Effective System Temperature

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

is the ratio of the gain response of the image and signal sidebands;

is the receiver noise temperature measured with hot and cold loads (and will generally differ for single and double sideband tunings);

is the antenna temperature of the sky (see below), and includes contributions from the atmosphere, antenna spillover, and cosmic microwave background;

is the rear spillover, scattering, and ohmic loss efficiency, given by:

where is the antenna power pattern and G is the maximum antenna gain (i.e., is the fraction of telescope power in the forward hemisphere).

is the forward spillover and scattering efficiency, given by:

where is a defined diffraction zone.

is the zenith optical depth of the atmosphere; and

A

is the number of airmasses at the observing elevation (given approximately by ).

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

is the mean temperature of the atmosphere (given approximately by . This quantity is frequency and weather dependent and can be given more accurately by atmospheric models.);

is the effective temperature of the rear spillover (also );

is the background temperature (usually taken to be the cosmic background temperature of 2.7 K).

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

Single Sideband = 0,

Double Sideband = 1.

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.