SSC/Structure/Polytropes/Analytic - Revision history

Joel2: /* Primary E-Type Solution */

2026-03-16T17:08:54Z

Primary E-Type Solution

@@ Line 1,001: / Line 1,001: @@
 -\frac{1}{ (3)^3 }
 \biggr]
 </math>
    </td>
@@ Line 1,006: / Line 1,028: @@
 </table>
+where, <math>P_c = K\rho_c^{6/5}</math>.
 </td></tr></table>
 The function of <math>\xi</math> inside the curly brackets of this last expression goes to unity as <math>\xi \rightarrow \infty</math>, so the integrated mass is finite even though the configuration extends to infinity.  Specifically, the total mass is,

Joel2: /* Primary E-Type Solution */

2026-03-16T16:58:10Z

Primary E-Type Solution

@@ Line 982: / Line 982: @@
    <td align="left">
 <math>
-- ~ 2\cdot 3^4 \cdot K   \rho_c^{6/5}
+\cdot 3^4 \cdot K   \rho_c^{6/5}
-\int_0^\xi \biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr] d\xi
+\biggl\{
 </math>
    </td>

Joel2: /* Primary E-Type Solution */

2026-03-16T16:44:07Z

Primary E-Type Solution

@@ Line 957: / Line 957: @@
 - ~\biggr( 2^3\cdot 3^7\pi \cdot G K  \biggr)^{1/2}  \rho_c^{8/5}
 \biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
 </math>
    </td>

Joel2: /* Primary E-Type Solution */

2026-03-16T16:24:16Z

Primary E-Type Solution

← Older revision		Revision as of 12:24, 16 March 2026
Line 955:		Line 955:
	<td align="left">		<td align="left">
	<math>		<math>
	- G ~~~3^{5/2}~~\biggr( ~~\frac{~~2\cdot 3^~~4 K^3}{~~\pi G~~^3}~~ \biggr)^{1/2} \rho_c^{-1/5} ~~\rho_c \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{4/5} \biggr]~~		- ~\biggr( 2^3\cdot 3^7\pi \cdot G K \biggr)^{1/2} \rho_c^{8/5}
	\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]		\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
	</math>		</math>

Joel2: /* Primary E-Type Solution */

2026-03-16T16:13:47Z

Primary E-Type Solution

← Older revision		Revision as of 12:13, 16 March 2026
Line 944:		Line 944:
	<td align="left">		<td align="left">
	<math>		<math>
	- G \biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} \rho_c \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{4/5} \biggr]		- G ~3^{5/2}\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} \rho_c \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{4/5} \biggr]
	\biggl[~\{ \xi^3 ( 3 + \xi^2 )^{-~~3/2} \}\biggr]~~		\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
	~~~\biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]~~
	~~~\biggl[\xi^{-2~~} \biggr]
	</math>		</math>
	</td>		</td>
Line 957:		Line 955:
	<td align="left">		<td align="left">
	<math>		<math>
	- G 3^{-3/2}\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{8/5} \biggr]		- G ~3^{5/2}\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} \rho_c \biggl[ \biggl( \frac{2\pi G}{3K_5}\biggr)\rho_c^{4/5} \biggr]
	~\biggl[ \xi ~~\biggl~~( 1 + ~~\frac{1}{3}~~\xi^2 ~~\biggr~~)^{-4} \biggr]		\biggl[~ \xi ( 3 + \xi^2 )^{-4} \biggr]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

	</table>		</table>

Joel2: /* n = 0 Polytrope */

2026-03-16T16:02:25Z

n = 0 Polytrope

← Older revision		Revision as of 12:02, 16 March 2026
Line 104:		Line 104:

	<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left" bgcolor="lightgreen">		<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left" bgcolor="lightgreen">
			<div align="center">'''n = 0 Polytrope'''</div>
	[[SSC/Structure/UniformDensity#Solution_Technique_1\|In particular]], after integrating the hydrostatic-balance equation,		[[SSC/Structure/UniformDensity#Solution_Technique_1\|In particular]], after integrating the hydrostatic-balance equation,
	<div align="center">		<div align="center">
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	<div align="center">		<div align="center">
	<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr)</math> .		<math>P_c = \frac{2\pi G}{3} \rho_c^2 R^2 = \frac{3G}{8\pi}\biggl( \frac{M^2_\mathrm{tot}}{R^4} \biggr)</math> .
	</div>		</div>
	</td></tr></table>		</td></tr></table>

Joel2: /* Primary E-Type Solution */

2026-03-15T21:53:39Z

Primary E-Type Solution

@@ Line 947: / Line 947: @@
 ~\biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]
 ~\biggl[\xi^{-2} \biggr]
 </math>
    </td>

Joel2: /* Primary E-Type Solution */

2026-03-15T21:45:00Z

Primary E-Type Solution

@@ Line 934: / Line 934: @@
 ~\biggl[ \rho_c \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]
 ~\biggl[ \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5} \xi \biggr]^{-2}
 </math>
    </td>

Joel2: /* Primary E-Type Solution */

2026-03-15T21:38:46Z

Primary E-Type Solution

← Older revision		Revision as of 17:38, 15 March 2026
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	<td align="left">		<td align="left">
	<math>		<math>
	- G \biggl[~~M_r~~\biggr]~\biggl[\~~rho~~\biggr]~\biggl[ r \biggr]^{-2}		- G \biggl[\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \}\biggr]
			~\biggl[ \rho_c \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]
			~\biggl[ \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5} \xi \biggr]^{-2}
	</math>		</math>
	</td>		</td>

Joel2: /* Primary E-Type Solution */

2026-03-15T21:35:03Z

Primary E-Type Solution

← Older revision		Revision as of 17:35, 15 March 2026
Line 894:		Line 894:
	<div align="center">		<div align="center">
	<math>		<math>
	\rho = \rho_c \Theta_H^5 = \rho_c \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-5/2} \, ,		\rho = \rho_c \Theta_H^5 = \rho_c \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,
	</math>		</math>
	</div>		</div>
Line 900:		Line 900:
	<div align="center">		<div align="center">
	<math>		<math>
	r = \biggl[ \frac{~~K_5~~}{2\pi G}\biggr]^{1 / 2}\rho_c^{-2/5} \xi		r = \biggl( \frac{3K_5}{2\pi G}\biggr)^{1 / 2}\rho_c^{-2/5} \xi
	</math>.		</math>.
	</div>		</div>
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	<td align="left">		<td align="left">
	<math>		<math>
	\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \} \, ,		\biggr( \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} ~\{ \xi^3 ( 3 + \xi^2 )^{-3/2} \} \, ,
	</math>		</math>
	</td>		</td>
Line 924:		Line 924:

	the RHS of the hydrostatic-balance relation can be written as,		the RHS of the hydrostatic-balance relation can be written as,
			<table align="center" cellpadding="8">

			<tr>
			<td align="right"><math>\mathrm{RHS} = - G \biggl[M_r\biggr]~\biggl[\rho\biggr]~\biggl[ r \biggr]^{-2}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			- G \biggl[M_r\biggr]~\biggl[\rho\biggr]~\biggl[ r \biggr]^{-2}
			</math>
			</td>
			</tr>
			</table>

	</td></tr></table>		</td></tr></table>
	The function of <math>\xi</math> inside the curly brackets of this last expression goes to unity as <math>\xi \rightarrow \infty</math>, so the integrated mass is finite even though the configuration extends to infinity. Specifically, the total mass is,		The function of <math>\xi</math> inside the curly brackets of this last expression goes to unity as <math>\xi \rightarrow \infty</math>, so the integrated mass is finite even though the configuration extends to infinity. Specifically, the total mass is,