You’ve got me attempting to do undergrad statistics at 2am after a couple of beers, so this may be a pile of nonsense. But the formula I get is as follows…
* By definition, SNR = Signal / sqrt(Sky_Noise^2 + Photon_Noise^2)
* Define the sky noise in an individual pixel to be sigma_sky. If the standard deviation of M sky pixels is measured to be sigma, then your best estimate of the sigma_sky = sigma*sqrt(M/(M-1)). The sqrt() term comes about because sigma is a slight underestimate of sigma_sky, since the same sample of M sky pixels was also used to calculate the mean sky level relative to which their standard deviation was measured. For large M, sigma_sky=sigma.
* The total noise in the summed brightness of all N pixels within your aperture is sigma_sky*sqrt(N) = sigma*sqrt(NM/(M-1)).
* This equals Sky_Noise in the SNR equation above.
* Now for photon noise. Define the number of photons collected from the star to be N_photon. By the definition of the gain, N_photon = S/G. The noise in N_photon is sqrt(S/G) since it is a Poisson process. Thus, the noise in S is sqrt(S/G)*G = sqrt(SG).
* Putting it all together…
SNR = S / sqrt(SG + sigma^2 * NM / (M-1))
* My formula is closer to yours than the AAVSO’s, but not quite the same.
* In my formula, SNR decreases as G gets larger. This must surely be correct? Higher gain cameras have more noise. In your formula, and the AAVSO’s, SNR increases with G. This doesn’t seem to make any sense.
* I think this formula behaves correctly when images are stacked. When you stack X images, sigma is divided by sqrt(X), and G is effectively divided by X. Net result: SNR increases as sqrt(X), as you expect.
* I’ve never done any aperture photometry in my life, so the above may or may not be complete nonsense.