  We present analytic radiative transfer solutions for the spectra of
unresolved, spherically symmetric, centrally heated, dusty sources. We find
that the dust thermal spectrum possesses scaling relations that provide a
natural classification for a broad range of sources, from low-mass protostars
to dusty galaxies. In particular, we find that, given our assumptions, spectral
energy distributions (SEDs) can be characterized by two distance-independent
parameters, the luminosity-to-mass ratio, $L/M$, and the surface density,
$\Sigma$, for a set of two functions, namely, the density profile and the
opacity curve. The goal is to use SEDs as a diagnostic tool in inferring the
large-scale physical conditions in protostellar and extragalactic sources, and
ultimately, evolutionary parameters. Our approach obviates the need to use SED
templates in the millimeter to far-infrared region of the spectrum; this is a
common practice in the extragalactic community that relies on observed
correlations established at low redshift that may not necessarily extend to
high redshift. Further, we demarcate the limited region of parameter space in
which density profiles can be inferred from the SED, which is of particular
import in the protostellar community as competing theories of star formation
are characterized by different density profiles. The functionality of our model
is unique in that in provides for a self-consistent analytic solution that we
have validated by comparison with a well-tested radiative transfer code (DUSTY)
to find excellent agreement with numerical results over a parameter space that
spans low-mass protostars to ultra-luminous infrared galaxies (ULIRGS).
