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Calculation of Atmospheric Emission Measured by Gregorian
Feed Horns
Paul F. Goldsmith
September 16, 2001
As we use higher frequencies, emission from the Earth's atmosphere becomes more sig-
nicant. This is important not only in determining system temperatures, but for initial
receiver calibration of the receiver temperatures and noise diodes.
The usual procedure is to take the receiver looking vertically in the receiver test laboratory.
Calibration is eected by measuring output when the input is looking at the `cold' sky
and at an `ambient load'. The ambient load is assumed to be at the local temperature,
but the question is what temperature to assign for the sky.
The atmospheric temperature is due to the low-frequency wings of molecular lines, pri-
marily those of H 2
O, but to a lesser extent of O 2
. The exact shape of these line wings is
hard to model, but there are standard programs that do this. One such is included in the
set of routines called "ASTRO" in the package "CLASS", and is called "ATM". To use
this program, you input the altitude, temperature, and precipitable water vapor (pwv) in
mm. The latter is the most variable quantity{it is the integral of the water content of the
atmosphere from starting altitude on up.
Luca Olmi ran an atmospheric model code, and generated the following values for pwv:
T(C) RH(%) pwv(mm)
28 80 44.0
25 70 32.7
30 85 52.1
Thus, we can expect pwv's between 30 and 50 mm at Arecibo. These are, not surprisingly,
considered large values. I have run the ATM code for several values of pwv with plausible
temperature, and generated the zenith sky temperature as a function of frequency between
1 and 15 GHz. I have added the Cosmic Background Radiation (CBR) set to be 2.74 K.
No other Galactic emission is included.
The sky temperature at 5 GHz is almost unaected by the water content of the atmo-
sphere, and is close to 5 K under all conditions. At 10 GHz the situation is somewhat
dierent, and we nd the results given in the following table:
pwv(mm) T sky (10GHz) @ zenith (K)
10 6.0
30 7.3
50 9.0
Thus, the zenith temperature could be vary between 6 K under exceptionally dry condi-
tions, and perhaps reach as high as 10K, under very humid conditions. These results do
not explicitly include the eect of clouds or precipitation.
One further question is the eective sky temperature resulting from the feed horn's radi-
ation pattern{it is not sensitive only to radiation from the zenith, but has a fairly wide
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acceptance pattern. The standard for Gregorian horns is to be down about 15 dB (relative
sensitivity = 0.03) at 60 degrees from boresight (in this case, the zenith).
A reasonable way to describe the main lobe of the radiation pattern of broad angle feed
horns, neglecting their sidelobes and other peculiarities, is as a cos n () distribution, where
 is the angle o{boresight. This is only applicable for angles between 0 degrees and 90
degrees. The exponent n is determined by the relative power P rel at a specic angle  rel
through the relationship
n = log(P rel )
log(cos rel ) : (1)
From our specication of P rel = 0.03 for  rel = 60 degrees, we nd that n = 5.06.
The antenna temperature measured is given by the convolution of the sky temperature
and the feed horn response pattern through
TA =
1
A
Z Z
T(
P
n(

d
: (2)
For the present azimuthally symmetric situation, assuming that the atmosphere is opti-
cally thin, we have
T(
= T () = T 0 =cos ; (3)
where T 0
is the atmospheric temperature at zenith. Putting in the expression for the feed
horn response, and using limits on the integral of 0 degrees and 90 degrees, we nd
TA =
2T
0
A
Z
cos n 1
sind =
2T 0
n
a
: (4)
The expression for the antenna solid angle is very similar:

A =
Z Z
P
n(

d
= 2
Z
cos n
sind =
2
n + 1 : (5)
Combining these we obtain the result that
TA = T 0
n + 1
n
: (6)
The result is actually not very sensitive to the tertiary edge taper (value of P rel , varying
from a factor of 1.30T 0
for P rel = -10 dB, to 1.20 for our standard value P rel = -15 dB,
to 1.12 for P rel = -25 dB. Thus, although the beam width of the feed horn certainly does
have an eect on the sky temperature measured, it is not a big contributor.
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Using the standard value and associated factor TA = 1:2T 0 , we obtain the following values
(note that you have to subtract the CBR values from previous table entries as they are
not scaled by factor 1.2, and then add them back in):
pwv(mm) TA (10GHz) @ zenith (K)
10 6.7
30 8.2
50 10.3
I think these values should be reasonably accurate, but I have not carried out any specic
tests of the atmospheric emission code. I would appreciate any comments on this, and on
the feed horn modeling as well. However, note also that we are not including large{angle
feed horn sidelobes, which can add in ground pickup. That is one reason why a \sky
cone" which I discussed with Edgar Castro seems like a good bet in almost any case.
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