Signal-to-noise ratio in photon counting

For source counts $S$ and independent background counts $B$ with Poisson distributions, the ideal counting standard deviation is $\sqrt{S+B}$, so the signal-to-noise ratio is $S/\sqrt{S+B}$. Detector noise and uncertainty in the estimated background add further variance. A larger sky aperture raises $B$, while better throughput raises $S$; both affect the faintness limit of a spectrograph.

Subtracting an exactly known mean sky level leaves source-plus-sky variance of a Poisson distribution $S+B$. If the sky per source pixel is estimated from $n_b$ equal-area, equally exposed independent sky pixels, its additional variance is $B/n_b$, so $R=S/\sqrt{S+B+B/n_b}$ before read and dark terms. The variance of removed background photons does not disappear with their mean.