Appendix/Ramblings/Radiation/RadHydro

Radiation-Hydrodynamics

Governing Equations

Hayes et al. (2006) — But Ignoring the Effects of Magnetic Fields

First, referencing §2 of 📚 J. C. Hayes, M. L. Norman, R. A. Fiedler, J. O. Bordner, P. S. Li, S. E. Clark, A. ud-Doula, & M.-M. Mac Low (2006, ApJ Suppl., Vol. 165, Issue 1, pp. 188 - 228) — alternatively see §2.1 of 📚 D. C. Marcello & J. E. Tohline (2012, ApJ Suppl., Vol. 199, Issue 2, article id. 35, 29 pp.) — we see that the set of principal governing equations that is typically used in the astrophysics community to include the effects of radiation on self-gravitating fluid flows includes the,

Poisson Equation

${\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho$

📚 ZEUS-MP (2006), p. 190, Eq. (15)

the,

Continuity Equation

$\frac{d\rho }{dt}+\rho \mathrm{\nabla }\cdot \overrightarrow{v}=0$

and — ignoring magnetic fields — a modified version of the,

Lagrangian Representation
of the Euler Equation,

$\frac{d\overrightarrow{v}}{dt}$

$=$

$-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }+\frac{1}{\rho }\left(\frac{\chi }{c}\right)\overrightarrow{F},$

plus the following pair of additional energy-conservation-based dynamical equations:

$\rho \frac{d}{dt}\left(\frac{e}{\rho }\right)+P\mathrm{\nabla }\cdot \overrightarrow{v}$	$=$	$c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}-4\pi {\kappa }_p{B}_p,$
$\rho \frac{d}{dt}\left(\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\right)$	$=$	$-[\mathrm{\nabla }\cdot \overrightarrow{F}+{\mathbf{𝐏}}_{\mathrm{s}\mathrm{t}}:\mathrm{\nabla }\overrightarrow{v}+c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}-4\pi {\kappa }_p{B}_p],$

where, in this last expression, ${\mathbf{𝐏}}_{\mathrm{s}\mathrm{t}}$ is the radiation stress tensor.

Various Realizations

First Law

By combining the continuity equation with the

First Law of Thermodynamics

$T\frac{ds}{dt}=\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)$

we can write,

$\rho T\frac{ds}{dt}$	$=$	$\rho \frac{dϵ}{dt}-\frac{P}{\rho }\frac{d\rho }{dt}$
	$=$	$\rho \frac{dϵ}{dt}+P\mathrm{\nabla }\cdot \overrightarrow{v}.$

Given that the specific internal energy $(ϵ)$ and the internal energy density $(e)$ are related via the expression, $ϵ=e/\rho$, we appreciate that the first of the above-identified energy-conservation-based dynamical equations is simply a restatement of the 1^st Law of Thermodynamics in the context of a physical system whose fluid elements gain or lose entropy as a result of the (radiation-transport-related) source and sink terms,

$\rho T\frac{ds}{dt}$

$=$

$c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}-4\pi {\kappa }_p{B}_p.$

Energy-Density of Radiation Field

By combining the left-hand side of the second of the above-identified energy-conservation-based dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the material) time derivative by its Eulerian counterpart, the left-hand side can be rewritten as,

$\rho \frac{d}{dt}\left(\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\right)$	$=$	$\frac{d{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}-\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\frac{d\rho }{dt}$
	$=$	$\frac{d{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}+E_{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{\nabla }\cdot \overrightarrow{v}$
	$=$	$\frac{\partial E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\partial t}+\overrightarrow{v}\cdot \mathrm{\nabla }{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}+E_{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{\nabla }\cdot \overrightarrow{v}$
	$=$	$\frac{\partial E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\partial t}+\mathrm{\nabla }\cdot (E_{\mathrm{r}\mathrm{a}\mathrm{d}}\overrightarrow{v}),$

which provides an alternate form of the expression, as found for example in equation (4) of 📚 Marcello & Tohline (2012).

Thermodynamic Equilibrium

In an optically thick environment that is in thermodynamic equilibrium at temperature, $T$, the energy-density of the radiation field is,

$E_{\mathrm{r}\mathrm{a}\mathrm{d}}$

$=$

$a_{\mathrm{r}\mathrm{a}\mathrm{d}}{T}^4,$

and each fluid element will radiate — and, hence lose some of its internal energy to the surrounding radiation field — at a rate that is governed by the integrated Planck function,

$B_p=\frac{\sigma }{\pi }{T}^4$

$=$

$\frac{c{a}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{4\pi }{T}^4,$

where, $\sigma \equiv \frac{1}{4}c{a}_{\mathrm{r}\mathrm{a}\mathrm{d}}$, is the Stefan-Boltzmann constant, and the radiation constant — which is included in an associated appendix among our list of key physical constants — is,

$a_{\mathrm{r}\mathrm{a}\mathrm{d}}$

$\equiv$

$\frac{8{\pi }^5}{15}\frac{k^4}{(hc{)}^3}.$

Also under these conditions, it can be shown that — see, for example, discussion associated with equations (12) and (18) in 📚 Marcello & Tohline (2012) —

${\mathbf{𝐏}}_{\mathrm{s}\mathrm{t}}:\mathrm{\nabla }\overrightarrow{v}$

$\to$

$\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\mathrm{\nabla }\cdot \overrightarrow{v},$

and,

$\overrightarrow{F}$

$\to$

$-\frac{1}{3}\left(\frac{c}{\chi }\right)\mathrm{\nabla }{E}_{\mathrm{r}\mathrm{a}\mathrm{d}},$

which implies,

$\left(\frac{\chi }{c}\right)\overrightarrow{F}$

$\to$

$-\mathrm{\nabla }{P}_{\mathrm{r}\mathrm{a}\mathrm{d}},$

where we have recognized that the radiation pressure,

$P_{\mathrm{r}\mathrm{a}\mathrm{d}}=\frac{1}{3}{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}$

$=$

$\frac{1}{3}{a}_{\mathrm{r}\mathrm{a}\mathrm{d}}{T}^4.$

Hence, the modified Euler equation becomes,

$\rho \frac{d\overrightarrow{v}}{dt}$

$=$

$-\mathrm{\nabla }(P+P_{\mathrm{r}\mathrm{a}\mathrm{d}})-\rho \mathrm{\nabla }\mathrm{\Phi },$

and the equation governing the time-dependent behavior of $E_{\mathrm{r}\mathrm{a}\mathrm{d}}$ becomes,

$\frac{\partial E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\partial t}+\mathrm{\nabla }\cdot (E_{\mathrm{r}\mathrm{a}\mathrm{d}}\overrightarrow{v})+\frac{1}{3}{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{\nabla }\cdot \overrightarrow{v}$

$=$

$-\mathrm{\nabla }\cdot \overrightarrow{F}-c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}+4\pi {\kappa }_p{B}_p.$

Optically Thick Regime

In the optically thick regime, the following conditions hold:

$c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}$	$\to$	$4\pi {\kappa }_p{B}_p,$
$E_{\mathrm{r}\mathrm{a}\mathrm{d}}$	$\to$	$a{T}^4,$
$\left(\frac{\chi }{c}\right)\overrightarrow{F}$	$\to$	$-\mathrm{\nabla }\left(\frac{a{T}^4}{3}\right),$
$\overrightarrow{\mathbf{𝐏}}:\mathrm{\nabla }\overrightarrow{v}$	$\to$	$\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\mathrm{\nabla }\cdot \overrightarrow{v}.$

Start with,

$Td{s}_{\mathrm{r}\mathrm{a}\mathrm{d}}=dQ$	$=$	$d\left(\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\right)+P_{\mathrm{r}\mathrm{a}\mathrm{d}}d\left(\frac{1}{\rho }\right)$
	$=$	$\frac{1}{\rho }d{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}+E_{\mathrm{r}\mathrm{a}\mathrm{d}}d\left(\frac{1}{\rho }\right)+P_{\mathrm{r}\mathrm{a}\mathrm{d}}d\left(\frac{1}{\rho }\right)$
	$=$	$\frac{1}{\rho }d(a{T}^4)+\frac{4}{3}a{T}^{4}d\left(\frac{1}{\rho }\right)$
	$=$	$\frac{4a{T}^3}{\rho }dT+\frac{4}{3}a{T}^{4}d\left(\frac{1}{\rho }\right)$
	$=$	$\frac{4aT}{3}[\frac{3T^2}{\rho }dT+T^{3}d(\frac{1}{\rho }\left)\right]$
	$=$	$\frac{4aT}{3}d\left(\frac{T^3}{\rho }\right)$
$\Rightarrow d{s}_{\mathrm{r}\mathrm{a}\mathrm{d}}$	$=$	$d\left(\frac{4a{T}^3}{3\rho }\right)$

Integrating then gives us,

$s_{\mathrm{r}\mathrm{a}\mathrm{d}}$

$=$

$\frac{4a{T}^3}{3\rho }+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}$

[Clayton68], Eq. (2-136)
[Shu92], Vol. I, §9, immediately following Eq. (9.22)

This also means that,

$\rho \frac{d}{dt}\left(\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\right)+\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\mathrm{\nabla }\cdot \overrightarrow{v}$	$=$	$\frac{d{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}-\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{\rho }\frac{d\rho }{dt}+\frac{E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\mathrm{\nabla }\cdot \overrightarrow{v}$
	$=$	$\frac{d{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}+\frac{4E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\mathrm{\nabla }\cdot \overrightarrow{v}$
	$=$	$\frac{4E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}[\frac{3}{4}\cdot \frac{d\ln E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}+\mathrm{\nabla }\cdot \overrightarrow{v}]$
	$=$	$\frac{4E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}[\frac{d\ln (E_{\mathrm{r}\mathrm{a}\mathrm{d}}{)}^{3/4}}{dt}+\mathrm{\nabla }\cdot \overrightarrow{v}]$
	$=$	$\frac{4E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}[\frac{d\ln T^3}{dt}-\frac{d\ln \rho }{dt}]$
	$=$	$\frac{4E_{\mathrm{r}\mathrm{a}\mathrm{d}}}{3}\left[\frac{d\ln (T^3/\rho )}{dt}\right]$
	$=$	$\frac{4a{T}^4}{3}\left(\frac{\rho }{T^3}\right)\left[\frac{d(T^3/\rho )}{dt}\right]$
	$=$	$\rho T\left[\frac{d{s}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}\right].$

Hence, the equation governing the time-dependent behavior of $E_{\mathrm{r}\mathrm{a}\mathrm{d}}$ becomes an expression detailing the time-dependent behavior of the specific entropy, namely,

$\rho T\frac{d{s}_{\mathrm{r}\mathrm{a}\mathrm{d}}}{dt}$

$=$

$-\mathrm{\nabla }\cdot \overrightarrow{F}-c{\kappa }_E{E}_{\mathrm{r}\mathrm{a}\mathrm{d}}+4\pi {\kappa }_p{B}_p.$

[Shu92], §9, Eq. (9.22)

Traditional Stellar Structure Equations

Hydrostatic Balance

$\frac{dP}{dr}=-\frac{G{M}_r\rho }{r^2}$

Mass Conservation

$\frac{d{M}_r}{dr}=4\pi r^2\rho$

Energy Conservation

$\frac{d{L}_r}{dr}=4\pi r^2\rho {ϵ}_{\mathrm{n}\mathrm{u}\mathrm{c}}$

Radiation Transport

$\frac{dT}{dr}=-\frac{3}{4a_{\mathrm{r}\mathrm{a}\mathrm{d}}c}\left(\frac{\kappa \rho }{T^3}\right)\frac{L_r}{4\pi r^2}$

M. Schwarzschild (1958), Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)
[Clayton68], Chapter 6, Eqs. (6-1), (6-2), (6-3a), (6-4a)
[HK94], Eqs. (1.5), (1.1), (1.54), (1.57)
[KW94], Eqs. (1.2), (2.4), (4.22), (5.11)
W. K. Rose (1998), Eqs. (2.27), (2.28), (2.xx), (2.80)
[P00], Vol. II, Eqs. (2.1), (2.2), (2.18), (2.8)
A. R. Choudhuri (2010), Chapter 3, Eqs. (3.2), (3.1), (3.15), (3.16)
D. Maoz (2016), §3.5, Eqs. (3.56), (3.57), (3.59), (3.58)

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