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\color{green}{-\lambda \nabla E
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frac{\partial e}{\partial t} + \nabla \cdot \left [ \left ( e + P \right ) \mathbf{v} \right ] = \color{red}{-\kappa_{0P}(4 \pi B-cE)} \color{green}{+\lambda \left ( 2 \frac{\kappa_{0P}}{\kappa_{0R}}-1 \right ) \mathbf{v} \cdot \nabla E} \color{blue}{-\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E
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frac{\partial E}{\partial t} \color{red}{ - \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E} = \color{red}{\kappa_{0P} (4 \pi B-cE)} \color{green}{-\lambda \left(2\frac{\kappa_{0P}}{\kappa_{0R}}-1 \right )\mathbf{v}\cdot \nabla E} \color{green}{-\nabla \cdot \left ( \frac{3-R_2}{2}\mathbf{v}E \right )} \color{blue}{+\frac{3-R_2}{2}\kappa_{0P}\frac{v^2}{c}E}
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For static diffusion, the terms in blue with v2/c can be dropped and the system can be split into the usual hydro update (black), radiative source terms (green) using time centered radiation energy, and a coupled implicit solve (red) for the radiation energy density and thermal energy density (ie temperature). If the opacity is not temperature dependent, then the implicit solve only involves the radiation energy density. If the opacity is not a linear function of the energy, then some form of sub-cycling would be required.
For now we will assume that kappa_{0P and kappa_{0R are constant over the implicit update. In this case we can solve the radiation energy equation:
frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B-cE