be small variations in the instrumental visibility,


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CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

Figure 5.3: Cumulative average visibility for the 75 s file used to make Figure 5.2. due to drifts in the alignment. Hence regular calibration observations are needed, to measure the changes in point-source response. Unresolved stars which are close to the target on the sky (ideally within few degrees) must be observed every 10-30 minutes. Even with such calibration measurements, calibration error is almost always the dominant uncertainty in the visibilities when 2-3r0 tilt-corrected apertures are used.

Observing sequence As the frequency of calibrator observations is increased, observing efficiency falls. This is because it takes several minutes to slew the telescopes between the target and the calibrator, acquire the star, and centre the fringe envelopes in the trolley sweeps, before fringe data can be recorded. Therefore the procedure outlined below has been adopted for COAST observations. Two experienced observers can perform one iteration in 30 minutes (i.e. taking 15 minutes between corresponding source and calibrator observations). However, it is often necessary to re-centre the fringes part-way through each half of the sequence, which increases the time required. 1. Acquire calibrator, centre fringes on all six baselines 2. Record 30-60 s of 1 kHz sampled data on each baseline separately 3. Record 60-100 s of 2.5 kHz sampled data on each triangle of baselines separately 4. Slew to source, centre fringes on all six baselines 5. Record 30-60 s of 1 kHz sampled data on each baseline separately 6. Record 60-100 s of 2.5 kHz sampled data on each triangle of baselines separately 7. Repeat


5.3. VISIBILITY AMPLITUDE ESTIMATION

67

The calibrator is acquired first so that the offsets from the predicted path delays can be determined when the fringe visibility is high. If the source is within 5Ö of the calibrator, the fringe envelope will usually still be within the sweep after switching to the source.

The following procedure is used for centring the fringe envelope within the trolley sweep when the fringes are difficult to see in the raw data. A short sequence of data (5-10 s) is taken in "local" mode. Each sweep is divided into five time segments, and the power spectrum of each segment calculated. The power spectra from corresponding segments of successive sweeps are added, to form a multi-summed power spectrum ("multi-sum" for short). The amount of fringe power in each segment indicates how much of the fringe envelope lies, on average, within a particular window of path delay. The "observe" program on the infrared PC can record data and calculate its multi-sum with just a single key-press from an observer. Two example multi-sums are shown in Figure 5.4. The lower plot is a multi-sum of three-baseline data, in which the three fringe envelopes overlap well. The time taken to scan through the fringe envelope is inversely proportional to the fringe frequency, so the lowest frequency peak can be seen in five time windows, whereas the highest frequency peak is only visible in the central window. Closure phase data files are longer than the 50 s used for single-baseline data, to improve the signal-to-noise of the measurement. The atmospheric path changes mean that the three fringe envelopes do not always overlap, hence many of the sweeps give no useful signal. One path-compensation trolley is in motion when recording single-baseline data at 1 3 Åm. The moving trolley executes a sawtooth motion (see Figure 3.2 on page 20) with 0.5 s period and 87 Åm amplitude in path delay, scanning fringes past the detector at 270 Hz. For three-baseline data, two trolleys are moving. Their motions are sawtooths with the same 0.5 s period and amplitudes of 173 Åm and 260 Åm respectively. The effective path variation on the baseline between the two moving trolleys has 87 Åm amplitude. The fringe frequencies for the three baselines are 270, 540 and 810 Hz. The same 0.5 s sweep period, long enough to always catch the entire envelope of 270 Hz fringes, is used for both types of data. This length is required for visibility amplitude measurements using this fringe frequency, but substantial observing time is wasted when recording closure phase data, as the three fringe envelopes usually only overlap in the middle 10-20% of the sweep (as in Figure 5.4). Currently, changing the sweep duration is very time-consuming, hence this is not done. The introduction of fringe envelope tracking will enable shorter sweeps to be used for both types of file.

5.3

Visibility amplitude estimation

The procedure used to estimate the visibility amplitude from single baseline data is the "integration method" discussed by Burns (1997). Consider the schematic power spectrum of such data in Figure 5.5. There are three features in the spectrum:

?

A fringe peak centred on 270 Hz. The peak is broadened from its natural width (set by the


68

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

Figure 5.4: Multi-summed power spectra (see text for explanation) of 5 s of singlebaseline data on the Mira variable star U Cyg (top), which had a J band magnitude of 2.5 at the epoch of observation, and 5 s of three-baseline data on Capella (bottom). The seven windows in the bottom plot only cover the central 50% of the sweep duration. The lower horizontal axes are frequency in Hertz, and the vertical axes are proportional to the power at each frequency.


5.3. VISIBILITY AMPLITUDE ESTIMATION

69

11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 F 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111 11111111111111111 00000000 00000000000000000 111111111 000000000 11111111 11111111111111111 00000000 00000000000000000 111111111 000000000 11111111 00000000 111111111 000000000 11111111 00000000a c d be f
Frequency
Figure 5.5: Schematic power spectrum of single-baseline data, to illustrate the integration method for visibility amplitude estimation. Only part of the low-frequency scintillation peak is shown. effective bandwidth) by fringe phase fluctuations due to the atmosphere

Power

? ?

A peak at 0 Hz due to the mean signal, broadened by scintillation A flat background, due to photon and read noise

In practice, the background has a negative slope in the region of the fringe peak, due to a contribution from the tail of the scintillation peak. The aim of the analysis procedure is to estimate the total fringe power above the noise background. This is done by calculating the integral
b

F
a



PÄ

Å



N d

Å

(5.1)

where the limits a b include the tails of the fringe peak. The background level subtracted, N , is estimated from the mean noise in windows either side of the peak:
Úd Úf
Ç
e

N

PÄÅd 2Äd cÅ
c

PÄÅd 2Ä f eÅ

(5.2)

These windows are chosen to be clear of the edges of the fringe peak, and equidistant from its centre. They must be wide enough for the calculation of N to be unaffected by the noise in the background, yet narrow enough for the background to be approximated by a straight line in the interval c f . The fringe power F is normalised by the square of the mean signal rate S, to givean estimateofthe ?? mean square visibility V 2 . There is an additional normalisation factor to take into account the fact that the fringes only occupy a small fraction of each sweep ( 0 09 for stellar observations through our J band filter):
?

V

2

?

F S
2

(5.3)


70

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

In practice the power spectrum is calculated by an FFT, which introduces other numerical factors into the above expression. The visibility is calculated for all four output channels (corresponding to the four outputs of the beam combiner). These visibilities are calibrated by dividing them by the equivalent estimates from the closest calibrator observation on the same baseline. The calibrated visibilities from the four channels are averaged. The "model fitting" method described by Burns is not used for J band data, because the fringe peaks are not well fit by a Gaussian function, except in very poor conditions, because the effective passband is asymmetric. In good conditions, the atmospheric coherence time may only be half the time taken to scan through the fringe envelope (which is 64 ms for 270 Hz fringes with the 12% effective bandwidth). The fringe peak will be broadened, but the underlying asymmetry will still be apparent.

5.4

Closure phase estimation

The closure phase is estimated from a data stream containing fringes on three baselines, at three different frequencies. The data stream is divided into short segments, chosen to be approximately twice the coherence time of the fringes1 , and corresponding to an integer number of fringes. For each baseline, the data segment is multiplied by appropriate sine and cosine weighting functions, and integrated. The phase of the weighting functions is calculated from the known trolley motion and the central wavelength of the passband employed. This integration yields the complex visibility of the fringes on each of three baselines, for each segment of data. The triple product for each segment is calculated, and corrected for photon noise bias (Wirnitzer 1985). We have not yet explored the regime where read noise bias becomes important. The triple product is averaged over all of the segments in the data file, and the argument of the mean triple product is adopted as the estimate of the closure phase. Woan and Duffett-Smith (1988) showed that this is the best way of averaging noisy phases. The closure phase estimates from the two channels (recall that only two pixels of the NICMOS detector can be read at 2.5 kHz) are also averaged in this way. Only those parts of each sweep containing significant fringe power on all three baselines should be analysed. These regions can be identified from a multi-sum of the data. The multi-sum also indicates how well the three fringe envelopes overlap, and hence whether there is likely to be a systematic error in the closure phase when the passband is asymmetric. Poor overlap also decreases the signal-to-noise, as there will only be significant signal (i.e. triple product amplitude) where all three fringe envelopes overlap. A closure phase can be calibrated by subtracting the closure phase measured for the calibrator on the same triangle of baselines. This is unwise for J band data, as there may be errors in
The coherence time is defined to be the time in which the autocorrelation of a data stream containing fringes falls by a factor of two.
1


5.5. CAPELLA OBSERVATIONS AND DATA REDUCTION

71

either the source or calibrator closure phases due to the effect of poor overlap with an asymmetric passband. The overlap will be different in the two cases, so the errors will be uncorrelated. There is no evidence for other systematic effects which will be common to source and calibrator, so subtracting the calibrator value will simply increase the noise on the closure phase measurement.

5.5

Capella observations and data reduction

The spectroscopic binary star Capella ( Aurigae) was observed with COAST on the night commencing 25 October 1997. Four elements of the COAST array were used, in a configuration with a 9 m maximum baseline. No aperture stop was employed in front of the infrared camera, in which the J band filter was installed. A total of five sets of observations were made for both Capella and the calibrator star Aurigae. Each set consisted of a single-baseline file, 30-75 s long, for each of the six interferometer baselines, and a three-baseline file (30-75 s) for each of the four different closure phases. The observations were made over a seven hour period, so Earth-rotation gave good Fourier plane, or uv, coverage (Figure 5.6).

Figure 5.6: uv coverage for Capella observations Visibility amplitudes and closure phases were estimated using the procedures outlined in Sections 5.3 and 5.4. The coherence times at 1.3 Åm were 15 ms, so 22 ms segments were used in the closure phase estimation. The calibrated visibility amplitudes were in the range 0.19-1.2. Their quality was limited by the calibration: the signal-to-noise in the summed power spectrum of 30 s of data from Capella (defined as the ratio of the height of the fringe peak to the RMS noise in the background) always exceeded 10:1, and was as high 2000:1 when the source was unresolved.


72

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

At the time, we were not experienced in recording data with the infrared system, so the delay between corresponding source and calibrator observations was sometimes over 50 minutes. Error bars equal to 10% of the visibility were assigned to the data. The calibrator closure phases were within a few degrees of zero when the three baselines were all well centred in the trolley sweeps. However, values of 0 ? 20Ö were obtained when the overlap was poor, yet good enough to yield significant triple-product amplitudes (about half of the values obtained for good overlap). The asymmetric passband clearly led to systematic errors in the closure phases. Therefore, for the reasons discussed in Section 5.4, the closure phases measured on Capella were not calibrated. Error bars of either 5Ö , 10Ö , or 20Ö were assigned to the closure phases on the basis of the degree of overlap of the three fringe envelopes. The degree of cross-talk between the baselines in a particular data file could be estimated by performing the standard closure phase analysis on sections of the data stream which only contained fringes on one or two baselines. The resulting triple product amplitudes were typically 1-2% of the values obtained from segments of identical length containing fringes on all three baselines. The two components of Capella emit unequal fluxes at a wavelength of 1.3 Åm, so a range of closure phase values were observed. These are in fact the first closure phase measurements at infrared wavelengths ( 1 Åm) from a separated-element interferometer.

5 .6

Image reconstruction

For any set of observations with a real interferometer, the Fourier plane will not be sampled evenly. From Fourier theory, the result of inverting the Fourier plane data to produce a map of the sky brightness distribution will be a "dirty map", i.e. the true brightness distribution convolved with the transform of the sampling function (known as the "dirty beam"). In practice, the dirty beam usually has large sidelobes, which makes the true sky brightness distribution difficult to infer from the dirty map. Several iterative processes for deconvolving the dirty map have been developed, for application to radio astronomy. The two most popular schemes are MEM (Sivia 1987) and CLEAN (Hogbom ? 1974). If the visibility phases are unavailable, as in VLBI and optical interferometry, the closure phase can be used as a constraint in the mapping process, via the technique of selfcalibration (Pearson and Readhead 1984). In this case the CLEAN procedure takes the form:

1. Take a trial sky distribution (the starting model). 2. Self-calibrate: assign phases to the measured visibility amplitudes, such that the 2 between the resulting complex visibilities and those of the model is minimised, and the phases around each triangle of baselines add up precisely to the measured closure phases. 3. Invert this data to produce a dirty map. Invert the sampling function to produce the dirty beam, and measure its central lobe. Use a Gaussian representation of the central lobe as the


5.6. IMAGE RECONSTRUCTION "CLEAN beam".

73

4. CLEAN: Iteratively subtract a fixed multiple ( 1) of the dirty beam, located at the peak flux position, from the dirty map. Continue for a fixed number of iterations, or until no feature remains in the residual map which exceeds a threshold flux. The CLEAN map consists a set of delta-functions at the peak flux positions, convolved with the CLEAN beam. 5. Construct a new model, consisting of a delta-function at each peak flux position. 6. Repeat from step 2, until the CLEAN map no longer changes significantly between successive cycles. This procedure makes assumptions about the data which do not usually apply to data from optical/NIR interferometers. The errors on the closure phases are assumed to be equal and insignificant. Neither is true of COAST data. Also, it is implicitly assumed in the self-calibration that corresponding visibility amplitudes and closure phases were measured at the same time. This would be possible at COAST if the detectors could be sampled more frequently. However, for the observing procedure described in Section 5.2.3, fifteen minutes may span the set of observations on all six baselines and four baseline triangles. Depending upon the source structure and baseline lengths, Earth rotation may change the visibilities and closure phases significantly over this time period. Hence self-calibration may not produce a set of phases which are appropriate to the true source morphology. An infrared image of Capella was reconstructed from the measurements described above. The "Difmap" software (Shepherd et al. 1994; Shepherd 1997) was used, which employs CLEAN deconvolution as part of a technique called difference mapping. The first cycle of the procedure is identical to that described above, but in subsequent cycles, the CLEAN deconvolution is performed on the difference map, which is the map obtained from the difference between the observed and model visibilities. The model components resulting from the deconvolution are added to the previous model. As the Fourier inversion to generate a map is linear, the difference map is equal to the original dirty map minus all of the accumulated CLEAN components. The finished map is the final difference map (which should just contain noise) plus all of the CLEAN components accumulated in previous cycles. Difference mapping allows the user to intervene part-way through the reconstruction process e.g. by deconvolving different parts of the map at different stages. The map of Capella is shown in Figure 5.7. This is the first infrared image from an interferometer with separated telescopes. It was obtained from a point-source starting model after four iterations of the self-calibrate, invert, CLEAN cycle within Difmap. As mentioned above, each set of six visibility and four closure phase measurements had to be assigned a single time of measurement. Both the visibilities and closure phases change rapidly with the hour angle of a resolved double source, so the assignment of times was carried out using the predicted separation and position angle of Capella (from the orbit of Hummel et al. 1994) to determine which quantity was changing most rapidly while each set was being measured. The time when this datum was measured was chosen as the nominal time for the entire set.


74

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS

Figure 5.7: Reconstructed image of Capella, from data taken on 25 October 1997 at a wavelength of 1.3 Åm. The contours are at -4, 4, 10, 20, 30, ..., 90% of the peak flux. The map has been restored with a circular beam for clarity. The binary structure of Capella is unambiguous in the map, but the array baselines were too short to resolve the stellar disk of either component. Capella is an unequal binary at this wavelength (with flux ratio 1.4:1), whereas it is nearly equal at 0 8 Åm (Hummel et al. 1994; Baldwin et al. 1996). The root-mean-square noise in the map is approximately 2% of the peak flux (i.e. the dynamic range is 50:1). We would expect this ratio to be given by I I V V
ä

NV

(5.4)

where I I is the dynamic range, V V is the mean fractional error in the visibilities, and NV is the number of visibility measurements. For these observations, NV 26 (the best map was obtained by excluding four points which were judged to have been poorly calibrated) and I I 0 1, hence we would expect I I 0 02, which is exactly the ratio observed. The separation and position angle of the two components in the map may be compared with predictions from the orbit of Hummel et al., which was determined from observations of Capella with the Mark III interferometer on 38 nights between 1988 and 1992. The predicted parameters, plus those measured from the map in Figure 5.7, as well as the results of fitting a two-component model to the data, are shown in Table 5.2. The parameters from the model fit discussed below are more accurate, but the position and separation from the map are in agreement with the previouslydetermined orbit. However, there is a disagreement at the 2 level between the position angle in the map and that obtained by model-fitting. In order to determine whether the discrepancy was due to the errors in the closure phases (ignored by the mapping algorithm), a data-set consisting of the measured


5.7. MODEL FITTING

75

visibility amplitudes and the closure phases predicted by the best-fit model was used to reconstruct an image. The position angle measured from this image was almost identical to that obtained from mapping with the original dataset, so the discrepancy cannot be due to a systematic effect in the closure phases. The discrepancy may just be the effect of noise in the measurements of the position angle.

5.7

Model fitting

Quantitative information about the astronomical target can be obtained by fitting simple models for sky brightness distribution. Such models can be fitted to COAST data using the software programs "modfit" and "erfit", which are two of the Caltech VLBI programs (Pearson 1991), modified by Dr. Joseph Lehar and by myself. Modfit finds the vector of model parameters x, for a simple model of the sky brightness distribution (consisting of delta-function, Gaussian and uniformly-bright disk components), which minimises
2

1 NV Ç N


ind

i1



NV

ÄVi



Vmod Äx 2 V

ÅÅ

2

Ç



i1



N

Äi



mod Äx 2

ÅÅ

2

(5.5)

where V and are the visibility amplitude and closure phase data2 , and Vmod ÄxÅ and mod ÄxÅ are the model predictions. and are normally chosen to be unity, but can be altered if the relative scaling of the visibility amplitude and closure phase errors is incorrect. All of the model parameters or a sub-set of parameters may be varied. An initial model must be supplied to specify the starting point for the search. It is advisable to try several different starting models, to increase the chance of finding a global minimum. A conjugate-gradient minimiser is used to find the approximate position of a minimum, then the program carries out a "brute-force" search of the parameter space in this region, to refine the position. Uncertainties in the model parameters are estimated by the program erfit, from the curvature of the 2 surface at the minimum, taking into account the correlations between different parameters. The error bars on the data are scaled so that the minimum value of 2 (as defined by Equation 5.5) is equal to unity. This removes the need to accurately determine the scale of the errors, which can be difficult for calibration errors in visibility amplitude data. However, if the approximate scale of the errors is known (as is the case with COAST IR data), the minimum value of 2 is a useful measure of the goodness-of-fit. Accurate values for the separation and position angle of the two components of Capella were obtained by fitting a two-component model for the sky brightness distribution. The components of the binary have angular diameters 6.4 mas and 8.5 mas (Hummel et al. 1994), hence both would have been essentially unresolved, even on the longest baseline. Consequently, the two components were modelled as delta-functions (the use of uniform disk components did not give a better fit). The minimisation yielded
2

2 min

1 9, with 2 for the visibilities and closure phases separately

N ind is the number of independent closure phase data. This may be less than N if more than one closure phase is assigned the same time of measurement, because some closure phases are linear combinations of others.


76

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS Table 5.1: Parameters for best-fit two-component model for Capella. Both components were modelled as delta-functions, and the position of the first component was fixed at the origin. Flux 0 386 0 533

? ?

r (mas) - 55 22

(Ö) - 54 51

0 014 0 014

?

0 24

?

0 24

Table 5.2: Comparison of observed separation and position angle for Capella with the predictions of the orbit derived from observations with the Mark III interferometer (Hummel et al. 1994). Separation (mas) Mark III prediction COAST map COAST model fit 55 0 55 ? 1 55 2 Position angle (Ö) 55 0 53 2 ? 0 5 54 5

?

02

?

02

being 2.7 and 0.9. These values indicate that the uncertainties in the closure phases (typically 10Ö) have been accurately estimated, but that the calibration errors in the visibilities are 16%, rather than the 10% assigned. The poor calibration is due to long periods (sometimes 50 minutes) between source and calibrator observations. The parameters of the best-fit model are listed in Table 5.1. Figure 5.8 is a plot of the visibilities and closure phases predicted by this model, against time on the night in question, with the observed data also shown. The separation and position angle from the model fit are entirely consistent with the values predicted from the orbit of Hummel et al. (1994) (see Table 5.2). The flux ratio FAa FAb from the best-fit model is 1 38 ? 0 06. The component Capella Aa is the cooler of the two, and appears brighter in the infrared. This enables us to identify the two components. Their relative positions, as inferred from the COAST data, are consistent with the well known orbit, which confirms that the closure phases from COAST have the correct sign.

5.7.1

Effective temperatures

Hummel et al. (1994) obtained effective temperatures for the two components from their luminosities (calculated using a distance from the orbital elements and a bolometric correction to the V band apparent magnitudes) and measured angular diameters. These were found to be consistent with the observed spectral types and B V and ÄV I ÅC colours. V J colours can be obtained from the COAST measurement of the J band flux ratio, the V band flux ratio from the Mark III observations, and the V and J magnitudes of both components


5.7. MODEL FITTING

Figure 5.8: Best-fit two-component model for Capella. The left-hand plot shows the visibility amplitude on all six baselines, and the right-hand plot shows the four closure phases. The model parameters are listed in Table 5.1. 77


78

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS Table 5.3: Synthetic V son (1989).



J colours for G- and K-type giant stars, from Bell and Gustafs-

Teff (K) 4500 5000 5500 6000

V



J

1.84-1.97 1.51-1.52 1.22-1.24 0.98-1.02

combined (from Johnson et al. 1966), for example: VAa



JAa

VAaÇAb



JAaÇ

Ab



2 5log10

1 Ç FJ 1 Ç FV

Aa Aa

FJ FV

Ab Ab

(5.6)

tions by Bell and Gustafsson (1989). Typical V J colours for model G- and K-type giant stars with effective temperatures from 4500 to 6000 K are shown in Table 5.3 (the synthetic colours were calculated for various values of the surface gravity and composition). The Teff grid is rather coarse, but nevertheless I can conclude that the measured V J colours are consistent with the more direct effective temperature determination of Hummel et al., who obtained Teff component Aa and Teff 5700 ? 100 K for Ab. 4940

The derived V J colours are 1 61 ? 0 03 for the component Aa, and 1 11 ? 0 04 for the component Ab. These may be compared to the colours obtained from synthetic spectral energy distribu-

?

50 K for the

5.8

Conclusions

These latest COAST measurements are entirely consistent with the orbit of Hummel et al. (1994), but do not permit the orbit to be refined. Much more data would be required for this, preferably using baselines long enough to resolve the disks of the component stars. The measured J band flux ratio provides independent confirmation of the effective temperatures derived by Hummel et al. The value of these observations is that they provide a stringent test of the COAST infrared correlator and its associated data reduction procedures. The Capella data demonstrate that a range of visibility amplitudes can be measured reliably. At least for bright targets such as Capella, their quality is limited by the calibration. An unequal binary star with a known orbit is an ideal target for testing the quality of the closure phases. We have identified errors of 10Ö which are caused by a combination of an asymmetric 1.3 Åm bandpass and imperfect overlap of the fringe envelopes. The Capella closure phase data prove that the error on an individual datum can be estimated accurately, and that there is no systematic effect on images reconstructed using a reasonable number of closure phase measurements. This source of error should be removed when more carefully-chosen infrared filters are used. COAST is the first interferometer to use separated telescopes to measure closure phases at IR wavelengths. An image of the expected quality was reconstructed using these data, proving that


5.8. CONCLUSIONS

79

COAST can image successfully in the infrared. For a simple binary target, only 26 visibility points were needed to produce a image. This bodes well for more demanding targets, such as resolved stellar disks. Observations of such a target, the red supergiant star Betelgeuse, are described in the next chapter.


80

CHAPTER 5. STELLAR OBSERVATIONS, DATA REDUCTION AND ANALYSIS


Chapter 6

The Wavelength-dependent Morphology of Betelgeuse
6 .1 Introduction

The red supergiant Betelgeuse ( Orionis) has the largest apparent diameter of any star other than the Sun. It is large enough to be resolved at the diffraction limit of conventional optical telescopes, and so has been the target of many high-resolution imaging experiments. Non-redundant masking (Baldwin et al. 1986) experiments by the Cambridge group (Buscher et al. 1990; Wilson et al. 1992, 1997; Tuthill et al. 1997) have revealed asymmetries on the stellar disk, which have been modelled as a small number (1-3) of bright features contributing 10-30% of the total flux, superimposed on a symmetric disk. Asymmetries have also been inferred from speckle interferometric data (Kluckers et al. 1997). The bright spots have been found in many different positions ? on the stellar disk, and their locations and brightnesses change on a timescale of weeks or months. The favoured explanation for these features is that they are the giant convective granules predicted by Schwarzschild (1975), who suggested that the evolution of these granules is the cause of the irregular photometric variability of red giants and supergiants. However, Wilson et al. (1997) found no relationship between variations in hotspot flux and changes in the total V band flux of Betelgeuse, although radial pulsation may also have affected the V band brightness. It is also unlikely that the bright spot found by conventional imaging in the ultraviolet with the Hubble Space Telescope (Gilliland and Dupree 1996) was directly caused by convection, as the emission must have originated from the chromosphere, where radiative energy transport dominates. Uitenbroek et al. (1998) derived an unexpectedly large rotation rate for Betelgeuse from spatially resolved spectroscopy with the Hubble Space Telescope, and asserted that the bright feature seen in the ultraviolet was located at the pole, and hence was the signature of a global shock wave caused by stellar pulsation. Betelgeuse exhibits emission in the thermal infrared from several expanding shells of dust and gas, the inner radius of the closest shell being 46 stellar radii1 (Danchi et al. 1994). Bloemhof et al.
1

Henceforth I shall use r? for the stellar radius.


82

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

(1985) found that the dust emission was asymmetric. Mass loss is evidently neither continuous nor isotropic, but the physical process driving the mass loss, and hence the origin of the asymmetry, is unknown. Asymmetries have also been identified in the chromospheric emission originating at 3-5r? (Hebden et al. 1986, 1987; Gilliland and Dupree 1996), and recently, in thermal emission at radio wavelengths, from high density gas with a temperature of 3500 K, located at 2r? (Lim et al. 1998). There is also evidence that dust can form at 4r? (Bester et al. 1996). No clear link between asymmetries at or near to the stellar surface and asymmetries in the circumstellar dust and gas has yet been established. In an attempt to pin down the physical mechanism responsible for the observed asymmetries in the stellar disk, a campaign of contemporaneous interferometric observations at COAST and the William Herschel Telescope (WHT) on La Palma was carried out in October and November 1997. The unique features of this programme were higher spatial resolution than previous observations, using COAST baselines up to 9 m in length, and extension of the observing wavelength into the near infrared.

6.2

Observations and data reduction

Interferometric observations of Betelgeuse were made during October and November 1997 in three different wavebands, using both COAST and the William Herschel Telescope. The centre wavelengths and widths of the passbands were 700/10 nm, 905/50 nm, and 1290/150 nm. Details of all of the observations are given in Table 6.1. Table 6.1: Log of observations at COAST and the WHT. Dates refer to the start of the night of each set of observations. Nvis and Ncl are the number of visibility and closurephase measurements made. An identical number of visibility measurements were made for the calibrator star. Each "measurement" corresponds to 30-100 seconds of observations. Date 97/10/21 97/10/24 97/10/31 97/11/11 97/11/12 97/11/15 97/11/16 97/11/21 Telescope COAST COAST COAST COAST COAST WHT WHT COAST Baseline range (m) 2.0-8.9 2.3-7.5 2 1 3 0 .3-6.9 .6-8.8 .1-6.9 .3-3.7 700 nm Nvis Ncl - - - - - 90 - - - - - - - 90 - - 905 nm Nvis Ncl - 10 5 - 13 - 60 29 - 4 1 - - - 60 9 1290 nm Nvis Ncl 18 - - 53 - - - - 7 - - 36 - - - -

0.3-3.7 1.6-7.7


6.2. OBSERVATIONS AND DATA REDUCTION

83

6.2.1

Measurements with COAST at 1.3 Åm

Betelgeuse was observed with COAST in the infrared on 97/10/21 and 97/11/11, through the 1.3 Åm J band filter. The configuration of the COAST array was the same as that used for the Capella observations described in the previous chapter, with a maximum baseline of 9 m. In order to make measurements on short baselines, observations were started four hours before Betelgeuse reached the meridian, so that the projected lengths of some of the baselines were very short. The elevation of the star was below 25Ö at these times. The observing procedures were as described previously, i.e. the visibility amplitude was measured on each baseline in succession, followed by measurements of each of the four closure phases. When the visibility was very low, on long baselines, the fringes were centred by boot-strapping: the fringes were centred on the two shortest baselines which formed a triangle with the desired long baseline. This automatically ensured that the fringe envelope was centred in the trolley sweep on the long baseline. For the initial observations on 97/10/21, Ori was used as the calibrator. This star is 15Ö lower in declination than Ori, hence there were large differences in elevation between source and calibrator at some times during the night, resulting in slightly uncertain calibration of some of the visibility amplitude measurements. A different calibrator, Å Gem, was used on 97/11/11. Although slightly resolved (the apparent diameter at 2.2 Åm is 13.5 mas, Di Benedetto and Rabbia 1987), this star is very bright in the infrared (J 0 8), which made finding and centring the fringes at extreme hour angles very easy. Because of this, the observing procedure (Section 5.2.3) was carried out very efficiently. The time between corresponding source and calibrator observations was typically 15-20 minutes, and always less than 30 minutes. The raw data streams were analysed using the methods described in Chapter 5, to yield four independent estimates of each uncalibrated visibility amplitude (from the four outputs of the beam combiner) and two independent estimates of each closure phase (from two of the beam combiner outputs). The visibility amplitudes were calibrated by dividing them by the visibility obtained from the corresponding calibrator observation, corrected for the finite size of the calibrator. The formal uncertainties in the calibrated visibilities were estimated from the standard deviation of the values from the four independent IR channels. Because of the brightness of Orionis, these formal uncertainties in the visibilities, due to photon and atmospheric noise, were negligible compared with the calibration errors, except when the calibrated visibility was below 0.1. The calibration errors were estimated (from the scatter in successive observations on the same baseline) to be 15% of the visibility for the data from 97/10/21, and 10% for the measurements from 97/11/11. The closure phases were not calibrated, for the reasons discussed in Section 5.4. They were assigned uncertainties between 5Ö and 20Ö, depending upon the degree of overlap of the three fringe envelopes and the value of the mean triple product amplitude.


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CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

6.2.2

Measurements with COAST at 905 nm

Near-contemporaneous measurements were made at a wavelength of 905 nm, using the visiblewavelength beam combiner and avalanche photodiode detectors at COAST. The passband was defined by interference filters mounted in front of three of the four APDs, all with central wavelength 905 nm and FWHM 50 nm. For these observations a stop was placed in front of each APD, to restrict the effective telescope aperture to 16 cm ( 2r0 at this wavelength). Data were recorded on four nights: 97/10/24, 97/10/31, 97/11/12, and 97/11/21. The array configuration was not optimal for observations of Betelgeuse at this wavelength. It was difficult to find fringes on one of the three shortest baselines in the array (the 5.5 m baseline between the Centre and West elements), except at very early or late hour angles. Consequently, most of the visibility amplitude and closure phase measurements were made on a single triangle of baselines, in which the longest baseline was 7 m. Only four visibility measurements were made on longer baselines, with the longest projected baseline being 7.8 m. The observing sequence was the same as that used for the IR observations. The visibility amplitude was measured on each baseline (on which fringes could be detected) separately. Afterwards, threebaseline files were recorded to measure the closure phases. Observations of Betelgeuse were interleaved with observations of unresolved calibrator stars. A number of different calibrators were used. Ori, which is close to Betelgeuse on the sky, was used whenever possible. In conditions of low atmospheric transmission, either Ori or CMi was used. All three calibrators would have been completely unresolved on the longest baseline at 905 nm. almost always give values of 0 ? 2Ö in good conditions. Hence, to make the most efficient use of the available observing time, no closure phase data was taken on the calibrator stars. Closure phase measurements on unresolved stars using the COAST visible-wavelength correlator

The standard data reduction procedures (Burns 1997; Burns et al. 1997), very similar to those used for the infrared data, were employed, to yield visibility amplitudes with formal errors in the range 2-10% of the values and closure phases with uncertainties of 5-10Ö (most of these measurements involved at least one baseline with a low fringe visibility). The calibrator visibilities appeared to be more sensitive to the seeing conditions at this wavelength than at 1.3 Åm. Depending upon the proximity on the sky and in time of the source and calibrator observations, calibration uncertainties of 10% or 20% were assigned to the data.

6.2.3

Measurements with the WHT

Observations were made of Betelgeuse on 15 and 16 November 1997 at the Ground-based High Resolution Imaging Laboratory (GHRIL) of the William Herschel Telescope, using the technique of non-redundant masking (Baldwin et al. 1986; Haniff et al. 1987). The apparatus differed slightly from that used in previous experiments, in that it allowed for the insertion of a polaroid in the pupil plane, in a fixed orientation with respect to the sky. Otherwise the set-up was equivalent to that


6.3. RESULTS

85

described by Buscher et al. (1990). The apparatus incorporated a pupil mask (in a re-imaged pupil plane), opaque except for a linear array of holes with non-redundant spacings, to select a number of interferometer baselines. The resulting fringe pattern was imaged onto a CCD detector, with the fringes aligned in the parallel direction. On-chip binning was used to compress the fringe pattern into a single row of pixels, allowing exposure times of 12 ms ( t0 ) to be used. The aperture mask and CCD were mounted on a motorised turntable, to allow the mask to be rotated to any desired angle with respect to the sky. A five-hole mask was used, which gave 10 baselines in the range 0.3-3.7 m. Measurements were made in two bandpasses: 700/10 nm as well as the same 905/50 nm band used at COAST. Consecutive observations were made with and without the polaroid in place at 700 nm, whereas observations at 905 nm were only made without the polaroid. I will only discuss the 700 nm data taken in unpolarised light in this chapter. In order to sample the Fourier plane well, the aperture mask was rotated relative to the sky and five thousand short-exposure images were recorded at each of six or nine different position angles, for both the source and the calibrator Ori. The image scale and orientation were determined by observations of two close visual binary stars with well-known orbits. The errors on the scale and orientation were a few percent and 2Ö respectively. The fringe data were reduced using standard procedures (Buscher et al. 1990; Haniff et al. 1987) to give estimates of the visibility amplitude on all 10 baselines and the closure phase on 10 (linear) triangles of baselines. The errors on the closure phases were just 1-3Ö, while the quality of the visibility amplitudes was again limited by the calibration. The calibration uncertainties were unusually large, being 30% of the visibility.

6.3
6.3.1

Results
Fourier data

Simple inspection of the visibilities and closure phases reveals that, at the epoch of observation, the apparent morphology of Betelgeuse at 1.3 Åm was very different to that seen at the shorter wavelengths. The infrared data (visibilities and closure phases measured on 97/11/11 are shown in Figure 6.1) shows the simplest behaviour: the visibility is approximately equal to one on short baselines, and falls monotonically with increasing baseline length, until it reaches a minimum at about 8 m. At the same point, the closure phase flips from 0Ö to 180Ö. This behaviour is characteristic of a symmetric disk. The visibility then rises again with increasing baseline. The visibilities beyond the minimum are all slightly below the values predicted by the uniform disk model which fits the shorter baseline data (the visibility curve for this model is plotted in Figure 6.1), which shows that the disk is limb-darkened. The smallest visibility amplitude measured was 0.025, at a baseline of 8.1 m. Such small visibilities are obviously incompatible with tens of per-cent of the total flux being emitted from unresolved features distributed asymmetrically on the stellar surface.


86

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

Betelgeuse 1290nm 1

0.8 Visibility amplitude

0.6

0.4

0.2 97/11/11 42.6mas UD 0 2000 4000 6000 Projected baseline /mm Betelgeuse 1290nm 210 180 Closure phase /degrees 150 120 90 60 30 0 0 2000 4000 6000 Longest projected baseline /mm 8000 97/11/11 8000

0

Figure 6.1: Visibility curve and corresponding closure phases for Betelgeuse at 1.3 Åm. Note that the visibility amplitude goes through a zero at ure phase changes from 0 to 180Ö, as expected for a disk model shown is the result of a fit to baselines 7 baselines beyond the null are systematically smaller than the same baseline as the clossymmetric disk. The uniform 5 m only. The visibilities on the predictions of this model,

which implies that the stellar disk appears darkened towards the limb.


6.3. RESULTS

87

The visibility data at 905 nm are plotted in Figure 6.2. The COAST data from 97/10/24, 97/10/31, and 97/11/12 were found to be self-consistent, and consistent with the WHT data at the same wavelength from 97/11/16 (the IR data sets from 97/10/21 and 97/11/11 are also consistent with each other). However, the visibilities measured on shorter baselines on 97/11/21 are not consistent with the other data, and imply a 7% (4) increase in apparent size between 16 November and 21 November. Both 905 nm visibility curves exhibit much more scatter on short baselines than the IR curve. Nevertheless, it is clear from the long baseline data that the departure from a uniform disk model is much greater than at 1.3 Åm. The small visibilities ( 3%) near the baseline corresponding to the secondary maximum of the uniform disk visibility function suggest a strongly limb-darkened radial intensity profile. In addition, the closure phases show an unambiguous signature of asymmetry. Most of the COAST measurements were on a single triangle of baselines. The values of this closure phase from 97/11/21 are plotted against the projected length of the longest baseline of the triangle in Figure 6.2. Its value is 20Ö on baselines shorter than the location of the visibility minimum near 5.5 m, and 130Ö beyond the minimum. Even if there were a constant bias in this closure phase, a transition of 180Ö, not 110Ö, would be expected if the stellar disk were reflectionsymmetric. The closure phase changes by less than 20Ö while the position angle of the longest baseline in the triangle changes by 30Ö, which indicates that the asymmetry is located near to the centre of the stellar disk. The unique feature of these data is that the small visibilities on long baselines constrain the amount of flux which is present in unresolved features. This can be seen from the results of model fitting, described in Section 6.3.3. The closure phases from the previous nights were measured when the projected lengths of the baselines were very short. Hence the largest value is 40Ö. The closure phase data are entirely consistent with the measurements from 97/11/21, but do not provide much information about the asymmetry. The WHT data at 905 nm only partially resolve the stellar disk. The average visibility on the longest (3.7 m) baseline is 0.35. Because of this, only a very large asymmetry could be detected from the closure phase values. In fact, all of the measurements are within 6Ö of zero, and most are smaller than 2Ö . However, the WHT data do not conflict with the asymmetry inferred from the COAST data at the same wavelength. The short-baseline data do provide one additional piece of information, which is that the visibility on very short baselines (0.3 m) is equal to unity. The stellar disk is unresolved on these baselines, hence there is no evidence for any large scale ( 0 2ÌÌ) structure. There is no evidence for extended structure at 700 nm or at 1.3 Åm either. This result is consistent with the observations made in October 1995 by Burns et al. (1997), who also failed to detect the dust shell formed in late 1994 (Bester et al. 1996), which had been detected at a wavelength of 700 nm by Wilson et al. (1997) in late 1994 and early 1995. The WHT observations at 700 nm, using the same 3.7 m maximum baseline, provide more resolution, but do not extend beyond the first minimum in the visibility curve. The observed visibilities, averaged over all position angles of the fringes, are plotted against baseline length in Figure 6.3. A uniform disk model is obviously a poor fit at this wavelength. The large number of accurate


88

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

Betelgeuse 905nm 1 Visibility amplitude 0.8 0.6 0.4 0.2 0 0 2000 4000 6000 8000 Projected baseline /mm Betelgeuse 905nm 210 180 150 120 90 60 30 0 0 Closure phase /degrees 97/11/21 97/11/21 47.8mas UD 1 Visibility amplitude 0.8 0.6 0.4 0.2 0 0

Betelgeuse 905nm 97/11/16 WHT 97/10/24-11/12 COAST 44.6mas UD

2000

4000

6000

8000

Projected baseline /mm

2000

4000

6000

8000

Longest projected baseline /mm

Figure 6.2: Visibility curves and closure phases for Betelgeuse at 905 nm. The data in the two left-hand plots is all from 97/11/21. Only closure phases measured using one triangle of baselines (that linking the Centre, East and North telescopes) are shown. The right-hand plot shows all the visibility data measured between 97/10/24 and 97/11/16. Each WHT point is the average of measurements at six different position angles. Note the consistency between the COAST and WHT data. The solid curve in this graph is the visibility curve for a uniform disk 7% (4) smaller than the UD which fits the short baseline data from 97/11/21.


6.3. RESULTS closure phase data (the errors are

89 2Ö ) show strong evidence for an asymmetric brightness dis-

tribution, with values as large as 30Ö. The closure phase on triangles including longer baselines varies slowly with position angle (see Figure 6.4 for an example). This behaviour again indicates that the asymmetric features in the brightness distribution are located within the stellar disk.
Betelgeuse 700nm 1 0.8 0.6 0.4 0.2 0 0 97/11/15 47mas UD 1000 2000 Projected baseline /mm 3000 4000

Figure 6.3: Visibility data for Betelgeuse at 700 nm. Each point is the average of nine measurements at different position angles. The visibility curve for a uniform disk with diameter 47 mas is shown for comparison.

Figure 6.4: Closure phases for Betelgeuse at 700 nm. The closure phases on two triangles of baselines are plotted, against the position angle of the fringes on the sky. The errors bars on the data are 2Ö.

Visibility amplitude


90

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

6.3.2

Images

An image was reconstructed from the 97/11/11 1.3 Åm COAST data, using difference mapping with CLEAN deconvolution, as described in Section 5.6. Four iterations of the self-calibrate, invert, CLEAN cycle within the Difmap software package were required, starting from a uniform disk trial model. The final image, as well as the uv coverage for the observations, are shown in Figure 6.5. The image shows the expected imprint of the elliptical beam (the shape of the CLEAN beam is indicated by the ellipse in the bottom-left corner of the map), but is otherwise a symmetric disk, as expected from the Fourier data. The noise level in the image (the RMS noise is 1% of the peak flux) is consistent with the number and precision of the visibility data. On the basis of the map, we can rule out the presence of a companion with a flux more than 0.9% of the total flux from Betelgeuse (i.e. a magnitude difference of 5.1), within 190 mas of the map centre (note that only a fraction of the map area is shown in Figure 6.5). Hence there is no evidence for the close companion reported by Karovska et al. (1986). There were insufficient uv points to reconstruct an image from the 905 nm COAST data from 97/11/21 alone. Combining the data from 97/11/21 with earlier data was deemed unwise, due to the apparent size change noted above. The combined data set from 97/10/24, 97/10/31, and 97/11/12 was also too small to produce a 905 nm map. The map in Figure 6.6 was reconstructed from the 700 nm WHT data, using a self-calibration algorithm incorporating MEM deconvolution (Sivia 1987) (a standard mapping package intended for radio VLBI data was used). The dynamic range of the image is low ( 20:1) because of the poor calibration of the visibility amplitude data. Nevertheless, it is clear that the star appears highly asymmetric, with the Eastern half of the disk appearing brightest, and the Northwest region appearing least bright. It is worth emphasising that the asymmetric features in the map are constrained by the high-quality closure phase data, because the reconstructed image is forced to fit the closure phases. The contrast between this map and the highly symmetric 1.3 Åm image in Figure 6.5 is striking. The apparent morphology is obviously very different at the two wavelengths. However, low-resolution images can only provide qualitative information. The change in apparent asymmetry with observing wavelength is confirmed by the results of model fitting, discussed below.

6.3.3

Asymmetries

The degree of asymmetry in the appearance of the stellar disk at the three wavelengths can be quantified by fitting simple models for the brightness distribution. The COAST data extend to spatial frequencies at which the visibility functions of uniformly-bright and limb-darkened disks can be distinguished from each other. To avoid introducing systematic errors in the parameters of the asymmetric model components, the model fitting software (modfit and erfit, described in


6.3. RESULTS

91

Section 5.7) was modified to allow limb-darkened disk components, with different degrees of limb-darkening, to be used. The new empirical model of Hestroffer (1997) was used for the limb-darkened intensity profiles, as it has only one free parameter , yet covers a wide range of limb-darkening, from uniform disk ( 0) to Gaussian-like profiles ( 8). Also, the profile with 1 is a good fit to intensity profiles calculated from recent model photospheres for cool giant stars (Hofmann and Scholz 1998). The model intensity profile is given by I Är
ä
Å

Å



(6.1)

where ÅÄrÅ 1 Är r0 Å2 is the cosine of the angle between the line of sight to the star and the stellar radius vector, with r0 being the radius of the stellar limb (at which the intensity is zero). Hence Å is equal to unity at the centre of the stellar disk, and zero at the limb. The visibility function is given by the Hankel transform of Equation 6.1, derived by Hestroffer: V ÄxÅ Ä
Ç 1ÅJ ÄxÅ Ä

x2

Å

(6.2)

Here x is the spatial frequency 2r0 B , where B is the length of the interferometer baseline and is the observing wavelength. The constant is given by 2 Ç 1, is the Gamma function, and J is the Bessel function of order . Plots of the visibility function for the cases 0 (solid line; this is a uniform disk), 2(dashed line) and 4 (dash-dot) are given in Figure 6.7, together with the corresponding intensity profiles. If 0, the Hestroffer profiles are fully-darkened: near the stellar limb, the intensity tends towards zero. Models consisting of such limb-darkened disks (henceforth abbreviated to FDD, for fully-darkened disk) with superposed bright or dark spots (modelled as narrow Gaussians), were fitted to the visibility amplitude and closure phase data. The observations with COAST and the WHT (summarised in Table 6.1) spanned an entire month. Wilson et al. (1997) found significant hotspot evolution in three weeks, so it was decided to restrict the comparative analysis to a subset of the available 905 nm and 1.3 Åm data. The 1.3 Åm data from 97/11/11 were used, together with the 700 nm data from 97/11/15. As discussed above, a 4 size change was detected at 905 nm between 16 November and 21 November. The data obtained prior to the size change are of limited value for determining the apparent asymmetry at 905 nm, as the data include only a few closure phases, with none measured on large baseline triangles. Hence the 97/11/21 data were used, on the assumption that the nature of the asymmetry did not change much between 97/11/11 and 97/11/21. The results of model fitting confirm that the apparent morphology at 1.3 Åm was highly symmetric. Fits of a model consisting of a FDD with 1 (whose diameter was a free parameter), plus one unresolved (10 mas FWHM) bright or dark feature were carried out. These led to the feature being placed very close to the centre of the disk, where it produces no closure phase signature. The 1 upper limit to the flux in an unresolved feature close to the disk centre is still just 6% of the total flux. This model does not fit the data any better than a model consisting of just a limb-darkened disk, so the value of 6% is definitely a hard upper limit to the flux in any unresolved feature. The limit on the brightness of a feature away from the disk centre is much smaller (see below).


92

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

Figure 6.5: Reconstructed image of Betelgeuse (top), and the corresponding uv-plane coverage (bottom), from data taken on 11 November 1997 at a wavelength of 1.3 Åm. North is up and East is to the left. The contours are at -2, 2, 5, 10, 20, 30, ..., 90, 99% of the peak intensity.


6.3. RESULTS

93

Figure 6.6: Reconstructed image of Betelgeuse, from WHT data taken on 15 November 1997 at a wavelength of 700 nm. North is up and East is to the left. The contours are at 10, 20, 30, ..., 90% of the peak flux. This image is highly asymmetric, in contrast with the infrared image from data taken four days earlier (Figure 6.5). Model fitting was also used to establish that there was no significant elliptical distortion of the stellar disk at 1.3 Åm. The axial ratio of the best-fit elliptical disk (with the radial intensity profile of a 1 FDD) is 0 99

?

0 01.

In contrast, symmetric limb-darkened disk models did not give adequate fits to the closure phase data measured at 905 nm with COAST. The addition of a single unresolved bright or dark spot could reproduce the simple closure phase signal described above. If the underlying FDD had 3 or 4, the model could also fit the visibilities reasonably well. The best-fit parameters for models incorporating a FDD by Equation 5.5) are also given. parameter) in order to constrain the WHT data at the same wavel with 4 are given in Table 6.2. The values of 2 (as defined It was necessary to fix the disk flux (rather than treat it as a free the visibility to be 1 on short baselines, for consistency with ength. The brightness distributions implied by the two different

models are similar (see Figure 6.8): the dark spot is 150Ö round from the bright spot, at a similar radius, and both the bright and dark features contribute just 3% of the total flux. The sizes of the bright and dark features are not well constrained by the 905 nm COAST data. The fit to the closure phases becomes slightly worse if the spot is resolved on long baselines (i.e. FWHM 12 mas), but the presence of a feature with a FWHM as large as 20 mas cannot be ruled

out. If this larger size is used, the flux in the feature must be increased to

8% of the total.


94

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

Figure 6.7: Visibility functions (top) and radial intensity profiles (bottom) for Hestroffer limb-darkened disk models, all with the same diameter, and limb-darkening parameters of 0 (solid line; this is a uniform disk), 2 (dashed line) and 4 (dash-dot). The visibility curves for 2 and 4 still have the nulls and secondary maxima apparent in the uniform disk curve, but the minima become shallower and the secondary maxima lower as is increased.


6.3. RESULTS Table 6.2: Best-fit one-spot models for Betelgeuse on 97/11/21 at 905 nm. The models consist of a limb-darkened disk with one (bright or dark) unresolved feature superimposed. The coordinates Är Å give the position of the unresolved feature with respect to the centre of the underlying disk ( is measured North through East). Hestroffer fullydarkened disk components are abbreviated to FDD. For Gaussian components, "diameter" refers to the FWHM. Parameters for which no uncertainty is quoted were fixed during the fitting process.
2

95

Component

Flux 0 95 0 027 ? 0 003

r (mas) 59

(Ö ) - 119 ? 5 - 31 ? 7

Diameter (mas) 71 0 ? 0 7 10 68 0 ? 0 8 10

4.3 FDD ( 4) Gaussian 3.9 FDD ( 4) Gaussian

?

-

10



0 95 - 0 029 ? 0 004 8 6 ? 1 1



As far as could be ascertained by repeating the model-fitting process with a range of starting models, the solutions in Table 6.2 are unique, given the assumed limb-darkening (and providing the features are unresolved). Previous authors (Wilson et al. 1992, 1997; Tuthill et al. 1997) have cited a potential source of ambiguity: the unresolved features can usually be moved further out on the disk and their flux contribution decreased, without changing the model closure phases. However, the small visibilities measured on long baselines with COAST, where the visibility due to the limb-darkened disk is small, imply that the flux from a single feature must be small. If there were several features, they could each contribute more flux, yet destructively interfere with each other to produce the small measured visibilities. There is no need to introduce further features to explain the data, but the limited Fourier plane coverage at 905 nm prevents the presence of additional features, at least in certain locations, from being excluded. The locations and fluxes of the bright and dark spots are slightly dependent upon the assumed limb-darkening. The 905 nm COAST data do not allow the limb-darkening and asymmetry to be determined independently, but the extremes of uniform and Gaussian intensity profiles for the disk can be ruled out. The Hestroffer limb-darkening model used here is a reasonable fit to the 830/40 nm visibility data obtained by Burns et al. (1997) for Betelgeuse in October 1995 (the dark for t dark closure phase data indicated that the star appeared symmetric at this epoch), with the limbening parameter equal to 3 7 ? 0 1. This suggests that models with 4 are reasonable he similar near-continuum bandpass centred on 905 nm. The use of models with less limbening does not allow the flux from an unresolved hot- or dark spot to be increased to more

than 5% of the total without mis-fitting the visibility amplitude data. Model fitting also helps to elucidate the nature of the asymmetry seen at 700 nm. In this case, models with one or two unresolved bright features cannot adequately fit the closure phase data. Three bright spots superimposed on a symmetric disk are sufficient however. The visibility data provide no information about the limb-darkening of the underlying disk. For consistency with the 905 nm models, the 700 nm models presented here incorporate FDDs with 4. Several


96

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE Table 6.3: Best-fit models for Betelgeuse on 97/11/15 at 700 nm. The models consist of a limb-darkened disk with one or more (bright or dark) unresolved features superimposed. The coordinates Är Å give the position of each unresolved feature with respect to the centre of the underlying disk ( is measured North through East). Hestroffer fullydarkened disk components are abbreviated to FDD. For Gaussian components, "diameter" refers to the FWHM. Parameters for which no uncertainty is quoted were fixed during the fitting process.
2

Component

Flux 0 68 0 08 0 13 0 09

3.0 FDD ( 4) Gaussian Gaussian Gaussian 3.1 FDD ( 4) Gaussian

? ? ? ?

r (mas) - 13 0

(Ö ) - 19

Diameter (mas) 76 ? 5 10 10 10 67 8 ? 0 5 10

0 07 0 02 0 02 0 02

?

11

?

4

59 12 9 ? 1 1

119 226 ? 4 - 59 6 ? 1 0

1 05 ? 0 05 - 0 06 ? 0 01 14 8 ? 0 4



minima, with comparable 2 values, were found in the models are all qualitatively similar, with hotspot fluxes possible to obtain a fit that was only marginally worse to be the preferred position for a bright feature inferred

eleven-dimensional parameter space. The between 7 and 16% of the total flux. It was by constraining the location of one feature from the 905 nm data (see Table 6.2). The

parameters of this 700 nm model are given in Table 6.3. Importantly, it was not possible to fit the data with any model in which the hotspot fluxes were just a few percent of the total. Closure phase signatures requiring models with three bright spots have been measured before for Betelgeuse, by Wilson et al. (1997) at a wavelength of 700 nm, and by Tuthill et al. (1997) at 633 nm. Tuthill et al. (1997) state that models with dark spots needed more free parameters to fit the data than those using bright spots. In contrast, these latest 700 nm data can be fit using a single dark feature, in a roughly similar (but definitely not identical) location to the dark spot needed to fit the 905 nm data. The parameters of this simpler model (with five parameters) are also listed in Table 6.3. I can be reasonably confident of having found a global minimum for 2 with this model. The sizes of the unresolved features, be they bright or dark, are not well constrained by the 700 nm data either. The upper limit for both types is 15 mas FWHM.

Are the data consistent with a single brightness distribution at the three wavelengths? In order to try to answer this question, models with both bright and dark features, fixed in the positions found from the 700 nm data, were fitted to the 97/11/21 905 nm data and to the 97/11/11 1.3 Åm data. The three-bright-spot model given in Table 6.3 cannot adequately fit the 905 nm data unless the fluxes of two of the three spots are reduced to less than 0.2% of the total. If the hotspot locations for 905 nm are taken from an alternative model for 700 nm, the fluxes of two of the features need to be 0 8% of the total. The other possible bright spot model for 700 nm can fit the 905 nm data

with two of the unresolved components contributing 1 3 ? 0 3% and 2 4 ? 0 5% of the total flux, and the third invisible. The 700 nm dark spot model cannot fit the 905 nm data at all unless the


6.3. RESULTS feature is moved inwards by

97 5 mas. There is no evidence for any of the bright or dark features 2% of the disk flux.

being visible at 1.3 Åm. The limits on their fluxes are all

I conclude that the appearance of Betelgeuse at the epoch of observation was very different at the three wavelengths, with a large asymmetry apparent at 700 nm, a small one at 905 nm, and no detectable asymmetry at 1.3 Åm. Figure 6.8 shows the brightness distributions for the models with dark features (top) and bright features (bottom) found to best fit the 700 nm (left) and 905 nm (right) data. Note that the brightness distributions for the two 700 nm models have a similar appearance, as do those for the two 905 nm models. Both 700 nm models show more deviation from a symmetric disk than their 905 nm counterparts, and are reasonably consistent with the 700 nm image in Figure 6.6.

6.3.4

Apparent sizes and limb-darkening

Cool stars, such as Betelgeuse and the Mira variable stars discussed in the next two chapters, have geometrically-extended atmospheres. This means that the radial extent of the region of the atmosphere from which photons can escape (the photosphere) is a substantial fraction of the stellar radius. Observations at different wavelengths probe different layers within the photosphere. This is in contrast to hotter stars such as the Sun, in which the photosphere is very thin compared to the size of the star. Such stars are said to have compact atmospheres. In these stars, the volume bounded by the photosphere contains most of the stellar mass, and so it is clear what is meant by the "radius of the star". Several different definitions of the radius of a star have been used in the literature. These are reviewed by Baschek et al. (1991), and include:

? ? ? ? ?

The density radius is the radius at which the density reaches a small value, characteristic of circumstellar or interstellar material. The mass radius is the radius which contains all but an arbitrary small fraction of the mass of the star. The optical depth radius, which in general is a function of observing wavelength, is the radius at which the optical depth at the observing wavelength is equal to some pre-chosen value of order unity. The intensity radius, again a function of wavelength, is the radius corresponding to the point on the observed stellar disk at which the intensity is equal to a pre-chosen small fraction of the peak intensity. The temperature radius is the radius at which the local kinetic temperature is equal to the effective temperature, i.e. T ÄRT Å ÄL Ä4R2 ÅÅ1 4. T

In a star with a geometrically-extended atmosphere, the different radii do not coincide with each other, and both the optical depth and intensity radii are strong functions of the observing wavelength. A wavelength-independent optical depth radius may be defined as the radius at which the


98

CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE

Figure 6.8: Brightness distributions for the best-fit models with dark features (top) and bright features (bottom). The models for the 700 nm data are on the left, and those for 905 nm are on the right. Note the greater asymmetry at the shorter of the two wavelengths. T strained by the with FWHM ..., 90% of the he brightest areas are shaded darkest. The sizes of the features are uncondata. To generate these plots, the features were assumed to be Gaussian 10 mas. North is up and East is to the left. The contours are at 10, 20, 30, peak brightness.


6.3. RESULTS

99

Rosseland mean (i.e. averaged over all wavelengths) optical depth is equal to unity (or sometimes 2 3). The Rosseland mean optical depth definition of radius is used by many workers, and is often chosen as the boundary radius in model atmospheres. Of these different radii, the only one which we have some hope of directly measuring is the intensity radius in a particular waveband. The directly observable property of the star is the centre-tolimb variation of intensity (CLV) in that waveband. At the wavelengths of strong absorption lines, the outermost layers of the photosphere can become optically thick, leading to a wider CLV being observed than at neighbouring continuum wavelengths. The shape (limb-darkening profile) of the CLV also varies with wavelength. An interferometer will measure the same visibility curve on short baselines (up to the first minimum of the visibility function of a uniform disk) for a small, uniformly-bright CLV, as for a larger, limb-darkened profile. The shape and extent of the CLV may only be determined uniquely from interferometric observations which span a large range of baseline length, i.e. at least several lobes of the visibility function of a uniform disk. Fitting simple models (with one or two variable parameters) for the limb-darkening, when the stellar diameter is unknown, is possible with visibility measurements in just the first two lobes of the visibility function. However, if these measurements are not made in a continuum bandpass, the observed diameter corrected for the observed limbdarkening will still be larger than the mean optical depth diameter, as a significant amount of the detected flux will have been emitted from the outer layers of the extended photosphere.

Betelgeuse measurements I have established that there was no detectable departure of the 1.3 Åm brightness distribution of Betelgeuse from a circularly-symmetric disk on 97/11/11. The visibility measurements on both sides of the first null in the visibility function could therefore be used to determine the angular diameter and the degree of limb-darkening independently of each other. This was done by fitting the Hestroffer fully-darkened disk model to the visibility data, with treated as a free parameter. The best-fitting model has a 2 per degree-of-freedom of 1.8. The plot of the region around the minimum in the 2 surface in Figure 6.9 indicates that the data do indeed constrain both and the disk diameter. The inferred values are 51 4 ? 0 8 mas for the zero-intensity disk diameter, and 1 3 ? 0 1. The quoted uncertainties take into account the correlation between the two parameters, which can be seen in the contour plot in Figure 6.9. Figure 6.9 also shows the visibility data, with the model visibility curve superimposed, and the inferred radial intensity profile. For comparison, the intensity profile for a Hestroffer disk with 4 and diameter 70 mas, similar to the disk components in the best-fitting models for the 905 nm data, is also shown. An equally good fit was obtained using the standard linear limb-darkening law I Är
Å

1



1 Ä1



Å

Å

(6.3)

with 1 1 25 ? 0 12 and a diameter of 52 6 ? 1 6 mas. This intensity function is very close to that of the Hestroffer model with a limb-darkening parameter of one. The intensity in fact goes


100 CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE slightly negative near the limb, which illustrates the fact that the linear model is unsuitable for strongly-limb-darkened disks.

6 .4
6.4.1

Discussion
Asymmetries

Convective hotspots Previous authors have explained the bright spots detected on the disk of Betelgeuse as convective granules, and have modelled these as areas of elevated temperature on the surface of the star. The latest 700 nm data can be interpreted in terms of three bright features, with fluxes similar to those of previously detected features. Model fitting to the Fourier data yields a "sky model", which for the models considered here, consists of a disk plus a number of superimposed components. The surface brightness at each point on the sky is given by the sum of contributions from all of the components of the sky model. The flux from the "spot" component in the sky model is not the same as the flux from the hot area of the stellar disk in the blackbody model. The distinction is illustrated by Figure 6.10. The ratio of the flux from a "spot" component of the sky model to that from the disk component at a particular wavelength , in terms of the parameters of the blackbody model, is given by f


A ÄPÄ Th Å PÄ T? PÄ T?Å

ÅÅ

(6.4)

where A is the ratio of the hotspot area to the disk area, PÄ T Å is the Planck function, Th is the hotspot temperature, and T? is the disk temperature. The ratio f1 f 2 is independent of the area of the hotspot. The 700 nm data can be modelled as a disk with areas of elevated temperature, with temperature excesses similar to those proposed by previous authors. None of the previous data have constrained the minimum size of the bright features, and the new measurements are no different in this respect. The data suggest that the upper limit to the area of a single hotspot is 20% of the disk area. Taking this as the hotspot area, the contrast of a representative hotspot at 700 nm ( f700 0 19) can be explained by a temperature excess of 450 K (I have assumed the disk temperature T? to be 3500 K). If the hotspot area were just 3% of the disk area, the hotspot would have to be 2000 K hotter than the disk. Could the observations at longer wavelengths be consistent with the same blackbody model as those at 700 nm? The three bright spots have similar fluxes to each other at 700 nm, yet the presence of only one of the features is consistent with the 905 nm data (this is the feature with f700 0 19). The difference in the contrast of this feature at the two wavelengths can be compared with the predictions of the blackbody model.


6.4. DISCUSSION

101

Figure 6.9: Best-fit two-parameter Hestroffer limb-darkened disk model for Betelgeuse at a wavelength of 1.3 Åm. The top left plot shows the visibility data (note that the signs of the visibility points have been inferred from the closure phase data), the best-fit interpolant, and the ?3 interpolants. The bottom left plot is the corresponding radial intensity profile. A contour plot of the negative logarithm of the posterior probability of the model (equal to 2 2 Ç constant) is shown in the top right. The contours are 68.3%,

90%, 95.4%, and 99% confidence intervals. For comparison, the bottom right plot shows the intensity profile for a Hestroffer FDD with 4 and zero-intensity diameter 70 mas, similar to the disk components in the best-fitting models for the 905 nm data.


102 CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE
11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 Th 11111 00000 11111 00000 11111 00000 11111 00000 00000 11111 00000 11111

I Blackbody model

T*

x

I Sky model

10000 01111 00000 11111 00000 11111

x

Figure 6.10: Illustration of the distinction between the hotspot in the "sky model" obtained from a fit to the Fourier data, and that in the blackbody model intended to represent the underlying physics. See text for explanation. Equation 6.4 with f 0 19 implies that f 0 13 for the same hotspot, whatever its size.

The measured value is 0 028 ? 0 003 for an unresolved feature, or 0 086 ? 0 009 for the maximum (20 mas FWHM) size. We would also expect the feature to be visible at 1.3 Åm, where it should contribute 0.08 of the disk flux. A superimposed feature with FWHM 20 mas emitting this fraction of the disk flux is certainly not present at 1.3 Åm. The conclusions are the same if disk temperatures of 3000 K or 4000 K are used. In each case, the blackbody hotspot model is ruled out. In short, there is no combination of disk and hotspot temperatures, and hotspot areas which will reconcile the blackbody model with the observations. The simple theory cannot account for the fairly typical asymmetry detected at 700 nm together with the lack of asymmetry seen at longer wavelengths. This conclusion is robust despite the ambiguities in modelling the 700 nm data. It is not only the convective cell model which is inconsistent with the observations. Any model in which the bright features are areas of elevated temperature at the continuum-forming photospheric layers are also ruled out.

700

905

Shock waves Uitenbroek et al. (1998) propose that the bright features are signatures of shock waves caused by pulsation. In their model, stellar rotation causes the radial density gradient to be steeper in the direction of the pole, so that the shock wave has the greatest effect there. However, the brightest features in Figure 6.8 are not located on their rotation axis. Uitenbroek et al. suggest that nonradial pulsation or the interaction of several shock waves could produce non-polar hotspots. Such complex behaviour would be very difficult to observe, and so the theory is hard to test.


6.4. DISCUSSION

103

The shock waves would become stronger as they reach regions of lower density, further from the centre of the star. Thus we would expect to see brighter features at wavelengths which probe the outer layers of the star. There is some evidence that the observed CLV is more extended at 905 nm than at 1.3 Åm (compare the two CLVs in Figure 6.9), and hence that the outer atmospheric layers appear more asymmetric. Despite the lack of a simple explanation for non-polar features, shock wave models are worthy of further consideration.

Dark features The dark spot interpretation of the Fourier data should also be considered. A natural explanation of dark features is that they are caused by clumps of opaque material, either in the outer layers of the photosphere, or further out. If such clumps were located near to the line of sight to the centre of the star, they would appear dark. However, a clump close to the stellar limb could scatter starlight into the line of sight, so that the feature would appear brighter than the disk. The inferred dark spot position at 700 nm is 15 mas from the disk centre, where limb-darkening is likely to have substantially reduced the disk intensity from its central value. Hence one might expect this feature to appear bright, not dark. If the same dark feature was detected at both 700 nm and 905 nm, then it must have moved by 5 mas on the stellar disk in six days. The distance to Betelgeuse, from its Hipparcos (ESA 1997) trigonometric parallax, is 131Ç36 pc, so the inferred speed of the feature is 190 km s 1 . This is 23 greater than the escape velocity, and so the apparent position shift cannot be due to any physical motion on the stellar surface. The bright spot interpretation will therefore be preferred if the asymmetries seen at 700 nm and 905 nm have the same physical origin. The asymmetries detected by Tuthill et al. (1997) could only be modelled by complicated arrangements of dark features, so if the latest asymmetries are caused by the same process, they should also be interpreted as bright features. In all, there appears to be ample evidence to reject the dark spot interpretation of the data.

A modified convective model for bright features If the features are bright, a modified theory incorporating convective granules could explain the latest observations. If convective plumes could make holes in a screen which is optically thick at 700 nm, but less so at 905 nm and 1.3 Åm, then the star would appear brighter at 700 nm for lines of sight through the holes. The resulting bright wavelengths. The numbers of hotspots and their observed. There is indirect evidence that large sphere: Lim et al. (1998) suggest that convection features would have less contrast at the longer locations would change over time, as has been convective plumes extend into the outer atmois the mechanism by which the asymmetrically-

distributed circumstellar gas they detect is transported from the photosphere. Would we expect to observe the screen directly? A circumstellar screen could scatter starlight into the line of sight. Any such screen which is significantly opaque at 700 nm must not extend


104 CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE beyond about 2r?, otherwise it would be resolved on the shortest WHT baseline. Hence a largescale screen is unlikely. A screen closer to the surface would cause the star to appear larger at wavelengths of high opacity. However, the lack of resolution at 700 nm does not allow a large, limb-darkened disk and a smaller uniform disk to be distinguished. Modelling of the 905 nm data suggests that a fully-darkened disk with 4 and zero-intensity diameter 70 mas is appropriate at this wavelength. This can be compared with 1 3 and a 51 mas diameter at 1.3 Åm (the two CLVs are shown in Figure 6.9), to give some evidence that the outer atmosphere is more opaque at 905 nm than 1.3 Åm. Hence the presence of partially-opaque material close to the surface of the star is a distinct possibility. The suggestion of a partially-opaque screen raises the questions of which materials could form such a screen, and dependence. The titanium oxide) in evidence that dust whether the opacity of any possible screen material has the correct wavelength two obvious candidate materials are circumstellar dust, and molecules (e.g. the outer parts of the photosphere. I shall consider dust first. There is some forms close to the star (Bester et al. 1996), so the lack of large-scale structure at

700 nm does not exclude the possibility of dust supplying the required opacity in November 1997. Danchi et al. (1994) inferred that there was no significant amount of dust within one arc-second of Betelgeuse, from 11 Åm visibilities obtained between 1988 and 1992. Bright features on the stellar disk were detected at wavelengths of 546 nm, 633 nm, 700 nm and 710 nm in this period (Buscher et al. 1990; Wilson et al. 1992; Tuthill et al. 1997). A screen of circumstellar dust is therefore unlikely to have been involved in the production of these features. The wavelength-dependence of the optical depth of a dust screen is straightforward to predict. We would expect grain sizes 1 Åm. When the observing wavelength is approximately equal to the grain size, the opacity is caused by Mie scattering, the cross-section for which goes as n , with n 1. If all but a fraction A of the disk area is screened by dust with optical depth , then model fitting to the visibility and closure phase data will yield a disk plus a superimposed feature with a fraction AÄ1 e Å f AÄe 1Å (6.5) e of the disk flux. If I assume that the optical depth due to the dust screen goes as 1 , then 700 1 gives f700 f 905 1 47 and f700 f 1290 2 38, independent of the hotspot area. Smaller values of 700 give less decrease in contrast with increasing wavelength. The data imply that f700 f 905 1 5and f700 f 1290 9 5. If the area seen through the hole is hotter than the rest of the disk, then Equation 6.5 becomes A ÄPÄ Th Å e PÄ e PÄ T?Å However, there is still no single combination of A, 700, and which will explain the measured f values. Hence holes in f T?
ÅÅ

(6.6)

Th (for T? in the range 3000-4000 K) a dust screen cannot account for the

observed change in asymmetry with wavelength. The remaining possibility for the screen material is titanium oxide. TiO is always present in the outer atmosphere of Betelgeuse, although the strength of the TiO absorption bands varies irregularly (see, for example, Morgan et al. 1997). A synthetic TiO spectrum (taken from J?rgensen


6.4. DISCUSSION

105

Figure 6.11: Synthetic TiO spectrum, for a red giant with Teff J?rgensen (1994).

3100 K, taken from

1994), for a slightly cooler star than Betelgeuse, is shown in Figure 6.11. None of the spectral features within the bands used for these observations are very strong in a star of spectral type M1 or M2. However, TiO is not only opaque at wavelengths at which spectral features can be identified. The spectra of cool stars are line-blanketed in the red and near-infrared. This means that there is significant absorption at all wavelengths over a wide spectral region, so that even between identifiable spectral features the stellar flux is lower than would be expected if the star radiated as a blackbody. Balega et al. (1982) measured the diameter of Betelgeuse in a number of narrow bands in the region of TiO absorption features, and found some evidence for larger diameters at wavelengths of strong absorption. Quirrenbach et al. (1993) measured the same effect in other cool stars. Also, studies using model atmospheres show that TiO opacity is significant, and has an important effect on the temperature structure of cool stars (Krupp et al. 1978; J?rgensen 1994). Despite this indirect evidence, model atmosphere calculations are probably required to determine whether the COAST and WHT observations could be explained by holes in a molecular screen. Previous experiments (Buscher et al. 1990; Wilson et al. 1992; Tuthill et al. 1997) have revealed only small differences in hotspot contrast between wavelengths of 700 nm (near-continuum) and 710 nm (strong TiO absorption). However, it is possible that there was significant leakage of continuum flux through the wings of the relatively broad (10 nm width) 710 nm filters, which due to the depression of the spectrum at 710 nm, may have amounted to a significant fraction of the TiO-band flux. The fact that the disk diameters measured at 700 nm and 710 nm did not


106 CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE differ substantially supports this suggestion. Future observations in narrower bands should reveal conclusively whether brighter features are seen at wavelengths of TiO absorption. If the model is correct, temporal variations in hotspot intensity should be correlated with changes in TiO band strength. Further experiments are needed to test both the TiO screen model and models involving shocks in the outer atmosphere.

6.4.2

Infrared limb-darkening

This is only the fourth time that interferometric data has allowed the observed limb-darkening curve for a star other than the Sun to be compared with the predictions of model atmospheres. Hanbury Brown et al. (1974) measured the correlation coefficient for Sirius (spectral type A1V) at 443 nm beyond its first minimum with the Intensity Interferometer, Quirrenbach et al. (1996) obtained visibility data beyond the first null in the visibility function of the K1 giant star Bootis at 550 nm, and Burns et al. (1997) made similar measurements for Betelgeuse at 830 nm. Both Hanbury Brown et al. and Quirrenbach et al. found reasonable agreement between the measured data and intensity profiles from model atmosphere calculations, which suggests that the models are adequate for hotter stars. The linear limb-darkening coefficient determined from the J band COAST measurements can be compared to values in the literature (with the caveat that the model atmospheres are more appropriate to giant stars than supergiants). Manduca (1979) obtained 1 0 52 from a leastsquares fit to the intensity profile (in a narrow passband centred on 1.2 Åm) of a model star with Teff 3750 K, log g 1 5 and solar composition. Van Hamme (1993) gives 1 0 49 for the J band limb-darkening coefficient of a model with Teff 3500 K, log g 0 5, and solar composition. The measured value for Betelgeuse is 1 1 25 ? 0 12. It is clear that Betelgeuse appears more strongly limb-darkened at 1.3 Åm than model atmospheres predict. The limb-darkening correction to the uniform disk diameter which would be inferred from the short baseline measurements is 21% (i.e. the diameter inferred from a uniform disk fit to shortbaseline data must be increased by 21% to obtain the zero-intensity diameter at the wavelength of observation). If neglected, this correction would lead to the effective temperature being overestimated by 10%. The degree of limb-darkening is expected to decrease with increasing wavelength (see, for example, Manduca 1979). The limb-darkening correction to the 11 Åm uniform disk diameter of 56 ? 1 The 11 Åm diameter is darkening. The discrep height of two different mas measured by Bester et al. (1996) should be negligibly small ( 1%). 9% larger than the diameter at 1.3 Åm, corrected for the observed limbancy could be due to a variation of opacity with wavelength, so that the layers of the photosphere are measured at the two wavelengths. Altern-

atively, a diameter change of this magnitude could be caused by stellar pulsation. The lack of asymmetry seen at 1.3 Åm suggests that future interferometric monitoring at IR wavelengths could be used to quantify the size changes associated with radial pulsation.


6.5. CONCLUSIONS

107

6.5

Conclusions

Near-contemporaneous interferometric observations of Betelgeuse have revealed very different apparent morphologies in three widely-separated wavebands. The visibility and closure phase data at a wavelength of 700 nm can be interpreted in terms of a disk with either three bright features or one dark feature superimposed. The dark spot interpretation is unlikely given that previous data have been incompatible with small numbers of dark features. A smaller asymmetry was detected at 905 nm, whereas the apparent brightness distribution at 1.3 Åm was highly symmetric. If the asymmetries are modelled as bright features superimposed on a circular disk, the change in contrast with wavelength is inconsistent with the bright features simply being areas of elevated temperature on the surface of Betelgeuse. Models in which the features are caused by the interaction of shock waves with material in the outer atmosphere cannot be ruled out however. A modified convective theory is suggested, which takes into account the geometrically-extended atmosphere of Betelgeuse. Convective plumes could cause TiO molecules in the outer photospheric layers to become dissociated in particular regions. These regions would appear bright at wavelengths where the TiO has significant opacity. The resulting hotspots would evolve in the same way as the underlying convective cells. Imaging in narrower bands, both at wavelengths of strong TiO absorption and in the continuum, is required to test this model. Model atmospheres should be used to predict the TiO opacity in the intended bands, so that we can be sure of probing different photospheric layers. The use of longer interferometer baselines will help constrain both the sizes of the features and the wavelengthdependence of the apparent stellar size.


108 CHAPTER 6. THE WAVELENGTH-DEPENDENT MORPHOLOGY OF BETELGEUSE


Chapter 7

Cyclic Variations in the Angular Diameter of Cygni
7 .1
7.1.1

Introduction
Mira variables

Mira variable stars are a sub-set of the class of Long-Period Variables (LPVs). LPVs are all evolved stars, located on the Asymptotic Giant Branch (AGB) of the Hertzsprung-Russell diagram. Mira variables are named after the prototype, Mira Ceti. Their defining property is that they exhibit regular, large-amplitude photometric variability, with amplitudes of 2.5-11 mag. at V , and periods in the range 100-700 days. This variability is due to stellar pulsations, believed to be radial. Whether most Miras pulsate in the fundamental radial mode (with a single node at the centre of the star, and an antinode at the boundary), or the first-overtone mode (which has a further node within the star) is still hotly debated. Part of the reason for the contention is that the fundamental properties (mass, luminosity, and radius) of Mira variables are difficult to measure. Mira variables have large radii ( 400R?) and very extended atmospheres. As an illustration, the diameter measured in strong molecular absorption bands can be more than twice the diameter seen in the continuum (Labeyrie et al. 1977; Bonneau et al. 1982). The physical diameters depend on poorly-known distances as as uncertain angular diameters. The luminosities of Miras depend on the same distances. masses of the main-sequence progenitors of Mira variables are believed to be 1-2M? (Wyat Cahn 1983), but Miras lose mass at rates of up to 10 4M? yr 1 (Knapp and Morris 1985), so current masses are likely to be at the low end of this range. Most stars with M 1M? are believed to evolve into LPVs in later life. Hence the study of Mira variables is important in understanding the later stages of stellar evolution. The rapid mass-loss from Miras enriches the interstellar medium, and the final mass of the star determines its ultimate fate. well The t and their


110

CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI

Linearised dynamical models of Mira variables have been used to predict period-mass-radius (PM R) relations for fundamental and first-overtone pulsators. The masses of Miras are reasonably well-constrained, as noted above, and their periods can easily be measured. Hence measurements of the physical radii of a sample of Miras should allow their pulsation modes to be determined. There are difficulties however. The radius in the PM R relation is the average radius, as the small-amplitude limit is implicitly assumed when the pulsation models are linearised. The precise definition of the radius used in the models must be considered (see the discussion in Chapter 6), together with how to convert observed quantities into this radius. The conversion from an angular size to a physical radius also requires the distance to the star. The fact that previous authors (Haniff et al. 1995; van Belle et al. 1996; van Leeuwen et al. 1997) have come to different conclusions about pulsation modes illustrates that the problems are non-trivial. This chapter deals with the problem of measuring the diameter changes of Mira variables. Such changes could seriously bias any determination of pulsation mode based on diameter measurements. The changes are also interesting in their own right, as tests of non-linear pulsation models. Both the amplitude of the pulsation and the visual phase of maximum radius are currently uncertain. The conversion of measured apparent diameters to standard optical depth radii in order to determine pulsation modes is treated in the following chapter.

7.1.2

Diameter changes of Miras

Diameter changes of Mira-variable stars have been measured previously by van Belle et al. (1996), who found variations with phase within a sample of one or two-epoch diameter measurements of 18 stars, and by Tuthill et al. (1995), who made seven observations of o Ceti spread over a threeyear period. The first direct detection of cyclic variations in the apparent diameter of an individual Mira was made by Burns et al. (1998, henceforth B98), who carried out 18 observations of R Leonis with COAST (Baldwin et al. 1998) and the William Herschel Telescope in a 16 month period. In this chapter recent COAST diameter measurements of Cygni are presented, which exhibit phasecoherent changes similar to the variations previously seen in R Leonis.

7.2
7.2.1

Observations and data reduction
Measurements with COAST

The Mira variable Cygni was monitored with COAST in two wavebands: a 50 nm-wide band centred on 905 nm, and the J band at 1.3 Åm. Observations were made on 22 nights covering more than a full pulsation cycle. The dates of the observations and other details are given in Table 7.1. The observations at 905 nm were made with the visible beam combiner and avalanche photodiode detectors. The 1.3 Åm measurements were made using the IR correlator.


7.2. OBSERVATIONS AND DATA REDUCTION

111

During the period over which the observations were made, four COAST telescopes were operational. The 40 cm primary mirrors were stopped down to 16 cm for the observations at 905 nm, to better match the seeing conditions (r0 10 cm at this wavelength). At 1.3 Åm, aperture sizes of either 24 cm or 40 cm were used. Observations of Cyg were interleaved with observations of a calibrator star, one of Cygni, 1 Cygni or Lyrae. The visibility amplitude was measured on all baselines on which fringes could be detected. For the reasons discussed in Section 5.2.3, each baseline was measured separately. If at least three baselines were measurable, and the atmospheric coherence time was sufficiently long, then closure phase measurements were also secured. The layout of the COAST array was changed in January 1998, increasing the maximum baseline from 9 m to 20 m. The longer baselines were ideal for observations of Cyg at 1.3 Åm, but at 905 nm fringes could only be detected on the shortest baseline. The standard COAST data reduction procedures, described by Burns et al. (1997), were used to obtain visibility amplitudes and closure phases at 905 nm. The equivalent procedures for the infrared system, which differ only in detail, were described in Chapter 5. Note that the calibrator star 1 Cyg was slightly resolved on the 20 m baseline at 1.3 Åm (1 Cyg was not used on long baselines at 905 nm). The relevant calibrated visibilities were corrected for this effect, using an adopted diameter for 1 Cyg of 4.8 mas, calculated from the relation between angular diameter

and V K colour for K-type giant stars of Di Benedetto (1993). This correction increased the Gaussian FWHM inferred from the 98/07/18 data by just 2%.

occasions when this calibrator was used. The photometric field of view was 20ÌÌ , and hence comfortably large enough to accept a seeing-limited image, yet small enough to limit the effect of the lunar background. The errors on the photometry, based on the scatter of repeated measurements, are typically 0.01-0.02 mag. The photometric measurements are plotted in Figure 7.4.

Because frequent calibrator observations were required, it was possible for photometry of Cyg relative to Lyr (which is a standard star for most photometric systems) to be obtained on most

7.2.2

Measurements with the WHT

Interferometric measurements were made in August 1997 at the WHT, using conventional aperturemasking apparatus (see, for example, Buscher et al. 1990). An aperture mask was placed in a reimaged pupil plane, to select a number of interferometer baselines. The resulting fringe patterns were sampled on a fast-readout CCD. Exposure times of 12 ms were used to freeze the seeing, together with a linear five-hole pupil mask which gave 10 baselines up to 3.7 metres in length. Measurements were made in the same 905/50 nm bandpass as that used at COAST, with Cyg serving as the calibration star. Good Fourier plane coverage was obtained by rotating the aperture mask relative to the sky, and recording five thousand short-exposure images at each of nine separate position angles. The image scale and orientation were determined by observations of two close binary stars with known orbits. Details of the observations are given in Table 7.1.


112

CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI

Table 7.1: Log of observations at COAST and the WHT. Dates refer to the start of the night of each observation. Where multiple dates are present this implies that data from a number of nights were combined to permit a reliable diameter estimate to be obtained. Note that the observations from 92/07/13 have been reported previously (Haniff et al. 1995). The visual phases were calculated using the ephemeris from the 1997 Observer's Handbook of the Royal Astronomical Society of Canada, which was checked against the information in AAVSO Bulletin 61. Nvis and Ncl refer to the number of visibility and closure-phase measurements made. An identical number of visibility measurements were made for the calibration source. Each "measurement" corresponds to 30-60 seconds of observations. Date(s) 92/07/13 97/07/19, 20 97/07/20 97/08/07 97/08/11 97/08/25, 27, 30 97/09/22 97/10/05 97/10/18 97/12/03 98/05/16 98/07/13 98/07/16 98/07/18 98/07/24 98/08/04 98/08/04 98/08/18 98/08/19 98/09/19 98/10/14 98/10/18 98/12/06 Telescope WHT COAST COAST COAST WHT COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST Pulsation phase 0 4 4 4 .3 .8 .8 .8 2 3 3 7 Baseline range (m) 0 4 4 3 .5-3.5 .0-8.4 .0-8.0 .5-8.2 905 nm Nvis 28 - 6 29 90 12 9 - 12 6 1 5 - - 9 6 - 9 - 20 10 9 26 Ncl 15 - - 25 90 - - - - - - - - - - - - - - - - - - 1290 nm Nvis - 11 - - - - 5 - - - - 1 32 - - 14 - - - - - - Ncl - - - - - - - - - - - - - - - - - - 17 - - - -

4.88 4.92 4.99 5.02 5.05 5.16 5.57 5.71 5.72 5.72 5.74 5.76 5 5 5 5 .7 .8 .8 .8 6 0 0 8

0.3-3.7 4.0-8.3 3.6-8.2 4.3-7.9 3.5-8.0 3.6-7.9 4.1 4.7-5.1 9.4 5.0-20.5 4.5-5.2 5.5-5.6 4.4-10.7 4.7-5.2 5.0-20.5 4.4-4.6 4.4 4.7-5.2 4.5-4.7

5.94 5.95 6.07


7.2. OBSERVATIONS AND DATA REDUCTION

113

Table 7.2: Apparent angular sizes (Gaussian full widths at half maximum intensity) and 1 errors for Cyg at 905 nm and 1290 nm. The WHT measurement from 92/07/13 is that of Haniff et al. (1995). Date 92/07/13 97/07/19 97/07/20 97/08/07 97/08/11 97/08/27 97/09/22 97/10/05 97/10/18 97/12/03 98/05/16 98/07/13 98/07/16 98/07/18 98/07/24 98/08/04 98/08/18 98/09/19 98/10/14 98/10/18 98/12/06 Pulsation phase 0.32 4.83 4.83 4.87 4.88 4.92 4.99 5.02 5.05 5.16 5.57 5.71 5.72 5.72 5.74 5.76 5.80 5.88 5.94 5.95 6.07 WHT COAST COAST COAST WHT COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST COAST Telescope Gaussian FWHM (mas) 905 nm 19 6 15 9 16 6 19 0 16 1 15 3 16 0 18 3 22 5 20 4

? ? ? ? ? ? ? ? ? ?

1290 nm - 13 9

15 04 01 04 02 02 13 4 02 02 19 04 12 3 12 2 05 02 0 0 0 0 4 6 08 2 12 1

-

?

11

- - - - - - - - -

-

?

08

- - 20 8 19 4 16 8 17 2 17 03 16 7 14 3

? ? ? ? ? ? ?

? ? ?

09 01 03

-

- - - - -

03


114

CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI

1

97/08/07 COAST 97/08/11 WHT 16.6mas Gauss 24.6mas UD

Visibility amplitude

0.8

0.6

0.4

0.2

0 0 2000 4000 6000 Projected baseline (mm) 8000

Figure 7.1: Visibility curve for Cyg at 905 nm. Vertical crosses are visibility amplitude measurements from COAST on 97/08/07, and diagonal crosses are the average of eight WHT measurements at different position angles, from 97/08/11. The best fitting Gaussian (solid line) and uniform disk (dashed line) models are also shown. The Gaussian model is clearly a better fit to the data.

7.3

Results

In order to characterise the apparent stellar size, both uniform disk and Gaussian models for the brightness distribution were fitted to the visibility amplitude data. At a wavelength of 905 nm, a Gaussian was a better fit to the long baseline data (i.e. when the visibility amplitude falls below 30%, see Figure 7.1 for an example), even near maximum light, and so only the results of the Gaussian fits are presented here. However, the phase-coherent modulation seen in the Gaussian diameters at 905 nm is also seen in the 905 nm uniform disk diameters, which suggests that most of the variation is due to real changes in the scale of the emitting region, rather than to changes in limb-darkening. The diameters inferred by fitting one-parameter models may be biased by departures of the stellar disk from circular symmetry. Asymmetric brightness distributions have previously been detected in Mira variables at optical wavelengths, and have been modelled as either elliptical disks (Karovska et al. 1991; Haniff et al. 1992; Wilson et al. 1992; Quirrenbach et al. 1992; Weigelt et al. 1996), or a few unresolved features superimposed on a circular disk (Haniff et al. 1995; Tuthill et al. 1999). Note that departures from reflection symmetry cannot be detected interferometrically if the phase information is missing or of poor quality, in which case an elliptical disk model would be chosen. In order to quantify any asymmetry, models consisting of a disk plus one or two unresolved features were fitted to those data sets including closure phases. The software described in Chapter 5


7.3. RESULTS

Figure 7.2: Best-fit Gaussian plus one unresolved feature model for Cyg at 905 nm on 97/08/07. 115


116

CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI

was used for this purpose. In the 97/08/07 905 nm data there is a clear modulation of the closure phase on two baseline triangles with the hour angle of the source (see Figure 7.2). The closure phase furthest from zero is 23Ö . Two bright unresolved features, contributing 9 and 6 per-cent of the total flux, are needed to fit the closure phase excursions. The underlying Gaussian disk in this model has a FWHM which differs by 6% from that in the one-parameter model. The 1.3 Åm data from late 1998 also provides evidence for asymmetry. The same circularlysymmetric Gaussian model is an excellent fit to the visibilities measured on 98/07/18 and 98/08/04. However, some of the closure phases from 98/09/19 are significantly non-zero, with values up to 27Ö. A modulation of closure phase with the hour angle of the source is evident on two of the four baseline triangles. The visibility amplitude data from 98/07/18 and 98/08/04 were combined with the closure phase data from 98/08/19, and a model consisting of a circular Gaussian disk plus one unresolved feature was fitted to this data set. This model was found to be a good fit to both the visibilities and closure phases, with the unresolved "hotspot" located in the outer part of the disk, at position angle 82Ö. The best estimate for the percentage of the total flux in the hotspot is only 1 2 ? 0 2%. The 1998 observations at 905 nm were all obtained using a single baseline, and the reader should note that an unresolved feature contributing 10% of the total flux would have biased the diameter by up to 15%, in either direction (although the feature would have to be very close to the stellar limb to increase the inferred diameter by this amount). Where the visibility amplitude was measured on two or more baselines (on which the source was resolved) with different orientations, elliptical Gaussian and uniform disk models were tried. In no case was evidence for significant ellipticity found. The apparent angular size of Cyg at 905 nm (Table 7.2, Figure 7.3) clearly varies in a phase coherent manner. The diameter increases slowly from phase 1.0 (maximum light) to phase 1.5 (minimum light), then decreases rapidly between phases 1.6 and 1.8. There is little change between phases 0.8 and 1.0 (this part of the cycle was observed twice). A similar slow diameter increase during the first half of the pulsation cycle is seen in the B98 measurements of R Leonis. The 1992 measurement of Haniff et al. (1995), made in an almost identical bandpass to the 905 nm band used for these observations, is consistent with the diameter modulation seen during 1997 and 1998, and therefore suggests that the variation in Cyg may be coherent over five cycles. Tuthill et al. (1999) found tentative evidence for an elongated morphology at a wavelength of 700 nm at the same epoch, but the asymmetry was slight: the axial ratio of their best-fit elliptical Gaussian model is 0 96 ? 0 07. The peak-to-peak amplitude of the diameter variation at 905 nm is approximately 45% of the smallest diameter. B98 found that the diameter of R Leonis varied by 35% at both 830 nm and 940 nm. Despite sparse coverage of pulsation phase at 1.3 Åm for Cyg, it is clear that the behaviour of the apparent diameter in this waveband is different to that in the 905 nm band. The ratio of the 1.3 Åm diameter at maximum light to that at phase 1.75 is slightly (but not significantly) greater than unity, whereas the same ratio at 905 nm is 0 74

?

0 02.


7.3. RESULTS

117

Gaussian FWHM (mas)

24 22 20 18 16 14 12 0.8 1 1.2 905nm 1997-8 Haniff et al (1995) 1.4 1.6 1.8 2 Pulsation Phase 1290nm 1997-8

Gaussian FWHM (mas)

24 22 20 18 16 14 12 0.8 1 1.2

1.4

1.6

1.8

2

Pulsation Phase

Figure 7.3: Apparent angular sizes (Gaussian full widths at half maximum intensity) and 1 errors for Cyg at 905 nm (top) and 1290 nm (bottom), plotted against the phase of the visual lightcurve (zero is maximum). The diagonal cross at phase 1 32 is the 1992 measurement at 902 nm of Haniff et al. (1995). The diameter increases slowly from phase 1.0 to phase 1.5, then decreases rapidly between phases 1.6 and 1.8. A similar slow diameter increase during the first half of the pulsation cycle is seen in the B98 measurements of R Leonis.
0 1 2 3 4 0.8 1 1.2 1.4 1.6 1.8 2 Pulsation Phase

Figure 7.4: Light curve for Cyg at 905 nm. The detected flux relative to that of Lyrae is plotted on a magnitude scale.

Relative Mag.


118

CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI

7 .4

Discussion

The emission detected from Cyg through the J band filter should be mostly continuum emission. The spectra of cool stars do have a wide absorption feature at 1.4 Åm due to H2 O, but almost all of the photons in the wavelength range of the feature will be absorbed by telluric water and thus not detected (evidence for this absorption in the power spectra of stellar fringes was given in Section 3.5.3). The fact that the 1.3 Åm diameter is always smaller than the 905 nm diameter at the same epoch supports this conclusion. The 905 nm bandpass is similar to the 830 nm and 940 nm filters used by B98, in that the transmitted flux is dominated by continuum emission, with only a small amount of atomic and molecular blanketing. There is however weak contamination by TiO absorption bands, whose strength varies with pulsation phase (Spinrad and Wing 1969). Hence the detected photons originate from several different layers within the extended photosphere of the star, whose relative contributions vary with phase. It is clear that the diameter we measure in these wavebands does not simply tell us the position of the continuum-forming layers. The strength of the TiO absorption bands in the spectrum of an M-type Mira define its spectral type. Cyg is classified as S-type, although its ZrO bands are much weaker than those due to TiO (Lockwood and Wing 1971). An "M-equivalent" spectral type can therefore be assigned on the basis of TiO band strength. Narrow-band spectrophotometry by Lockwood and Wing (1971) shows that this spectral type is earliest close to visual maximum. The spectral type increases until minimum light is reached, then the latest type ( M9) isretaineduptophase 0 7, after which it falls rapidly. This variation of TiO band strength with pulsation phase is well correlated with the diameter changes at 905 nm, which suggests that a substantial fraction of the apparent diameter variation at 905 nm is caused by changes in TiO opacity in the outer photospheric layers, and not by physical motions of the photosphere. The diameter measurements at 1.3 Åm are more likely to trace continuum emission, and show much smaller variations. The 905 nm diameter variations are also well correlated with the 905 nm light curve. The photometric amplitude in this band is 3.0 mag., compared with the amplitude in the continuum at 1.04 Åm of 2.3 mag. (Lockwood and Wing 1971). The larger amplitude at 905 nm is probably due to variations in TiO absorption in this band. The size of the effect indicates that TiO contamination in the 905 nm band is significant. Hofmann et al. (1998) have calculated the apparent diameters of six model Mira variables covering a wide range of effective temperatures, and both fundamental and first-overtone pulsation, at various wavelengths and pulsation phases. The physical size of their model stars is largest at phase 0.2, therefore the apparent diameter in the continuum is larger at maximum than at minimum light for all six of the model series. At near-continuum wavelengths increased contamination by molecular bands tends to enlarge the star near minimum. Despite this effect, five of the six model series still appear larger at maximum, the exception being the P series (mean Teff 2860 K, fundamental mode pulsation), whose apparent diameter at 820 nm and 920 nm is 15-30% larger at minimum than at maximum. The change in a Gaussian fit between minimum and maximum should


7.5. CONCLUSIONS

119

be slightly smaller than this, if the Hofmann et al. predictions for phase-dependent limb-darkening are correct (higher resolution would be required to actually measure the change in limb-darkening with phase). These differences are much less than the 45% observed for Cyg at 905 nm, which suggests that either the treatment of molecular opacity in the models is inadequate, or the physical size of Cyg is largest near minimum light.

7.5

Conclusions

Phase-coherent variations in the apparent diameter of Cygni have been detected. The amplitude at a wavelength of 905 nm is 45% of the smallest diameter. Any variation in the diameter at 1.3 Åm has a much smaller amplitude. These measurements may be explained in two different ways: 1. The continuum diameter of Cyg is largest near minimum light. The apparent diameter at 905 nm is further enlarged near minimum by an increase in TiO opacity. 2. The continuum diameter of Cyg is largest nearer to maximum light. At 905 nm, the star appears largest near photometric minimum because of a huge increase in TiO opacity, larger than predicted by the latest models. Infrared observations near to minimum light should enable the correct interpretation to be determined. Regardless of the phase-dependence of the 1.3 Åm diameter, the amplitude at this wavelength is small, which means that any diameter measurement in this band will be close to the average diameter needed to infer the pulsation mode. A programme to determine the pulsation modes of a small sample of Miras from their 1.3 Åm diameters is described in the next chapter.


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CHAPTER 7. CYCLIC VARIATIONS IN THE ANGULAR DIAMETER OF CYGNI


Chapter 8

Effective Temperatures, Linear Radii, and Pulsation Modes of Mira Variables
8 .1 Introduction

This chapter is concerned with measuring the "average" angular diameters of a sample of nearby Mira variables, in order to determine their effective temperatures, and, in conjunction with their distances, linear radii and hence pulsation modes. Such measurements are best made in the nearinfrared, for two reasons. First, the outer atmospheres of Miras are optically thin at IR wavelengths longer than 1 Åm. Hence the conversion of a measured angular diameter to the "photospheric" angular diameter (recall that this may defined in several different ways) involves smaller correction factors and is less dependent upon the details of model atmosphere calculations. Second, the measurements of Cygni presented in the previous chapter show that the 1.3 Åm diameter varies much less with pulsation phase than the diameter measured in a near-continuum band below 1 Åm. Thus the average diameter is easier to measure in the infrared. If nearby Miras are observed at IR wavelengths, only separated-element interferometers have sufficiently long baselines to fully resolve them. Until now, such observations have been made with a single baseline (Ridgway et al. 1992; van Belle et al. 1996), leaving the measured diameter sensitive to the effect of asymmetries, the presence of which appears to be ubiquitous at optical wavelengths (Tuthill et al. 1999). Closure phase measurements in the IR will enable the apparent asymmetries to be quantified, so that the true disk diameter can be determined.

8.2

Observations and data reduction

Observations were made during 1997 and 1998 of six nearby oxygen-rich Mira variables (o Ceti, U Herculis, Cygni, T Cephei, R Cassiopeiae and T Cassiopeiae) with COAST in the J band, using the infrared correlator. The Miras were selected on the basis of their large apparent angular diameters. Details of the observations are given in Table 8.1.


122

CHAPTER 8. MIRA VARIABLES

The observations before January 1998 were made using an array layout with a 9 m maximum baseline. The resolution of COAST in this configuration at 1.3 Åm was found to be insufficient for even the largest Miras in the sample, and so the configuration was changed to one with a maximum baseline of 20 metres. It was possible to find fringes on the 20 m baseline for all six stars, by bootstrapping. Once fringes were found on a long baseline on which the calibrated visibility was a few per-cent, a 60 second integration typically gave a signal-to-noise of 10 in the accumulated power spectrum. For the observations in 1998, closure phase measurements were interleaved with visibility amplitude measurements on those nights when the atmospheric coherence time at 1.3 Åm was longer than about 15 ms. No closure phases were measured for Cyg on 98/07/18 and 98/08/04 due to the short coherence times on those nights, so closure phases only were measured on 98/08/19. Visibility amplitudes and closure phases were extracted from the raw data using the procedures described in Chapter 5. Some of the calibrator stars used were slightly resolved on the longer baselines of the array. The calibrated visibilities were corrected for this effect using the calibrator diameters in Table 8.3. In no case did the correction change the inferred Mira diameter by more than 6%, and a 10% uncertainty in the adopted calibrator diameters is completely insignificant. Wherever possible, be very close to the Cep (Dyck et al. calculated from the infrared measurements of the calibrator J band diameters). Such measurements 1998; Di Benedetto 1993). Diameters relation between angular diameter and V diameters were used (which should have been published for Cet and for Her, Cyg and Cas were K colour for K-type giant stars of

Di Benedetto (1993). The diameter for Cas inferred in this way is consistent with the visible wavelength measurement of Mozurkewich et al. (1991).

8.3
8.3.1

Results
Simple models

The first stage in the analysis of the visibility amplitude and closure phase data was to fit simple one-parameter models for the stellar brightness distribution. These model fits allow comparison of repeated observations of the same star, and comparison of the observations described here with results in the literature. The three circularly symmetric models used were a uniform disk, a Gaussian disk, and the linear ( 1) fully-darkened disk model of Hestroffer (1997). As discussed in Chapter 5, the major source of uncertainty in the visibility amplitude measurements was calibration error, due to changes in seeing conditions between the observations of source and calibrator. This uncertainty was modelled as a constant fractional error in the visibilities, typically 10% of the visibility, but occasionally 20% where indicated by the scatter in repeated measurements on the same baseline. Where visibility measurements had been made on long baselines, one or more of the simple models were found to be obviously poor fits to the long baseline points, indicating that the true brightness distribution of the star differed significantly from that of the model, either because the star was


8.3. RESULTS

123

Table 8.1: Log of observations at COAST. Dates refer to the start of the night of each observation. Where multiple dates are present this implies that data from a number of nights were combined to permit a reliable diameter estimate to be obtained. The visual phases were calculated using the ephemerides (periods are given in the first column) from the 1997 Observer's Handbook of the Royal Astronomical Society of Canada, which were checked against the information in AAVSO Bulletin 61. Nvis and Ncl refer to the number of visibility and closure-phase measurements made. A similar number of visibility measurements were made for the calibration source. Each "measurement" corresponds to 30-100 seconds of observations. Star & period (d) o Cet 332 U Her 406 Cyg 407 97/10/21 98/08/10, 19 97/07/25 98/08/06 97/07/19, 20 97/10/05 98/07/18 98/08/04 98/08/19 T Cep 390 R Cas 431 98/07/23 98/08/07 97/10/18, 21 98/07/21 98/08/05 98/08/06 98/08/19 Date(s) Pulsation phase 0.76 1.67 0.91 0.83 0.83 1.02 1.72 1.76 1.80 0.20 0.24 0.52 1.16 1.20 0.58 0.60 Baseline range (m) 3.6-8.4 5.6-13.7 6.5-7.0 3.5-20.4 4.0-8.4 4.3-7.9 5.0-20.2 4.4-10.7 5.0-20.5 5.5-19.9 5.5-19.8 7.0 4.9-20.5 4.8-20.5 5.5-20.4 5.5-20.5 6 11 3 6 11 5 32 14 - 26 16 4 25 18 12 6 - - - - - - - - 17 10 7 - 3 4 7 4 Cet Cet Ser Her Lyr Lyr Cyg Lyr, Cyg Nvis Ncl Calibrator(s)

Cep Cep Cas Cas Cas Cas Cas

T Cas 445


124

CHAPTER 8. MIRA VARIABLES

not circularly symmetric, or because the centre-to-limb brightness profile of the star was different from that of the model. More appropriate models are discussed later, but in this preliminary analysis, to allow comparisions to be made with other observations, these long-baseline points were removed so that a satisfactory fit could be obtained. If fits to the complete data sets had been used, the inferred diameters would have been biased. The small visibilities on long baselines have the smallest error bars (because most of the uncertainty is calibration error, which is a fixed fraction of the visibility), and the models would therefore be forced to fit these points and mis-fit the shorter-baseline data. The observed visibility curves for the four stars which were fully resolved by COAST (Figures 8.1 and 8.2) differ substantially from each other at long baselines. The visibility function for T Cep has the minimum (near 17.5 m), and corresponding abrupt change in closure phase expected for a sharp-edged disk, whereas the curve for Cyg is very well fit by a Gaussian profile for the star. There is no discontinuity in the closure phases for Cyg. The curves for R Cas and o Cet are harder to interpret, due to limited or missing closure phase data.

8.3.2

Asymmetries

The data were examined for evidence of departures from circular symmetry by a procedure similar to that employed by Tuthill et al. (1994) and HST95, consisting of inspection of the visibilities and closure phases, and fitting of simple models with a small number of free parameters. Reconstruction of images from the Fourier data was not possible for any of the three fully-resolved stars for which closure phase data was taken, as the available mapping packages were unable to cope with data sets in which not all the closure phases were measured (T Cep and R Cas) or the closure phases were measured on a different night from the corresponding visibilities ( Cyg). Two initial tests for asymmetries were performed. First, closure phases significantly different from zero or 180Ö imply a departure from reflection symmetry. The presence of unresolved features on the stellar disk is usually indicated by a slow modulation of the closure phase on a particular baseline triangle with the orientation of the triangle. Secondly, the visibilities measured using the longer-baseline array configuration gave a direct measure of the degree of elongation of the source, as the baseline between the Centre and North telescopes and that between the East and West telescopes had very similar lengths (sufficient to resolve these targets), but were oriented at right angles to each other. Model fitting, using the methods described in Chapters 5 and 6, was performed on those data sets with sufficient measurements of both visibility amplitude and closure phase to provide information about asymmetries. Data sets taken within a few weeks which appeared to be consistent with each other were combined first. The models consisted of either a circular limb-darkened disk (Gaussian or Hestroffer FDD) with one or two unresolved bright features (modelled as narrow Gaussians) superimposed, or an elliptical limb-darkened disk. As before, 2 was used to measure the goodness-of-fit. The best-fit model parameters are given in Tables 8.4 and 8.5. I will now discuss the results of this analysis for each star in turn.


8.3. RESULTS

125

T Cep 1 Visibility amplitude 0.8 0.6 0.4 0.2 0 0 98/07/23 98/08/07 18.5mas UD 5000 10000 15000 20000 Projected baseline /mm T Cep Closure phase /degrees 0 -30 -60 -90 -120 -150 -180 0 98/07/23 98/08/07 5000 10000 15000 20000 Longest projected baseline /mm Closure phase /degrees 30 40 20 0 -20 -40 0 98/08/19 5000 Visibility amplitude 1 0.8 0.6 0.4 0.2 0 0

Chi Cyg

98/07/18 98/08/04 12.1mas Gauss 5000 10000 15000 20000 Projected baseline /mm Chi Cyg

10000

15000

20000

Longest projected baseline /mm

Figure 8.1: J band visibility curves (top) and corresponding closure phases (bottom) for T Cep (left) and Cyg (right). The visibility amplitude for T Cep goes through a minimum, and at a similar baseline the closure phase changes from 0 to 150 degrees. In contrast, there is no null in the visibility function for Cyg, which is very well fit by a Gaussian model for the intensity profile. The closure phases for Cyg are all 30Ö .

R Cas 1 Visibility amplitude 0.8 0.6 0.4 0.2 0 0 5000 10000 15000 20000 Projected baseline /mm 98/07/21 98/08/05 21.5mas UD 1 Visibility amplitude 0.8 0.6 0.4 0.2 0 0 5000

o Cet 98/08/10,19 29.1mas UD 19.4mas Gauss

10000

15000

Projected baseline /mm

Figure 8.2: J band visibility curves for R Cas (left) and o Cet (right). Uniform disk models (solid lines) are poor fits to the long baseline visibilities for both stars.


126

CHAPTER 8. MIRA VARIABLES

Table 8.2: Best-fitting one-parameter models and 1 errors for the sample of Mira variables. The errors include a contribution from the 1% uncertainty in the effective wavelength of observation. The models are uniform disk (UD), Hestroffer fully-darkened disk (FDD) with 1, and Gaussian (for which the FWHM is given). For some stars in the sample, some or all of the models were obviously a poor fit to the visibilities on long baselines. In those cases, which are indicated in the table by parentheses around the best-fit model parameter, the long-baseline data were removed prior to re-fitting the model, to avoid biasing the diameter (see discussion in text). Star o Cet Date 97/10/21 98/08/10, 19 97/07/25 98/08/06 97/07/19, 20 97/10/05 98/07/18 98/08/04 T Cep 98/07/23 98/08/07 R Cas 97/10/18, 21 98/07/21 98/08/05 T Cas 98/08/06 98/08/19 Pulsation phase 0 76 1 67 0 91 0 83 0 83 1 02 1 72 1 76 0 20 0 24 0 52 1 16 1 20 0 58 0 60 UD (mas) 28 7 Ä29 1 FDD (mas)
Å

? ?

06 05

32 6 Ä33 2

? ?

Gaussian (mas)
Å

06 05

18 8 Ä19 4

? ?

04 04

Å

U Her

11 9 22 3 21 9
Ä

? ? ? ? ? ? ? ? ? ? ? ?

03 16 11 02 05 02 02 09 03 03 03 03
Å Å Å

unresolved 13 5 ? 0 4 25 1 24 7
Ä

77

?

03 11 08 01 03 03 03 06 03 03
Å Å Å Å

Cyg

18 2 19 3 18 6 18 5 25 8

20 7 21 8 21 8 21 7 29 0

? ? ? ? ? ? ? ? ? ? ?

18 13 03 05 03 03 10 03 03 04 03
Å Å Å

13 9 13 4 12 2 12 1
Ä Ä

? ? ? ? ? ? ? ? ?

13 6

13 8 16 1

Ä Ä

21 1 21 9

Ä Ä

24 4 25 3

Ä15

5 Ä16 6 84 78

12 7 12 0

14 4 13 6

? ?

03 02

Table 8.3: Adopted uniform disk diameters for calibrator stars Calibrator Cet Her Cyg Cep Cas UD (mas) 11 6 37 48 56 55


8.3. RESULTS o Ceti

127

No closure phase data was taken for o Cet, and there were insufficient visibility measurements to draw reliable conclusions about any possible departures from circular symmetry. Circularlysymmetric models gave adequate fits to the visibilities on baselines shorter than 10 m. The visibilities on longer baselines were removed for the subsequent determination of the photospheric diameter, as the visibility due to the disk is small at long baselines, and so undetected unresolved features could have significantly affected these data points.

U Herculis Only a small number of visibility amplitude measurements were made for U Her, and hence no conclusions could be drawn about the morphology of this star. Circularly-symmetric uniform disk or Gaussian models were found to be good fits to the data.

Cygni The apparent asymmetry of Cyg at 1.3 Åm was discussed in the previous chapter (the same 1.3 Åm Cyg data is used here and in Chapter 7). To summarise, evidence was found for the presence of a small asymmetry in late 1998. This asymmetry was modelled as an unresolved feature superimposed on a circular disk, contributing just 1 2 ? 0 2% of the total flux. The departure from circular symmetry was very small, as shown by the elliptical Gaussian fit, which yielded an

axial ratio of 0 99 ? 0 02. The very slight asymmetry detected for Cyg at this epoch will not significantly bias the photospheric diameter inferred on the basis of circular symmetry.

T Cephei The visibilities measured with the Centre-North and East-West baselines on 98/07/23 and 98/08/07 indicate a slight elongation of the star: taken separately, measurements on the two baselines imply uniform disk diameters of 19 7 ? 0 2 mas and 20 3 ? 0 2 mas respectively. Closure phases had been measured on two of the four baseline triangles on 98/07/23 and 98/08/07. No closure phase measurements were attempted on triangles involving the baseline between the North and West elements, because the visibility on that baseline (close to the minimum in the visibility curve) was undetectable or very small at all hour angles. None of the closure phases on the smaller triangle were significantly different from zero, but the mean of the four measurements on the larger triangle (which included the 20 m baseline) was 152 ? 5Ö, significantly different from the value of 180Ö expected for a centro-symmetric object. Two different models with single unresolved features (contributing 1 and 2% of the flux), located on opposite sides of the disk, could fit the data. An elliptical disk with its major axis oriented along position angle 96Ö and an axial ratio of 0 93 ? 0 02 was a good fit to the visibilities and


128

CHAPTER 8. MIRA VARIABLES

to the closure phases on the shorter triangle. In an attempt to determine the magnitude of the departure from reflection symmetry implied by the closure phases on the larger triangle, a model consisting of this best-fit elliptical disk, plus a single unresolved feature was then fitted to the data. Only the fluxes of the two components and the hotspot position were allowed to vary. The best-fit model, with just 1% of the total flux in the unresolved component, was a good fit to all of the data points. Hence it is clear that the main component of the asymmetry in T Cep was a slight elongation along position angle 96Ö , and so an elliptical disk model is adequate for inferring the photospheric size of the star.

R Cassiopeiae An elongation was obvious from the visibility data taken on both 98/07/21 and 98/08/05. All but one of the closure phases had been measured on the smallest triangle, and these were consistent with a value of zero. Elliptical limb-darkened disk models were a good fit to the data from both epochs. The axial ratios at the two epochs were identical (0.92) and the position angles of the major axis were similar, at 105 ? 7Ö and 130 ? 8Ö respectively. However, the major axis was 3% larger on 98/08/05 than on 98/07/23 (a 2 change).

Models consisting of various circularly-symmetric limb-darkened disks plus one unresolved feature were fitted to the 98/07/21 and 98/08/05 data sets separately. The best-fit models of this type did not appear to fit any real structure in the data which could not be fit by an elliptical disk model, and are only presented in Table 8.4 to indicate how well the flux in an (undetected) unresolved feature is constrained by the small number of closure phase measurements. In an alternative procedure, unresolved features were superimposed on the best-fit elliptical disk models for the 98/07/21 and 98/08/05 data. The fluxes of both components and the position of the unresolved component were allowed to vary. The "hotspot" fluxes in the final best-fit models were only 0.8% and 0.7% of the total, which clearly indicates that elliptical disks were good models for R Cas at these epochs.

T Cassiopeiae All of the closure phases measured on T Cas were consistent with a value of zero, even on triangles including the longest baseline. The visibility fell to 20% on this baseline, and so any departure from circular symmetry was small. There was no evidence from the visibility data for ellipticity.

8.3.3

Photospheric diameters

The best-fit diameters of the simple models discussed above are an indication of the apparent angular size of the object in the waveband used for the observations, but do not necessarily represent the "true" angular size of the star. The different possible definitions of the stellar radius were discussed in Chapter 6. I wish to determine the radius which appears in the period-mass-radius


8.3. RESULTS

129

Table 8.4: Best-fit models consisting of a limb-darkened disk with one unresolved feature superimposed. The coordinates Är Å give the position of the unresolved feature with respect to the centre of the underlying disk ( is measured North through East). In the case of the Gaussian disk, "diameter" refers to its FWHM. Star Date(s) Component Flux 1 04 0 013

Cyg 98/07/18, Gaussian 98/08/04, 19 Spot T Cep 98/07/23, 98/08/07 R Cas 98/07/21

? ? ? ?

r (mas)

(Ö ) - 82

Diameter (mas) 12 28

0 02 - 0 002 11 8 ? 1 5 0 03 - 0 005 5 6 ? 1 2 0 03 0 004 0 04 0 006 1 4 - 11

?

? ?

0 10

3 21 9

- 02

FDD ( 1) 0 88 Spot 0 020 FDD ( 2) 1 00 Spot 0 017 FDD ( 2) Spot 0 94 0 080



- 74 ? 5

- 27 4

? ? ? ?

?

2

- 59 3 ? 0 4 - 12

?
-

02

R Cas 98/08/05

?

-

03

7

?

32 1

?
-

04

Table 8.5: Best-fit elliptical limb-darkened disk models. Star Cyg Date(s) 98/07/18, 98/08/04, 19 T Cep 98/07/23, 98/08/07 R Cas R Cas 98/07/21 98/08/05 FDD ( 2) FDD ( 2) 28 6 29 5 FDD ( 1) 23 0 Disk type Gaussian Major axis (mas) 12 2

? ? ? ?

Axial ratio 0 99

02

? ? ? ?

Major axis PA (Ö ) 78

0 02

?

88

04

0 93

0 02

96

?

7

03 03

0 92 0 92

0 01 0 02

105 130

? ?

7 8


130

CHAPTER 8. MIRA VARIABLES

(PM R) relations. The linear pulsation models of Fox and Wood (1982) (used later to obtain the PM R relations) use an optical depth radius to define the boundary of the star. Hence I shall follow previous workers in defining the stellar radius (or photospheric radius) R to be the distance from the centre of the star at which the Rosseland mean optical depth equals unity. An intensity radius could be obtained from a measured CLV (but note that the intensity radius is wavelength-dependent). To determine both the shape and extent of the CLV with an interferometer would require visibility measurements covering a wide range of spatial frequencies. However, even an intensity radius measured in the continuum will not necessarily correspond to the Rosseland optical depth radius R in a Mira (Hofmann et al. 1998, henceforth HSW98). The model CLVs in Figure 8.3 illustrate this: the positions of the steep intensity gradients in the 1.3 Åm CLVs do not correspond to the Rosseland radii. Hence the CLV in the specific bandpass used for the observations, for a model star for which R can be calculated, must be used to convert the visibility amplitude measurements into a photospheric angular diameter R . The precise method used in this work is detailed below. CLVs for a subset of the Mira models described by Bessell et al. (1996) and HSW98 were computed by Michael Scholz. I selected models with luminosities, radii and effective temperatures likely to be close to those of the sample stars, and including both fundamental and first-overtone mode pulsators. The parameters of the model stars (taken from Tables 1 and 2 in HSW98) are given here for reference, in Table 8.6. The CLV calculations were made for a number of bandpasses which have been used at COAST for observations of Miras. As discussed in Section 3.5.3, water vapour in the Earth's atmosphere absorbs almost all of the stellar radiation at wavelengths longer than 1.36 Åm which would otherwise be transmitted by the J band filter. The model CLVs were calculated for two different spectral responses at 1.3 Åm. The first was simply the transmission curve of the J band filter. The second was identical below 1.35 Åm, but then cut off sharply and had zero response above 1.36 Åm. The cut-off spectral response is much closer to the true spectral response of the instrument, and only the CLVs calculated for this transmission curve were used in the subsequent analysis. Use of the alternative CLVs would decrease the inferred photospheric diameters, but only by a maximum of 4%. Figure 8.3 shows two of the CLVs calculated for the realistic spectral response. The two CLVs in the figure are for different phases of the first-overtone-pulsating E model series. The CLVs for the other models considered here have similar shapes. Overplotted are Hestroffer FDD models with diameters obtained from fits to the CLVs in the Fourier domain, normalised to the same zerospacing visibility. As noted by HSW98, the Hestroffer intensity profiles are good representations of the model-predicted Mira CLVs, which are intermediate between uniform disk and Gaussian profiles. Previous workers (Ridgway et al. 1992; van Belle et al. 1996) have determined photospheric angular diameters from interferometric data by fitting a uniform disk model to the visibilities then applying a scaling factor derived from an appropriate model photosphere. Wherever circularlysymmetric models gave adequate fits to the data, the more direct method of HST95 was employed.


8.3. RESULTS

131

Table 8.6: Fundamental properties of Mira models from Bessell et al. (1996) and HSW98. The E, P and M models labelled as "static" are the non-pulsating parent models from which the other, time-resolved models are derived. Model E E8300 E8380 E8560 P P71800 = P05 P73200 = P10 M M96400 = M05 M97600 = M10 Pulsation Mode static 1st overtone 1st overtone 1st overtone static fundamental fundamental static fundamental fundamental Visual phase - 0.83 1.00 1.21 - 0.50 1.00 - 0.50 1.00 L L? 6310 4790 6750 7650 3470 1650 5300 3470 1470 4910 R R? 366 425 399 428 241 289 248 260 242 309 Teff K 2700 2330 2620 2610 2860 2160 3130 2750 2310 2750

1.2 1 Intensity 0.8 0.6 0.4 0.2 0 0 0.5 1

E8300 CLV FDD (=2) Intensity

1.2 1 0.8 0.6 0.4 0.2 0

E8380 CLV FDD (=1)

1.5

2

0

0.5

1 r/R

1.5

2

r/R

Figure 8.3: J band centre-to-limb intensity profiles (solid lines) for two Mira models from Bessell et al. (1996). In both plots, the radial coordinate is in units of the Rosseland mean radius R. The models represent a star pulsating in the first-overtone mode, at two different phases. The phase of the E8300 model (left) is 0.83, and that of the E8380 model (right) is 1.00. The effective temperatures of these models are 2330 K and 2620 K respectively (other model parameters are listed in Table 8.6). A general feature of all of the model series is that the wing-like features in the outer parts of their J band CLVs are more pronounced in cooler models. The Hestroffer profiles shown (dashed lines) were obtained from fits to visibilities calculated from the model CLVs, and hence are normalised to the same zero-spacing visibility.


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CHAPTER 8. MIRA VARIABLES

The Hankel transform of the model CLV was taken (i.e. circular symmetry was assumed), and the resulting visibility function fitted to the observed visibility data, wherever possible using the closure phases to determine the sign of the measured visibility. This procedure gave the photospheric angular diameter R directly, and also allowed comparison of the model visibility curves with the data. In this way, the shapes of the model CLVs could be compared with the true CLVs. The Cyg visibility data from 98/07/18 were found to be incompatible with all of the model CLVs except that of the P05 model, which is the coolest of the set. Assuming the apparent bolometric magnitude near to mean light to be the mean bolometric magnitude in Table 8.10 (van Leeuwen et al. 1997), the angular diameter derived from the P05 CLV implies an effective temperature of

2670 ? 60 K. This is much higher than the effective temperature of the P05 model itself (2160 K). I chose not to use the P05 model to determine photospheric diameters, since it is probably too cool to represent any of the sample stars at the observed phases. The P10 CLV is almost identical to the M10 CLV, and so the P series was ignored for the purpose of obtaining photospheric diameters. It was necessary to truncate the 98/07/18 Cyg data at a baseline of 14 m to obtain adequate fits to the visibility functions of the E8300 and M05 models. In the case of T Cep, and of R Cas on 98/07/21 and 98/08/05, the apparent elliptical shape of the star necessitated the use of scaling factors to obtain R (due to limitations of the analysis software), but as elliptical uniform disks were poor fits to the data, scaling factors appropriate for analytical limb-darkened disk models were calculated. These scaling factors were determined by fitting the visibility function for the analytical limb-darkening profile to the visibility function of the model CLV (calculated as above), up to the maximum baseline in the real data. Factors to convert from Hestroffer FDD fits (with limb-darkening parameters of 1 and 2) to the Rosseland optical depth diameter are given in Table 8.7. To calculate the photospheric diameter, assuming a particular model CLV, the appropriate scaling factor was applied to the major and minor axes of the best-fit elliptical limb-darkened disk model. The photospheric diameter was then calculated (following HST95) as the diameter of a sphere having the same volume as the prolate spheroid with major and minor axes as measured i.e. Ämajor axisÅ ? Äaxial ratioÅ2 3. FDDs with values of 1 and 2, chosen because elliptical disks with such CLVs gave good fits to the data for T Cep and R Cas respectively, were obviously poor fits (assuming similar error bars to those on the real data) are indicated by blank entries models E8300 and P05 (which the observations of T Cep, and to the visibility curves for some of the model CLVs. These cases in Table 8.7. Hence I can conclude that the CLVs derived from are two of the coolest models in Table 8.6) are incompatible with that the P05 CLV is incompatible with the 98/07/21 and 98/08/05

observations of R Cas. These conclusions should not be too surprising, as the phases of the two models are very different from those of the real stars at the epochs of observation. Photospheric angular diameters R , derived on the assumptions of both fundamental mode and first-overtone mode pulsation, are presented in Table 8.8. These are averages of the values obtained using each of the model CLVs listed in the table (the models chosen were the E or M models at the most appropriate phases). It is important to stress that the photospheric diameters obtained using the E and M series CLVs differ by only a few per-cent, which indicates that R is only


8.3. RESULTS Table 8.7: Scaling factors to obtain the photospheric angular diameter R from an angular diameter derived from a limb-darkened disk fit. See text for details. Photospheric model E8300 E8380 E8560 P05 P10 M05 M10 Analytical disk model FDD ( 1) FDD ( 2) - 0.900 0.925 - 0.902 0.946 0.902 0.840 0.807 0.830 - 0.809 0.851 0.809

133

weakly model-dependent. This is in sharp contrast to the results of HST95 at optical wavelengths, who found 15% difference between the diameters inferred using two different sets of models. Because the differences between the two sets of diameters in Table 8.8 are entirely negligible, I will only use the photospheric angular diameters obtained using the E series models in the calculations that follow.

8.3.4

Variation with phase

The variation of Mira diameters with pulsation phase was discussed in the previous chapter. The two stars with multiple measurements at dissimilar phases, Cyg and R Cas, exhibit changes in R of 20 ? 6% and 20 ? 4% respectively. T Cep has a very small photometric amplitude in the infrared compared with Miras of similar period (Lockwood and Wing 1971; Strecker 1973), hence one might expect its diameter variations to be unusually small. The amplitude of the pulsation in o Cet, U Her and T Cas is unknown, but is unlikely to be much greater than the 20% seen for Cyg and R Cas. Mean photospheric angular diameters (weighted means of the values in Table 8.8) are presented in Table 8.9. In those cases where there was more than one measurement, the uncertainty quoted includes a contribution from the standard deviation of the mean diameter.

8.3.5

Effective temperatures

Mean effective temperatures for the sample stars (Table 8.10) were calculated using the relation Teff 2341 Fbol 2 R
14

(8.1)

where the bolometric flux Fbol is in units of 10 8 erg cm 2 s 1 and R is in mas. The mean bolometric magnitudes of van Leeuwen et al. (1997) were used together with the mean photospheric angular diameters R derived from the first-overtone-pulsating E models.


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CHAPTER 8. MIRA VARIABLES

Table 8.8: Photospheric angular diameters R , determined using the CLVs of model stars pulsating in either the fundamental (5th column) or first-overtone mode (7th column). Star o Cet Date(s) 97/10/21 98/08/10, 19 Pulsation phase 0.76 1.67 0.83 0.83 1.02 1.72 1.76 0.22 fundamental Model(s) R (mas) M05, M10 30 2 M05 31 4 M10 M10 M10 12 1 22 6 22 1 1st overtone Model(s) E8300 E8300 E8300 E8300 E8380 E8300 E8300 E8560 R (mas) 30 4 31 0 12 5 23 0 21 8 19 2 20 0 20 3 09 05 03 12 12 05 04 07

? ? ? ? ? ? ?

? ? ? ? ? ? ? ?

06 05 03 12 12 02 02 05

U Her 98/08/06 Cyg 97/07/19, 20 97/10/05 98/07/18 98/08/04 T Cep 98/07/23, 98/08/07 R Cas 97/10/18, 21 98/07/21 98/08/05 T Cas 98/08/06 98/08/19

M05, M10 19 2 M05, M10 19 9 M05, M10 20 2

? ? ? ? ? ?

0.52 1.16 1.20 0.58 0.60

M05 M10 M10 M05 M05

26 9 21 8 22 5 13 7 12 8

0 9 E8300, E8560 26 7 03 E8560 22 3 04 E8560 23 1 04 03 E8300 E8300 13 4 12 6

? ? ? ? ?

09 03 04 03 03

Table 8.9: Mean values of photospheric angular diameters from Table 8.8. Star o Cet U Her Cyg T Cep R Cas T Cas fundamental R (mas) 31 1 12 1 20 0 20 2 22 4 13 1 1st overtone R (mas) 30 8 12 5 19 7 20 3 22 9 13 0

? ? ? ? ? ?

06 03 08 07 16 05

? ? ? ? ? ?

03 03 09 05 13 04


8.4. DISCUSSION Table 8.10: Mean effective temperatures. Apparent bolometric magnitudes are from van Leeuwen et al. (1997). Spectral types are the "normal" types at visual phase 0.25 (i.e. mean light) from the tables of Lockwood and Wing (1971). The full range of spectral types measured by Lockwood and Wing follows in parentheses. Star oCet Cyg TCep RCas m 0 70 1 39 1 50 1 40

135

? ? ? ?

bol

Teff (K) 2560 2730 2620 2520

0 10 0 10 0 10 0 10

? ? ? ?

Spectral type M7.5 M8.5 M8 M8.5 (M4.5-M9.2) (M2-M9.0) (M5.5-M8.8) (M6.0-M10)

60 88 67 91

The range of mean spectral type in Table 8.10 is only one sub-class, so I can take the mean of the four Teff values, 2610 ? 50 K, to be a robust measurement of the effective temperature of a Mira of spectral type M8III. The reader should note that the spectral type of a Mira varies by 5 sub-classes over a pulsation period (see Table 8.10), and so the effective temperature will also vary substantially.

8.3.6

Linear radii

The apparent angular diameters in Table 8.9 were converted into linear radii using two different estimates of the distance to each star. The Hipparcos (ESA 1997) trigonometric parallax provides a direct measurement of the distance, albeit with a large uncertainty for all but the nearest Miras. Alternative distances were calculated from the empirical (period, K band luminosity) relation of Feast et al. (1989) for Miras in the Large Magellanic Cloud. Taking the distance to the LMC to be a recent Cepheid distance (Feast 1995), the PL relation becomes M
K



3 47 log P

Ç

0 91

(8.2)

where MK is the absolute K magnitude and P is the pulsation period in days. Mean apparent K magnitudes for all but T Cas were taken from van Leeuwen et al. (1997) or Catchpole et al. (1979). K band photometry for T Cas with good phase coverage does not exist in the literature, so the mean of the two measurements of Dyck et al. (1974) (made at approximate phases 0.38 and 0.46) was used. Errors on the distances were calculated assuming errors of 0.1 mag. in the mean apparent K magnitudes (0.2 mag. for T Cas) and an uncertainty of 0.1 mag. in the distance modulus of the LMC, plus the observed scatter in the PL relation of 0.13 mag. The final percentage errors in the distances from the period-luminosity relation were 9% (13% for T Cas).

8 .4
8.4.1

Discussion
Asymmetries and limb-darkening

These near-infrared data rule out the presence of the large asymmetries which have been seen at wavelengths below 1 Åm in almost all studies capable of detecting them (Karovska et al. 1991;


136

CHAPTER 8. MIRA VARIABLES Table 8.11: Periods, distances, and mean linear radii. Linear radii were calculated using distances from the Hipparcos (ESA 1997) parallaxes (left columns) and also using distances from the Feast et al. (1989) period-luminosity relation (right columns). For those stars with parallaxes smaller than the uncertainty in the parallax, 1 lower limits to the distances, and hence to the radii derived from these distances, are given. These cases are indicated by asterisks. Star o Cet U Her Cyg T Cep R Cas T Cas Period (d) 332 406 407 390 431 445 Hipparcos d pc R R? Ç21 424Ç70 128 15 50 339* 455* 106 210 107 PL d pc 117 365 175 187 193 299

Ç Ç Ç

18 13 39 29 14 11

224 458 263

Ç Ç Ç

38 27 85 63 38 31

602*

842*

? ? ? ? ? ?

11 33 16 17 17 39

R R? 387 ? 37 490 ? 46 371 408 475 418

? ? ? ?

37 38 51 56

Haniff et al. 1992; Wilson et al. 1992; Quirrenbach et al. 1992; Haniff et al. 1995; Weigelt et al. 1996; Tuthill being located wavelengths. of Miras: the et al. 1999). The difference in apparent morphology may be due to the asymmetries in the tenuous outer layers of the atmosphere, which are nearly transparent at IR Nevertheless, good Fourier plane coverage is still desirable for infrared observations slight elongations detected for T Cep and R Cas could have led to systematic errors

of a few per-cent in single baseline diameter measurements. Errors of this magnitude may be large enough to mask the true continuum pulsation. Unambiguous comparisons of observed and model limb-darkening were possible for Cyg, T Cep and R Cas. The CLVs at phase 0.2 of T Cep and R Cas are well represented by Hestroffer FDDs with values of 1 and 2 respectively. These analytical curves are compatible with the CLVs of the HSW98 M and E series models at the appropriate phases. The general agreement between observations and model CLVs is in contrast to the results of HST95 at optical wavelengths, who found that the observed limb-darkening profiles were generally more Gaussian than those of the models. A Gaussian CLV was observed at 1.3 Åmfor Cyg at phase 0.72. A roughly Gaussian CLV is seen in the P model series at minimum light, although not in the otherwise very similar M series. Both the P0 Cyg model minor 5 and M05 models have effective temperatures which are much lower than that inferred for at phase 0.72 on the basis of recent phase-averaged photometry, so it is not clear that either is appropriate, although the difference between the P05 and M05 CLVs is an indication that changes in the stellar parameters can lead to large changes in the observed limb-darkening at

cool phases. Gaussian CLVs have been predicted (Scholz and Takeda 1987; HSW98) and observed (e.g. HST95) in regions of the stellar spectrum which are moderately contaminated by molecular absorption features. However, the 1.3 Åm bandpass used for these observations is not expected to be contaminated. The model CLVs calculated for the wider, slightly water-contaminated bandpass described above do not have significantly more Gaussian shapes either. The discrepancies between


8.4. DISCUSSION

137

models and observations commonly seen at visible wavelengths, as well as in the IR for Cyg, may be symptoms of insufficient molecular opacity in the model photospheres, or may be due to scattering by dust grains (which are not included in the models).

8.4.2

Previously-published angular diameters

Photospheric angular diameters derived from infrared interferometric observations have been published for three of the sample stars (the observations were all in the K band). Ridgway et al. (1992) measured a uniform disk diameter of 36 1 ? 1 4 mas for o Cet at a phase of 0.9. Correction for atmospheric extension and limb-darkening using the E series factors from HSW98 gives a photospheric diameter R of 34 7 ? 1 3 mas (the scaling used by Ridgway et al. was incorrect), 13% larger than the COAST measurements at phases 0.76 and 0.67. T Cas and U Her were observed with IOTA in 1995 (van Belle et al. 1996), yielding photospheric angular diameters of 10 4 ? 0 8 mas for U Her at phase 0.03, and 14 2 ? 0 7 mas for T Cas at phase 0.30. The value for T Cas is in excellent agreement with the COAST measurement in Table 8.8, whereas the COAST measurement of U Her is 16% larger than the IOTA measurement at a similar phase. However, if K band scaling factors from the M series photospheric models used here (van Belle et al. used monochromatic 2.2 Åm scaling factors from Scholz and Takeda 1987) are applied to the IOTA UD values, the discrepancies become 4% for T Cas and 8% for U Her, both within the quoted error bars. Clearly the new COAST diameter measurements agree reasonably well with previous infrared observations. The discrepancies are only 10%, and do not suggest any systematic error. It is interesting to compare the values in the right-hand column of Table 8.9 with the mean photospheric diameters of HST95, who used the same E series photospheric models to correct visible wavelength observations for atmospheric extension and limb-darkening. The diameters derived from the infrared data are smaller for all of the four stars common to both samples, by an average of 24%. In fact the discrepancy is only 8% for o Cet and R Cas, but 16% for T Cep and 32% for Cyg. The huge discrepancy for Cyg was probably caused by the 45% diameter variations at 0.9 Åm not seen in the infrared (see Chapter 7): the HST95 measurements at 833 nm and 902 nm which were averaged to give the final "mean" diameter were both made at phase 0.32. The same effect is likely to have enlarged the diameters of the other three stars ( 35% diameter variations at 830 nm and 940 nm were found for R Leo by Burns et al. 1998, so such variations may be ubiquitous) compared with their IR diameters. The effect may be smaller for o Cet, R Cas, and T Cep because of smaller amplitude diameter variations at 0.9 Åm.

8.4.3

Effective temperatures

The mean effective temperatures derived for the sample stars (Table 8.10) are consistent with the (Teff , spectral type) relation for Miras of van Belle et al. (1996). This relation has a large amount of scatter, which may be due to the use of spectral types measured by Lockwood and


138

CHAPTER 8. MIRA VARIABLES from K

Wing (1971) in much earlier cycles, or to the use of an empirical relation to obtain m

bol

band flux. However, a large part of the scatter is likely to be intrinsic -- in Mira variables the outer photospheric layers where molecular lines are formed are decoupled from the deeper layers which emit most of the total flux (Spinrad and Wing 1969). For this reason, a colour temperature measured between two continuum points is likely to be a much better indicator of Teff than the spectral type. The new effective temperatures from COAST confirm the result of van Belle et al. that Miras are significantly cooler than normal giant stars of the same spectral type. Effective temperature scales for late-type giants, based on diameters from lunar occultations (Ridgway et al. 1980) and interferometry at IR wavelengths (Di Benedetto 1993; Dyck et al. 1996, 1998) all predict effective temperatures for M8 stars of 3000 K, compared with 2610 ? 50 K for the mean Teff of the four Miras in Table 8.10. Effective temperatures for individual stars in the sample have been published by Ridgway et al. (1992) and HST95, based on the diameter measurements mentioned in the previous section. The discrepancies between these temperatures and those in Table 8.10 are mostly due to the large diameter differences already discussed.

8.4.4

Linear radii and pulsation modes

Linear radii are plotted against pulsation period in Figure 8.4. Also shown are period-mass-radius (PM R) relations based on the linear pulsation models of Fox and Wood (1982). These PM R relations are similar to those of Ostlie and Cox (1986). I have chosen to use the Fox and Wood results because Ostlie and Cox have no models with first-overtone periods 245 d. For the fundamental mode PM R relation, I have used the parameterisation given by Wood (1990): log P



2 07

Ç

1 94 logÄR R?

Å



0 9logÄM M?

Å

(8.3)

where P is the pulsation period in days, R is the average stellar radius and M is the stellar mass. The first-overtone lines in Figure 8.4 are given by the pulsation equation: Q PÄM M?
Å
12

Ä

R R?Å

32

(8.4)

with values of the pulsation "constant" Q taken from the results of Fox and Wood (1982) for "cool disk stars", i.e. using what is now believed to be a realistic effective temperature scale. I have taken the mass range for the sample Miras to be 1 0 M M? 1 5. The masses of Mira variables are well constrained to be 1M? (see, for example, Wyatt and Cahn 1983). Masses greater than 1 5M? are very unlikely given the high mass-loss rates of long-period variables (Knapp and Morris 1985). Ya'ari and Tuchman (1996) argue that the PM R relations from linear models are invalid, as their non-linear hydrodynamic models do not always retain the pulsation periods predicted by linear analysis. However, the same non-linear models exhibit cyclic radius variations of 80%, which


8.4. DISCUSSION

139

800 700 600 500 400 300 200 300 350 400 P /d 450 500 M 1 5 M? 1st overtone M 1 0 M? M 1 5 M? fundamental M 1 0 M? R R?

800 700 600 500 400 300 200 300 350 400 P /d 450 500 M 1 5 M? 1st overtone M 1 0 M? M 1 5 M? fundamental M 1 0 M? R R?

Figure 8.4: Stellar radius plotted against pulsation period for the six stars observed with COAST. The lines are theoretical predictions for 1 0M? and 1 5M? stars pulsating in the fundamental and first-overtone modes (see text for details). Linear radii calculated using distances from Hipparcos parallaxes are shown in the upper plot, whereas the lower plot uses distances from the Feast et al. (1989) period-luminosity relation.


140

CHAPTER 8. MIRA VARIABLES

are highly unlikely in real Miras given the results of Chapter 7, and so the other predictions of these models should be treated with caution. The points in the upper plot in Figure 8.4 are linear radii calculated using distances from Hipparcos trigonometric parallaxes. The parallaxes are a small fraction of the V band apparent angular diameters of these stars, so the asymmetries likely to have been observed in the V band will have affected the measurements. However the timescale for the evolution of asymmetries is 1 month (Tuthill et al. 1999), much shorter than the time over which the stars were observed by the Hipparcos satellite. Hence the effect of asymmetries is likely to have been an increase in random, rather than systematic, error. I have assumed that the error bars on the parallaxes include this contribution to the random error. The validity of this assumption is supported by the fact that independent parallax measurements for Cyg (Stein 1991) and R Leonis (Gatewood 1992) agree with the values from Hipparcos. The uncertainties in the angular diameters are completely insignificant compared with the uncertainties in the Hipparcos distances ( 20% for the nearest Miras). If the predictions of the linear pulsation models are correct, it is clear that not all of the sample stars can be pulsating in the same mode. Cyg and R Cas are clearly identified as fundamental-mode pulsators, whereas o Cet and T Cep must be pulsating in the first-overtone mode. The error bars for T Cas and U Her are enormous, but the 1 lower limits on the radii exclude fundamental mode pulsation. As well as having small linear radii, Cyg and R Cas are anomalous in other respects: they have lower luminosities (van Leeuwen et al. 1997) and greater amplitudes at 1.04 Åm (Lockwood and Wing 1971) than other Miras of similar period ( Cyg also has an extraordinarily large range in spectral type). The non-linear models of Bessell et al. (1996) and HSW98 exhibit precisely these differences between fundamental and first-overtone pulsators, which suggests that the conclusions reached on the basis of PM R relations are robust. The inferred pulsation modes for o Cet, Cyg, T Cep and R Cas are in fact the same as those obtained by van Leeuwen et al. (1997) using the HST95 angular diameters and Hipparcos distances. That van Leeuwen et al. arrived at the same conclusions despite using angular diameters which were systematically larger than those used here is mostly due to the large uncertainties in the distances. The empirical period-luminosity relation for LMC Miras of Feast et al. (1989) may be used to give more accurate distances to the sample stars, with the caveat that Cyg and R Cas do not fit the PL relation. The PL distances to o Cet and T Cep (Table 8.11) and to the other oxygen-rich Miras considered by van Leeuwen et al. are consistent with the trigonometric parallaxes. Linear radii calculated using the PL distances are shown in the lower part of Figure 8.4. If these distances are not systematically wrong for T Cas and U Her, then both are confirmed as first-overtone pulsators. The inferred pulsation mode for o Cet and T Cep is still first-overtone. Note that R Cas is moved into the first-overtone region by adopting this distance, and that Cyg now falls between the fundamental and first-overtone regions. Van Belle et al. (1996) used an inhomogeneous set of distance estimates for the 18 Miras in their sample. I have calculated distances from the PL relation (Equation 8.2) used here for those stars


8.4. DISCUSSION

141

800 700 600 500 400 300 200 250 300 350 P /d 400 450 500 M 1 5 M? 1st overtone M 1 0 M? M 1 5 M? fundamental M 1 0 M? R R?
1

Figure 8.5: Stellar radius plotted against pulsation period for the nine normal Miras from van Belle et al. (1996) for which mean K band photometry, used here to calculate distances from the period-luminosity relation, is available from the literature. The lines are the same as those in Figure 8.4. for which mean K magnitudes were available from van Leeuwen et al. or Catchpole et al. (1979)1 . Linear radii were calculated from these distances and the IOTA K band uniform disk angular diameters, corrected for limb-darkening and extension using the E series factors from HSW98. These radii are plotted against pulsation period in Figure 8.5. Most of the Miras in the sample were observed at a single epoch, so the radii could be up to 10% different from the mean values. The COAST results for Cyg, T Cep and R Cas suggest that the presence of asymmetries could have introduced errors of a few per-cent. Systematic errors due to the use of a uniform disk model will be of a similar magnitude. The use of a homogeneous set of distances has reduced the range of linear radii from that found by van Belle et al. However, the data are still not consistent with a single pulsation mode for a mass range 1 0 M M? 1 5. Five of the nine stellar radii plotted in Figure 8.5 are consistent

with first-overtone pulsation, and the other four points lie within 2 of the first-overtone region. Despite the observation that not all local Mira variables fit the same PL relation, it seems likely that a large fraction of the local Miras are first-overtone pulsators. The observation that the 79 LMC Miras considered by Feast et al. obey a tight PL relation strongly suggests that most are pulsating in the same mode. Excluding Cyg and R Cas, van Leeuwen et al. found the zero-point of the K band PL relation (c.f. Equation 8.2) for local Miras to be 0 88 ? 0 18 (assuming the slope from the LMC). The LMC distance modulus from Cepheids, 18 57 ? 0 1, is very close to the value of 18 60 ? 0 18 which follows from the assumption that the PL relation for LMC Miras has the same zero-point as that for local Miras. This contradicts the theoretical result of Wood (1990), disputed by Feast and collaborators (Feast 1996; Whitelock et al. 1994),
I have excluded the symbiotic star R Aqr from this sub-set


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CHAPTER 8. MIRA VARIABLES

that local Miras are intrinsically fainter than LMC Miras because of lower metallicity. Thus if the local Miras apart from Cyg and R Cas are first-overtone pulsators, if the same PL relation applies to local Miras as to those in the LMC, and if the commonly-adopted distance modulus to the LMC is correct, then the LMC Miras must also be first-overtone pulsators.

8.5

Conclusions

The COAST measurements described in this chapter show that the apparent asymmetries in the shape of the stellar disk are small when Miras are observed in a broad near-infrared band. The limb-darkening profiles of three stars could be inferred from the data. These are the first measurements of the infrared limb-darkening of Mira variables. Two of the profiles agree reasonably well with predictions from the models of HSW98. A uniform disk model is a poor fit to the visibility curves of all four stars which were fully resolved by COAST, even at spatial frequencies below the first visibility minimum. The mean effective temperatures of four Mira variables were found to be consistent with the scale of van Belle et al. (1996), i.e. about 400 K cooler than normal giant stars of the same spectral type. By measuring the limb-darkening profile and degree of asymmetry of three stars, by at least partially accounting for the effect of variability on the measured diameter, and by using trigonometric parallaxes as direct distance measurements, reliable identifications of pulsation mode were possible for o Cet, Cyg, T Cep and R Cas. Cyg and R Cas were identified as fundamental mode pulsators, in agreement with indirect evidence, whereas the more typical Miras o Cet and T Cep were found to be first-overtone pulsators. T Cas and U Her probably also pulsate in the first-overtone mode.


Chapter 9

Conclusions
The aims of my PhD. project were to give COAST the capability of imaging in the infrared, to prove that the instrument worked, and then to do real astrophysics using COAST at IR wavelengths. In this final section of my thesis, I will attempt to summarise the important points put forward in each chapter, in the context of the overall aims listed above. The reasons why it is preferable to do interferometry at IR rather than optical wavelengths were explained in Chapter 1. The most important reason is that atmospheric turbulence has less effect on interference fringes at longer wavelengths, so that wider apertures and longer integration times can be used. Chapters 2-4 dealt with the infrared correlator at COAST, specifically its two components: the pupil plane beam combiner (Chapter 3), and the NICMOS3 infrared camera (Chapters 2 and 4). The important developments which allowed the IR correlator to be used for astronomy were the development of a procedure for aligning the four-way beam combiner using infrared light, and increasing the maximum sampling rate of the detector to 2.5 kHz, without increasing the read noise. The crucial property of the finished correlator is that the visibility losses and path errors due to the instrument are much smaller than those introduced by the atmosphere above the telescopes. In Chapter 5 I described how the working COAST infrared system is used to measure visibility amplitudes and closure phases, and how an image of the astronomical target is reconstructed from such measurements. Observations of the binary star Capella at a wavelength of 1.3 Åm were presented. Both the reconstructed image and model fits to the Fourier data are consistent with the well-known orbit. This provided a vital test of the correlator hardware and the data reduction procedures, and proved that COAST can successfully measure closure phases at IR wavelengths. COAST is in fact the first interferometer to use separated telescopes to measure closure phases at wavelengths above 1 Åm. It is also the first to image an astronomical target by aperture synthesis at IR wavelengths. Closure phase measurements were a vital part of the astronomical programmes described in the final three chapters, as were measurements on long baselines (up to 20 m), which provided angular


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resolution of 13 mas. Closure phase measurements on long baselines are very sensitive to asymmetries in the appearance of stellar disks. No such asymmetries were found for the red supergiant Betelgeuse at 1.3 Åm (Chapter 6), whereas contemporaneous observations at shorter wavelengths did indicate a definite asymmetry. The sharp decrease in apparent asymmetry with observing wavelength meant that the explanation of the asymmetries as hotspots (i.e. areas of elevated temperature on the surface of the star, caused by convection) could be ruled out, unless the hotspots create holes in a screen which has significant opacity at visible wavelengths but almost none in the infrared. Chapter 7 described an experiment to monitor the angular diameter of the Mira variable Cygni. The diameter changes seen in a near-continuum band centred on 905 nm confirm the detection (for R Leonis) by Burns et al. (1998) of 40% cyclic diameter variations in bands slightly contaminated by molecular absorption features. Diameter measurements made at 1.3 Åm over the same period indicate that a substantial fraction of the apparent diameter variation at shorter wavelengths is probably due to variations in TiO band strength, rather than to changes in the "true" diameter. Future monitoring in carefully-chosen IR bands, with the high precision already achieved, should allow the motion of the continuum-forming layers to be measured. The measurement of only small changes in the infrared apparent diameter of Cygni showed that angular diameters measured in the infrared could be used to obtain good approximations to the "average" angular diameters of a sample of Mira variables. A programme to perform such measurements with COAST was described in Chapter 8. Photospheric models were needed to relate the apparent diameters measured in a broad IR waveband to a standard definition of the stellar radius, but these conversions are only weakly model-dependent for diameters measured in the near-infrared. Distance estimates to the stars were used to convert angular diameters to physical diameters, which were used to infer the radial pulsation modes of the six stars in the sample. Four of the six Miras were fully resolved by COAST, and for three of these it was established that the apparent asymmetry at 1.3 Åm was small. The lack of asymmetry in the IR, compared with the large asymmetries typically seen at optical wavelengths, is similar to that observed in the red supergiant Betelgeuse. Both types of star have very extended atmospheres, the outer layers of which become more transparent in the infrared. It is natural to think that most of the asymmetries originate in the outer atmospheric layers. The evidence for a high degree of symmetry at 1.3 Åm allowed visibility measurements on long baselines to be used to determine the degree of limb-darkening of Betelgeuse and the three Mira variables T Cep, R Cas and Cyg. All were found to have significant limb-darkening, with Betelgeuse and Cyg in particular being more strongly limb-darkened than model atmospheres predict. This illustrates the potential of IR interferometry for testing the latest stellar models.


9.1. THE FUTURE

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9.1

The future

At the time of writing, a filter wheel is being installed in the dewar of the COAST IR camera. This will have space for five interference filters, and will permit the observing waveband to be changed in a matter of seconds. This will facilitate observations of cool stars in a number of narrow infrared bands, located both in the continuum and within strong absorption features. Such measurements will probe a range of atmospheric layers, in order to characterise the true pulsation of LPVs, and identify the cause of the asymmetries seen in cool stars. A new design of hybrid infrared array detector should be available from Rockwell in the near future. It is claimed that these 128 ? 128 pixel arrays can be read out at pixel rates of up to 10 MHz, with sub-electron read noise. If the performance of the devices even approaches that predicted, their use will lead to substantial improvements in the limiting magnitudes of interferometers operating at IR wavelengths. The high frame rates should allow fringes to be sampled on many baselines at once, using either pupil plane or image plane beam combination. This capability is vital for the planned successor to COAST, the Large Optical Array (LOA, see Baldwin et al. 1998), which would achieve its best limiting magnitude in the near-infrared. LOA would have 15 telescopes, six to eight of which might feed each beam combiner. Infrared detector technology has come a long way in a few decades. Optical and near-IR interferometry has evolved in parallel over a similar period of time. The technology is now sufficiently advanced to imagine that five years from now, aperture synthesis at IR wavelengths could produce the first image of the inner region of an AGN.


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