Physics of Radiation Transfer
| tau = l \kappa=\frac{l}{\lambda_p
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| beta = \frac{u}{c
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| tau << 1 | streaming limit
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| tau >> 1 \mbox{, } \beta \tau << | static diffusion limit
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| tau >> 1 \mbox{, } \beta \tau >> | dynamic diffusion limit
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Equations of Radiation Hydrodynamics
| frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) =
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| frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P+\mathbf{G
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| frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = c G^
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| frac{\partial E}{\partial t} + \nabla \cdot \mathbf{F} = -c G_0
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| frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = -\mathbf{G
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where
| E=\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu
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| mathbf{F}=\int_0^\infty d \nu \int d \Omega \mathbf{n} I(\mathbf{n}, \nu
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| \mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu
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and
| G^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right
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| \mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ] \mathbf{n
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Simplifying assumptions
- If the flux spectrum of the radiation is direction-independent then we can write the radiation four-force density in terms of the moments of the radiation field
| ^0=\gamma [ \gamma^2 \kappa_{0E} + (1-\gamma^2)\kappa_{0F}]E-\gamma \kappa_{0P}a_RT_0^4-\gamma(\mathbf{v} \cdot \mathbf{F}/c^2)[\kappa_{0F}-2\gamma^2(\kappa_{0F}-\kappa_{0E})] - \gamma^3(\kappa_{0F}-\kappa+{0E})(\mathbf{vv}) : \mathbf{P}/c^
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| mathbf{G} = \gamma \kappa_{0F}(\mathbf{F}/c)-\gamma \kappa_{0P}a_RT_0^4 (\mathbf{v}/c) - \left [ \gamma^3( \kappa_{0F} - \kappa_{0E})( \mathbf{v}/c)E + \gamma \kappa_{0F}(\mathbf{v}/c) \cdot \mathbf{P} \right ] + \gamma^3 ( \kappa_{0F} - \kappa_{0E}) \left [ 2 \mathbf{v} \cdot \mathbf{F}/c^3 - ( \mathbf{vv}) : \mathbf{P}/c^3 \right ] \mathbf{v}
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where
| kappa_{0P | comoving-frame Planck function weighted opacity
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| kappa_{0E | comoving-frame radiation energy weighted opacity
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| kappa_{0F | comoving-frame radiation flux weighted opacity
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- If the radiation has a blackbody spectrum then kappa_{0E}=\kappa_{0P
- If the radiation is optically thick, then
mathbf{F_0}(\nu_0) \propto -\nabla E_0(\nu_0)/\kappa_0(\nu_0) \propto -[\partial B(\nu_0, T_0) / \partial T_0](\nabla T_0)/\kappa_0(\nu_0
which implies that | kappa_{0F}^{-1}=\kappa_{0R}^{-1}=\frac{\int_0^\infty d \nu_0 \kappa_0(\nu_0)^{-1}[\partial B(\nu_0,T_0)/\partial T_0]}{\int_0^\infty d \nu_0 [\partial B(\nu_0, T_0)/\partial T_0]
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- In the optically thin regime, \mathbf{F}_0(\nu_0)| \rightarrow cE_0(\nu_0 so we would have kappa_{0F}=\kappa_{0E however assuming a blackbody temperature in the optically thin limit may be any more accurate than assuming that kappa_{0F}=\kappa_{0R
Flux limited diffusion
The flux limited diffusion approximation drops the radiation momentum equation in favor of
mathbf{F}_0=-\frac{c\lambda}{\kappa_{0R}}\nabla E_
where lambd is the flux-limiter
| lambda = \frac{1}{R} \left ( \coth R - \frac{1}{R} \right )
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| =\frac{|\nabla E_0|}{\kappa_{0R}E_0
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which corresponds to a pressure tensor
| mathbf{P}_0=\frac{E_0}{2}[(1-R_2)\mathbf{I}+(3R_2-1)\mathbf{n}_0\mathbf{n}_0
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| _2=\lambda+\lambda^2R^
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If we Lorentz boost the comoving terms into the lab frame and keep terms necessary to maintain accuracy we get:
| ^0=\kappa_{0P} \left ( E-\frac{4 \pi B}{c} \right ) + \left ( \frac{\lambda}{c} \right ) \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \mathbf{v} \cdot \nabla E - \frac{\kappa_{0P}}{c^2} E \left [ \frac{3-R_2}{2}v^2 + \frac{3R_2-1}{2}(\mathbf{v} \cdot \mathbf{n})^2 \right ] + \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \kappa_{0P} \left ( E - \frac{4 \pi B}{c} \right )
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| mathbf{G} = -\lambda \nabla E + \kappa_{0P} \frac{\mathbf{v}}{c} \left ( E - \frac{4 \pi B}{c} \right ) - \frac{1}{2} \left ( \frac{v}{c} \right ) ^2 \lambda \nabla E + 2 \lambda \left ( \frac{\kappa_{0P}}{\kappa_{0R}} - 1 \right ) \frac{(\mathbf{v} \cdot \nabla E )\mathbf{v}}{c^2
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which if we plug back into the gas equations and keep terms necessary to maintain accuracy we get:
| frac{\partial }{\partial t} \left ( \rho \mathbf{v} \right ) + \nabla \cdot \left ( \rho \mathbf{vv} \right ) = -\nabla P-\lambda \nabla
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| frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = -\kappa_{0P}(4 \pi B-cE) + \lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E - \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}
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| frac{\partial E}{\partial t} - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E = \kappa_{0P} (4 \pi B-cE) - \lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E + \frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )
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