From Galaxy Clusters to Ultra-Faint Dwarf Spheroidals: A Fundamental
  Curve Connecting Dispersion-supported Galaxies to Their Dark Matter Halos
  We examine scaling relations of dispersion-supported galaxies over more than
eight orders of magnitude in luminosity by transforming standard fundamental
plane parameters into a space of mass (M1/2), radius (r1/2), and luminosity
(L1/2). We find that from ultra-faint dwarf spheroidals to giant cluster
spheroids, dispersion-supported galaxies scatter about a one-dimensional
"fundamental curve" through this MRL space. The weakness of the M1/2-L1/2 slope
on the faint end may imply that potential well depth limits galaxy formation in
small galaxies, while the stronger dependence on L1/2 on the bright end
suggests that baryonic physics limits galaxy formation in massive galaxies. The
mass-radius projection of this curve can be compared to median dark matter halo
mass profiles of LCDM halos in order to construct a virial mass-luminosity
relationship (Mvir-L) for galaxies that spans seven orders of magnitude in
Mvir. Independent of any global abundance or clustering information, we find
that (spheroidal) galaxy formation needs to be most efficient in halos of Mvir
~ 10^12 Msun and to become inefficient above and below this scale. Moreover,
this profile matching technique is most accurate at the high and low luminosity
extremes (where dark matter fractions are highest) and is therefore quite
complementary to statistical approaches that rely on having a well-sampled
luminosity function. We also consider the significance and utility of the
scatter about this relation, and find that in the dSph regime observational
errors are almost at the point where we can explore the intrinsic scatter in
the luminosity-virial mass relation. Finally, we note that purely stellar
systems like Globular Clusters and Ultra Compact Dwarfs do not follow the
fundamental curve relation. This allows them to be easily distinguished from
dark-matter dominated dSph galaxies in MRL space. (abridged)
