Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S26 / 3 / c

The integrated spectrum combines stars moving toward and away from us within the galaxy, broadening their absorption by Doppler shifts. Assuming instrumental and individual-stellar widths have been removed or are negligible, the line-of-sight velocity dispersion isThe angular half-light radius at the low-redshift distance is $R_e\simeq 2.35$ kpc. For a pressure-supported, approximately isotropic equilibrium spheroid with a standard structural virial coefficient of order five, useThe coefficient represents assumed density profile, aperture and orbital structure; a different reasonable mass estimator changes the result by an order-unity factor. Treating $D$ as a luminosity distance and using $D_A=D_L/(1+z{)}^2$ would reduce the radius and mass by about four percent here.

The distance modulus is about $34.93$. The magnitude within $R_e$ therefore corresponds to $M_{V,\mathrm{half}}\simeq -19.23$ andDoubling the half-light luminosity givesThis ratio is compatible with an old stellar population plus possible dark matter. It is larger than a solar-like mass-to-light ratio, but does not by itself isolate a unique dark-matter fraction: stellar age, initial mass function, anisotropy and the virial coefficient are not specified. Counting the quoted half-light magnitude as the total would incorrectly double the inferred ratio.