8.5.1 Simplest case

Let's assume that $T_{load}\simeq T_{gr}\simeq T_{atm}$; then:

 

$$\eta_f T_{gr} e^{-\tau}$$ M_load-M_atm =

$$\Delta T_{A}^* e^{-\tau}$$ M_sou-M_atm =

$$\Delta T_{A}^* \frac{M_{sou}-M_{atm}}{M_{load}-M_{atm}} \eta_{f} T_{gr}$$ = (8.31)

Note that in Eq.8.31, the measurement of the antenna temperature includes the atmospheric opacity correction, but does not depend explicitly on an assumption on the atmospheric opacity. We can write:

$$\Delta T_A^*= \frac{M_{sou}-M_{atm}}{M_{load}-M_{atm}}{\bf T_{cal}}$$ (8.32)

where we define T_cal as $T_{cal}= \eta_f T_{gr} \simeq \eta_f T_{atm}$.