Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 3 / ii

The result is in the broad range expected for an old, relatively faint stellar population, but is high compared with many observed stellar mass-to-light ratios of a few solar units in optical bands. The comparison must specify the band and whether the measured mass includes dark matter: for example, early-type galaxy dynamical measurements find an $I$-band normalization near $3.8$ solar units at velocity dispersion $200\mathrm{km}{\mathrm{s}}^{-1}$, including a contribution from dark matter. That is an illustrative observational comparison, not a direct equality with a bolometric stellar calculation.

For an old stellar population, an upper surviving main sequence mass near $M_{\odot }$ is reasonable because more massive stars have shorter lifetimes; low-mass main sequence stars also contain much of the surviving stellar mass. However, red giants and other evolved stars can contribute a large fraction of the light despite their small numbers, so omitting them generally makes this calculated mass-to-light ratio too large. Omitting stellar remnants has the opposite effect on the inferred mass, and a single cubic mass-luminosity relation is only approximate across the stated range. Finally, a single $m^{-3}$ initial mass function down to $0.1M_{\odot }$ is steeper and more low-mass-heavy than commonly adopted distributions, which flatten at low mass. Ten solar units is a useful toy-model estimate, not a precision stellar-population prediction.