  The kinematics of stars and planetary nebulae in early type galaxies provide
vital clues to the enigmatic physics of their dark matter halos. We fit
published data for fourteen such galaxies using a spherical, self-gravitating
model with two components: (1) a Sersic stellar profile fixed according to
photometric parameters, and (2) a polytropic dark matter halo that conforms
consistently to the shared gravitational potential. The polytropic equation of
state can describe extended theories of dark matter involving self-interaction,
non-extensive thermostatistics, or boson condensation (in a classical limit).
In such models, the flat-cored mass profiles widely observed in disc galaxies
are due to innate dark physics, regardless of any baryonic agitation. One of
the natural parameters of this scenario is the number of effective thermal
degrees of freedom of dark matter (F_d) which is proportional to the dark heat
capacity. By default we assume a cosmic ratio of baryonic and dark mass.
Non-Sersic kinematic ideosyncrasies and possible non-sphericity thwart fitting
in some cases. In all fourteen galaxies the fit with a polytropic dark halo
improves or at least gives similar fits to the velocity dispersion profile,
compared to a stars-only model. The good halo fits usually prefer F_d values
from six to eight. This range complements the recently inferred limit of
7<F_d<10 (Saxton & Wu), derived from constraints on galaxy cluster core radii
and black hole masses. However a degeneracy remains: radial orbital anisotropy
or a depleted dark mass fraction could shift our models' preference towards
lower F_d; whereas a loss of baryons would favour higher F_d.
