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Spectral Line Calibration Techniques with Single Dish Telescopes
K. O'Neil NRAO - GB

A Quick Review

A Quick Rev iew

A Quick Rev iew

The Rayleigh-Jeans Approximation
· Planck Law for Bl ackbody radi ati on: B=
2h c
2 3

Antenna Temperature
· Antenna t heorem : Ae A = r 2 · Measured flux: S= = · 2kT

1 e
2 h /kT

-1

· If ~GHz, oft en h << kT. T ayl or seri es gives: B=
2kT c
2

A A A

=

2kT
2

2 2kT A
e

· Sourc e flux i n Rayl eigh J eans lim it:
2k

S=



2

sT(, )d
2kT
2

Tem perat ure: TA = (Ae/ 2) Tsrc(, )Pn(, )d = (r/ A) Tsrc(, )Pn(, )d

· If brightness tem perat ure i s c onst ant across sourc e: S=
S
16

B= Brightnes s, = frequenc y ; h = 6.626 x 10-

J s; k = 1.380 x 10-23 J K-1 ;T = temperature

A = area; = Solid angle of tel. pattern; r = frac tional power trans mis s ion; = wa vele ngth; S= Flux, k = 1.380 x 10-23 J K-1 , T = temperature; P = antenna power pattern

A Quick Rev iew

A Quick Rev iew

Antenna Temperature
· T em perat ure: TA = (Ae/ 2) Tsrc(, )Pn(, )d = (r/ A) Tsrc(, )Pn(, )d · Poi nt Sourc e: TA = (r/ A) Tsrc(, ) d = r ( S/ A )T

Minimal Detectable Temperature
Set by the syst em noi se Tsys = TA + (1/)TR + TLP[1/ -1] Sensiti vity is rm s noi se of syst em : T
av g r ms

= KS T

sys

/ ( t n)

B S

rms

= (2k/ 2) KS Tsys / ( t n) = (2k/Ae) KS T T
min sys

rms

/ ( t n)

· Sourc e > B eam , T=Tconst TA = (r Tconst/ A) Pn(, ) d = r (

const

/ A )

~ 5 X T

b

r ms

TR = rec eive r temperature; TLP = trans miss ion line temperature; = effic ienc y trans miss ion; KS = telesc ope s ens itivity c onstant (~ 1);
r = fractional power of antenna transm ission; A = antenna solid angle; Pn(, ) = antenna power pattern; b = s olid angle subtended by main beam and side lobes

n = pre-detec tion bandwidth (Hz ); t = integration time, one rec ord; n= number of rec ords

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A Quick Rev iew

Antenna Temperature
· T el esc ope observes a poi nt sourc e (fl ux density S) · T el esc ope feed repl ac ed with m atched l oad (resi st or) · Load t em perat ure adj ust ed until pow er rec ei ved equal s pow er of the sourc e · This i s equal t o the Ant enna T em perat ure

Determining the Source Temperature

Determ in ing T

source

Determ in ing T

source

Measured Intensities
·T
mea s

Measured Intensities
ON +T OFF
everything else

Arbitrary Units

T

mea s

=T

so urce

+T

e verything e lse

c hannel

Arbitrary Units

(, ,az,za) = T src(,,az,za) + T RX + T gr(za,az) + Tcel(, , t) + T CMB + T atm(z a)

Ts

ource

T

everything else

c hannel

Determ in ing T

source

Determ in ing T

source

Relative Intensities
ON - OFF ) - (T

Relative Intensities
(ON ­ OFF)/OFF ) - (Teverything else)]/ Te

(T

source

+T

everything else

everything else

)

[(T

source

+ Te

verything else

verything else

Arbitrary Units

c hannel

% Tsy

s

c hannel

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Cho os ing the Bes t Off

Choosing the Best Off
Ts = (O N ­ O FF) O FF ???

Baseline Fitting with Best Fit Line

ourc e

???

Image on right c ourtes y of C. Cons elic e

Cho os ing the Bes t Off

Cho os ing the Bes t Off

Baseline Fitting with Best Fit Line

Frequency Switching

· Sim plest & m ost effici ent m ethod · Not feasibl e i f:
Line of interest is l arge compared with bandpass Standing waves in data Cannot r eadily fit bandpass

· Errors are prim aril y from quality of fit

Ra w spec tra

Cho os ing the Bes t Off

Cho os ing the Bes t Off

Frequency Switching

Frequency Switching
· Allow s for rapid switch betw een ON & OFF observati ons · Does not require m oti on of t el esc ope · Can be very effici ent · Disad · · · vant ages: Frequency of line of int erest m ust be known Syst em m ust be st abl e W ill not w ork with tim e or frequency varying baseli nes

Cali bra te d spec tra

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Cho os ing the Bes t Off

Cho os ing the Bes t Off

Position Switching

Position Switching
· Little a pri ori inform ati on needed · Typic all y gi ves very good results · Disadvant ages: Syst em m ust be st abl e in tim e Requi res re-pointing the t el esc ope Result s in tim e off sourc e Sky positi on m ust be c arefull y chosen Sourc e m ust not be t oo ext ended · Best result s if the sam e sky (AZ, EL) positi on used

ON Source

OFF Source

Cho os ing the Bes t Off

Cho os ing the Bes t Off

Beam Switching
· Sam e idea as positi on switching · Rem oves need t o m ove t el esc ope · Disadvant ages/ Caveats: Requi res hardw are t o exist Sky positi on m ust be c arefull y chosen Sourc e m ust not be ext ended beyond throw

Beam Switching ­ 2 Beams
· Sam e idea as positi on switching · Rem oves need t o m ove t el esc ope · Always on sourc e! · Disadvant ages/ Caveats: Requi res additi onal hardw are Sky positi on m ust be c arefull y chosen Sourc e m ust not be ext ended beyond beam separat i on

Cho os ing the Bes t Off

Cho os ing the Bes t Off

Baseline Fitting with an Av erage Fit

Position Switching on Strong Continuum

ON Source 1

OFF Source 1

Alter na ti ve if freque nc y s wi tc hi ng is no t a n o ptio n May lose detai le d i nfor matio n for i ndivi dua l fi ts Sys te m mus t be very sta ble

ON Source 2

OFF Source 2

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Cho os ing the Bes t Off

Cho os ing the Bes t Off

Position Switching on Strong Continuum
· Possibl y onl y alt ernati ve i f T > few x T

Position Switching on Strong Continuum

s rc

sys

· Designed to rem ove residual st anding waves · Result: [On() ­ Off()]source1 [On() ­ Off()]source2

[(On ­ O ff) ]1 [(On ­ O ff) ]2

[(On ­ O ff) /Off]1 [(On ­ O ff) /Off]2

R=

Standard (On ­ O ff) /O ff

From ATOM 2001 -02 by Ghos h & Salter

From ATOM 2001 -02 bu Ghos h & Salter

Cho os ing the Bes t Off

Position Switching on Strong Continuum

Determining T
[(T + Te

sourc e

source

verything else

(ON ­ OFF)/OFF ) - (Teverything else)]/ Te

verything else

Standard (On ­ O ff) /O ff

Result

=

Ts Ts

ourc e yst em

[(On ­ O ff) /Off]1 [(On ­ O ff) /Off]2 [(On ­ O ff) ]1 [(On ­ O ff) ]2
From ATOM 2001 -02 by Ghos h & Salter

Units are: % Syst em T em perat ure

Need t o det erm ine syst em tem perat ure t o c ali brat e dat a

Determining System Temperature

Determ in ing Sys te m Temper a ture

·T

mea s

(, ,az,za) = T src(,,az,za) + T RX + T gr(za,az) + Tcel(, , t) + T CMB + T atm(z a)
so urce

T

mea s

=T

+T

syste m

5


Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

Theory
Measure v arious components of T
Decreasing Confidence

1 - Noise Diodes

sys:

T RX TCMB Tcel(, , t) Tatm(za) T gr(za,az)



C an W el Can Ca n Ca n

be readil y measured/monitor ed l known (2.7 K) be determined from other (tel.) measurem ents be determined from other (tel.) measur ements be c alcul ated

Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

1 - Noise Diodes

1 - Noise Diode Mea surement Considerations
· Frequency dependenc e

Tsrc/Tsys = (ON ­ OFF)/OFF . . . Tdiode/ Tsys = (On ­ Off) / Off Tsys = Td
i ode *

Off/(On ­ Off)
Lab meas urements of the GBT L-Band c alibration diode, tak en from work of M. Stennes & T. Dunbrac k - February 14, 2002

Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

1 - Noise Diode Mea surement Considerations
· Tim e stability

1 - Noise Diode Mea surement Considerations
· Accuracy of m easurem ent s:

Typicall y measur ed against another diode or other calibrator Errors inherent in i nstruments used to measure both diodes Measur ements often done in lab. H ave num erous loss es through path from diode i njecti on to bac k ends



2

measured value

=

2

standard cal

+

2

instrumental error

+

2

loss uncertainties

6


Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

1 - Noise Diode Mea surement Considerations

The Y-Factor (Two Diodes)

· Frequency dependence · Tim e stability · Accuracy of measurements

2 measured value

T1 + T Y= T +T 2
+
2 loss uncertainties

off off

Toff =

T1 - YT Y- 1

2

=

2

standard cal

+

2

instrumental error

2

total

=

2

freq. dependence

+

2 stability

+ 2

measured value

+

2 conversion error

· Can be more accurate than j ust one diode · Ignores effects of the antenna

Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

2 - Hot & Cold Loads

2 - Hot & Cold Loads

· Same idea as two diodes · Takes antenna into account · True temperature measurement (no conversion)

Cooling System Tcold

Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

2 - Hot & Cold Loads

2 - Hot & Cold Loads

Absorber (Th H ot Load Thot

ot/cold

)

T

o ff

=

T1 - YT Y- 1

2

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Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

2 - Hot & Cold Loads

2 - Hot & Cold Loads

· Sam e idea as two di odes · T akes ant enna i nt o acc ount · True t em perature m easurem ent (no c onversi ons)

Requi res a reli abl e l oad abl e t o enc om pass the rec eiver, Requi res a reli abl e l oad abl e t o enc om pass the rec eiver, with response fast enough for on-the-fl y m easurem ents with response fast enough for on-the-fl y m easurem ents

Determ in ing Sys te m Temper a ture

Determ in ing Sys te m Temper a ture

3 - Astronomical Measurements
· Use sourc es with w ell det erm ined fluxes for c alibrati on · Easy to obt ai n high spectral frequenc y resol uti on · Uses sam e hardw are as observati ons

Determining T
Theory:

sys

Needs detail ed understanding of telesc ope & structure Atmosphere & ground sc atter m ust be stable and understood

Noise Diodes:
Can be fired rapidly to m onitor temperature Requires no `los t' time Depends on accurate meas urements of diodes

Hot/Cold Loads:
Requi res ext rem ely reli abl e m easurem ents of sourc e fl ux
Can be v er y acc urate Observations not possible w hen l oad on Must be in mm range for on-the-fl y m eas urements

Astronom ical Measurements:
Error will alw ays be dom inat ed by sourc e error
Can be v er y acc urate Uses the s ame hardware as astronomical measur ements Must know source fluxes extremel y w ell

Determining T Ts =

sourc e

ourc e

(O N ­ O FF) O FF

Tsy

st em

Determining Telescope Response

Blank S ky or other

From di odes, Hot/ Col d loads, etc.

Telescope response has not been accounted for!

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Telesco pe Re spo nse

Telesco pe Re spo nse

Telescope Response
· Main B eam Brightness: TMB = · Flux Density: S= 2k T(,) P (,)d n 2

1 - Ideal Telescope

beam

T

measured

Accurat e gai n, tel esc ope response c an be m odel ed Can be used t o det erm ine the fl ux densit y of `standard' c ontinuum sourc es Not practic al in c ases where t el esc ope is non-ideal

Units: W m -2 Hz-1 or Jy (1 J y = 10-22 W m-2 Hz-1)

(bloc ke d a per ture, ca bli ng/e lectro nics losses, gro und ref lectio n, e tc)

= Solid angle of tel. pattern;

beam

= teles c ope effic ienc y ; = wa vele ngth; S= Flux, k = c onstants, T = temperature; P = antenna power pattern

Telesco pe Re spo nse

Telesco pe Re spo nse

1 - Ideal Telescope

2 - `Bootstrapping'

Observe sourc e with pre-det erm ined fluxes Det erm ine t el esc ope gai n

T

source

= (ON ­ OFF) T OFF

1
system

GAIN

GAIN =

OFF Tsys (ON ­ OFF) Tso

tem urce

Telesco pe Re spo nse

Telesco pe Re spo nse

2 -`Bootstrapping'

3 - Pre-determined Gain Values
Pre-det erm ined Gain curves:

· Useful when gain is not readily modeled · Offers ready means for determ ining telescope gain · Requires flux of calibrator sources be known in advance · Not practical if gain changes rapidly with position

· Allow s for accurat e represent ati on of gai n at all positi ons · Saves observing tim e · Can be only practic al soluti on

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Telesco pe Re spo nse

Telesco pe Re spo nse

3 - Pre-determined Gain Values

3 - Pre-determined Gain Values

Average Ga in [(pol A+po lB) /2]:
gai navg(az, za, f=1415M Hz) = 10.99 9 - 0.10291 * za + 0.01343 57*( za-14) 2 - 0.00717 45*( za-14 )3 - 5.2154 x1008*cos(a z) - 1. 3225x10-0 7*si n(az) + 1.1642x10-08 *cos(2* a z) - 7.3761 -07*si n(2*a z) - 0.20 990*cos( 3*a z) 0.098026*si n(3*a z) gai navg(az, za, f=1175M Hz) = 11.37 8 - 0.08130 4* za - 0.0 26763*( za-14)2 - 0.0 026350*( za-14)3 + 1.0319 x1006*cos(a z) - 3. 1292x10-0 7*si n(az) - 7.5973x 10-07*cos (2*a z) - 1. 9372x10-0 7*si n(2*a z) - 0.171 80*cos(3 *a z) 0.046071*si n(3*a z) gai navg(az, za, f=1300M Hz) = 11.26 5 - 0.09514 5* za + 0.0042 48*( za-14) 2 - 0.00667 83*( za-14 )3 + 7.2271x1007*cos(a z) + 9.089 7x10-07*si n(a z) + 4.3958 x10-07*co s(2*a z) - 8.1956x10- 07*si n(2*a z) - 0.22 135*cos( 3*a z) 0.074295*si n(3*a z) gai navg(az, za, f=1375M Hz) = 11.11 4 - 0.10412 * za + 0.02391 5*( za-14)2 - 0.009493 8*( za-14) 3 - 8.3447x 1007*cos(a z) + 1.072 9x10-06*si n(a z) - 4.5 402x10-08 *cos(2* a z) - 1.3411 x10-07*si n(2*a z) - 0.22827* cos(3*a z) 0.080216*si n(3*a z) gai navg(az, za, f=1550M Hz) = 10.78 6 - 0.10748 * za + 0.01926 5*( za-14)2 - 0.007553 0( za-14)3 - 7.8976x1 007*cos(a z) - 6. 5565x10-0 7*si n(az) - 7.4506x 10-08*cos (2*a z) - 4. 1723x10-0 7*si n(2*a z) - 0.209 72*cos(3 *a z) 0.14330*si n(3*a z)

Telesco pe Re spo nse

3 - Pre-determined Gain Values
Pre-det erm ined Gain values:

Determining T Ts = (O N ­ O FF) OFF

sourc e

ourc e

Tsy

st em

Allow s for accurat e represent ati on of gai n at all positi ons Saves observing tim e Can be only practic al soluti on Caveat: Observers shoul d always check the predicted gain duri ng observati ons against a num ber of cali brat ors!

1 GAIN

Blank S ky or other

Theoretic al, or Observati onal From di odes, Hot/ Col d loads, etc.

Great, you're done? !

Othe r Iss ues

A Few Other Issues

Pointing

Result s in reducti on of t el esc ope gain Typic all y c an be c orrect ed in t el esc ope pointi ng m odel or offset

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Othe r Iss ues

Othe r Iss ues

Focus

Side Lobes*
· Allows in extraneous or unexpected radiati on

· Can res ult i n fals e detections, over-estimates of fl ux, incorrect gai n determination

Result s in reducti on of t el esc ope gain Can be c orrect ed m echanic all y if rcvr/subrefl ect or c an be adjust ed

· Solution is to full y understand shape and v arianc e in side lobes

Bea m

Othe r Iss ues

Othe r Iss ues

Imag e fro m AT OM 99-02, Heiles

Comatic Error
· Sub-refl ect or shi ft ed perpendic ul ar from m ain beam · Result s in an offset betw een the beam and sky poi nting

Astigmatism
· Deform ities in the refl ect ors

Can result in fal se det ecti ons, over-estim at es of flux, i nc orrect gain det erm inati on Sol uti on i s t o full y underst and beam shape
Imag e fro m AT OM 99-02, Heiles

T he E n d

List of usef ul ref erenc es pp 31 0-3 11 in bo ok

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