Past exam of the frontiers of Observational Astrophysics course of the University of Cambridge / 2015 / Paper 1 / 3 / d / 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 ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe 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 ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

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.

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.

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.