SSC/Structure/BiPolytropes/Analytic51/Pt2 - Revision history

Joel2: /* UV Plane */

2026-11-06T20:40:28Z

UV Plane

← Older revision		Revision as of 15:40, 6 November 2026
Line 1,009:		Line 1,009:

	===Envelope===		===Envelope===
			The properties of the envelope of equilibrium models are,
			<table border="0" align="center" cellpadding="8">

			<tr>
			<td align="right"><math>\phi</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			A \biggl[ \frac{\sin(\eta - B)}{\eta} \biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>- \frac{d\phi}{d\eta}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{A}{\eta^2}\biggl[ \sin(\eta-B) - \eta \cos(\eta-B) \biggr]
			</math>
			</td>
			</tr>
			</table>

			Hence, we find that,
			<table border="0" align="center" cellpadding="8">

			<tr>
			<td align="right"><math>V_\mathrm{Ch67}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{\eta}{\phi} \biggl(- \frac{d\phi}{d\xi} \biggr)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{1}{A}\biggl[ \frac{\eta^2}{\sin(\eta-B)} \biggr]
			\frac{A}{\eta^2}\biggl[ \sin(\eta-B) - \eta \cos(\eta-B) \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\biggl[ 1 - \eta \cot(\eta-B) \biggr] \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"><math>U_\mathrm{Ch67}</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\eta \phi \biggl(-\frac{d\phi}{d\eta}\biggr)^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			A\sin(\eta-B)\biggl\{ \frac{A}{\eta^2}\biggl[ \sin(\eta-B) - \eta \cos(\eta-B) \biggr] \biggr\}^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\eta^2 \biggl[ \frac{\sin(\eta - B)}{\sin(\eta-B) - \eta \cos(\eta-B)} \biggr]
			</math>
			</td>
			</tr>
			</table>

			===Radial Displacement Function===

			Try,
			<table border="0" align="center" cellpadding="8">
			<tr>
			<td align="right"><math>x \equiv \biggl[ a - \frac{b}{U_\mathrm{Ch67}}\biggr] </math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			a - \frac{b}{\eta^2} \biggl[ 1 - \eta \cot(\eta-B) \biggr]
			</math>
			</td>
			</tr>
			</table>

	=Related Discussions=		=Related Discussions=

Joel2: /* UV Plane */

2026-11-06T18:37:46Z

UV Plane

← Older revision		Revision as of 13:37, 6 November 2026
Line 847:		Line 847:
	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="right"><math>U_\mathrm{Sch}</math></td>		<td align="right"><math>V_\mathrm{Sch}</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	+ \frac{d\ln ~~m_0~~}{d\ln r} = \frac{~~4\pi r^3~~ \rho}{~~M_r~~} \, ,		+ \frac{d\ln P}{d\ln r} = \frac{G \rho M_r}{Pr} \, ,
	</math>		</math>
	</td>		</td>
Line 857:		Line 857:

	<tr>		<tr>
	<td align="right"><math>V_\mathrm{Sch}</math></td>		<td align="right"><math>U_\mathrm{Sch}</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	+ \frac{d\ln P}{d\ln r} = \frac{G \rho ~~M_r~~}{Pr} \, .		+ \frac{d\ln m_0}{d\ln r} = \frac{4\pi r^3 \rho}{M_r} \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

	</table>		</table>

Line 942:		Line 943:
	</table>		</table>

	Compare with …		Compare with (p. 105, Eq. 115 of) Ch67 …

	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">

Joel2: /* UV Plane */

2026-11-06T18:30:52Z

UV Plane

← Older revision		Revision as of 13:30, 6 November 2026
Line 847:		Line 847:
	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="right"><math>U</math></td>		<td align="right"><math>U_\mathrm{Sch}</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left">		<td align="left">
Line 857:		Line 857:

	<tr>		<tr>
	<td align="right"><math>V</math></td>		<td align="right"><math>V_\mathrm{Sch}</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left">		<td align="left">
Line 872:		Line 872:

	<tr>		<tr>
	<td align="right"><math>V</math></td>		<td align="right"><math>V_\mathrm{Sch}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
Line 908:		Line 908:

	<tr>		<tr>
	<td align="right"><math>U</math></td>		<td align="right"><math>U_\mathrm{Sch}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
Line 947:		Line 947:

	<tr>		<tr>
	<td align="right"><math>V</math></td>		<td align="right"><math>V_\mathrm{Ch67} = \frac{V_\mathrm{Sch}}{6}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
Line 977:		Line 977:

	<tr>		<tr>
	<td align="right"><math>U</math></td>		<td align="right"><math>U_\mathrm{Ch67}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
Line 1,001:		Line 1,001:
	<td align="left">		<td align="left">
	<math>		<math>
	3 \biggl( 1 + \frac{\xi^2}{3}\biggr)^{-1}		3 \biggl( 1 + \frac{\xi^2}{3}\biggr)^{-1} \, .
	</math>		</math>
	</td>		</td>

Joel2: /* UV Plane */

2026-11-06T18:23:37Z

UV Plane

@@ Line 902: / Line 902: @@
    <td align="left">
 <math>
-\biggl(1 + \frac{\xi^2}{3}  \biggr)^{-1} \xi^2 \, .
+\biggl(1 + \frac{\xi^2}{3}  \biggr)^{-1} \xi^2 \, ;
 </math>
 </td>
 </tr>
 <tr>
@@ Line 944: / Line 941: @@
 </tr>
 </table>
 ===Envelope===

Joel2: /* UV Plane */

2026-11-06T17:55:40Z

UV Plane

@@ Line 918: / Line 918: @@
 \biggl[ \rho_0 \biggl(1+\frac{\xi^2}{3} \biggr)^{-5/2} \biggr]
 \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggl(\frac{6}{\pi}\biggr)^{1/2} \xi^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2}\biggr]^{-1}
 </math>
 </td>

Joel2: /* UV Plane */

2026-11-06T17:43:10Z

UV Plane

← Older revision		Revision as of 12:43, 6 November 2026
Line 908:		Line 908:
	</table>		</table>

			<table border="0" align="center" cellpadding="8">

			<tr>
			<td align="right"><math>U</math></td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			4\pi \biggl[ K_c^{1/2} G^{-1/2} \rho_0^{-2/5} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi\biggr]^3
			\biggl[ \rho_0 \biggl(1+\frac{\xi^2}{3} \biggr)^{-5/2} \biggr]
			\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggl(\frac{6}{\pi}\biggr)^{1/2} \xi^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2}\biggr]^{-1}
			</math>
			</td>
			</tr>
			</table>
	===Envelope===		===Envelope===

Joel2: /* UV Plane */

2026-11-06T17:31:28Z

UV Plane

@@ Line 893: / Line 893: @@
 \biggl[ \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3} \biggr]^{-1}
 \biggl[\biggl( \frac{6}{\pi}\biggr)^{1/2} \xi^3 \theta^3\biggr]
 </math>
 </td>

70.180.56.157: /* Core */

2026-11-06T00:55:30Z

Core

@@ Line 880: / Line 880: @@
 \biggl[ K_c\rho_0^{6/5} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3} \biggr]^{-1}
 \biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggl( \frac{6}{\pi}\biggr)^{1/2} \xi^3 \theta^3\biggr]
 </math>
 </td>

70.180.56.157: /* Core */

2026-11-06T00:47:03Z

Core

← Older revision		Revision as of 19:47, 5 November 2026
Line 876:		Line 876:
	<td align="left">		<td align="left">
	<math>		<math>
	\frac{G \~~rho M_r~~}{Pr}		G\biggl[\biggl(\frac{3}{2\pi}\biggr)^{1/2}\xi~ K_c^{1/2} G^{-1/2} \rho_0^{-2/5}\biggr]^{-1}
			\biggl[ \rho_0 \biggl(1 + \frac{\xi^2}{3} \biggr)^{-5/2} \biggr]
			\biggl[ K_c\rho_0^{6/5} \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3} \biggr]^{-1}
			\biggl[ K_c^{3/2} G^{-3/2} \rho_0^{-1/5} \biggl( \frac{6}{\pi}\biggr)^{1/2} \xi^3 \theta^3\biggr]
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>




	===Envelope===		===Envelope===

Joel2: /* UV Plane */

2026-11-06T00:29:07Z

UV Plane

@@ Line 866: / Line 866: @@
 </tr>
 </table>
 =Related Discussions=