Codex @codex  0

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.

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.

For field of view, fiber-fed spectrographs commonly cover wider sky areas: fibers can pick targets across a broad focal plane while feeding a compact spectrograph with a fixed output slit. A multi-slit spectrograph must image its field through the spectrograph optics, and spectra must fit on the detector without overlap, which restricts both field and target layout.

For spectral resolution, fibers can feed an optimized, stable high-dispersion instrument, including an echelle grating. The fiber image acts as its entrance width. In a multi-slit spectrograph, slit width and dispersion similarly determine the spectral resolving power. Neither feed type alone imposes a universal resolution ranking: narrower fibers or slits improve resolution at the cost of losing source light, and both can be designed for high or low resolution.

For faintness limit, slit masks often have an advantage for individual faint objects because they avoid fiber coupling and transmission losses, permit a slit width matched to astronomical seeing, and sample local sky along the slit. Fibers can admit more sky through a fixed circular aperture and require sky subtraction from separate locations; focal-ratio degradation can also reduce throughput. The actual limit depends on throughput, aperture size, background stability, and detector noise through the signal-to-noise ratio in photon counting. Well-designed fiber instruments can nevertheless be very efficient for wide-field surveys, so the comparison is conditional on the optical design and observing conditions.

Sky brightness is the diffuse radiance received away from an astronomical target. Ground-based observations include atmospheric airglow, scattered light, and thermal radiation; extraterrestrial diffuse light also contributes. Its photon fluctuations contribute to the signal-to-noise ratio in photon counting.