  We calculate the mean-field thermodynamics of a spherically trapped Fermi gas
with unequal spin populations in the unitarity limit, comparing results from
the Bogoliubov-de Gennes equations and the local density approximation. We
follow the usual mean-field decoupling in deriving the Bogoliubov-de Gennes
equations and set up an efficient and accurate method for solving these
equations. In the local density approximation we consider locally homogeneous
solutions, with a slowly varying order parameter. With a large particle number
these two approximation schemes give rise to essentially the same results for
various thermodynamic quantities, including the density profiles. This
excellent agreement strongly indicates that the small oscillation of order
parameters near the edge of trap, sometimes interpreted as spatially
inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov states in previous studies of
Bogoliubov-de Gennes equations, is a finite size effect. We find that a bimodal
structure emerges in the density profile of the minority spin state at finite
temperature, as observed in experiments. The superfluid transition temperature
as a function of the population imbalance is determined, and is shown to be
consistent with recent experimental measurements. The temperature dependence of
the equation of state is discussed.
