Aperture photometry with an estimated background ($\mathrm{Var}\hat{S}=S+n(b+d)t+nM{r}^2+n^2{\sigma }_{\hat{B}}^2$)

For total source electrons $S$, $n$ aperture pixels, sky-plus-dark rate $b+d$, time $t$, $M$ reads with rms $r$, and an independent mean-background estimate of variance ${\sigma }_{\hat{B}}^2$ per pixel, the aperture variance is $S+n(b+d)t+nM{r}^2+n^2{\sigma }_{\hat{B}}^2$ before additional calibration terms. The last factor is $n^2$ because one uncertain background mean is subtracted from all aperture pixels. Pixel correlations require covariance terms.