# Radiometer Equation Applied to Interferometers

### Reference Material

Recall that for a single dish, we had:

${\displaystyle {\frac {S}{N}}={\frac {T_{src}}{T_{rms}}}={\frac {T_{src}}{T_{sys}}}{\sqrt {\tau \Delta \nu }}\,\!}$

where:

• ${\displaystyle T_{src}}$ is the signal of the source you’re observing
• ${\displaystyle T_{sys}}$ is your system temperature
• ${\displaystyle T_{rms}}$ is the noise in your system, or the RMS fluctuations in your system temperature
• ${\displaystyle \Delta \nu }$ is the bandwidth of your correlator (in Hz)
• ${\displaystyle \tau }$ is integration time (seconds)

In some sense, this equation applies just as well to interferometers, so long as you are treating all the antennas in the array together as the “dish”. If you really want to break it down by antenna, though, we just need a couple more steps.

The first key is to recognize that, applied to an interferometer, the radiometer equation is really concerned with the synthesized beam — that is, the beam you get when you phase all of your antennas together. But for an interferometer, because we get to see the correlation of each antenna pair separately, we make a choice to define ${\displaystyle T_{sys}}$ to be noise on each correlation pair (i.e. visibility). All this really means is that we have chosen to use the primary beam of individual antenna elements as the angular area in our gain/noise temperature calculations.

So applying the Radiometer Equation to an interferometer really boils down to figuring out the difference in beam area between the primary beam of an individual element, and the synthesized beam of the array. Unfortunately, beam areas can be rather tricky to calculate (the full-width-half-max method is woefully inadequate for sparse arrays). But fortunately, there’s an easier way.

We know that, however long our observation is, and whatever our phase center is, we will always be adding ${\displaystyle N(N-1)/2}$ visibilities together (where ${\displaystyle N}$ is the number of antennas) with some phasor applied to each number. When dealing with noise, we don’t really care what the phasor is; we know that for Gaussian noise, averaging that many samples (with equal weight) will beat down noise as the square root. Finally, there is one more factor to deal with. All our visibilities are complex numbers, but we know the sky to be real-valued, so we get to throw out the half of our noise that ends up in the imaginary part of the sum. Another way of saying this exact same thing is that, for a real valued sky, we get two uv-samples: one at (u,v), and one at (-u,-v). In either case, we can effectively pretend we actually have ${\displaystyle N(N-1)}$ independent samples. So at the end of the day, the radiometer equation for an interferometer is:

${\displaystyle {\frac {S}{N}}={\frac {T_{src}}{T_{rms}}}={\frac {T_{src}}{T_{sys}}}{\sqrt {N(N-1)\tau \Delta \nu }}\,\!}$

It’s worth stressing that this only applies if you equally weight all of your visibility samples. If you start using weighting schemes other than natural weighting, you need to take those weights into account in your radiometer equation (and they can only hurt you).