1. IRAM Observing School 2007 Clemens Thum IRAM, Grenoble, France Lecture 2 : Fundamentals continued calibration efficiencies beam shape observing modes (single pixel heterodyne Rxs)

2. C. Thum 2/2 V amb = G (T amb + T Rx ) V sky = G (T sky + T Rx ) V ON = G (T sky + T Rx + T A ) V OFF = G (T sky + T Rx )  V cal = V amb – V sky = G (T amb – T sky ) = G (T amb – T amb (1 – e -  )) = G T amb e -   V sig = V ON – V OFF = G T A = G T A ' e -  Calibration – 1 : basic chopper wheel method Basic idea: T amb V sig V cal T A ' = T A ' is the antenna temperature corrected for atmospheric extinction Note the difference to the y-factor method: T-scale in K, measurement of a second load at different (cold) temperature avoided receiver noise temperature is not obtained

3. C. Thum 2/3 T cal V sig V cal T A ' = Calibration – 2 : chopper wheel method – better approximations Zero order: T cal = T amb First order: Second order: Higher orders: History of chopper wheel method: invented by Penzias & Burrus in 1973 (ARAA 11, 51) important contributions by NRAO staff (e.g. Kutner & Ulich 1981, ApJ 250, 341) a modified version is implemented at IRAM telescopes: - IRAM report by S. Guilloteau (1987) - “Radio Astronomy Techniques” (D. Downes, IRAM report No. 151, 1988) - detailed description by C. Kramer in “30m Manual” by W. Wild (1994) T amb ≠ T atm and g s ≠ g i T cal = (T amb – T atm )(1 + g i ) e -  s + T atm (1+ g i e  s -  i )  F < 1.0 (forward efficiency) V = G (  F T sky + (1 -  F ) T amb + T Rx ) 3 K background temperature is not always negligible Rayleigh-Jeans approximation not always valid Preconditions: Atmosphere is well behaved receiver gain is sufficiently constant T A ' T A = TA'TA' FF

4. C. Thum 2/2 calibration – 3 : implementation at 30m telescope T cal V ON - V OFF  V cal T A ' = basic equation: T cal T sys = V OFF  V cal introduce the system temperature: T A ' = T sys on - T sys off calibration factor: T cal =( T amb - T sky ) 1 + g i  e e  = 1 : T sys is on the T A ' scale (not meaningful)  =  F : T sys is on the T A scale (antenna temperature)  =  mb : T sys is on the T mb scale (main beam brightness temperature) User interface PaKo knows two efficiency parameters: F eff and B eff :  set F eff to  F (always)  set B eff to a value in the range from  F to  mb, depending on source size w/r main beam Remarks: - T mb is the scale physically most meaningful - often observers use the T A scale, worry about coupling efficiency later - historically, moon efficiency was also used

5. C. Thum 2/2 calibration – 4 : calibration measurements at the 30m telescope Chopper wheel method is extended to include a measurement on a cold load input: T HOT sensor T COLD lookup table T AMB meteo station Pressure meteo station Humidity meteo station image gain observer output: T RX T CAL using atm. model T SYS “ pwv “ opacity “

6. C. Thum 2/4 efficiencies – 1 : forward efficiency  F V sky = G (  F T sky + (1 -  F ) T amb + T Rx ) Now we calibrate and move the antenna in elevation  (skydip) : T A (  ) =  F T atm ( 1 – e -  (  ) ) + ( 1 -  F ) T amb Remark: - since  F can be precisely measured, T A is a very useful concept - T A is the antenna temperature corrected for: (i) atmospheric extinction and (ii) spillover (power not coming from the sky) - when do we make skydips ? only when we are desperate !

7. C. Thum 2/5 How do we determine  A ? observation of point sources of known flux density easy with big single dish, difficult with interferometer sum up all known losses example: 30m telescope at 1.3mm ohmic losses 0.95 ( ≤ 1% loss per reflection) aperture blocking 0.90 (subreflector, quadrupod) illumination 0.95 misalignment 0.98 surface leakage 0.99 surface roughness 0.50 (rms = /16) +.... --------------------------------------  A ≃ 0.40  A = 2 k A TA'TA' S efficiencies – 2 : aperture efficiency  A Definition:  A = A eff A where A is the physical surface area of the antenna (30m: 707 m 2 ) Flux density of a point source: w = k T A ' = ½ S A eff = ½  A A S

8. C. Thum 2/2 efficiencies – 2: measurement of aperture efficiency ( measured by Juan Peñalver in August 2007 ) Ruze's formula: (  = rms surface roughness)

9. C. Thum 2/6 efficiencies – 3 : beam efficiency  mb Definition:  mb = AA  mb  mb = main beam solid angle, where P (  ) ≥ 50 % = ∬ mb P( ,  ) d   A = beam solid angle, integrated over 4  = ∬ 4  P( ,  ) d  what is the brightness temperature of a source filling the main lobe? but what if the source is  smaller than  mb ?  larger than  mb ? A eff 2 k S = TA'TA' 2 2 k S = T mb  mb A eff AA 2 = T b = T mb = TA'TA'  mb T b = T mb SS  mb (correction for beam dilution) T b < T mb ( since  mb   F )

10. C. Thum 2/2 Rayleigh-Jeans correction Rayleigh-Jeans correction factor c : mm (and submm) wavelengths are special in radio astronomy because the Rayleigh- Jeans approximation does not always hold Questions: Observations are made against the CMB of 3 K. How much flux does it contribute in a MAMBO beam? Can MAMBO detect CMB fluctuations? Which temperature do we get when scanning across a planet ? P

11. C. Thum 2/2 The Jy/K ratio (from IRAM web page) antenna dependent quantities: T A, T A ', T A, T mb physical quantities: S,, T b remarks: - we need to specify the T-scale - the Jy/K ratio is very frequency dependent - simplest way to get to Jy (recommended) note that another useful relation:

12. C. Thum 2/2 efficeincies – 5 : how do we determine  mb ? (a) measure  mb,  A : impractical (b) measure T A ', T mb : not simple (c) sufficiently accurate fudge:  A A A eff  A = 2 =  mb  mb ⇒  mb =  A A  mb 2 Conclusion:  mb can be measured as accurately as  A Remarks: the calibration scale depends on the beam width through  mb the beam width is not a smooth function of frequency therefore needs to be measured. Feed horns are designed for one frequency.

13. C. Thum 2/2 beam shape and sidelobes : the error beam derived from scans across the moon's limb (Greve et al. 1998) decomposition into sidelobe components (at 88 GHz)

14. C. Thum 2/2 error beam : why worry ? M51 CO (2 1) Schuster et al. 2007

15. C. Thum 2/2 observing modes: why switching ? T sys = T A + T sky + T spillover + T rx T A ≪ T sys (most of the time) T A ≤ 1 K T sky ~ 30 K (at 3mm) T spillover ~ 20 K T Rx ~ 100 K (at 3mm) The enemies: - sky emission fluctuations - receiver gain fluctuations Solutions for single pixel receivers: most of the time - position switching (SWtotal, ONOFF) - wobbler switching (SWwobbler, ONOFF - frequency switching (SWfrequency, TRACK) some of the time - on-the-fly (fast scanning)

16. C. Thum 2/2 Dicke switching conclusion: balanced receivers are insensitive to gain fluctuations but: switching must be faster than the fluctuations an exact balancing is not always possible, e.g. on-the-fly and: spectroscopic observations are relatively insensitive to gain fluctuations

17. C. Thum 2/2 observing modes: wobbler switching new consideration: spectroscopic baseline affected by gain fluctuations Rx bandpass is not flat affected by reflections inside the telescope (receiver does not absorb all incoming power) T A,i ' = T sys,i on - T sys,i off WSwitch: 3 cycles interwoven wobbler period of 4 sec cancels most fluctuations telescope nodding : 1min removes standing waves Anti-nodding: 2 min removes slow linear drifts FFT #1 – FFT#2 spectrum

18. C. Thum 2/2 Observing modes: respective merits Wobbler switching: problems: throw limited by hardware to ±2', gain drops off axis, slow at high speeds and large throws Position switching: pros: large offsets possible (but not reasonable) cons: baselines often poor for too large offsets Frequency switching: pros: time efficient, spares the search for emission-free reference position cons: baselines usually poor, only reasonable for narrow lines (dark clouds) Fast scanning: pros: very useful for mapping, time efficient cons: depends very much on atm. and Rx/Be stability Warning: mesospheric lines !

19. C. Thum 2/2 Baseline ripple Fabry-Perot effect between reflecting surfaces separation = /2 feed horn does not absorb all incoming power (r F ≃ 0.4) reflected wave may interfere with original wave amplitude of modulation: 1 + 4 r r F best known ripples: 7.5 MHz between subreflector and Rx 15.5 MHz between Vertex and Rx 29.2 MHz between Rx and calibration unit 90 MHz between Rx and Martin-Puplett

20. C. Thum 2/2