Editing SSC/Structure/BiPolytropes/Analytic1.53/Pt2 (section)

===Handling Radiation Transport===

Here we begin with the [[PGE/FirstLawOfThermodynamics#Example_B|familiar expression for the radiation flux]],  
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<math>\vec{F}_\mathrm{rad}</math>
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<math>=</math>
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<math>- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) </math>
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<math>~=</math>
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<math>-\chi_\mathrm{rad} \nabla T \, ,</math>
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[<b>[[Appendix/References#Shu92|<font color="red">Shu92</font>]]</b>], Vol. I, &sect;2, p. 17, Eq. (2.17)
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  <td align="left" colspan="2">and &nbsp; &nbsp;[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (67)
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where [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] refers to 
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<math>\chi_\mathrm{rad}</math>
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<math>\equiv</math>
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<math>
\frac{4c a_\mathrm{rad} T^3}{3\kappa \rho} \, ,
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[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, p. 57, Eq. (68)
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as the coefficient of ''radiative'' conductivity.  When modeling spherically symmetric configurations, the radiation flux has only a radial component, that is, <math>\vec{F}_\mathrm{rad} = \hat{e}_r(F_r)</math>.  And, as pointed out in the context of Eq. (170) on p. 214 of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] <font color="darkgreen">&hellip; the quantity <math>L_r \equiv 4\pi r^2 F_r</math>, which is the net amount of energy crossing a spherical surface of radius <math>r</math>, is generally introduced instead of <math>F_r</math>.</font>  We therefore have,
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  <td align="right"><math>\frac{L_r}{4\pi r^2}</math></td>
  <td align="center"><math>=</math></td>
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- \frac{c}{3\rho\kappa_R} \frac{d}{dr} (a_\mathrm{rad}T^4)
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<math>\Rightarrow ~~~\frac{dT}{dr}
  </math>
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  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{3}{4ca_\mathrm{rad}} \frac{\rho\kappa_R}{T^3} \frac{L_r}{4\pi r^2}</math></td>
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[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter V, Eq. (171)<br />
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], &sect;6, Eq. (6-4a)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;9.1, Eq. (9.6)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;7.1, Eq. (7.8)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;5.2, Eq. (5.15)
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Dimensional Analysis:
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  <td align="right"><math>\frac{\kappa_R L}{ca_\mathrm{rad}}</math></td>
  <td align="center"><math>\sim</math></td>
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\frac{rT^4}{\rho} \sim m^{-1}\ell^4 ( {^\circ}K)^4 \, .
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NOTE:  This is consistent with the opacity, <math>\kappa_R \sim (\ell^2 m^{-1})</math>.
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