Implementation of the radiating sphere problem of Graziani (2008), as
described in Swesty and Myra (2009).

The problem puts a hot sphere inside an low temperature ambient
medium, characteristized by an opacity that goes like 1/nu**3.  The
radiation from the hot sphere produces a prompt spectrum away from the
sphere.  An analytic solution is provided for the spectrum a some
radius, r, from the sphere at time t.

To implement the problem, we do not model the sphere.  Rather the
inner boundary in 1-d spherical coordinates serves as the sphere, with
Dirichlet BCs giving the radiation energy emitted from the sphere.

This is a multigroup radiation test -- no hydrodynamics is done.

Our strategy is to initialize the ambient material with a radiation
field corresponding to an early-time analytic solution to the problem
(at early times, the analytic solution is just the blackbody spectrum
of the ambient medium).  These group-dependent initial conditions for
the radiation field are computed using analytic.f90, with the
observation time (t_obs) set to a small value (e.g. 1.e-20).  

At the moment, the group structure is not exposed in the Castro
initrad routine, so the initial values for the radiation field are
hardcoded in there.  In the future, if the group structure (centers,
weights) are passed into initrad, then the initial conditions can
be computed directly in there from the Planck function (multiplied
by dnu -- the group-bin weight to get a radiation energy density).


Auxiliary files in Tools/: 

  analytic.f90: computes the analytic solution, using Eqs. 76 and 77
                from Swesty and Myra (2009).  Swesty and Myra do not
                give their coefficient for their opacity.  The value
                in analytic.f90 comes directly from Eric.

  radbc.f90: compute the values for the group-dependent Dirichlet
             boundary conditions, corresponding to the hot sphere.  This
             is simply the blackbody (Planck) function evaluated at the
             group-centers, with the sphere's temperature.  Note, the
             final numbers are multiplied by the group weights, since
             Castro stores the energy in each group, not the energy / Hz
             (or eV).

  fradsphere.f90: dumps out the radiation energy density for each
                  group at a specified distance from the origin.



references:

Graziani, F., 2008, Computational Methods in Transport: Verification
and Validation, ed. F. Graziani (Berlin:Springer), 151

Swesty, F. D. and Myra, E. S., 2009, ApJS, 181, 1.

