SSC/Structure/OtherAnalyticModels - Revision history

Joel2: /* Equilibrium Structure */

2024-09-07T19:24:12Z

Equilibrium Structure

← Older revision		Revision as of 15:24, 7 September 2024
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	</table>		</table>
	</div>		</div>
	Note that the total mass is obtained by setting <math>r = R</math> in the expression for <math>M_r(r )</math>, namely,		<span id="TotalMass">Note that the total mass</span> is obtained by setting <math>r = R</math> in the expression for <math>M_r(r )</math>, namely,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* Equilibrium Structure */

2024-08-17T00:39:49Z

Equilibrium Structure

← Older revision		Revision as of 20:39, 16 August 2024
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	Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, {{ Prasad49 }} determines that,		<span id="Pressure">Hence</span>, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, {{ Prasad49 }} determines that,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* Equilibrium Structure */

2024-08-10T19:01:28Z

Equilibrium Structure

← Older revision		Revision as of 15:01, 10 August 2024
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	</tr>		</tr>
	</table>		</table>
	So, the enthalpy is given by the expression,		<span id="SphericalEnthalpyProfile">So, the enthalpy is given by the expression,</span>

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

Joel2: /* Equilibrium Structure */

2024-07-30T22:46:28Z

Equilibrium Structure

← Older revision		Revision as of 18:46, 30 July 2024
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	</table>		</table>

	This matches the gravitational potential [[ParabolicDensity/GravPot#ParabolicPotential\|derived in a separate chapter]] using ~~EFE's~~ <math>A_i</math> and <math>A_{ij}</math> coefficients.		<font color="red">This matches the gravitational potential</font> [[ParabolicDensity/GravPot#ParabolicPotential\|derived in a separate chapter]] using some of the <math>A_i</math> and <math>A_{ij}</math> coefficients adopted by [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>]; see also the [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#For_Spheres_(aℓ_=_am_=_as)\|raw derivation of the relevant coefficients]].

Joel2: /* Equilibrium Structure */

2024-07-30T22:37:58Z

Equilibrium Structure

← Older revision		Revision as of 18:37, 30 July 2024
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	</tr>		</tr>
	</table>		</table>
	Note that in terms of Cartesian coordinates, this expression becomes,		<span id="ParabolicPotential">Note that in terms of Cartesian coordinates,</span> this expression becomes,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>

			This matches the gravitational potential [[ParabolicDensity/GravPot#ParabolicPotential\|derived in a separate chapter]] using EFE's <math>A_i</math> and <math>A_{ij}</math> coefficients.

Joel2: /* Equilibrium Structure */

2024-07-30T22:25:15Z

Equilibrium Structure

← Older revision		Revision as of 18:25, 30 July 2024
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2		\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
	- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
	\, .		\, .
	</math>		</math>
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	<td align="left">		<td align="left">
	<math>		<math>
	- \biggl( \frac{GM}{10 R} \biggr)		- \biggl( \frac{GM_\mathrm{tot}}{4 R} \biggr)
	\biggl[5\biggl(\frac{r}{R}\biggr)^2 - \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr]		\biggl[5\biggl(\frac{r}{R}\biggr)^2 - \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr]
	+ C \, .		+ C \, .
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{~~7GM~~}{20 R} \, .		\frac{7GM_\mathrm{tot}}{8 R} \, .
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	-\biggl( \frac{GM}{10 R} \biggr)		-\biggl( \frac{GM_\mathrm{tot}}{4 R} \biggr)
	\biggl[5\biggl(\frac{r}{R}\biggr)^2 - \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr]		\biggl[5\biggl(\frac{r}{R}\biggr)^2 - \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr]
	+ \frac{7GM}{20 R}		+ \frac{7GM}{8 R}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{GM}{20 R}		\frac{GM_\mathrm{tot}}{8 R} \biggl[7 - 10\biggl(\frac{r}{R}\biggr)^2 + 3\biggl(\frac{r}{R}\biggr)^4\biggr]
	\biggl[7 - 10\biggl(\frac{r}{R}\biggr)^2 + 3\biggl(\frac{r}{R}\biggr)^4\biggr] \, .		\, .
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	-~\frac{GM}{~~20R~~} \biggl\{27 - 10 \biggl(\frac{r}{R}\biggr)^2		\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
	+ 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
	+		+
	\frac{GM}{20 R} \biggl[7 - 10\biggl(\frac{r}{R}\biggr)^2 + 3\biggl(\frac{r}{R}\biggr)^4\biggr]		\frac{GM_\mathrm{tot}}{8 R} \biggl[7 - 10\biggl(\frac{r}{R}\biggr)^2 + 3\biggl(\frac{r}{R}\biggr)^4\biggr]
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	-~\frac{GM}{R} \, .		-~\frac{GM_\mathrm{tot}}{R} \, .
	</math>		</math>
	</td>		</td>

Joel2: /* Equilibrium Structure */

2024-07-30T21:57:57Z

Equilibrium Structure

← Older revision		Revision as of 17:57, 30 July 2024
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	</tr>		</tr>
	</table>		</table>
	Let's choose a constant, <math>C_\Phi</math>, such that the potential is <math>-GM/R</math> at the surface <math>(r/R = 1)</math>, that is,		Let's choose a constant, <math>C_\Phi</math>, such that the potential is <math>-GM_\mathrm{tot}/R</math> at the surface <math>(r/R = 1)</math>, that is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>- \frac{GM}{R}</math>		<math>- \frac{GM_\mathrm{tot}}{R}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{GM}{4R} \biggl\{ 5 - \frac{3}{2}\biggr\}		<math>\frac{GM_\mathrm{tot}}{4R} \biggl\{ 5 - \frac{3}{2}\biggr\}
	+ C_\Phi		+ C_\Phi
	</math>		</math>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{~~7GM~~}{8R} - \frac{GM}{R} = -\frac{~~15GM~~}{8R} \, .		<math>-\frac{7GM_\mathrm{tot}}{8R} - \frac{GM_\mathrm{tot}}{R} = -\frac{15GM_\mathrm{tot}}{8R} \, .
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	-~\frac{GM}{~~20R~~} \biggl\{27 - 10 \biggl(\frac{r}{R}\biggr)^2		\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2
	+ 3\biggl(\frac{r}{R}\biggr)^4\biggr\}		- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
	\, .		\, .
	</math>		</math>
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	<math>		<math>
	-~\frac{\pi G\rho R^2}{15}		-~\frac{\pi G\rho R^2}{15}
	\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)		\biggl\{15 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
	+ \frac{3}{R^4}\biggl(x^2 + y^2 + z^2\biggr)\biggl(x^2 + y^2 + z^2\biggr)\biggr\}		+ \frac{3}{R^4}\biggl(x^2 + y^2 + z^2\biggr)\biggl(x^2 + y^2 + z^2\biggr)\biggr\}
	</math>		</math>
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	<math>		<math>
	-~\frac{\pi G\rho R^2}{15}		-~\frac{\pi G\rho R^2}{15}
	\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)		\biggl\{15 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
	+ \frac{3}{R^4}		+ \frac{3}{R^4}
	\biggl( x^4 + x^2y^2 + x^2z^2 + y^2x^2 + y^4 + y^2z^2 + x^2z^2 + y^2z^2 + z^4\biggr)		\biggl( x^4 + x^2y^2 + x^2z^2 + y^2x^2 + y^4 + y^2z^2 + x^2z^2 + y^2z^2 + z^4\biggr)
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	<math>		<math>
	-~\frac{\pi G\rho R^2}{15}		-~\frac{\pi G\rho R^2}{15}
	\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)		\biggl\{15 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
	+		+
	\frac{6}{R^4}\biggl(x^2y^2 + x^2z^2 + y^2z^2\biggr) + \frac{3}{R^4}\biggl( x^4 + y^4 + z^4 \biggr)		\frac{6}{R^4}\biggl(x^2y^2 + x^2z^2 + y^2z^2\biggr) + \frac{3}{R^4}\biggl( x^4 + y^4 + z^4 \biggr)
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	<math>		<math>
	-~\pi G\rho R^2		-~\pi G\rho R^2
	\biggl\{~~\frac{9}{5}~~ - \frac{2}{3R^2} \biggl(x^2 + y^2 + z^2\biggr)		\biggl\{1 - \frac{2}{3R^2} \biggl(x^2 + y^2 + z^2\biggr)
	+		+
	\frac{2}{5R^4}\biggl(x^2y^2 + x^2z^2 + y^2z^2\biggr) + \frac{1}{5R^4}\biggl( x^4 + y^4 + z^4 \biggr)		\frac{2}{5R^4}\biggl(x^2y^2 + x^2z^2 + y^2z^2\biggr) + \frac{1}{5R^4}\biggl( x^4 + y^4 + z^4 \biggr)

Joel2: /* Equilibrium Structure */

2024-07-30T21:39:24Z

Equilibrium Structure

← Older revision		Revision as of 17:39, 30 July 2024
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	<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>		<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>
	</div>		</div>
	where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[SSCpt2/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,		where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[SSCpt2/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
			Note that the total mass is obtained by setting <math>r = R</math> in the expression for <math>M_r(r )</math>, namely,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>M_\mathrm{tot}</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>\frac{4\pi\rho_c R^3}{3} \biggl[\frac{2}{5}\biggr]
			=
			\frac{8\pi\rho_c R^3}{15}
			</math>
			<math>\Rightarrow</math>
			<math>
			2\pi \rho_c = \frac{15 M_\mathrm{tot}}{4R^3}
			\, .
			</math>
			</td>
			</tr>
			</table>


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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{GM}{~~10R~~} \biggl\{ 5 \biggl(\frac{r}{R}\biggr)^2		<math>
			\frac{G M_\mathrm{tot}}{4R} \biggl\{ 5 \biggl(\frac{r}{R}\biggr)^2
	- \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr\}		- \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr\}
	+ C_\Phi \, .		+ C_\Phi
			\, .
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{GM}{~~10R~~} \biggl\{ 5 - \frac{3}{2}\biggr\}		<math>\frac{GM}{4R} \biggl\{ 5 - \frac{3}{2}\biggr\}
	+ C_\Phi		+ C_\Phi
	</math>		</math>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>-\frac{7GM}{~~20R~~} - \frac{GM}{R} = -\frac{~~27GM~~}{~~20R~~} \, .		<math>-\frac{7GM}{8R} - \frac{GM}{R} = -\frac{15GM}{8R} \, .
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{GM}{~~10R~~} \biggl\{ 5 \biggl(\frac{r}{R}\biggr)^2		<math>
			\frac{G M_\mathrm{tot}}{4R} \biggl\{ 5 \biggl(\frac{r}{R}\biggr)^2
	- \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr\}		- \frac{3}{2}\biggl(\frac{r}{R}\biggr)^4\biggr\}
	-\frac{~~27GM~~}{~~20R~~}		- \frac{15 GM_\mathrm{tot}}{8R}
	</math>		</math>
	</td>		</td>

Joel2: /* Equilibrium Structure */

2024-07-30T00:40:05Z

Equilibrium Structure

@@ Line 825: / Line 825: @@
++
 \frac{6}{R^4}\biggl(x^2y^2 + x^2z^2  + y^2z^2\biggr) + \frac{3}{R^4}\biggl( x^4 + y^4 + z^4 \biggr)
 \biggr\}
 \, .

Joel2: /* Equilibrium Structure */

2024-07-28T19:58:40Z

Equilibrium Structure

← Older revision		Revision as of 15:58, 28 July 2024
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	</tr>		</tr>
	</table>		</table>
			Note that in terms of Cartesian coordinates, this expression becomes,

			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\Phi_\mathrm{grav}</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-~\frac{\pi G\rho R^2}{15}
			\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
			+ \frac{3}{R^4}\biggl(x^2 + y^2 + z^2\biggr)\biggl(x^2 + y^2 + z^2\biggr)\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-~\frac{\pi G\rho R^2}{15}
			\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
			+ \frac{3}{R^4}
			\biggl( x^4 + x^2y^2 + x^2z^2 + y^2x^2 + y^4 + y^2z^2 + x^2z^2 + y^2z^2 + z^4\biggr)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-~\frac{\pi G\rho R^2}{15}
			\biggl\{27 - \frac{10}{R^2} \biggl(x^2 + y^2 + z^2\biggr)
			+
			\frac{6}{R^4}\biggl(x^2y^2 + x^2z^2 + y^2z^2\biggr) + \frac{3}{R^4}\biggl( x^4 + y^4 + z^4 \biggr)
			\biggr\}
			\, .
			</math>
			</td>
			</tr>
			</table>