With respect to the signal sideband, the effective system temperature,
, referred to a perfect telescope above the earth's
atmosphere, is given by (c.f. Ulich & Haas 1976; Kutner &
Ulich 1981)
where
The definition given in Equation 1 is on the
scale as defined by Kutner & Ulich (1981). Some
observatories prefer the
scale which differs from
by
the
factor. The conclusions of this paper mostly depend
on ratios in which
divides out, so the difference in
definitions is not important.
In Equation 1, the numerator corresponds to the various sources of noise present, whereas the denominator represents the scaling factors that account for signal losses through telescope inefficiencies and atmospheric attenuation. The antenna temperature of the sky is given by the sum of noise contributions due to sky, antenna, and cosmic microwave background emission
where
To properly calculate Equations 1 and 4, the temperatures used should be the equivalent Rayleigh-Jeans temperatures of the point on the Planck blackbody curve corresponding to the same frequency.1 This correction factor is given by (see Ulich & Haas 1976)
For simplicity of notation, we will retain the symbol ``T''
for temperatures, but in calculations T should be replaced by
J(,T).
Real receivers, whether they are intended to be single or double
sideband, have varying ratios of sideband gain ,
i.e. single sideband systems have imperfect rejection so that
, while double sideband systems often have
slightly unequal gain ratios so that
.
However, these are usually minor effects in terms of system
temperature, so we will consider the two cases of principal interest
Furthermore, we will take:
where is the image termination physical
temperature, which is, for example,
K for the quasi-optical
image termination of the 1.3mm dual-channel receiver on the NRAO 12m
telescope. For quasi-optical systems the image termination
temperature is also increased by optics losses, such as vacuum
windows, grids, etc. before the beam terminates on an absorber.
Thus, the system temperature of a single sideband system is
The SSB system temperature of a double sideband system is
The factor
is the difference between single and double sideband
system temperatures, and can be thought of as the double sideband
penalty. As is apparent, if the termination temperature of the image
sideband, , exceeds
, the double sideband ``penalty''
becomes an advantage. The product of the spillover efficiencies,
, is an upper limit on the conventional beam efficiency,
. Through arrangement of optics and by redirecting spillover
between the ground and the sky, one can transfer losses between
and
. It is most advantageous to maximize
since it
contributes to both an ambient temperature noise term and a loss
factor.
contributes only a loss factor and is a simple
scaling factor.