= Solution
The <optical telescope> focuses the sky onto an <optical slit>; the <collimator> turns the selected <light> into a beam incident on the reflection <diffraction grating>; the <optical camera> images each diffracted direction onto the <photodetector>. The drawing also shows the monochromatic <optical slit> image for uniform <optical slit> illumination in the geometric, slit-limited approximation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-338-spectrograph.png]
{title=Reflection-grating spectrograph and slit-limited line profile}
Near the chosen <wavelength> $\lambda$, a small <wavelength> increment moves the image by $q\,\Delta\lambda$, where $q$ is the magnitude of the <linear spectral dispersion>. An <optical slit> image of physical width $p$ therefore has an apparent spectral width $p/q$. With one <optical slit> width as the adopted separation criterion,
$$
\boxed{\Delta\lambda=\frac{p}{q},\qquad R=\frac{\lambda q}{p}.}
$$
Uniform illumination gives a top-hat monochromatic profile of width $p$; an unresolved stellar image need not illuminate the <optical slit> uniformly, so its profile follows the actual illumination convolved with the instrument response. Negligible <optical slit> <diffraction> and <optical camera> <optical aberrations> do not remove the finite <diffraction grating> <diffraction> width. The formula is the slit-limited result when that width is small compared with $p$, rather than a universal exact line-profile criterion.
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