System temperature

We start by deriving the sky temperature with:

$$T_{sky} = \left(T_{hot}+T_{rec}\right)\frac{P_{sky}}{P_{hot}}-T_{rec}-T_{spill}$$ (52)

Estimating $T_{spill}$ requires knowledge of the forward efficiency $\eta_f$. It is also used to compute the sky emission $T_{emi}$:

$$T_{emi} = \frac{T_{sky}}{\eta_f}$$ (53)

The water vapor content variable of an atmospheric model predicting sky brightness ( $T_{emi,s}^{atm}$ and $T_{emi,i}^{atm}$) and opacity ( $\tau_{atm,s}$ and $\tau_{atm,i}$) in the signal and image sideband is then varied in order to minimize the following quantity:

$$T_{emi} - \frac{T_{emi,s}^{atm}+g_{im}T_{emi,i}^{atm}}{1+g_{im}}$$ (54)

The system temperatures for the signal $T_{sys,s}$ and image $T_{sys,i}$ sidebands are then computed as follows:

$$T_{sys,s} = \frac{e^{\tau_s}\left(T_{hot}-T_{sky}-T_{spill}\right)(1+g_{im})}{\eta_f}\frac{P_{sky}}{P_{hot}-P_{sky}}$$ (55)

$$T_{sys,i} = \frac{e^{\tau_i}\left(T_{hot}-T_{sky}-T_{spill}\right)(1+\frac{1}{g_{im}})}{\eta_f}\frac{P_{sky}}{P_{hot}-P_{sky}}$$ (56)

where:

$$T_{sky}+T_{spill} = (1-\eta_f) \left(\frac{T_{emi,s}^{atm}+g_{im}T_{emi,i}^{atm}}{1+g_{im}} \right)+\eta_f T_{ground}$$ (57)