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We examine scaling relations of dispersion-supported galaxies over more thaneight orders of magnitude in luminosity by transforming standard fundamentalplane 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 clusterspheroids, dispersion-supported galaxies scatter about a one-dimensional"fundamental curve" through this MRL space. The weakness of the M1/2-L1/2 slopeon the faint end may imply that potential well depth limits galaxy formation insmall galaxies, while the stronger dependence on L1/2 on the bright endsuggests that baryonic physics limits galaxy formation in massive galaxies. Themass-radius projection of this curve can be compared to median dark matter halomass profiles of LCDM halos in order to construct a virial mass-luminosityrelationship (Mvir-L) for galaxies that spans seven orders of magnitude inMvir. Independent of any global abundance or clustering information, we findthat (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 luminosityextremes (where dark matter fractions are highest) and is therefore quitecomplementary to statistical approaches that rely on having a well-sampledluminosity function. We also consider the significance and utility of thescatter about this relation, and find that in the dSph regime observationalerrors are almost at the point where we can explore the intrinsic scatter inthe luminosity-virial mass relation. Finally, we note that purely stellarsystems like Globular Clusters and Ultra Compact Dwarfs do not follow thefundamental curve relation. This allows them to be easily distinguished fromdark-matter dominated dSph galaxies in MRL space. (abridged)

From Galaxy Clusters to Ultra-Faint Dwarf Spheroidals: A Fundamental Curve Connecting Dispersion-supported Galaxies to Their Dark Matter Halos