Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 338 / 2 / a / i

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.

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,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.