Low opacity approximation

To outline the relative importance of all parameters, I will now derive an approximate formula for the calibration factor $Tcal$ in 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

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