Aperture dependence of mass-to-light ratio (${\rm Y}(<R)=M(<R)/L(<R)$)

A dynamical mass-to-light ratio must specify its aperture and whether the mass includes dark matter. A growing dark halo can increase ${\rm Y}(<R)$ outside the bright stellar body even when the stellar ratio is constant. Comparing a tidal-radius satellite mass with a large-aperture host mass is different from comparing both within their half-light radii. From $M\propto {\sigma }^{2}R$ and $L\propto {\sigma }^4$, the inferred ratio scales as $R/{\sigma }^2$, so equal ratios additionally require homologous radius scaling.