&= 2.4 \times 10^5 \mathrm{\; s} \left(\frac{T_\mathrm{gas}}{100 \mathrm{\; K}} \right) \left(\frac{T_\mathrm{rad}}{100 \mathrm{\; K}} \right)^{-4} \left(\frac{\kappa}{10 \mathrm{\; cm}^2 / \mathrm{g}} \right)^{-1} \;,\end{aligned}$$ which is also much shorter than the dynamical time.

In this case, the condition of radiative equilibrium allows us to compute the dust temperatures by balancing radiative heating and cooling, $$\begin{aligned}
&4 \pi \int_\lambda { \rho \,  \kappa_\mathrm{abs} (\lambda) \, B_\lambda (T_\mathrm{dust}) \, d\lambda} \nonumber \\ &= \int_{\omega,\lambda}{\rho \,\kappa_\mathrm{abs} (\lambda) I_\lambda \, d\omega \, d\lambda} \;.\end{aligned}$$

For most calculations, the photons are emitted isotropically at the edge of the 0.1 pc sphere surrounding the origin. The effect of anisotropic emission is treated in section \[AnisotropicResults\]. We follow the photon propagation for time intervals of $5 \times 10^9$ seconds, at which point we update the temperature of the dust in each grid zone. We treat dust as present everywhere where the dust temperature is below 1400 Kelvin. The dust temperatures are updated until convergence is obtained at the one percent level, which typically takes fewer than 40 iterations if the initial dust temperature is set to 100 Kelvin in every zone.

Finally, for estimating the dynamics of the gas based on the radiation pressure on the dust, we assume perfect 