Estimating $T_{sys}$

For an unresolved source of known brightness $I$, in the limit $T_a \ll T_{sys}$, $\rho_{ij}^\circ=I$ and Eq. 1 can be written as

$$\rho_{ij}^{Obs} = I g_ig_j^\star \approx  I\sqrt{{\eta_i\eta_j} \over {T_{sys_i} T_{sys_j}}}$$ (11)

where $\eta_i=A_e/{2k_b}$, $A_e$ is the effective area of the dish, $k_b$ is the Boltzman's constant and

$$\left\vert g_i\right\vert = \sqrt{\eta_i \over T_{sys_i}}$$ (12)

Hence, knowing $\eta_i$, $T_{sys_i}$ can be estimated from the amplitude of the antenna dependent complex gains.

All contributions to $\rho_{ij}^{Obs}$, which cannot be factored into antenna dependent gains, will result in the reduction of $\vert g\vert$. $\eta$ remaining constant, this will be indistinguishable from an increase in the effective system temperature. Since majority of later processing of interferometry data for mapping (primary calibration, bandpass calibration, SelfCal, etc.) is done by treating the visibility as a product of two antenna based numbers, this is the effective system temperature that will determine the noise in the final map (though, as a final step in the mapping process, baseline based calibration can possibly improve the noise in the map).

In the normal case of no significant baseline based terms ( $\epsilon_{ij}$) in $X_{ij}$, the system temperature as measured by the above method will be equivalent to any other determination of $T_{sys_i}$.

$T_{sys}$ can also be determined by recording interferometric data for a strong point source with and without an independent noise source of known temperature at each antenna. In this case

$$T_{sys_i} = T_{n_i}({{g_i^{ON}}^2 \over {g_i^{OFF}}^2 - {g_i^{ON}}^2})$$ (13)

where $g_i^{ON}$ and $g_i^{OFF}$ are the antenna dependent gains with and without the noise source of temperature $T_n$. Note that $\eta_i$ does not enter this equation. Also, $T_n$ should be such that $\sqrt{T_a/(T_n+T_{sys})} \ge 0.2$ to ensure that the correlated signal is measured with sufficient signal-to-noise ratio (in this case, $\ge 0.04$).