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Physics of Magnetic Stars, 2007, pp. 315­324

Propagation of pulsation waves in roAp atmospheres
M. Sachkov1 , T. Ryab chikova
1 2 3 4

1,2

, O. Ko chukhov3 , D. Lyashko

4

Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya 48, 119017 Moscow, Russia Department of Astronomy, University of Vienna, Turkenschanzstrasse 17, A-1180 Wien, Austria ¨ Department of Astronomy and Space Physics, Uppsala University Box 515, SE-751 20 Uppsala, Sweden Tavrian National University, Yaltinskaya 4, 330000 Simferopol, Ukraine

Abstract. We present a detailed analysis of the vertical cross-section of the roAp pulsation mo des in eight rapidly oscillating (roAp) stars basing on sp ectroscopic time-series observations. We prop ose to use the phase-amplitude diagram as the first step in the interpretation of roAp pulsational observations. Such an approach has an advantage of b eing suitable to compare the b ehaviour of different elements, which is imp ossible for studies of phase / amplitude dep endence on line intensity. Although the overall pulsational b ehaviour of roAp stars is not identical, we found certain common features. The phase shifts of the RV curves are arranged in the following sequence: the lowest RV amplitudes are detected in the layers of Eu ii (and Fe in 33 Lib) line formation, then pulsations go through the layers where the Hcore, Nd and Pr lines are formed, the RV amplitude reaches a maximum and after that, in most stars, shows a decrease of the amplitude. The phases of RV curves of the first ions are always followed by the second ones. The largest phase shifts are detected in the Tb iii and Th iii lines. A pulsational variability of the Th iii lines has b een detected and studied for the first time. In the atmospheres of roAp stars with the pulsation frequency b elow the cut-off frequency, pulsations have a standing wave character in the deep er layers and then b ehave like a running wave in the outer layers.

Key words: lations

stars: atmospheres ­ stars: chemically p eculiar ­ stars: magnetic fields ­ stars: oscil-

1

Intro duction

More than 30 co ol Ap stars exhibit high-overtone, low-degree, non-radial p-mo de pulsations with p erio ds in the range of 6­21 minutes (Kurtz & Martinez 2000), with their observed pulsation amplitudes mo dulated according to the visible magnetic field structure. These rapid ly oscil lating Ap (roAp) stars are key ob jects for asteroseismology, which presently is the most p owerful to ol for testing theories of stellar structure and evolution. The van Ho of effect (van Ho of & Struve 1953) ­ phase lag b etween radial velo city curves of lines of different elements and ions ­ is one of the most interesting phenomenon in roAp stars. It is interpreted as the consequence of the wave propagation through the atmosphere: starting from the inner part of the star, the wave encounters first the deep est layers and then the outermost ones during its propagation. The time it takes to cover this distance pro duces the observed phase lag. This effect yields a unique p ossibility for the vertical atmospheric structure analysis. At the same time, reconstruction of the roAp vertical at315


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mospheric structure encounters many difficulties: one has to take into account vertical stratification effects, construct a self-consistent stratified atmospheric mo del and then estimate the influence of the surface element distribution and of the magnetic field. To constrain the vertical cross-section of an Ap-star atmosphere, we need to know the formation depth of sp ectral lines. This requires a stratification analysis of stellar atmosphere with NLTE effects taken into account. Up to now such an analysis has b een made for one star only, HD 24712 (Sachkov et al. 2006). We analyzed the H core, Nd, Pr, Ca, Sr, Ba and Fe lines (see b ottom panel of Fig. 6). The abundance distribution of the different elements allows us to calculate the formation depth of individual sp ectral lines and then to construct a distribution of pulsation amplitudes and phases with optical depth. It was shown that a non-adiabatic mo del for axisymmetric non-radial pulsations in the presence of a dip ole magnetic field (Saio 2005) roughly explains amplitude and phase distribution up to log 5000 = -4: amplitude and phase increase towards the outer layers (Sachkov et al. 2006), which gives an evidence for pulsation wave propagating through the stellar atmosphere. In this way, the later in time the pulsation wave reaches its maximum, the higher in the atmosphere a chemical element is concentrated. An agreement of the observed amplitude and phase distribution with the theoretical predictions encourages us to use pure observables -- amplitude-phase diagrams -- for an analysis of the vertical structure of p-mo des and for a study of the pulsation wave propagation. Therefore, the aim of this analysis is to derive from observations a complete picture of the depth-dep endence of amplitudes and phases of propagating waves without estimating line depth formation itself and then to correlate the resulting vertical mo de cross-section with other pulsational characteristics. For our analysis we chose a sample of slowly rotating roAp stars. A full pap er describing our investigation has b een submitted to Astronomy and Astrophysics.

2

Observations and data reduction

The ESO Archive facility was used to search for and retrieve science exp osures and the resp ective calibration frames. The main observational dataset analyzed in our study consists of observations of 6 roAp stars obtained with the UVES instrument at the ESO VLT b etween Octob er 8, 2003 and March 12, 2004 in the context of the observing program 072.D-0138. The red 600 nm UVES dataset was complemented by the observations of HD 24712 obtained on Novemb er 11, 2004 in the DDT program 274.D-5011. A detailed description of the acquisition and reduction of these data was given by Ryab chikova et al. (2007a ). For the roAp star HD 201601 ( Equ) we analysed 70 sp ectra obtained on 19 August, 2003 with the NES sp ectrograph attached to the 6-m telescop e of the Sp ecial Astrophysical Observatory of the ec A Russian Academy of Sciences. These ´ helle sp ectra cover a region of 4250­6000 ° and have a typical S/N of 80. The data were recorded by Ko chukhov et al. (2004), who searched for rapid magnetic field variability in Equ. We refer readers to the latter pap er for the details on the acquisition and reduction of the time-series observations at the SAO 6-m telescop e. A detailed description of the observations made for each target is presented in Table 1. All UVES sp ectra were reduced and normalized to the continuum level with a routine sp ecially develop ed by one of us (DL) for a fast reduction of sp ectroscopic time-series observations. A sp ecial ec mo dification of Vienna automatic pip eline for ´ helle sp ectra pro cessing (Tsymbal et al. 2003) was develop ed. All bias and flat field images were median averaged b efore calibration. The scattered light was subtracted by using the 2-D background approximation. For cleaning cosmic ray hits we used an algorithm which compares the direct and reversed sp ectral profiles. To determine the ec b oundaries of ´ helle orders, the co de used a sp ecial template for each order p osition in each row across the disp ersion axes. The shift of the row sp ectra relative to the template was derived by a cross-correlation technique. The wavelength calibration was based on a single ThAr exp osure


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recorded immediately after each stellar time-series. The calibration was p erformed by the usual 2-D approximation of the disp ersion surface. An internal accuracy of 30­40 m s -1 was achieved by using several hundred ThAr lines in all ´ helle orders. The final step of continuum normalization and ec merging of ´ helle orders was carried out by transformation of the flat field blaze function to the ec resp onse function in each order. Table 1: Log of time-series observations of roAp stars. The columns give the star's name, helio centric Julian dates for the middle time of sp ectroscopic monitoring, numb er of observations, exp osure and overhead times, p eak signal-to-noise ratio of individual sp ectra and information ab out the telescop e and instrument with which the data were obtained.

Star HD HD HD HD HD HD HD HD 12932 19918 24712 101065 122970 134214 137949 201601

HJD (2450000+) 2921.66383 2921.56756 3321.70077 3071.71895 3069.75168 3070.81710 3071.80455 2871.51383

Numb er of exp osures 69 69 92 111 111 111 111 70

Exp osure time (s) 80 80 50 40 40 40 40 80

Overhead time (s) 25 25 22 25 25 25 25 42

Peak S/N 90 100 300 180 160 260 350 80

Telescop e/Instrument (observing mo de) VLT/UVES (600 nm) VLT/UVES (600 nm) VLT/UVES (390+580 nm) VLT/UVES (600 nm) VLT/UVES (600 nm) VLT/UVES (600 nm) VLT/UVES (600 nm) SAO 6-m/NES (425­600 nm)

The global continuum normalization was improved by iterative fit of a smo othing spline function to the high p oints in the average sp ectrum of each roAp star. With this pro cedure we have corrected an underestimate of the continuum level, unavoidable in the analysis of small sp ectral regions of the crowded sp ectra of co ol Ap stars. A correct determination of the absolute continuum level is imp ortant for retrieving unbiased amplitudes of radial velo city variability, when the centre-ofgravity metho d is used. In addition to the global continuum correction, sp ectroscopic time-series were p ost-pro cessed to ensure a homogeneity in the continuum normalization of individual sp ectra. The extracted sp ectra were divided by the mean, the resulting ratio was heavily smo othed and then used to correct continua in individual sp ectra. Without this correction a spurious amplitude mo dulation of pulsation in variable sp ectral lines may arise b ecause of the inconsistent continuum normalization. Post-pro cessing of the ´ helle sp ectra of HD 201601 was executed consistently with the pro cedure ec adopted for the main dataset.

3

Fundamental parameters of program stars

Fundamental parameters of the program stars are given in Table 2. For five stars effective temp eratures Teff , surface gravities log g , and mean surface magnetic fields B s were taken from the literature. For the 3 remaining stars, HD 12932 HD 19918 and HD 134214, atmospheric parameters were derived using Str¨ omgren photometric indices (Hauck & Mermillio d 1998) with the calibrations by Mo on & Dworetsky (1985) and by Napiwotzki et al. (1993) implemented in the TEMPLOGG co de (Rogers 1995). In addition, Geneva photometric indices (Burki et al. 2005) 1 with the calibration of Hauck & North (1993) were used for effective temp erature determination. The colour excesses were
1

http://obswww.unige.ch/gcpd/ph13.html


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HD numb er 12932 19918 24712 101065 122970 134214 137949 201601

Teff (K) 7620 8110 7250 6600 6930 7315 7550 7700

log g 4.15 4.34 4.30 4.20 4.10 4.45 4.30 4.20

Table 2: Fundamental parameters of target stars. ve sin i Bs P Reference -1 ) (km s (kG) (min) 3.5 1.7 11.633 this pap er 3.0 1.6 11.052 this pap er 5.6 3.1 6.125, 6.282 Ryab chikova et al. 2.0 2.3 12.171 Cowley et al.2000 4.5 2.3 11.187 Ryab chikova et al. 2.0 3.1 5.690 this pap er 2.0 5.0 8.271,4.136, 9.422 Ryab chikova et al. 1.0 4.1 12.20 Ryab chikova et al.

1997 2000 2004 2002

estimated from the reddening maps by Lucke (1978). In Table 2 we present average values of the effective temp eratures derived with three different calibrations. A typical disp ersion is ±150 K. For all stars, but for HD 101065, rotation velo cities were estimated by fitting line profiles of the magnetically insensitive Fe i 5434.5, 5576.1 lines. The magnetic sp ectrum synthesis co de SYNTHMAG (Ko chukhov 2007) was used in our calculations. Atomic parameters of sp ectral lines were extracted from the VALD (Kupka et al. 1999) and DREAM (Bi´ emont et al. 1999) databases, supplemented with the new oscillator strengths for La ii (Lawler et al. 2001), Nd ii (Den Hartog et al. 2003), Nd iii (Ryab chikova et al. 2006), Sm ii (Lawler et al. 2006), Gd ii (Den Hartog et al. 2006). We confirmed the rotation velo cities derived previously for HD 24712, HD 122970, HD 137949, and HD 201601. The high sp ectral resolution of the present data allows us to improve the rotation velo city in HD 101065 using partially resolved Zeeman patterns in numerous lines of the rare-earth (REE) elements. The value of the magnetic field, 2.3 kG (Cowley et al. 2000), was confirmed.

4

Radial velo city measurements

To study the selective pulsational amplitudes in sp ectral lines of different chemical elements/ions, one has to b e very careful in the choice of lines for pulsation measurements. For this purp ose, we have synthesized the observed sp ectral region for each star with the mo del atmosphere parameters and magnetic field values from Table 2. Abundances for HD 24712, HD 101065, HD 122970, HD 137949, and HD 201601 were taken from the pap ers cited in the last column of Table 2. For the remaining three stars abundances were estimated in this pap er. The radial velo cities were measured with a centre-of-gravity technique. We used only unblended or minimally blended lines. In some cases where the line of interest was partially overlapping with the nearby lines, the unblended central part of the line was integrated, therefore some lines were not measured b etween the continuum p oints. This usually leads to lower pulsation amplitudes if we have strong variations of the pulsation signal across a sp ectral line (see Equ ­ Sachkov et al. 2004, and HD 99563 ­ Elkin et al. 2005). Bisector radial velo city measurements were p erformed for the H core. Two strongest Th iii lines at 5376.13 and 6599.48 have b een measured for RV pulsational variability for the first time in the sp ectra of a few stars from our sample.

5

Perio d analysis

Although a detailed frequency analysis is not a primary goal of this pap er, we rep eated it here using a discrete Fourier transformation (DFT). The p erio d corresp onding to the highest amplitude value was then improved by the sine wave least-squares fitting of RV data to the pulsation p erio d, amplitude and phase as free parameters. The sine wave was removed from the data and then Fourier


PROPAGATION OF PULSATION WAVES IN ROAP ATMOSPHERES

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2000

Bisector amplitude, m/s

Bisector phase

1500

HD HD HD HD HD HD HD HD

101065, 122970, 24712, 134214, 137949, 12932, 201601, 19918,

Teff=6600 Teff=6910 Teff=7250 Teff=7500 Teff=7550 Teff=7620 Teff=7700 Teff=8110

K K K K K K K K

1 0.8 0.6 0.4 0.2 0

1000

500

0 0.1

0.2

Residual intensity

0.3

0.4

0.5

0.1

0.2

Residual intensity

0.3

0.4

0.5

Figure 1: Bisector measurements in the H core of the program stars. The RV amplitudes are shown in the left panel and pulsation phases (based on the main p erio ds from Table 2) in the right panel. analysis was applied to the residuals. This pro cedure was rep eated for all frequencies with a signalto-noise ratio ab ove 5. For each sp ectral line we estimated the probability of the variation with a given p erio d using the prescription by Horne & Baliunas (1986). For lines with the probabilities higher than 0.999 we calculated a weighted average value of the pulsation p erio d. This value was used for subsequent determination of amplitude and phase. If a star had one dominant p erio d, a further RV analysis was made with this p erio d. In the cases of more than one dominant p erio d, simultaneous fit with up to three p erio ds was p erformed. RV variation was approximated by the expression:
3

V = V0 (t - t0 ) +
i=1

Vi cos {2 [(t - t0 )/Pi - i ]}.

(1)

Here the first term takes into account p ossible drift of the sp ectrograph's zero p oint. For all stars, but for HD 24712, HJD of the first exp osure of the star at a given night was chosen as a reference time t0 . For HD 24712 HJD=2453320.0 was used as a reference time. V j i and j i are amplitude and phase of the RV variability with the i-th p erio d (i max = 3), resp ectively. With the minus sign in front of i a larger phase corresp onds to a later time of the RV maximum. This phase agreement is natural when discussing effects of the outward propagation of pulsation waves in the atmospheres of roAp stars. The p erio ds participating in the RV amplitude and phase analysis are given in the seventh column of Table 2. If more than one p erio d was used in fitting pro cedure, we marked by italics the p erio d for which RV amplitudes and phases were calculated.

6

Bisector measurements in the H core

Fig. 1 shows bisector amplitudes (left panel) and phases (right panel) across the core of the H line calculated with the pulsation p erio ds from Table 2. Where two p erio ds or the main p erio d and its first harmonic are resolved, simultaneous fit with two frequencies was done. In all program stars an increase of the bisector amplitude two or more times from the transition region to the deep est part of the core is observed to b e in full agreement with the results obtained by Kurtz et al. (2005a ). Bisector RV changes are accompanied by small phase changes. Only in three stars phase changes exceed the measurement errors. These are the stars with the shortest pulsation p erio ds close to cut-off frequencies: HD 24712, HD 132214 and HD 137949 (33 Lib). NLTE calculations show that in the atmosphere of a normal star with T eff b etween 7000 and 8000 K the H core is formed at


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1250

1500

1000

RV amplitude, m/s

750

RV amplitude, m/s

H core Nd Pr Dy Tb La Eu Y Th

1250

1000

H core Nd Pr Dy Tb La Eu Y Th

750

500

500

250

250

0 0.6

0.7

0.8

Pulsation phase

0.9

1

1.1

1.2

0 0.7

0.8

0.9

Pulsation phase

1

1.1

Figure 2: HD 101065 (left panel) and HD 122970 (right panel). These stars with the lowest effective temp eratures show practically constant pulsation phase throughout most of the atmosphere. Phase shifts observed in the layers of Tb iii and Th iii line formation.
2000
2500

1500

RV amplitude, m/s

RV amplitude, m/s

H Nd Pr Dy Tb La Eu Y

2000

1500

H core Nd Pr Dy Tb Eu Y

1000

1000

500

500

0 0.5

0.6

0.7

0.8 0.9 Pulsation phase

1

1.1

0 0.5

0.6

0.7

0.8

0.9 1 Pulsation phase

1.1

1.2

1.3

Figure 3: HD 201601 (left panel) and HD 19918 (right panel). These stars show the largest RV amplitudes. In b oth stars we observed first an increase of the RV amplitudes with approximately constant phases (standing wave) and after that the pulsation wave is transforming into a running one. -5 < log 5000 < -2 (Mashonkina, private communication). However, in all program stars a core wing anomaly is present (Cowley et al. 2000), which was explained by a temp erature bump b elow log 5000 = -4 (Ko chukhov et al. 2002). This change in the atmospheric structure may lead to an upward shift in formation depth of the base of the H core (Mashonkina, private communication).

7

Phase-amplitude diagrams

In all stars we have detected pulsational variability in the lines of rare-earth elements, which show a maximum radial velo city (RV) amplitude. Our analysis is thus based primarily on REE lines. In addition, for HD 137949 we used the pulsation signal present in Fe lines. From 14 rare-earth elements we primarily used those with the observed lines in the first and second ionization stages (Pr, Nd, Tb. Dy). We also analyzed Y ii lines where the low amplitude pulsations were registered. This element is interesting by its pulsation phases: in all stars where pulsations were detected in Y ii lines, pulsation phases corresp ond to those for Pr iii lines which have the highest amplitudes.


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600

1000

500

RV amplitude, m/s

RV amplitude, m/s

400

H Nd Pr Dy Tb La Eu Y Th III

900 800 700 600 500 400 300 200 100

H core Nd Pr Dy Tb La Eu Y Th

300

200

100

0

0

0.1

0.2

0.3

0.4

Pulsation phase

0.5

0.6

0.7

0.8

0.9

1

0

0.6

0.7

0.8

0.9

Pulsation phase

1

1.1

1.2

1.3

1.4

1.5

Figure 4: HD 24712 (left panel) and HD 134214 (right panel). These stars have the shortest pulsation p erio ds close to or b elow the limit defined by the acoustic cut-off frequency. Phase changes for all elements are explained by a running wave through the whole atmosphere. Running waves in short-p erio d roAp stars are predicted to exist by non-adiabatic pulsation theory (Saio, private communication).
3000 600
H core Nd II Nd III Pr II Pr III Tb III Eu II Y II Dy II Dy III

2500

1500

RV amplitude, m/s
1 1.1 Pulsation phase 1.2 1.3

RV amplitude, m/s

2000

H core Nd Pr Dy Tb Y Eu

500

400

300

1000

200

500

100

0 0.8

0.9

0

0

0.1

0.2

Pulsation phase, in periods

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5: HD 12932 (left panel) is an example of the star with a standing wave b ehaviour in the layers of the H core formation. HD 137949 (right panel) shows the most complex pulsational b ehaviour among all roAp stars. In Figures 2­5 we show phase-amplitude plots for stars in our sample. Finally, for the first time we have found pulsations in doubly-ionized thorium lines in four co olest roAp stars of our sample: HD 101065, HD 122970, HD 24712 and HD 134214. Similar to REEs, thorium shows characteristic abundance anomaly: a 1­2 dex difference in the element abundance derived from the lines of the first and second ions. We attribute this anomaly to a strong vertical stratification similar to REEs -- a layer with 4­5 dex overabundance ab ove log 5000 = -4. At present thorium is the heaviest element with this kind of stratification which shows measurable pulsation amplitudes. Although the overall pulsational b ehaviour in roAp stars is different, we found certain common features. The phase shifts of the RV curves are arranged in the following sequence: - the lowest RV amplitudes are detected in the layers where the Eu I I (and Fe in 33 Lib) lines form, then they go through the layers where the H core, Nd and Pr lines are formed, reach maximum and after that, show a decrease of the amplitude in most stars; - the maximum RV of the second REE ions is always delayed relative to that of the first ones;


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1 0.9 0.8 0.7

Pulsation phase

0.6 0.5 0.4 0.3 0.2 0.1 0 -7
H core Nd lines Pr lines Ca resonance line cores Sr 2 4215 (core) Y lines Saio's model

-6

-5

-4

log5000

-3

-2

-1

0

-4
Nd

-6

Fe Ca

Sr

log(Nel/Ntot)

-8

Pr

Ba

-10

-12 -6

-4

-2 log5000

0

2

Figure 6: Upp er panel: distribution of pulsation phases with optical depth in HD 24712. Dashed curve shows prediction of non-adiabatic theoretical pulsation mo del by Saio (private communication). Bottom panel: stratification of chemical abundances used in NLTE calculations of line formation depths for HD 24712.


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- the largest phase shifts are detected in Tb I I I and Th I I I lines; - in the atmospheres of roAp stars with the pulsation frequency b elow the cut-off frequency pulsations have a standing wave character in the deep er layers and then b ehave like a running wave in the outer layers. In three stars: HD 24712, HD 134214, which have pulsation frequency close to (or even higher than) the cut-off frequency, the pulsation wave b ehaves like the running one from the deep er layers; - Y I I lines show the lowest detectable RV amplitudes. However, their phases differ by 0.5 p erio d from other weakly pulsating lines.

8

HD 24712

HD 24712 is the b est studied star, for which an analysis of the line formation depths was p erformed taking into account NLTE effects and stratification. Abundance distribution of a few elements is shown in Fig. 6 (b ottom panel) (see also Ryab chikova et al. 2007b). According to NLTE analysis, sp ectral lines of Nd and Pr are formed at nearly the same layers in the stellar atmosphere. Hence the phase shift b etween RV curves of individual lines of these elements (Fig. 6, upp er panel) can not b e explained in terms of their different vertical distribution. The difference in horizontal distribution of these elements (see Luftinger et al., this conference) may give rise to the observed phase shifts. ¨
Acknowledgements. Resources provided by the electronic databases (VALD, SIMBAD, NASA's ADS) are acknowledged. This work was supported by the research grants from RFBI (04-02-16788a, 06-02-16110a, 0602-16379a), from the Swedish Kungliga Fysiografiska S¨ lskapet and Royal Academy of Sciences (grant No. al 11630102), and from Austrian Science Fund (FWF-P17580).

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