NATIONAL RADIO ASTRONOMY OBSERVATORY e/o KITT PEAK NATIONAL OBSERVATORY P. 0. BOX 4130 TUCSON, ARIZONA 85717

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1 NAIONAL RADIO ASRONOMY OBSERVAORY e/o KI PEAK NAIONAL OBSERVAORY P. 0. BOX 4130 UCSON, ARIZONA ELEPHONE IDS OFFICE BOX 2 EDGEMON ROAD GREEN BANK. WES VIRCIXIA *4944 MaC-h 20, 1973 CHARLOESVILLE. VIRGINIA ELEPHONE o04-4o > ELEPHONE ffx twx o4F2 MEMORANDUM From: B.L. Ulich Subject: Absolute hermal Calibration of Spectral Line Observations he thermal calibration signal in a millimeter wavelength spectral line receiver is generally derived from a rotating half-disk covered with absorbent material. When the chopper wheel is rotated, the synchronously detected power is proportional to the difference in the temperature of the absorbent material and the sky brightness temperature. We now derive the absolute temperature change seen by the receiver feed horn when the chopper is rotated, and thus the equivalent antenna temperature of the calibration signal. Since millimeter wavelength spectral line receivers are double sideband, we allow the possibility of different gains and -atmospheric optical depths at the signal and image frequencies. When the absorbent material is in front of the feed horn, the antenna temperature is: A = VAMB + G I AMB t-0 where. = Antenna temperature looking at absorbent material ( K) IaMB Gg Gj = Ambient temperature ( K) = Receiver gain in signal sideband a Receiver gain in image sideband OPERAED BY ASSOCIAED UNIVERSIIES, INC., UNDER CONRAC WIH HE NAIONAL SCIENCE FOUNDAION

2 page 2 Since the receiver is near the ground, we can safely assume that the absorbent material is at ambient temperature. When the chopper wheel is not in front of the feed horn, the antenna temperature is: V = G S (L L S;Y (SIGNAL) (1 - N L ) ^} + Gj {n L SKY (IMAGE) + (1- il L ) amb ) (2) ' o where = Antenna temperature looking at sky ( K) r)^ = Antenna loss efficiency g^y = Sky brightness temperature ( K) he antenna loss efficiency includes ohmic loss, blockage by spars and radiometer, and spillover. It is assumed that all absorbed, blocked, or spilled over radiation falls on a blackbody at ambient temperature. he synchronously detected chopper wheel antenna temperature is given by CAL = A " A G s\ { AMB - SKY (SIGNAL)} + G L\ { AMB - W IMAGE ) } C3) where = Chopper wheel antenna temperature ( K) Neglecting the small effect of the 2.7 K cosmic background radiation, the sky brightness temperature is adequately modeled by (Falcone t ad., 1971, and Ulich, 1973) SKY = V " E X ) ( 4 )

3 page 3 where g^y = Sky brightness temperature ( K) XM = Mean atmospheric temperature ( K) = Zenith optical depth of atmosphere x = Air mass (~secant of zenith angle) he mean atmospheric temperature depends slightly on the zenith optical depth (Kislyakov, 1966) since for large only the atmosphere near the antenna is sampled. In this analysis, however, we assume that is independent of. he sky brightness temperatures in the two sidebands are given by sky (SIGNAL) = M ( 1 - e" SX ) (5) and SKY (IMAGE) = M (1 - e"* 1 *) (6) where g^y(signal) = Sky brightness temperature in signal sideband ( K) SKY^IMAGE^ = brightness temperature in image sideband ( K) tg tj = Zenith optical depth in signal sideband = Zenith optical depth in image sideband Inserting Eq.5 and Eq.6 into Eq.3 we get for the chopper wheel antenna temperature CAL = G S n L * AMB _GX V 1 " e ^ * R X + G I\ ( AMB - V 1 - E " 1» W he antenna temperature of a spectral line in the signal sideband is given by " - S X C8) A ~ G S n B n L B

4 page 4 f '. 0 where = Antenna temperature of the spectral line ( K) Dg g = Beam coupling efficiency = Brightness temperature of the spectral line ( K) he beam coupling efficiency is the normalized convolution of the antenna power pattern and the source brightness distribution. For sources very large compared to the half power beamwidth, such as the sky itself, rig =! Note that the chopper wheel calibration and the line observation are assumed to be made at the same air mass x. he directly measured quantity in a spectral line observation is the ratio R of the peak line intensity to the chopper wheel calibration signal R - A a CAL _nx = G S ri B n L B e ~ (9) CAL where R = Ratio of spectral line peak intensity to chopper wheel calibration signal Solving for g we get Rp So X r.- _ t r cx a t b = n BG sn L l G s r L {t AMB - V 1 e * G I"L {t AMB - V 1 - "***»] n [a + I) ( t amr " M' e V + t M B - G s * gi V Cx 8 " xx)x] (10)

5 page 5 We define the absolute calibration temperature C as C = M + p^we^s " I^X. S M n cx (11) + (1 + P)( AMB - V E G s where C = Absolute calibration temperature ( K) hus t b = f (12) B he first term of C is the constant term independent of optical depth and air mass. he second term corrects for the different gains and zenith optical depths of the two sidebands. he last term accounts for the fact that the sky is always colder than the chopper wheel and thus the sky fails to emit enough radiation to completely correct for absorption of the spectral line radiation. he mean atmospheric temperature ^ exhibits a small dependence on the surface ambient temperature. he relationship derived by Altshuler et al. (1968) is M " K12 AMB " 50 ( 13 > Substituting Eq.13 into Eq.ll we get G I ( S -V* C =" (^"AMB ' 50 ' + 4 C1 ' 12aMB ' 50)6 G. x (14) (1 g~)( AMB )e u Eq.14 should be used to derive the calibration temperature when Gj/Gg, g, j. and x are known.

6 3/20/75 page 6 If G I G S * C If G I ^ 12 amb + (100 Gg and tg (15) C (2.24 amb -100) + (100 0,24 AMB- )e (16) Eq.15 is valid when the signal and image gains are equal. Eq.16 holds for equal gains and equal optical depths in both sidebands. Note that in the center of a broad atmospheric window g and j are very nearly identical, but close to an atmospheric absorption line they can be quite different. In this case, the second term in Eq.14 may become appreciable, and values of C larger than 2^g are possible. he traditional chopper wheel calibration temperature of 400 K is incorrect, and the published antenna (or bright- ness) temperatures for the 1? 1 f\ C 0 J - 1 -> 0 transition at 2.6 MM are considerably in error.

7 page 7 REFERENCES ALSHULER, E.E., FALCONE, V.J., AND WULFSBERG, K.N. (1968). Atmospheric effects on propagation at millimeter wavelengths. IEEE Spectrum 5_, FALCONE, V.J., WULFSBERG, K.N., AND GIELSON, S. (1971). Atmospheric emission and absorption at millimeter wavelengths. Radio Science 6, KISLYAKOV, A.G. (1966). Effective path length and mean temperature of the atmosphere. Radiofizika ULICH, B.L. (1973). Absolute brightness tempeature measurements at millimeter wavelengths. he University of exas at Austin, Electrical Engineering Research Laboratory echnical Report No. NGL