ParabolicDensity/Spheres/Structure - Revision history

Joel2: /* Spherically Symmetric Equilibrium Structure */

2024-09-03T17:47:58Z

Spherically Symmetric Equilibrium Structure

@@ Line 122: / Line 122: @@
 </table>
-===Barotropic Profiles===
+===Effective Barotropic Relations===
 By replacing <math>r/R</math> with <math>\rho/\rho_c</math>, we obtain analytic expression for, respectively, the pressure-density and enthalpy-density (effective [[SR#Barotropic_Structure|barotropic]]) relations that are relevant in this ''parabolic'' configuration.  Specifically,
@@ Line 198: / Line 198: @@
    </td>
 </tr>
 </table>

Joel2: /* Spherically Symmetric Equilibrium Structure */

2024-09-03T17:28:15Z

Spherically Symmetric Equilibrium Structure

@@ Line 24: / Line 24: @@
 <math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>
 </div>
-where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star.
+where, <math>\rho_c</math> is the central density and <math>R</math> is the radius of the star.
 ===Radial Profiles===
-In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|related discussion]] we derived the following expressions that describe analytically various structural properties of these configurations.
+In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|related discussion]] we have derived the following expressions that describe analytically various structural properties of this equilibrium configuration.
 <table border="0" cellpadding="5" align="center">
@@ Line 83: / Line 83: @@
 \biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr]
 \, ;</math>
    </td>
 </tr>
@@ Line 109: / Line 120: @@
    </td>
 </tr>
 </table>

Joel2: /* Spherically Symmetric Equilibrium Structure */

2024-09-03T16:56:40Z

Spherically Symmetric Equilibrium Structure

← Older revision		Revision as of 12:56, 3 September 2024
Line 24:		Line 24:
	<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.~~ In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|related discussion]] we derived the following expressions that describe analytically various structural properties of these configurations.		where, <math>\rho_c</math> is the central density and, <math>R</math> is the radius of the star.

			===Radial Profiles===
			In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution\|related discussion]] we derived the following expressions that describe analytically various structural properties of these configurations.

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 65:		Line 68:
	\, ;		\, ;
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>P(r)</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>\frac{4\pi G\rho_c^2 R^2}{15}
			\biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2
			\biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr]
			\, ;</math>
	</td>		</td>
	</tr>		</tr>
Line 92:		Line 110:
	</tr>		</tr>
	</table>		</table>


	=See Also=		=See Also=

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Joel2: Created page with "FORCETOC =Parabolic Density Distribution=

Part I: Gravitational Potential
Part II: Spherical Structures
2024-09-03T16:52:29Z

Created page with "FORCETOC  =Parabolic Density Distribution= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="25%"><br />Part I: Gravitational Potential </td> <td align="center" bgcolor="lightblue" width="25%"><br />Part II: Spherical Structures </td> <td align="center" bgcolor="ligh..."

New page
FORCETOC

=Parabolic Density Distribution=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/GravPot|Part I:   Gravitational Potential]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Spheres/Structure|Part II:   Spherical Structures]]
 
</td>
<td align="center" bgcolor="lightblue" width="25%"><br />[[ParabolicDensity/Axisymmetric/Structure|Part III:   Axisymmetric Equilibrium Structures]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[ParabolicDensity/Triaxial/Structure|Part IV:   Triaxial Equilibrium Structures (Exploration)]]
 
</td>
</tr>
</table>

==Spherically Symmetric Equilibrium Structure==
In an article titled, "Radial Oscillations of a Stellar Model," {{ Prasad49full }} investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
<div align="center">
<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>
</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. In a [[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|related discussion]] we derived the following expressions that describe analytically various structural properties of these configurations.

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

<tr>
<td align="right">
<math>M_r(r)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{4\pi\rho_c r^3}{3} \biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2 \biggr] \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>~g_0(r) \equiv \frac{G M_r(r) }{r^2} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4\pi G \rho_c r}{3}
\biggl[1 - \frac{3}{5} \biggl( \frac{r}{R} \biggr)^2\biggr] \, ;</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Phi_\mathrm{grav}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{G M_\mathrm{tot}}{8R} \biggl\{- 15 + 10 \biggl(\frac{r}{R}\biggr)^2- 3\biggl(\frac{r}{R}\biggr)^4\biggr\}
\, ;
</math>
</td>
</tr>
</table>

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>

=See Also=

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