10.2.1 Low opacity approximation and implication for T_cal

When the opacity of the atmosphere is weak ( $\tau_\nu < 0.2$) and equal in both image and signal bands, T_cal is mostly dependent of T_atm and both of them can be considered as independent of $\tau_\nu$ and hence w.

In the conditions mentioned above, $\tau_\nu$ can be eliminated from Eq.10.5. The equation becomes:

$$T_{cal}^L = \frac{\eta_F(1+G^{UL})\times T_{atm}} {\eta_B\times(1-\eta_F\times\frac{T_{cab}-T_{atm}}{T_{cab}-T_{emi}^L})}$$ (10.11)

(details about the derivation of Eq.10.11 are given in the documentation ``Amplitude Calibration'' by S.Guilloteau). In Eq.10.11, the unknown is T_atm, the physical temperature of the absorbing layers. T_atm is mostly dependent on the outside temperature, pressure and site altitude and weakly on $\tau_\nu$. For this reason, T_cal and T_sys remain correct even if w and hence $\tau_\nu$ are not properly constrained.

Figures 10.1 and 10.2 illustrate this point. Thick lines correspond to the exact equation (Eq.10.5) and dashed lines to the approximation (Eq.10.11). The comparison between Eq.10.11 and 10.5 was done for three common cases 1) at 87 GHz, with G^UL=10^-2, 2) at 115 GHz, with G^LU=0.5and at 230 GHz, with G^UL=0.5. For the 15-m dishes, the forward efficiencies used are $\eta_f =0.93$ at 3mm and $\eta_f = 0.89$ at 1.3mm. Fig.10.1 is done for a source at elevation=20⁰ and Fig.10.2 for a source at elevation=60^o.

  

Figure 10.1: Calibration temperature as function of water vapor (or opacity) at 87, 115 and 230 GHz for a source at 20 degrees elevation. Parameters are taken for the Bure interferometer.

  

Figure 10.2: Calibration temperature as function of water vapor (or opacity) at 87, 115 and 230 GHz for a source at 60 degrees elevation. Parameters are taken for the Bure interferometer.

The following points can be deduced from these figures:

1.

As long as T_sky^L=T_sky^U, the equation 10.11 remains valid even at high frequencies > 200 GHz and for w> 5 mm.

2.

This comes from the fact the T_atm is mostly independent of the atmospheric water vapor content.

3.

As soon as $T_{sky}^L \neq T_{sky}^U$, the equation 10.11 is not valid. Note also that the error is about constant with the opacity because T_atm is mostly independent of the atmospheric water vapor content. Moreover at 115 GHz, the atmospheric opacity is dominated by the 118 GHz Oxygen line and cannot be below 0.2, the amount of opacity added by the water vapor is small. T_calremains mostly constant with w.

At mm wavelengths, the derivation of the T_cal (or T_sys) using an atmospheric model is then quite safe.