Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S26 / 3 / 4 / Solution

Use a Tully-Fisher relation $L\propto v_{\max }^4$. A simple justification assumes comparable mean surface brightness, so $L\propto R^2$, and comparable total mass-to-light ratio, so $M\propto L$. The virial scaling $v^2\sim GM/R$ then gives $v^4\propto M^2/R^2\propto L$. Calibrating to the previous galaxy yieldsThis is a rough luminosity estimate: the empirical slope depends on passband and population, and the assumed constant surface brightness and mass-to-light ratio are approximations rather than exact properties of every spiral galaxy.