= Solution
{c}
The <optical telescope> supplies collecting area and spatial selection, the spectral selector supplies a passband, and the <photodetector> converts received <radiant flux> into a measured signal. For an extended source, a useful schematic photon count is
$$
N_s\simeq\eta\tau A\Omega\int_{\rm band}\frac{L_\lambda\lambda}{hc}\,d\lambda,
$$
where $A\Omega$ is the <etendue>, $\tau$ the integration time, $\eta$ the net transmission and <quantum efficiency>, and $L_\lambda$ the <spectral radiance>. An illustrative independent-noise model gives
$$
\mathrm{SNR}\simeq\frac{N_s}{\sqrt{N_s+N_b+N_d+\sigma_{\rm read}^2}}.
$$
Here $N_b$ and $N_d$ denote background and dark counts, and $\sigma_{\rm read}$ is readout <standard deviation>. This makes the roles of the three components explicit.
A larger <optical telescope> collects more photons at fixed angular pixel size and improves the <diffraction-limited resolution>. Its limits include <optical aberrations>, alignment, launch size, <stray light> and pointing. Conservation of <etendue> constrains simultaneous collecting area and angular acceptance. Increasing spatial resolution by narrowing a pixel's <solid angle> reduces its photons.
A narrower spectral passband or higher <spectral resolving power> separates diagnostic features and rejects unwanted background, but usually reduces photons per channel and hence <signal-to-noise ratio>. Selector transmission, leakage, slit width, detector format and the required time cadence limit the trade. A wider band helps only insofar as it adds useful signal rather than mostly background.
The <photodetector>'s <quantum efficiency>, dark current, read noise, dynamic range and response time determine conversion losses and added noise. Cooling can suppress thermal background and dark current, but cannot remove <photon shot noise>. Longer integration improves a stationary signal's <signal-to-noise ratio> until motion, saturation, drift or required cadence becomes limiting.
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