The fundamental manifold of spheroids

We present a unifying empirical description of the structural and kinematic properties of all spheroids embedded in dark matter halos. We find that the intracluster stellar spheroidal components of galaxy clusters, which we call cluster spheroids (CSphs) and which are typically 100 times the size of normal elliptical galaxies, lie on a ''fundamental plane'' as tight as that defined by elliptical galaxies (rms in effective radius of similar to 0.07) but having a different slope. The slope, as measured by the coefficient of the log sigma term, declines significantly and systematically between the fundamental planes of ellipticals, brightest cluster galaxies (BCGs), and CSphs. We attribute this decline primarily to a continuous change in M-e/L-e, the mass-to-light ratio within the effective radius re, with spheroid scale. The magnitude of the slope change requires that it arise principally from differences in the relative distributions of luminous and dark matter, rather than from stellar population differences such as in age and metallicity. By expressing the M-e/L-e term as a function of sigma in the simple derivation of the fundamental plane and requiring the behavior of that term to mimic the observed nonlinear relationship between logM(e)/L-e and log sigma, we simultaneously fit a two-dimensional manifold to the measured properties of dwarf elliptical and elliptical galaxies, BCGs, and CSphs. The combined data have an rms scatter in log r(e) of 0.114 (0.099 for the combination of ellipticals, BCGs, and CSphs), which is modestly larger than each fundamental plane has alone, but which includes the scatter introduced by merging different studies done in different filters by different investigators. This ''fundamental manifold'' fits the structural and kinematic properties of spheroids that span a factor of 100 in sigma and 1000 in re. While our mathematical form is neither unique nor derived from physical principles, the tightness of the fit leaves little room for improvement by other unification schemes over the range of observed spheroids.