Shielding Functions

Overview

The routines in shield.F90 compute attenuated photodissociation and photoionization rates for key species in photodissociation regions (PDRs), accounting for:

• Dust extinction and scattering

• Molecular self-shielding

• Mutual shielding by H₂

• Line overlap effects

• Temperature-dependent shielding (for CI)

These functions follow well-established prescriptions from the literature, primarily:

• Federman, Glassgold & Kwan (1979)

• van Dishoeck & Black (1988)

• Lee et al. (1996)

• Kamp & Bertoldi (2000)

• Savage & Mathis (1979)

• Wagenblast & Hartquist (1989)

All rates scale linearly with the incident FUV field \(G_0\) (in Draine units).

General Form of Photoreaction Rates

All photorates computed here follow the generic structure:

\[k = k_0 \, G_0 \, f_{\mathrm{shield}} \, f_{\mathrm{dust}}\]

where:

• \(k_0\) is the unattenuated rate

• \(f_{\mathrm{shield}}\) accounts for line shielding

• \(f_{\mathrm{dust}}\) accounts for dust extinction and scattering

H₂ Photodissociation Rate

H2PDRATE(K0, G0, AV, NH2)

This function computes the H₂ photodissociation rate including self-shielding and dust attenuation.

\[k_{\mathrm{H_2}} = k_0 \, G_0 \, f_{\mathrm{H_2}}(N_{\mathrm{H_2}}) \times f_{\mathrm{dust}}(A_V, \lambda)\]

Self-Shielding

Self-shielding is computed using the Federman et al. (1979) formalism via H2SHIELD1.

The Doppler linewidth is determined by turbulent broadening:

\[\Delta \nu_D = \frac{v_{\mathrm{turb}}}{\lambda}\]

where a representative wavelength of \(\lambda = 1000\,\mathrm{Å}\) is assumed.

CO Photodissociation Rate

COPDRATE(K0, G0, AV, NCO, NH2)

The CO photodissociation rate includes CO self-shielding, H₂ shielding, and dust extinction:

\[k_{\mathrm{CO}} = k_0 \, G_0 \, f_{\mathrm{CO}}(N_{\mathrm{CO}}, N_{\mathrm{H_2}}) \times f_{\mathrm{dust}}(A_V, \bar{\lambda})\]

The mean wavelength \(\bar{\lambda}\) of the dissociating bands is computed using LBAR following van Dishoeck & Black (1988).

CI Photoionization Rate

CIPDRATE(K0, G0, AV, KAV, NCI, NH2, TGAS)

The CI photoionization rate follows Kamp & Bertoldi (2000), Equation (8):

\[k_{\mathrm{CI}} = k_0 \, G_0 \, e^{-\tau_{\mathrm{CI}}}\]

with optical depth:

\[\tau_{\mathrm{CI}} = K_{AV} A_V + 1.1 \times 10^{-17} N_{\mathrm{CI}} + 0.9 \, T_{\mathrm{gas}}^{0.27} \left( \frac{N_{\mathrm{H_2}}}{1.59 \times 10^{21}} \right)^{0.45}\]

This formulation accounts for:

• Dust extinction

• CI line shielding

• Overlapping H₂ absorption lines

• Temperature-dependent shielding effects

SI Photoionization Rate

SIPDRATE(K0, G0, AV, KAV)

Currently implemented as a dust-only attenuated rate:

\[k_{\mathrm{SI}} = k_0 \, G_0 \, e^{-K_{AV} A_V} / 2\]

This routine is a placeholder and does not yet include line shielding.

H₂ Self-Shielding (Analytic)

H2SHIELD1(NH2, DOPW, RADW)

Implements the Federman, Glassgold & Kwan (1979) shielding function:

\[f_{\mathrm{H_2}} = J_D + J_R\]

where:

• \(J_D\) is the Doppler core contribution

• \(J_R\) is the radiative wing contribution

The optical depth at line centre is:

\[\tau_D = N_{\mathrm{H_2}} \, f_{\mathrm{para}} \, \frac{\pi e^2}{m c} \, \frac{f}{\sqrt{\pi} \Delta \nu_D}\]

A fixed ortho–para ratio of 1 is assumed (\(f_{\mathrm{para}} = 0.5\)).

H₂ Shielding (Tabulated)

H2SHIELD2(NH2)

Alternative H₂ shielding based on tabulated values from Lee et al. (1996), Table 10.

• Shielding factors are interpolated using cubic splines.

• Includes shielding by both H₂ and atomic H.

• Used mainly for validation or comparison.

CO Shielding

COSHIELD(NCO, NH2)

Computes combined CO self-shielding and H₂ shielding using two-dimensional spline interpolation over the tables of van Dishoeck & Black (1988), Table 5.

Interpolation is performed in:

\[\log_{10}(N_{\mathrm{CO}}), \quad \log_{10}(N_{\mathrm{H_2}})\]

Dust Scattering Attenuation

SCATTER(AV, LAMBDA)

Computes attenuation due to dust scattering and absorption, following Wagenblast & Hartquist (1989) and Flannery et al. (1980).

The optical depth at wavelength \(\lambda\) is:

\[\tau_\lambda = \frac{A_V}{1.086} \times \frac{\tau(\lambda)}{\tau(V)}\]

The attenuation factor is piecewise:

• For \(\tau_\lambda < 1\):

  \[f = a_0 \, e^{-k_0 \tau_\lambda}\]

• For \(\tau_\lambda \ge 1\):

  \[f = \sum_{i=1}^5 a_i \, e^{-k_i \tau_\lambda}\]

The coefficients assume:

• Dust albedo = 0.3

• Scattering asymmetry parameter = 0.8

Extinction Curve

XLAMBDA(LAMBDA)

Computes the ratio:

\[\frac{\tau(\lambda)}{\tau(V)}\]

using spline interpolation over the extinction curve of Savage & Mathis (1979).

Mean CO Band Wavelength

LBAR(NCO, NH2)

Computes the mean wavelength of the 33 CO-dissociating bands:

\[\bar{\lambda} = (5675 - 200.6 W) - (571.6 - 24.09 W) U + (18.22 - 0.7664 W) U^2\]

where:

\[U = \log_{10}(N_{\mathrm{CO}}), \quad W = \log_{10}(N_{\mathrm{H_2}})\]

The wavelength is constrained to:

\[913.6\,\mathrm{Å} \le \bar{\lambda} \le 1076.1\,\mathrm{Å}\]

Summary

The shielding and photorate routines in shield.F90 provide a physically consistent and literature-calibrated treatment of UV attenuation in PDRs, accounting for:

• Line self-shielding

• Mutual shielding

• Dust extinction and scattering

• Temperature-dependent effects

They form a critical link between radiative transfer, chemistry, and thermal balance in the 3D-PDR framework.