Dust Temperature Treatment

This file documents the calculation of the dust temperature in the CALCULATE_DUST_TEMPERATURES subroutine of the 3D-PDR code.

The dust temperature is computed self-consistently for each grid particle by balancing heating from the local far-ultraviolet (FUV) radiation field, reprocessed infrared (IR) radiation, and the cosmic microwave background (CMB). The adopted formulation is based primarily on the work of Hollenbach, Takahashi & Tielens (1991; hereafter “HTT91”), with important modifications to account for attenuation of the re-emitted IR radiation into the cloud.

Physical Motivation

The dust temperature plays a central role in the thermal and chemical evolution of photodissociation regions. In 3D-PDR it affects:

  1. Gas cooling via far-infrared dust emission coupled to line radiative transfer;

  2. Gas–grain collisional heating or cooling;

  3. H2 formation through the temperature-dependent sticking probability;

  4. Freeze-out, evaporation, and surface chemistry on grains.

Accurate modelling of the dust temperature is therefore essential for a realistic description of both the gas energetics and the chemistry.

Adopted Formalism

The dust temperature calculation follows the treatment of HTT91, who derive analytic expressions for dust heating by FUV photons.

For each particle, an initial contribution to the dust emission is computed from:

  • the local isotropic FUV radiation field, and

  • the cosmic microwave background (CMB), included as a temperature floor.

This is expressed as a sum of fifth powers of temperature, consistent with radiative equilibrium of dust grains.

Directional FUV Heating

The directional dependence of the radiation field is explicitly taken into account by looping over all rays associated with a particle.

For each ray:

  • The incident FUV field at the cloud surface is converted from Draine units to Habing units by a factor of 1.71.

  • A characteristic surface dust temperature \(T_0\) is computed as

    \[T_0 = 12.2 \, G_0^{0.2}\]

    where \(G_0\) is the local FUV field in Habing units.

  • For rays with visual extinction \(A_V > 1\), the temperature is attenuated according to

    \[T_0 \propto A_V^{-0.4}\]

This prescription follows the infrared-only dust temperature approximation of Rowan-Robinson (1980).

The contribution from each ray is added to the total dust emission using the modified HTT91 formulation, which depends on the effective dust optical depth at 100 μm.

Infrared Attenuation

In the original HTT91 treatment, the infrared radiation re-emitted by dust at the cloud surface is not attenuated as it propagates deeper into the cloud. This leads to unrealistically high dust (and therefore gas) temperatures at large depths for high-density, high-FUV models.

To address this issue, 3D-PDR introduces attenuation of the reprocessed IR radiation following Rowan-Robinson (1980). The dust temperature decreases with depth as

\[T_{\rm dust} \propto r^{-0.4}\]

where \(r\) corresponds to the depth into the cloud, normalized such that \(A_V \simeq 1\) marks the outer FUV-processed layer.

This modification prevents excessive heating deep inside the cloud and yields more realistic dust and gas temperatures at high extinctions.

Final Dust Temperature

After summing all directional and isotropic contributions, the dust temperature is obtained by converting the total emission intensity to a temperature:

\[T_{\rm dust} = I^{1/5}\]

where \(I\) is the accumulated emission term.

Temperature Limits

Two safeguards are applied:

  • Lower limit: The dust temperature is not allowed to fall below 10 K. Lower values would strongly suppress H2 formation by preventing molecule desorption from grain surfaces.

  • Upper limit: If the computed dust temperature exceeds 1000 K, the code terminates, as such values are considered unphysical in the context of PDR modelling.

Summary

The dust temperature treatment in 3D-PDR:

  • Accounts for both isotropic and directional FUV heating;

  • Includes heating from the CMB;

  • Incorporates attenuation of reprocessed infrared radiation;

  • Avoids unphysical dust and gas temperatures at large cloud depths;

  • Provides a physically motivated and numerically stable dust temperature for use in the thermal and chemical balance.