Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 338 / 3 / b

Treat $Q$ and $B$ as mean detected counts, or as photon counts with unit quantum efficiency. Let the independent patch measurements be $X\sim \mathrm{Poisson}(Q+B)$ and $Y\sim \mathrm{Poisson}(fB)$. For the specified unweighted background subtraction $\hat{Q}=X-Y$,The first error is a fixed bias of an estimator; the last expression is the sum of independent photon shot noise variances. Since $Q,B$ are already patch totals, no extra factor of the pixel count $n$ is needed, and read noise is neglected.

To obtain the systematic-error ceiling requested in the following clause, define the accuracy measure $Z$ using total root-mean-square error relative to the true source count. By the bias-variance decomposition of mean squared error,For $f\simeq 1$, the shot-noise term is approximately $Q+2B$, but the mismatch term must be retained.

There is a terminology qualification: the usual variance-based signal-to-noise ratio in photon counting is $Q/\sqrt{Q+(1+f)B}$ and does not include a fixed bias as noise. The printed next-part limit requires the root-mean-square accuracy convention above. A deterministic background mismatch contributes $b^2$ to mean squared error, not to the statistical variance.

Put $Q=R_{Q}T$ and $B=R_{B}T$ in the root-mean-square accuracy measure:For a fixed uncorrected $f\ne 1$ and $R_B>0$, the squared bias eventually dominates the shot-noise terms. ThereforeThe PDF omits the absolute value. Its expression is the positive accuracy ratio only if $f<1$; for $f>1$ the residual background changes sign, but an error magnitude and a signal-to-noise ratio in photon counting remain nonnegative. The printed signed formula can instead be read as source divided by signed bias.

For exact background matching $f=1$, there is no systematic-error signal-to-noise ceiling: $Z=R_Q\sqrt{T/(R_Q+2R_B)}$ grows without bound in this idealized model. With $R_B=0$ the same conclusion holds. Limits in which $f$ itself approaches one with exposure time are different from the fixed-mismatch limit used here.

Increasing exposure reduces fractional photon shot noise as $T^{-1/2}$, but a fixed background mismatch grows in proportion to the signal. The fractional systematic error is $|1-f|R_B/R_Q$, independent of exposure. The systematic-error signal-to-noise ceiling can therefore be poor for a faint source in a bright sky even when random fluctuations are tiny.

Better background subtraction requires matching sky location and time, correcting detector response, dithering or modelling spatial background variations. If $f$ were known exactly, one could instead use $X-Y/f$, which is unbiased and has variance $Q+B+B/f$. The ceiling arises from an uncorrected or unknown mismatch in the prescribed subtraction, not an unavoidable property of measuring two patches.

To reach an accuracy ratio $Z_{\ast }$ at all, a necessary condition in the fixed-mismatch model isEquality allows only an asymptotic approach; finite exposure adds positive random error. More exposure improves precision, but only better calibration removes the fixed background bias.