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# Radiative equilibrium estimates of dust temperature and mass in high-redshift galaxies

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### Abstract

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.

### Author and article information

###### Journal
27 April 2020
###### Article
2004.12612