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Radiation-Hydrodynamics

Whitworth's (1981) Isothermal Free-Energy Surface
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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 \, .

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} \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} \, .

Start with,

~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], §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)

Traditional Stellar Structure Equations

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}


Radiation Transport

~\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)


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Whitworth's (1981) Isothermal Free-Energy Surface

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