Converting brightness temperatures to flux densities

Once the brightness temperature is known for each filter frequency, the integrated flux density of the planet in question, $S_{int_\nu}$ is calculated as

$$S_{int_\nu} = \frac{2 h \nu^3}{c^2}\frac{ \Omega_p}{\exp{\frac{h \nu}{k T_{b_\nu}}} - 1}$$

Then, with the assumptions that the planet is a flat disk with constant flux distribution across its disk, and that the beam of the telescope is a Gaussian of HPBW $\theta_{beam_\nu}$, the beam-corrected flux density of the planet at Earth, $S_{beam_\nu}$ is given by

$$S_{beam_\nu} = 1.133\ {\theta_{beam_\nu}}^{2/\Omega_p}\ [1 -... ...{1}{1.133\ {\theta_{beam_\nu}}^{2/\Omega_p} }})]\ S_{int_\nu}.$$

The values of the HPBW, $\theta_{beam_\nu}$, are supplied in the same look-up file used for the other planets' brightness temperatures.

The implication of the flat-disk assumption for the planet is that the observed HPBW, $\theta_{obs}$, is Gaussian and given by

$${\theta_{obs_\nu}}^2 = {\theta_{beam_\nu}}^2 + \frac{\ln 2}{2}\ {(2 D_{semi})}^2$$

for $2 D_{semi} < \theta_{beam_\nu}$, as discussed originally by Baars (1973, IEEE trans. on Antennas and Propogation, 21, 461).