Signal to Noise Ratio

Since the signals^3.15 in a radio telescope are random in nature, the output of a total power detector attached to a radio telescope too will show random fluctuations. Supposing a telescope with system temperature T$_{\rm sys}$, gain G, and bandwidth $\Delta \nu$ is used to try and detect some astrophysical source. The strategy one could follow is to first look at a `blank' part of the sky, and then switch to a region containing the source. Clearly if the received power increases, then one has detected radio waves from this source^3.16. But given that the output even on a blank region of sky is fluctuating, how can one be sure that the increase in antenna temperature is not a random fluctuation but is indeed due to the astrophysical source? In order to make this decision, one needs to know what the rms is in the fluctuations. It will be shown later^3.17, that for a total power detector with instantaneous rms T$_{\rm sys}$, the rms after integrating a signal of bandwidth $\Delta \nu$ Hz for $\tau$ seconds is^3.18 T $_{\rm sys}/\sqrt{\Delta\nu \tau}$. The increase in system temperature is just GS, where S is the flux density of the source. The signal to noise ratio is hence

$${\rm snr} = {{\rm GS}\sqrt{\Delta\nu \tau} \over {\rm T}_{\rm sys}}$$

This is the fundamental equation for the sensitivity of a single dish telescope. Provided the signal to noise ratio is sufficiently large, one can be confident of having detected the source.

The signal to noise ratio here considers only the `thermal noise', i.e. the noise from the receivers, spillover, sky temperature etc. In addition there will be random fluctuations from position to position as discussed below because of confusion. For most single dish radio telescopes, especially at low frequencies, the thermal noise reaches the confusion limit (see Section 3.4) in fairly short integration times. To detect even fainter sources, it becomes necessary then to go for higher resolution, usually attainable only through interferometry.

Footnotes

... signals^3.15

Apart from interference etc.

... source^3.16

Assuming of course that you have enough spatial resolution to make this identification

... later^3.17

Chapter 5

... is^3.18

This can be heuristically understood as follows. For a stochastic proccess of bandwidth $\Delta \nu$, the coherence time is $\sim 1/\Delta \nu$, which means that in a time of $\tau$ seconds, one has $\Delta\nu\ \tau$ independent samples. The rms decreases as the square root of the number of independent samples.