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, radius, and luminosity. The radius variable r(1/2) is the deprojected (three-dimensional) half-light radius, the mass variable M-1/2 is the total gravitating mass within this radius, and L-1/2 is half the luminosity. 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 mass-radius-luminosity relation transitions from M1/2 similar to r(1/2)(1.44) similar to L-1/2(0.30) for the faintest dwarf spheroidal galaxies to M-1/2 similar to r(1/2)(1.42) similar to L-1/2(3.2) for the most luminous galaxy cluster spheroids. The weakness of the M-1/2 - L-1/2 slope on the faint end may imply that potential well depth limits galaxy formation in small galaxies, while the stronger dependence on L-1/2 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 CDM halos in order to construct a virial mass-luminosity relationship (M-vir-L) for galaxies that spans seven orders of magnitude in M-vir. Independent of any global abundance or clustering information, we find that (spheroidal) galaxy formation needs to be most efficient in halos of M-vir similar to 10(12) M-circle dot and to become inefficient above and below this scale. Moreover, this profile matching technique for deriving the M-vir-L 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 such as 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.