SSC/Structure/BiPolytropes/Analytic1.53 - Revision history

Jet53man: /* Power-Law Density Distribution */

2023-12-08T20:32:50Z

Power-Law Density Distribution

← Older revision		Revision as of 16:32, 8 December 2023
Line 1,442:		Line 1,442:
	In astrophysically relevant configurations, the exponent on <math>\xi</math> must be negative, which means that we are confined to models for which <math>n < \tfrac{7}{2}</math>.		In astrophysically relevant configurations, the exponent on <math>\xi</math> must be negative, which means that we are confined to models for which <math>n < \tfrac{7}{2}</math>.

			Now, from our [[SSC/Structure/PowerLawDensity#Derivation\|associated discussion of power-law density distributions]] in polytropes, we discovered that hydrostatic balance can be established at all radial positions within a spherically symmetric configuration for power-law density distributions of the form,
			<div align="center">
			<math>
			\frac{\rho}{\rho_c} \propto \xi^{- 2n/(n-1)}
			</math>
			</div>
			This matches our just-derived point-source model if,
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

	<tr>		<tr>
	<td align="right"><math>~~\frac{\mathcal{A}}{a_n^3} \biggl(\frac{13-4n}{13 - 2n} \biggr) \biggl[ \frac{T}{T_\mathrm{norm}}\biggr]^{~~(2n-13)~~/2 }~~ </math></td>		<td align="right"><math>6n/(2n-13)</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left"><math>- 2n/(n-1)</math>
	<math>
	~~3\xi^3 \, ,~~
	</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~where,~~
	~~<table border="0" align="center" cellpadding="5">~~

	<tr>		<tr>
	<td align="right">		<td align="right"><math>\Rightarrow ~~~ 6(n-1)</math></td>
	<math>		<td align="center"><math>=</math></td>
	~~\xi~~		<td align="left"><math>2(13 - 2n)</math>
	</math>
	</td>		</td>
	~~<td align="center"><math>\equiv~~</~~math></td~~>		</tr>
	~~<td align="left">~~
	<~~math>~~		<tr>
	~~\frac{r}{a_n} \, ,~~		<td align="right"><math>\Rightarrow ~~~ n</math></td>
	~~</math>~~		<td align="center"><math>=</math></td>
	~~</td>~~		<td align="left"><math>\frac{16}{5} \, ,</math>
	~~<td align="center">      and,      </td~~>
	<td align="right">
	<math>
	T_\~~mathrm{norm}~~
	</math>
	</td>
	<td align="center"><math>~~\equiv~~</math></td>
	<td align="left">
	<math>
	~~\biggl[~~\frac{~~\mathcal{A}~~}{~~a_n^3~~}\~~biggr]^{2/(13-2n)}~~
	\, .
	</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			which ''is'' less than <math>\tfrac{7}{2}</math>, so it is an astrophysically viable result.

	==Our Derivation==		==Our Derivation==

Jet53man: /* Power-Law Density Distribution */

2023-12-08T19:01:21Z

Power-Law Density Distribution

@@ Line 1,391: / Line 1,391: @@
 ===Power-Law Density Distribution===
-In an [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|accompanying discussion]], we have demonstrated that power-law density distributions can provide analytic solutions of the Lane-Emden equation, although the associated boundary conditions do not naturally conform to the boundary conditions that are suitable to astrophysical configurations.  We have just shown that the point-source envelope configuration appears to admit a power-law temperature solution; specifically, setting the integration constant to zero, our result gives,
+In an [[SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|accompanying discussion]], we have demonstrated that power-law density distributions can provide analytic solutions of the Lane-Emden equation, although the associated boundary conditions do not naturally conform to the boundary conditions that are suitable to astrophysical configurations.  We have just shown that the point-source envelope configuration appears to admit a power-law temperature (alternatively, enthalpy) solution. Via the polytropic relation, <math>H = K\rho^{1 / n}</math>, we can convert to the density-radius relation,
 <table border="0" align="center" cellpadding="5">

Jet53man: /* Adopting Enthalpy (instead of Temperature) */

2023-12-08T18:16:40Z