Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 3 / vi / Solution

The host mass inside $D$ follows from the same singular isothermal sphere:Using the supplied host luminosity givesThis treats essentially all the stated optical light as contained in that aperture; a measured luminosity profile would refine that convention. Keeping consistent unrounded quantities,Thus the dwarf ratio is about twenty times the parent ratio; the rough equal-density tidal convention would give a factor thirty.

There is no logical contradiction in two objects sharing $L\propto {\sigma }^4$ but having different mass-to-light ratios. A virial theorem scaling gives $M\propto {\sigma }^{2}R$, so that $M/L\propto R/{\sigma }^2$. Equal ratios additionally require homologous relevant radii scaling as $R\propto {\sigma }^2$. Here the satellite's tidal radius scales as ${\sigma }_d$ in the fixed host field, not as ${\sigma }_d^2$. Equating the two ratios would require $r_t=D({\sigma }_d/{\sigma }_h{)}^2\simeq 56\mathrm{pc}$, far smaller than the derived tidal radius.

The heuristic derivation of the Faber-Jackson relation with common mass-to-light ratio and common surface brightness made auxiliary structural assumptions; the empirical relation alone cannot enforce them. Furthermore, the aperture dependence of mass-to-light ratio matters: a satellite's whole tidal mass and a host's $50\mathrm{kpc}$ halo mass are not homologous half-light-aperture masses. Dark matter need not follow stellar light, and a dwarf spheroidal galaxy is not automatically a scaled giant elliptical galaxy. The model calculates total gravitating mass: reading its density literally as stars alone would imply an implausibly large purely stellar ratio, exposing the limits of that interpretation.