Signal-to-noise ratio in photon counting
= 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>.