8.5 Correction for atmospheric absorption, T_A^*

By analogy with the Rayleigh-Jeans approximation, $I= 2kT/\lambda^2$, which strictly applies to long wavelengths, the mm-wave radio astronomers have introduced the concept of ``radiation'' or ``effective'' temperatures, which scale linearly with the detected power.

The noise power detected by the telescope is the sum of the power received by the antenna, ${\mathcal{W}}_{A}$, and of the noise generated by the receiver and transmission lines, ${\mathcal{W}}_{rec}$.

Using Nyquist's relation ${\mathcal{W}} = kT \Delta\nu$, ${\mathcal{W}}_{A}$ and ${\mathcal{W}}_{rec}$ can be expressed in terms of the temperatures T_A and T_rec of two resistors, located at the end of the transmission line, which would yield noise powers equal to ${\mathcal{W}}_{A}$ and ${\mathcal{W}}_{rec}$, respectively:

$${\mathcal{W}}_A + {\mathcal{W}}_{rec} = k T_A \Delta\nu + k T_{rec} \Delta\nu = k (T_A + T_{rec}) \Delta\nu .$$

T_A is called the ``antenna temperature'' and T<sub>rec</sub> the ``receiver temperature''. T_A becomes T_load when the receiver horn sees a load, instead of the antenna, and T_gr when it sees the ground. It should be noted that T_load and T_gr are not stricto sensus equal to the load and ground physical temperatures, but are only ``Rayleigh-Jeans'' equivalent of these temperatures (they are proportional to the radiated power). For ambient loads, they approach however closely the physical temperature, since $h\nu /k\simeq 11$ K at $\lambda= 1.3$ mm.

When observing with the antenna a source and an adjacent emission-free reference field, one sees a change $\Delta T_A= T_A(sou) - T_A(ref)$ in antenna temperature.

Because of the calibration method explained below, it is customary, in mm-wave astronomy, to replace $\Delta T_A$, the source antenna temperature, by $\Delta T_A^*$, the source antenna temperature corrected for atmosphere absorption and spillover losses. Both are related through:

$$T_A = (1-\eta_f) T_{gr} + \eta_f (T_{sky}+ \Delta T_A^* e^{-\tau})$$

where $\tau$ is the line-of-sight atmosphere opacity. $\eta_f$ and $\eta_b$ are the forward and beam efficiency factors, which denote the fractions of the power radiated by the antenna on the sky and in the main beam, respectively (they are typically of the order of 0.9 and 0.7).

The source equivalent ``radiation temperature'' T<sub>R</sub> (often improperly called ``brightness temperature'' and therefore denoted T_MB when it is averaged over the main beam) and $\Delta T_A^*$ are related through

$$\Delta T_A^* =\int _{sou}T_R {\mathcal{A}}(x,y)dxdy$$

where ${\mathcal{A}}(x,y)$ is the antenna power pattern. For a source smaller than the main beam, $\Delta T_A* = \eta_b T_{MB}$.

When observing a small astronomical source with an antenna temperature $\Delta T_{A}'>> T_{BG}=2.7$ K, located at an elevation el, one detects a signal $\cal V$(of scale: ${\cal G}$ volt or counts per Kelvin):

$$\frac{{\cal V}_{sour}}{\cal G}=M_{sour}= T_{rec}+ (1-\eta_{f})T_{gr}+\eta_{f} T_{sky}+ \eta_{b} \Delta T_{A}'e^{-\tau}$$ (8.29)

This signal can be compared with the signals observed on the blank sky (T_atm), close to the source, and to that observed on a hot load (T_load):

 

$$T_{rec} +(1-\eta_{f})T_{gr}+\eta_{f} T_{sky}$$ M_atm = (8.30)

$$(1-e^{-\tau})T_{atm}$$ T_sky =

M_load = T_rec + T_load

here, we have neglected the cosmologic background and assume, in a first step, that the receiver is tuned single sideband.