SSC/Structure/BiPolytropes/51RenormaizePart3 - Revision history

Jet53man: /* Switch from Core Mass to Total Mass */

2023-11-14T03:16:30Z

Switch from Core Mass to Total Mass

← Older revision		Revision as of 23:16, 13 November 2023
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	<math>		<math>
	G^{-9}K_c^{10} M_\mathrm{tot}^{-6} \biggl\{		G^{-9}K_c^{10} M_\mathrm{tot}^{-6} \biggl\{
	\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-12}		\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-4}
	\biggl( \frac{2}{\pi} \biggr)~~^{3}~~ \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s^6		\biggl( \frac{2}{\pi} \biggr) \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s^2
	\biggr\} \, ;		\biggr\}^3 \, ;
	</math>		</math>
	</td>		</td>

Jet53man: /* Based on Pressure-Truncated n = 5 Polytrope */

2023-11-14T02:54:39Z

Based on Pressure-Truncated n = 5 Polytrope

@@ Line 1,617: / Line 1,617: @@
 ==Switch from Core Mass to Total Mass==
-Let's now switch from the core mass to the total mass, drawing the following expression from [[#Throughout_the_Envelope|above]] &hellip;
+Now with the bipolytropic model in mind, let's switch from the core mass to the total mass, drawing the following expression from [[#Throughout_the_Envelope|above]] &hellip;
 <table border="0" align="center"  cellpadding="5">
@@ Line 1,628: / Line 1,628: @@
    </td>
    <td align="left">
-<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s</math>
+<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2}
    </td>
 </tr>

Jet53man: /* Switch from Core Mass to Total Mass */

2023-11-14T02:35:20Z

Switch from Core Mass to Total Mass

← Older revision		Revision as of 22:35, 13 November 2023
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	==Switch from Core Mass to Total Mass==		==Switch from Core Mass to Total Mass==
			Let's now switch from the core mass to the total mass, drawing the following expression from [[#Throughout_the_Envelope\|above]] …
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

Jet53man: /* Based on Pressure-Truncated n = 5 Polytrope */

2023-11-14T02:32:43Z

Based on Pressure-Truncated n = 5 Polytrope

← Older revision		Revision as of 22:32, 13 November 2023
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	This leads to panel (a) of [[SSC/Structure/PolytropesEmbedded#Fig3\|Figure 3 from that discussion]]; also shown here, on the right.		This leads to panel (a) of [[SSC/Structure/PolytropesEmbedded#Fig3\|Figure 3 from that discussion]]; also shown here, on the right.

			==Switch from Core Mass to Total Mass==

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

			<tr>
			<td align="right">
			<math>M_\mathrm{tot}</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)_s</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Jet53man: /* Chieze Normalization */

2023-11-14T02:23:48Z

Chieze Normalization

← Older revision		Revision as of 22:23, 13 November 2023
Line 1,535:		Line 1,535:
	=Based on Pressure-Truncated n = 5 Polytrope=		=Based on Pressure-Truncated n = 5 Polytrope=

	==~~Chieze~~ Normalization==		==Chieze87 Normalization==

	In a [[SSC/Structure/PolytropesEmbedded#Chieze's_Presentation\|subsection]] of our [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|separate discussion of pressure-truncated polytropes]], we highlighted the published work of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)]. It can [[SSC/Structure/PolytropesEmbedded#Example_Sequences\|readily be shown]] that his expressions for <math>P_e</math>, <math>R_\mathrm{eq}</math>, and <math>M_\mathrm{tot}</math> — in our terminology, <math>P_i</math>, <math>r_i</math>, and <math>M_\mathrm{core}</math> — are identical to the expressions we [[#Throughout_the_Core\|presented above]] in the context of the n = 5 core of our bipolytrope. Specifically,		In a [[SSC/Structure/PolytropesEmbedded#Chieze's_Presentation\|subsection]] of our [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|separate discussion of pressure-truncated polytropes]], we highlighted the published work of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)]. It can [[SSC/Structure/PolytropesEmbedded#Example_Sequences\|readily be shown]] that his expressions for <math>P_e</math>, <math>R_\mathrm{eq}</math>, and <math>M_\mathrm{tot}</math> — in our terminology, <math>P_i</math>, <math>r_i</math>, and <math>M_\mathrm{core}</math> — are identical to the expressions we [[#Throughout_the_Core\|presented above]] in the context of the n = 5 core of our bipolytrope. Specifically,

Jet53man: /* Chieze Normalization */

2023-11-14T02:22:50Z

Chieze Normalization

Show changes

Jet53man: /* See Also */

2023-11-14T01:50:53Z

Jet53man: /* New Normalization */

2023-11-13T22:57:11Z

New Normalization

← Older revision		Revision as of 18:57, 13 November 2023
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\biggl( \frac{K_e}{K_c} \biggr) </math>		<math>\biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m} </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-4}_i</math>		<math>\rho_0^{-4/5}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m-2} \theta^{-4}_i</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>\Rightarrow ~~~ \rho_0</math>		<math>\Rightarrow ~~~ \biggl[ \biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m}\biggr]^{5 / 4} </math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>\rho_0^{-1}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{5(m-2)/4} \theta^{-5}_i</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~ \rho_0 </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 360:		Line 372:
	<td align="left">		<td align="left">
	<math>		<math>
	\biggl[ \biggl( \frac{K_e}{K_c} \biggr)~~^{-1}~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{~~- 2~~} ~~\theta^{-4}_i~~ \biggr]~~^{5 / 4}~~		\underbrace{\biggl[ \biggl( \frac{K_e}{K_c} \biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{m}\biggr]^{- 5 / 4}}_{\rho_\mathrm{alt}}
	=		\biggl( \frac{\mu_e}{\mu_c} \biggr)^{5(m-2)/4} \theta^{-5}_i
	~~\biggl[ \biggl( \frac{K_e}{K_c} \biggr)~~^{-5 / 4} ~~\biggl( \frac{\mu_e}{\mu_c~~} ~~\biggr)^~~{~~-5 / 2} \theta^{-5}_i \biggr]~~
	=
	\rho_\mathrm{~~norm~~}~~\biggl[~~ \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-5 / 2} \theta^{-5}_i ~~\biggr]~~
	~~\, .~~
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			<font color="red"><b>ABANDONED</b></font>
	</td></tr></table>		</td></tr></table>
	<!-- END alternate01 -->		<!-- END alternate01 -->

Jet53man: /* New Normalization */

2023-11-13T19:53:53Z

New Normalization

@@ Line 333: / Line 333: @@
 </tr>
 </table>
 Hence, throughout the core, we have,

Jet53man: /* Maximum Total Mass */

2023-11-13T19:14:15Z

Maximum Total Mass

← Older revision		Revision as of 15:14, 13 November 2023
Line 1,451:		Line 1,451:

	<tr>		<tr>
	<td align="center" colspan="13">Maximum Total Mass ''a la'' {{ Yabushita75 }}</td>		<td align="center" colspan="14">Maximum Total Mass ''a la'' {{ Yabushita75 }}</td>
	</tr>		</tr>

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	<td align="center"><math>M_\mathrm{tot}/M_\mathrm{norm}</math></td>		<td align="center"><math>M_\mathrm{tot}/M_\mathrm{norm}</math></td>
	<td align="center">[[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models\|LAWE]]</td>		<td align="center">[[SSC/Stability/BiPolytropes#Equilibrium_Properties_of_Marginally_Unstable_Models\|LAWE]]</td>
			<td align="center">[[SSC/Stability/BiPolytropes/51Models#Stability\|Implicit<br />Scheme]]</td>
	</tr>		</tr>

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	<td align="center">3.4698691</td>		<td align="center">3.4698691</td>
	<td align="center" bgcolor="yellow">1.6686460157</td>		<td align="center" bgcolor="yellow">1.6686460157</td>
			<td align="center" bgcolor="yellow">1.6639103365</td>
	</tr>		</tr>
	</table>		</table>

@@ Line 1,532: / Line 1,532: @@
 </tr>
 </table>
 =See Also=