Low opacity approximation

To outline the relative importance of all parameters, I will now derive an approximate formula for the calibration factor Tcalin the limiting (but frequent) case where the opacities are weak and equal in the image and signal bands.

$$T_{sky} = \tau * T_{atm}$$ (25)

where T_atm is the physical temperature of the absorbing layers. Thus

$$T_{emi} = F_{eff} * \tau * T_{atm} + (1-F_{eff}) * T_{cab}$$ (26)

$$\tau = (T_{emi} - (1-F_{eff}) * T_{cab}) / ( F_{eff} * T_{atm} )$$ (27)

$$Tcal = C_{eff} * (T_{load} - T_{emi}) (1+Gain\_i) / ((1-\tau) * B_s)$$ (28)

Eliminating $\tau$ gives

$$Tcal = C_{eff} \frac{(T_{load}-T_{emi}) * (1+Gain\_i) * T_{at... ...f}} {B_s * (F_{eff} * (T_{atm}-T_{cab}) + T_{cab} - T_{emi})}$$ (29)

Although T_cab is a weighted average of the physical temperature of the cabin and the outside temperature, it is not very different from T_load. Hence we can simplify :

$$Tcal = C_{eff} \frac{F_{eff} * (1+Gain\_i) * T_{atm}} {B_s * ( 1 - F_{eff} * \frac{T_{cab}-T_{atm}}{T_{cab}-T_{emi}})}$$ (30)

It is important to note that in this formula, T_atm is dependent mostly on the outside temperature and pressure (and site altitude of course), and weakly on $\tau$ because T_atm is the (physical) temperature of the absorbing layers. T_emi is the effective temperature seen by the antenna, and hence small compared to T_cab. The parameters are F_eff, B_eff and $Gain\_i$. Setting reasonable numbers :

T_cab = 290 K, T_atm = 240 K, T_emi = 50 K, and C_eff = 1,

yield

$$Tcal = \frac{240 * (1+Gain\_i) * F_{eff}}{B_s * (1 - 0.2*F_{eff})}$$ (31)