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.