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

Use the classical Faber-Jackson relation $L\propto {\sigma }^4$ as the explicitly requested extrapolation. The supplied host fixes its normalization:With the mass derived using circular-orbit Jacobi truncation of two singular isothermal spheres, the dynamical mass-to-light ratio isHence to one significant figureThe rough equal-density truncation convention gives ${{\rm Y}}_d\simeq 6\times 10^2{M}_{\odot }/L_{\odot }$ instead. Both conclusions indicate a very large dynamical mass-to-light ratio compared with a normal old stellar population.

Within their bright stellar bodies, typical giant elliptical galaxies have optical dynamical mass-to-light ratios of order a few to ten in solar units, with variations in band, initial mass function, and dark matter. Large outer apertures can give substantially larger ratios. Many dwarf spheroidal galaxies have much larger inferred ratios, from tens to hundreds and, for some very faint systems, still larger values. Thus the computed ratio is plausible as the scale of a very dark-matter-dominated faint dwarf, not as a purely stellar ratio. Dynamical inference depends on member identification, binary motions, equilibrium, and the assumed mass profile.

The stipulated Faber-Jackson relation is not a reliable universal extrapolation down to every dwarf spheroidal galaxy. Also, the fundamental plane of elliptical galaxies contains radius and surface brightness as well as velocity dispersion; the two-variable Faber-Jackson relation is a projection of that information, not an interchangeable definition of the full plane.