User:Tohline/Appendix/Ramblings/RadiationHydro
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RadiationHydrodynamics
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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 selfgravitating fluid flows includes the,
the,
and — ignoring magnetic fields — a modified version of the,
Lagrangian Representation
of the Euler Equation,



plus the following pair of additional energyconservationbased dynamical equations:






where, in this last expression, is the radiation stress tensor.
Various Realizations
First Law
By combining the continuity equation with the
we can write,






Given that the specific internal energy and the internal energy density are related via the expression, , we appreciate that the first of the aboveidentified energyconservationbased 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 (radiationtransportrelated) source and sink terms,



EnergyDensity of Radiation Field
By combining the lefthand side of the second of the aboveidentified energyconservationbased dynamical equations with the continuity equation, then replacing the Lagrangian (that is, the material) time derivative by its Eulerian counterpart, the lefthand side can be rewritten as,












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, , the energydensity of the radiation field is,



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,



where, , is the StefanBoltzmann constant, and the radiation constant — which is included in an associated appendix among our list of key physical constants — is,



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) —



and,



which implies,



where we have recognized that the radiation pressure,



Hence, the modified Euler equation becomes,



and the equation governing the timedependent behavior of becomes,



Optically Thick Regime
In the optically thick regime, the following conditions hold:












Start with,





















Integrating then gives us,



D. D. Clayton (1968), Eq. (2136)
[Shu92], §9, immediately following Eq. (9.22)
This also means that,
























Hence, the equation governing the timedependent behavior of becomes an expression detailing the timedependent behavior of the specific entropy, namely,



[Shu92], §9, Eq. (9.22)
Traditional Stellar Structure Equations
Hydrostatic Balance

Mass Conservation

Energy Conservation

Radiation Transport

M. Schwarzschild (1958), Chapter III, §12, Eqs. (12.1), (12.2), (12.3), (12.4)
D. D. Clayton (1968), Chapter 6, Eqs. (61), (62), (63a), (64a)
[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
 Euler equation viewed from a rotating frame of reference.
 An earlier draft of this "Euler equation" presentation.
© 2014  2019 by Joel E. Tohline 