Radio Telescopes and their use

The previous sections have described in some detail the process in which molecules emit radiation, however, in astronomy the key objective is the detection of this radiation after it has travelled the huge distances between the emission source and the Earth. For the types of object being described here the radiation will have typically travelled anywhere between a few hundred and a few thousand parsecs. Therefore the signal received on Earth even by a large telescope will be extremely weak. The operation of a typical radio telescope (the JCMT) will be described in the next section. This section will concentrate on the methods used to describe the signals received.

At its most basic level the telescope counts the photons that fall on the dish^2.3. These photons carry a certain amount of energy which defines their frequency (via $E=h\nu$). The receiver on the telescope is capable of counting photons at a particular frequency and can therefore tell how much energy is received at that frequency. Most detectors do not actually return a reading saying how many photons have been detected but rather enable the total energy collected to be measured. This is known as the radiation intensity, generally denoted $I_\nu$ or $J_{\nu}$ (the $\nu$ because it is at a specific frequency, $I_\nu$ is generally the actual intensity of a source and $J_{\nu}$ is generally the observed intensity). This is defined as the amount of energy received (or transmitted) per unit area, per solid angle, per unit frequency, per second (ie. ${\rm J\,m^{-2}\,sr^{-1}\,Hz^{-1}\,s^{-1}}$).

The total luminosity of a source (generally $L$) is the total power output of a source (ie. is measured simply in Watts). At a distance, $d$, from a point source the flux may be defined as the power passing through a unit area. Assuming the source emits isotropically this is the luminosity of the source divided by the surface area of a sphere of radius $d$, ie.

$$F=\frac{L}{4\pi d^2} \index{flux}$$

This can of course also be written as a frequency, specificaly as $F_\nu=\frac{B_\nu}{4\pi d^2}$ where $B_\nu$ is the luminosity per unit frequency and is thus related to the total luminosity by $L=\int^\infty_0 B_\nu \, d\nu$. This in turn can be related to the radiation intensity by

$$F'_\nu=\int I_\nu \, d\Omega$$

where the integral is over the solid angle of the source. Note that $F'_\nu$ will only be exactly the same as $F_\nu$for a point source, however, as long as the source subtends a 'small' solid angle as viewed by a telescope they will be approximately the same.

If the source is a black body then the radiation intensity will depend only on the source's excitation temperature and will be given by the Planck function

$$\index{planck!function} I_\nu=\frac{2h\nu^3}{c^2\left(e^{\frac{h\nu}{kT_{\rm EX}}}-1\right)}$$ (2.28)

From this the brightness temperature is defined by applying the Rayleigh-Jeans approximation that $h\nu \ll kT$ so that $e^{\frac{h\nu}{kT}}-1 \approx \frac{h\nu}{kT}$ to give

$$I_\nu=\frac{2k\nu^2}{c^2}T_B$$ (2.29)

Note that since this is the definition of brightness temperature it is unimportant whether the Rayleigh-Jeans approximation holds or not (and it is not valid for radio wavelengths). Since equations 2.30 and 2.31 are both a measure of the observed intensity they can be set equal to each other to give

$$T_B=\frac{h\nu}{k}\frac{1}{e^{\frac{h\nu}{kT_{\rm EX}}}-1} \index{temperature!brightness}$$ (2.30)

This is the definition of the observed intensity (ie. $J_\nu=T_B$).