Atmospheric Model

Evaluating $Tcal$ thus requires to determine both $T_{emi}$ and $\tau$. Assuming the receiver temperature is known, the noise power received on the sky $T_{emi}$ is given by

$$T_{emi} = \frac{(T_{load} + Trec) * Mean\_atm}{Mean\_load} - Trec$$ (7)

and is related to the sky emission temperature by

$$T_{sky} = \frac{T_{emi} - (1.-F_{eff}) * T_{cab}}{F_{eff}}$$ (8)

where the second formula corrects for the antenna coupling to the sky ($F_{eff}$ is the forward efficiency). Strictly speaking, eq. (8) assumes that the rear sidelobes of the antenna look into the receiver cabin kept at a constant temperature $T_{cab}$, with low ohmic losses. $T_{cab}$ can however be chosen as a weighted average of the focal cabin temperature and the outside temperature for all rearward losses.

In case of a double side-band receiver and single side-band signal, $T_{sky}$ is a sum of the contribution of the two receiver bands weighted by the sideband gain ratio (gain in the image band divided by gain in the signal band).

$$T_{sky} = \frac{T_{sky\_s} + T_{sky\_i} * Gain\_i} {(1. + Gain\_i)}$$ (9)

and a similar equation for $T_{emi}$. Note that this assumes $F_{eff}$ is identical in the two side bands.

To determine the opacity $\tau$, the trick is to model the atmospheric emission to derive the transmission. There is some hope that it can work (at least in reasonably good conditions) because the transmission is dominated by a few constituants, among which only the water vapor varies significantly with time. Hence, if the atmosphere can be modelled by a small number of layers, it is possible to derive the transmission from the emission. The atmospheric model used (ATM, J. Cernicharo) is derived from a ``Standard atmosphere'' and the knowledge of the Atmospheric pressure $P_{amb}$ and outside Temperature $T_{amb}$ (and the altitude of the site). Together with the season in the year, these parameters give a good approximation of the physical temperature of the absorbing layers. By a minimization routine, with the amount of precipitable water vapor in millimeters as variable, the best model fitting the measured $T_{sky}$ is found. Then, for each band (signal and image), the total zenith opacities are computed by summing the opacities due to Oxygen and Water, with a small empirical correction factor for other minor constituents.

The zenith opacities are used together with the elevation (number of air masses) to compute $Tcal$, the SSB scaling factor

$$Tcal = \frac{(T_{load} * (1. + Gain\_i) - T_{emi\_s} - Gain\_i * T_{emi\_i})} {B_s * e^{-Tau\_s * Air\_mass}}$$ (10)

where $Tau\_s$ is the zenith opacity in the signal sideband.