I feel like the discussion of the maximum of SNR (signal-to-noise ratio) will benefit my thesis, because my boring mathematical derivation can easily occupy some space to lengthen the number of pages.
To make things easy, I assume that the signal of the observed target can be described by a Gaussian distribution, namely,
where is some referenced intensity, is the radius to the centre of the source, and describes its FWHM. The integrated intensity of the source within an aperture is then calculated from
After doing some simple algebra the result is simply
The SNR of the source is given by the following equation:
For simplicity, we assume the sky background remains constant across the targeted signal source, so can be regarded as a combination constant value related to the sky background intensity and the readout noise of the CCD. To obtain the maximum SNR, we ought to find out when , which can be transformed to
With some algebra, this equation can be changed to
where , and . Unfortunately, this is a transcendental equation so I cannot find out its solution analytically. Instead, let me use the numerical method.
Assuming the targeted source is very faint, which means that , I obtain . Conversely, if the source is extremely bright, i.e., , then we have .
Anyway, the best-SNR radius as a function of is plotted in the following figure. Note that I have converted the radius in FWHM by using .
In conclusion, using FWHM looks to be a good choice.