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.

中文翻译：

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高红移星系中尘埃温度和质量的辐射平衡估计

估计高 $z$ 星系中尘埃的温度和质量对于讨论早期宇宙中尘埃的起源至关重要。然而，这受到红外光谱能量分布的有限采样的影响。在这里，我们提出了一种推导星系中尘埃温度和质量的算法，假设尘埃处于辐射平衡。我们为三种几何体制定了算法：薄球壳、均质球体和块状球体。我们还分别讨论了灰尘在紫外线和红外线波长下的质量吸收系数 $\kappa_{\rm UV}$ 和 $\kappa_{\rm IR}$ 的影响。例如，我们将算法应用于 $z=7.5$ 处的一个正常的、多尘的恒星形成星系 A1689zD1，其中尘埃连续体中的三个数据点可用。使用 $\kappa_{\rm UV}=5。0\times10^4$ cm$^2$ g$^{-1}$ 和 $\kappa_{\rm IR}=30(\lambda/100\mu m)^{-\beta}$ cm$^2 $ g$^{-1}$ 和 $\beta=2.0$，我们得到了 38--70~K 的尘埃温度和 $10^{6.5-7.3}$ M$_\odot$ 考虑的三种几何形状. 我们仅从尘埃连续体中的单个数据点获得了相似的温度和质量，这表明该算法对于红外观测有限的高 z 星系很有用。在块状球体的情况下，温度变得等于通常修改的黑体拟合的温度，因为描述块状的附加参数用作调节器。最适合的团块参数为 $\xi_{\rm cl}=0.1$，如果团块大小为 $\，则对应于这个高 $z$ 星系中团块的体积填充因子的 $\sim10$\% sim10$ pc，类似于本地宇宙中的巨型分子云。