an approximate frequency-integrated transfer equation

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We look for a frequency-integrated transfer equation such that it reduces to a desired set of frequency-integrated moments. This is the form of the frequency-integrated transfer equation used by the Athena++ code. See also Foundations of Radiation Hydrodynamics by Mihalas & Mihalas section 82 for related ideas.

0.1 Definitions

The specific energy density of radiation \(E_\nu \), the radiative energy flux \(\vec {F}_\nu \), and radiation pressure tensor \(\mathbf {P}_\nu \), whose components are given by \begin {align} &J_\nu = \frac {1}{4\pi } \int I_\nu (\hat {n}) d\Omega && E_{\nu } = \frac {1}{c}\int I_\nu (\hat {n}) d\Omega = \frac {4\pi J_\nu }{c}\\ &H^{i}_\nu = \frac {1}{4\pi } \int \mu _{i}I_\nu (\hat {n}) d\Omega && F^i_{\nu } = \int \mu _{i}I_\nu (\hat {n}) d\Omega = 4\pi H^i_\nu \\ &K^{ij}_\nu = \frac {1}{4\pi } \int \mu _i\mu _j I_\nu (\hat {n}) d\Omega && P^{ij}_{\nu } = \frac {1}{c} \int \mu _i\mu _j I_\nu (\hat {n}) d\Omega = \frac {4\pi K^{ij}_\nu }{c} \end {align}

where \(\mu _i = \hat {n}\cdot \hat {x}_i\), with \(\hat {x}_i\) some basis vector. Frequency-integrated quantities will be denoted by dropping the \(\nu \) subscript, \(B \equiv \int B_\nu d\nu = \sigma T^4/\pi \) for example. Furthermore define Planck opacity \begin {equation} \kappa _p \equiv \frac {1}{B}\int \kappa _{a,\nu } B_\nu d\nu \end {equation} and Rosseland mean opacities \begin {equation} \kappa _s \equiv \left (\frac {\int \partial B/\partial T d\nu }{\int \frac {\partial B/\partial T}{\kappa _{s,\nu }}d\nu }\right )^{-1} \end {equation} \begin {equation} \kappa _a \equiv \left (\frac {\int \partial B/\partial T d\nu }{\int \frac {\partial B/\partial T}{\kappa _{a,\nu }}d\nu }\right )^{-1} \end {equation}

0.2 Moments of the transfer equation

The time-dependent transfer equation is \begin {equation} \frac {1}{c}\frac {\partial I_\nu }{\partial t} + \hat {n} \cdot \nabla I_\nu = \left (\sigma _{a,\nu } + \sigma _{s,\nu }\right ) \left (S_\nu - I_\nu \right ) \end {equation} We can plug in the source function \(S_\nu \) for isotropic scattering, \begin {equation} S_\nu = \frac {\sigma _{a,\nu }B_\nu }{\sigma _{a,\nu } + \sigma _{s,\nu }} + \frac {\sigma _{s,\nu }J_\nu }{\sigma _{a,\nu } + \sigma _{s,\nu }} \end {equation} to get \begin {equation} \frac {1}{c}\frac {\partial I_\nu }{\partial t} + \hat {n} \cdot \nabla I_\nu = \sigma _{a,\nu } B_\nu + \sigma _{s,\nu } J_\nu -\left (\sigma _{a,\nu } + \sigma _{s,\nu }\right ) I_\nu \end {equation}

0.2.1 Zeroth moment transfer equation

We can take the zeroth moment of this equation, \begin {equation} \underbrace {\int \frac {1}{c}\frac {\partial I_\nu }{\partial t} d\Omega }_{\frac {4\pi }{c}\frac {\partial J_\nu }{\partial t}}+\underbrace {\int \hat {n} \cdot \nabla I_\nu d\Omega }_{\nabla \cdot \vec {F}_\nu }= \underbrace {\int \sigma _{a,\nu } B_\nu d\Omega }_{4\pi \sigma _{a,\nu } B_\nu } + \underbrace {\int \sigma _{s,\nu } J_\nu d\Omega }_{4\pi \sigma _{s,\nu }J_\nu }- \underbrace {\int \left (\sigma _{a,\nu } + \sigma _{s,\nu }\right ) I_\nu d\Omega }_{4\pi \left (\sigma _{a,\nu } + \sigma _{s,\nu }\right )J_\nu } \end {equation} to get the frequency-dependent zeroth moment transfer equation \begin {equation} \label {eq:0nu} \frac {\partial E_\nu }{\partial t} + \nabla \cdot \vec {F}_\nu = 4\pi \sigma _{a,\nu } (B_\nu - J_\nu ) \end {equation} We can try to get a frequency-integrated form by integrating each term over frequency: \begin {equation} \int \frac {\partial E_\nu }{\partial t} d\nu +\int \nabla \cdot \vec {F}_\nu d\nu = \int 4\pi \sigma _{a,\nu } (B_\nu - J_\nu ) d\nu \end {equation} because \(B_\nu \) is a known function, we can take care of the \(B_\nu \) integral by leveraging the Planck opacity, but the \(J_\nu \) term can’t be integrated over without solving the full frequency-dependent transfer. \begin {equation} \frac {\partial E}{\partial t} +\nabla \cdot \vec {F}= 4\pi \sigma _p B -4\pi \int \sigma _{a,\nu } J_\nu d\nu \end {equation} But maybe we can approximate \(\sigma _{a,\nu }\) with a frequency-integrated opacity that isn’t so bad. To figure out what frequency-integrated opacity we should use we look at the frequency-dependent zeroth moment transfer equation (8) and its limiting behavior. At high optical depth \(I_\nu \rightarrow B_\nu \) and the source term \(4\pi \sigma _{a,\nu }(B_\nu - J_\nu )\rightarrow 0\) and the precise value of \(\sigma _{a,\nu }\) is not so important. At low optical depth however, \(B_\nu \) and \(J_\nu \) can be very different. To maximize correctness in all regimes then we ought to use a frequency-integrated opacity which is more concerned with getting the right absorption in the optically thin regime i.e. the Planck opacity \(\sigma _p\equiv \rho \kappa _p\). With this in mind, our approximately correct frequency-integrated zeroth moment equation is: \begin {equation} \boxed { \frac {\partial E}{\partial t} +\nabla \cdot \vec {F}= 4\pi \sigma _p (B-J) } \end {equation}

0.2.2 First moment transfer equation

We can also take the first moment of the transfer equation \begin {equation} \begin {split} &\underbrace {\int \frac {\mu _i}{c}\frac {\partial I_\nu }{\partial t} d\Omega }_{\frac {4\pi }{c}\frac {\partial H^i_\nu }{\partial t}}+\underbrace {\int \mu _i\hat {n} \cdot \nabla I_\nu d\Omega }_{4\pi \nabla _j \cdot \mathbf {K^{ij}}_\nu }\\ &= \underbrace {\int \mu _i\sigma _{a,\nu } B_\nu d\Omega }_{0} + \underbrace {\int \mu _i\sigma _{s,\nu } J_\nu d\Omega }_{0}- \underbrace {\int \mu _i\left (\sigma _{a,\nu } + \sigma _{s,\nu }\right ) I_\nu d\Omega }_{4\pi \left (\sigma _{a,\nu } + \sigma _{s,\nu }\right )H^i_\nu } \end {split} \end {equation} to get a vector equation for the frequency-dependent first order transfer equation, \begin {equation} \frac {1}{c}\frac {\partial \vec {F}_\nu }{\partial t} + c\nabla \cdot \mathbf {P_\nu } =- (\sigma _{a,\nu } + \sigma _{s,\nu }) \vec {F}_\nu \end {equation} We again attempt to frequency integrate this \begin {equation} \int \frac {1}{c}\frac {\partial \vec {F}_\nu }{\partial t}d\nu + \int c\nabla \cdot \mathbf {P_\nu }d\nu =-\int (\sigma _{a,\nu } + \sigma _{s,\nu }) \vec {F}_\nu d\nu \end {equation} but are stuck with the source term because we don’t a priori know the form of \(F_\nu \) \begin {equation} \frac {1}{c}\frac {\partial \vec {F}}{\partial t} + c\nabla \cdot \mathbf {P} =-\int (\sigma _{a,\nu } + \sigma _{s,\nu }) \vec {F}_\nu d\nu \end {equation} We opt for opacities that recover the appropriate fluxes in the diffusion limit by taking Rosseland mean opacities \(\sigma _a \equiv \rho \kappa _a\) and \(\sigma _s \equiv \rho \kappa _s\). (Note: it is not entirely precise to use the split scattering and absorption Rosseland means when typically the the Rosseland mean includes both scattering and absorption, i.e. \(1/\sigma _a + 1/\sigma _s \neq 1/(\sigma _a + \sigma _s)\)) In any case we make the first-moment frequency integrated equation to look like \begin {equation} \boxed { \frac {1}{c}\frac {\partial \vec {F}}{\partial t} + c\nabla \cdot \mathbf {P} = -(\sigma _a + \sigma _s)\vec {F}} \end {equation}

0.3 Frequency-integrated form of transfer equation

We now look for a frequency integrated form of \begin {equation} \frac {1}{c}\frac {\partial I_\nu }{\partial t} + \hat {n} \cdot \nabla I_\nu = \sigma _{a,\nu } B_\nu + \sigma _{s,\nu } J_\nu -\left (\sigma _{a,\nu } + \sigma _{s,\nu }\right ) I_\nu \end {equation} such that two boxed equations are recovered when taking moments. We can supply our frequency integrated transfer equation with arbitrary coefficients \((X,Y,Z)\) and then take moments to figure out the required coefficients. \begin {equation} \frac {1}{c}\frac {\partial I}{\partial t} + \hat {n} \cdot \nabla I = XB + Y J + Z I \end {equation} Frequency Integrated Zeroth Moment: \begin {equation} \frac {\partial E}{\partial t} + \nabla \cdot \vec {F} = XB + (Y + Z ) J \end {equation} Frequency Integrated First Moment: \begin {equation} \frac {1}{c}\frac {\partial \vec {F}}{\partial t} + c\nabla \cdot \mathbf {P} = Z\vec {F} \end {equation} which upon comparing with the boxed moment equations yields the conditions: \begin {align*} X &= \sigma _p\\ Y+Z &= -\sigma _p\\ Z &= -\left (\sigma _a+\sigma _s\right ) \end {align*}

so that the frequency integrated transfer equation ought to be: \begin {equation} \boxed {\frac {1}{c}\frac {\partial I}{\partial t} + \hat {n} \cdot \nabla I = \sigma _pB + (\sigma _a + \sigma _s - \sigma _p) J_\nu -\left (\sigma _a +\sigma _s\right ) I_\nu } \end {equation} if we would like it to recover the frequency-integrated zeroth and first moments given above upon taking appropriate integrals.