Governing Equations

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

First, referencing §2 of J. C. Hayes et al. (2006, ApJS, 165, 188 - 228) — alternatively see §2.1 of D. C. Marcello & J. E. Tohline (2012, ApJS, 199, 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

 $\nabla^2 \Phi = 4\pi G \rho$

Hayes et al. (2006), p. 190, Eq. (15)

the,

Continuity Equation

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

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

Lagrangian Representation
of the Euler Equation,

 $~\frac{d\vec{v}}{dt}$ $~=$ $~ - \frac{1}{\rho}\nabla P - \nabla \Phi + \frac{1}{\rho}\biggl(\frac{\chi}{c}\biggr) \vec{F} \, ,$

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

 $~\rho \frac{d}{dt} \biggl( \frac{e}{\rho}\biggr) + P\nabla \cdot \vec{v}$ $~=$ $~ c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, ,$ $~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)$ $~=$ $~ - \biggl[ \nabla \cdot \vec{F} + \bold{P}_\mathrm{st}:\nabla{\vec{v}} + c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \biggr] \, ,$

where, in this last expression, $~\bold{P}_\mathrm{st}$ is the radiation stress tensor.

Various Realizations

First Law

By combining the continuity equation with the

First Law of Thermodynamics

 $T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)$

we can write,

 $~\rho T\frac{ds}{dt}$ $~=$ $~ \rho \frac{d\epsilon}{dt} - \frac{P}{\rho} \frac{d\rho}{dt}$ $~=$ $~ \rho \frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} \, .$

Given that the specific internal energy $~(\epsilon)$ and the internal energy density $~(e)$ are related via the expression, $~\epsilon = e/\rho$, we appreciate that the first of the above-identified energy-conservation-based dynamical equations is simply a restatement of the 1st 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,

 $~\rho T \frac{ds}{dt}$ $~=$ $~c\kappa_E E_\mathrm{rad} - 4\pi \kappa_p B_p \, .$

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,

 $~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr)$ $~=$ $~ \frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho}~\frac{d\rho}{dt}$ $~=$ $~ \frac{dE_\mathrm{rad}}{dt} + E_\mathrm{rad}\nabla\cdot \vec{v}$ $~=$ $~ \frac{\partial E_\mathrm{rad}}{\partial t} + \vec{v}\cdot \nabla E_\mathrm{rad}+ E_\mathrm{rad}\nabla\cdot \vec{v}$ $~=$ $~ \frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) \, ,$

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

 $~E_\mathrm{rad}$ $~=$ $~a_\mathrm{rad}T^4 \, ,$

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,

 $~B_p = \frac{\sigma}{\pi}T^4$ $~=$ $~\frac{ca_\mathrm{rad}}{4\pi} T^4 \, ,$

where, $~\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}$, is the Stefan-Boltzmann constant, and the radiation constant — which is included in an associated appendix among our list of key physical constants — is,

 $~a_\mathrm{rad}$ $~\equiv$ $~\frac{8\pi^5}{15}\frac{k^4}{(hc)^3} \, .$

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

 $~ \bold{P}_\mathrm{st} :\nabla{\vec{v}}$ $~\rightarrow$ $~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, ,$

and,

 $~\vec{F}$ $~\rightarrow$ $~- \frac{1}{3}\biggl(\frac{c}{\chi}\biggr) \nabla E_\mathrm{rad} \, ,$

which implies,

 $~\biggl(\frac{\chi}{c}\biggr) \vec{F}$ $~\rightarrow$ $~-\nabla P_\mathrm{rad} \, ,$

where we have recognized that the radiation pressure,

 $~P_\mathrm{rad} = \frac{1}{3}E_\mathrm{rad}$ $~=$ $~\frac{1}{3}a_\mathrm{rad}T^4 \, .$

Hence, the modified Euler equation becomes,

 $~\rho ~ \frac{d\vec{v}}{dt}$ $~=$ $~ - \nabla (P+P_\mathrm{rad}) - \rho \nabla \Phi \, ,$

and the equation governing the time-dependent behavior of $~E_\mathrm{rad}$ becomes,

 $~\frac{\partial E_\mathrm{rad}}{\partial t} + \nabla\cdot (E_\mathrm{rad} \vec{v}) + \frac{1}{3}E_\mathrm{rad} \nabla \cdot \vec{v}$ $~=$ $~ - \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, .$

Optically Thick Regime

In the optically thick regime, the following conditions hold:

 $~c\kappa_E E_\mathrm{rad}$ $~\rightarrow$ $~4\pi \kappa_p B_p \, ,$ $~E_\mathrm{rad}$ $~\rightarrow$ $~aT^4 \, ,$ $~\biggl(\frac{\chi}{c}\biggr) \vec{F}$ $~\rightarrow$ $~- \nabla \biggl(\frac{aT^4}{3} \biggr) \, ,$ $~ \vec{\bold{P}}:\nabla{\vec{v}}$ $~\rightarrow$ $~\frac{E_\mathrm{rad}}{3} \nabla \cdot \vec{v} \, .$

 $~Tds_\mathrm{rad} = dQ$ $~=$ $~ d\biggl(\frac{E_\mathrm{rad}}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)$ $~=$ $~ \frac{1}{\rho}~d E_\mathrm{rad} + E_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr) + P_\mathrm{rad~}d\biggl( \frac{1}{\rho} \biggr)$ $~=$ $~ \frac{1}{\rho}~d (aT^4 ) + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr)$ $~=$ $~ \frac{4aT^3}{\rho}~dT + \frac{4}{3} aT^4~d\biggl( \frac{1}{\rho} \biggr)$ $~=$ $~ \frac{4aT}{3} \biggl[ \frac{3T^2}{\rho}~dT + T^3~d\biggl( \frac{1}{\rho} \biggr) \biggr]$ $~=$ $~ \frac{4aT}{3} ~d\biggl( \frac{T^3}{\rho} \biggr)$ $~\Rightarrow ~~~ ds_\mathrm{rad}$ $~=$ $~ ~d\biggl( \frac{4aT^3}{3\rho} \biggr)$

Integrating then gives us,

 $~s_\mathrm{rad}$ $~=$ $~ ~\frac{4aT^3}{3\rho} + \mathrm{const.}$

D. D. Clayton (1968), Eq. (2-136)
[Shu92], Vol. I, §9, immediately following Eq. (9.22)

This also means that,

 $~\rho \frac{d}{dt} \biggl( \frac{E_\mathrm{rad}}{\rho}\biggr) + \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}$ $~=$ $~ \frac{dE_\mathrm{rad}}{dt} - \frac{E_\mathrm{rad}}{\rho} \frac{d\rho}{dt} + \frac{E_\mathrm{rad}}{3} \nabla\cdot\vec{v}$ $~=$ $~ \frac{dE_\mathrm{rad}}{dt} + \frac{4E_\mathrm{rad}}{3} \nabla\cdot\vec{v}$ $~=$ $~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{3}{4} \cdot \frac{d\ln E_\mathrm{rad}}{dt} + \nabla\cdot\vec{v} \biggr]$ $~=$ $~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln (E_\mathrm{rad})^{3/4}}{dt} + \nabla\cdot\vec{v} \biggr]$ $~=$ $~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln T^3}{dt} - \frac{d\ln\rho}{dt} \biggr]$ $~=$ $~\frac{4E_\mathrm{rad}}{3} \biggl[ \frac{d\ln (T^3/\rho)}{dt} \biggr]$ $~=$ $~\frac{4aT^4}{3} \biggl( \frac{\rho}{T^3}\biggr) \biggl[ \frac{d(T^3/\rho)}{dt} \biggr]$ $~=$ $~ \rho T\biggl[ \frac{ds_\mathrm{rad}}{dt} \biggr] \, .$

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

 $~\rho T~\frac{ds_\mathrm{rad}}{dt}$ $~=$ $~ - \nabla \cdot \vec{F} - c\kappa_E E_\mathrm{rad} + 4\pi \kappa_p B_p \, .$

[Shu92], §9, Eq. (9.22)

Hydrostatic Balance

 $~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}$

Mass Conservation

 $~\frac{dM_r}{dr} = 4\pi r^2 \rho$

Energy Conservation

 $~\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon_\mathrm{nuc}$

 $~\frac{dT}{dr} = - \frac{ 3 }{ 4a_\mathrm{rad} c} \biggl(\frac{ \kappa \rho }{ T^3 }\biggr) \frac{ L_r }{ 4\pi r^2 }$

M. Schwarzschild (1958), Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)
D. D. Clayton (1968), 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)

Related Discussions

 © 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation