= Solution
Measure counts in electrons so that <photon> and detector <noise> can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms <read noise> $r$ per pixel, and an independently estimated mean background per pixel of <variance> $\sigma_{\widehat B}^2$, an ideal aperture-sum budget is
$$
\boxed{\operatorname{Var}(\widehat S)
\simeq S+n(b+d)t+nMr^2+n^2\sigma_{\widehat B}^2
+\sigma_{\rm cal}^2,\qquad
{\rm SNR}=\frac{S}{\sqrt{\operatorname{Var}(\widehat S)}}.}
$$
The last term represents the propagated <calibration> uncertainty if it can be treated as a random <variance>; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is $\sigma_{\widehat B}^2=[(b+d)t+Mr^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.
\b[<Photon> counting and sky.] Source <photon shot noise> has <variance> $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky <photons>, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better <seeing> or <adaptive optics> can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky <photons> already recorded, and the sky estimate adds uncertainty of its own.
\b[Dark and <read noise>.] Thermal <dark current> contributes a mean and shot <noise>. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic <noise>. At fixed total integration, many short exposures add more read <variance> than fewer long exposures, so use exposures long enough to become sky limited when saturation, <cosmic rays> and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic <read noise>.
\b[<Calibration> and non-Poisson effects.] Pixel sensitivity differences, illumination gradients, fringes, nonlinearity, imperfect charge transfer and electronic pattern <noise> leave <calibration> errors. High-count flat fields, stable operating conditions, linearity <calibration>, dithering and appropriate detector models help, while the finite signal-to-noise of <calibration> frames propagates into the result. Cosmic-ray hits, hot pixels, satellite trails and scattered-light artefacts are not ordinary stationary Gaussian <noise>: identify or mask them, often using several exposures. Overaggressive clipping can remove real transient or compact-source signal.
<Atmospheric scintillation> adds correlated intensity fluctuations, especially in short bright-star measurements; <seeing> changes and guiding errors move <flux> between pixels or apertures. Differential photometry using appropriate reference <stars> and stable apertures reduces common variations but does not guarantee removal of chromatic or position-dependent effects. For a constant flux-limited source in independent stationary <noise> the final signal-to-noise usually grows as $\sqrt t$ once background limited; a systematic floor or correlated sky can stop that improvement. Distinguish improvement of <calibration>, which removes reproducible bias, from elimination of intrinsically random <photon> arrivals, which is impossible.
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