Radiative equilibrium estimates of dust temperature and mass in
  high-redshift galaxies
  Estimating the temperature and mass of dust in high-$z$ galaxies is essential
for discussions of the origin of dust in the early Universe. However, this
suffers from limited sampling of the infrared spectral-energy distribution.
Here we present an algorithm for deriving the temperature and mass of dust in a
galaxy, assuming dust to be in radiative equilibrium. We formulate the
algorithm for three geometries: a thin spherical shell, a homogeneous sphere,
and a clumpy sphere. We also discuss effects of the mass absorption
coefficients of dust at ultraviolet and infrared wavelengths, $\kappa_{\rm UV}$
and $\kappa_{\rm IR}$, respectively. As an example, we apply the algorithm to a
normal, dusty star-forming galaxy at $z=7.5$, A1689zD1, for which three data
points in the dust continuum are available. Using $\kappa_{\rm
UV}=5.0\times10^4$ cm$^2$ g$^{-1}$ and $\kappa_{\rm IR}=30(\lambda/100\mu
m)^{-\beta}$ cm$^2$ g$^{-1}$ with $\beta=2.0$, we obtain dust temperatures of
38--70~K and masses of $10^{6.5-7.3}$ M$_\odot$ for the three geometries
considered. We obtain similar temperatures and masses from just a single data
point in the dust continuum, suggesting the usefulness of the algorithm for
high-$z$ galaxies with limited infrared observations. In the clumpy-sphere
case, the temperature becomes equal to that of the usual modified black-body
fit, because an additional parameter describing the clumpiness works as an
adjuster. The best-fit clumpiness parameter is $\xi_{\rm cl}=0.1$,
corresponding to $\sim10$\% of the volume filling factor of the clumps in this
high-$z$ galaxy if the clump size is $\sim10$ pc, similar to that of giant
molecular clouds in the local Universe.
