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 / S2 / 2 / d

Let $I$ denote a characteristic surface brightness and ${\rm Y}=M/L$ the relevant dynamical mass-to-light ratio. With comparable profiles, $L\propto I{R}^2$ and $M\propto V^{2}R/G$. Eliminating $R$ givesIf both $I$ and ${\rm Y}$ are independent of speed, converting luminosity to magnitude gives $M_V=\text{constant}-10{\log }_{10}V$. The empirical coefficient $-8$ instead means $L\propto V^{3.2}$, and thereforeFor constant surface brightness this requires ${\rm Y}\propto V^{0.4}$; for constant mass-to-light ratio it requires $I\propto V^{0.8}$. A mixture of these changes, including varying stellar populations or dark-matter fractions, can give the same observed slope. The simple derivation assumes comparable characteristic radii and structural coefficients.