SSC/Structure/PolytropesEmbedded - Revision history

Jet53man: /* Equilibrium Sequences */

2022-10-10T15:38:03Z

Equilibrium Sequences

← Older revision		Revision as of 11:38, 10 October 2022
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	</td></tr></table>		</td></tr></table>


	<ul><li>		<ul><li>
	External Pressure vs. Volume (fixed mass):		External Pressure vs. Volume (fixed mass); displayed in panel "a" of [[#Fig3\|Figure 3]]:
	</li></ul>		</li></ul>

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	<ul><li>		<ul><li>
	Mass vs. Radius (fixed external pressure):		Mass vs. Radius (fixed external pressure); displayed in panel "b" of [[#Fig3\|Figure 3]]:
	</li></ul>		</li></ul>

Line 2,186:		Line 2,187:

	<ul><li>		<ul><li>
	Mass vs. Central Density (fixed external pressure):		Mass vs. Central Density (fixed external pressure); displayed in panel "c" of [[#Fig3\|Figure 3]]:
	</li></ul>		</li></ul>

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	<ul><li>		<ul><li>
	Mass vs. Central Density (fixed radius):		Mass vs. Central Density (fixed radius); displayed in panel "d" of [[#Fig3\|Figure 3]]:
	</li></ul>		</li></ul>

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	<tr>		<tr>
	<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>		<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
	<td align="center" colspan="1" rowspan="4">		<td align="center" colspan="1" rowspan="4">(a)<br />
	[[File:N5Sequence01B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence01B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="4">		<td align="center" colspan="1" rowspan="4">(b)<br />
	[[File:N5Sequence02B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence02B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="4">		<td align="center" colspan="1" rowspan="4">(c)<br />
	[[File:N5Sequence03B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence03B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="4">		<td align="center" colspan="1" rowspan="4">(d)<br />
	[[File:N5Sequence04B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence04B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>

Jet53man: /* Example Sequences */

2022-10-10T15:30:41Z

Example Sequences

← Older revision		Revision as of 11:30, 10 October 2022
Line 2,343:		Line 2,343:


	<div align="center" id="~~Fig1~~">		<div align="center" id="Fig3">
	<table border="1" align="center" cellpadding="8" width="1050px">		<table border="1" align="center" cellpadding="8" width="1050px">
	<tr>		<tr>
	<td align="center" colspan="6">		<td align="center" colspan="6">
	<b>Figure 1:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)		<b>Figure 3:</b>   Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Equilibrium Sequences */

2022-10-10T00:12:33Z

Equilibrium Sequences

@@ Line 2,572: / Line 2,572: @@
 <li>
-External Pressure vs. Volume (fixed mass):
+Mass vs. Central Density (fixed radius):
    <ol type="a">
    <li>
-   Maximum <math>P_e</math> (green circular marker) &hellip;
+   Maximum mass (??? circular marker) &hellip;
 <table border="0" align="center" cellpadding="5">
@@ Line 2,588: / Line 2,588: @@
    <td align="left">
 <math>
-\frac{d}{d\xi} \biggl\{ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr\}
+\frac{d}{d\xi} \biggl\{\xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2} \biggr\}
 </math>
    </td>

Jet53man: /* Example Extrema */

2022-10-09T23:56:32Z

Example Extrema

@@ Line 2,398: / Line 2,398: @@
 ====Example Extrema====
-<ol><li>
+<ol>
 External Pressure vs. Volume (fixed mass):
    <ol type="a">
@@ Line 2,568: / Line 2,569: @@
    </li>
    </ol>
-</li></ol>
+</li>
 ==Additional, Numerically Constructed Polytropic Configurations==

Jet53man: /* Example Extrema */

2022-10-09T23:51:59Z

Example Extrema

@@ Line 2,500: / Line 2,500: @@
 <math>
 \frac{d}{d\xi}\biggl\{ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9} \biggr\}
 </math>
    </td>

Jet53man: /* Example Extrema */

2022-10-09T23:45:35Z

Example Extrema

@@ Line 2,416: / Line 2,416: @@
 <math>
 \frac{d}{d\xi} \biggl\{ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr\}
 </math>
    </td>

Jet53man: /* Equilibrium Sequences */

2022-10-09T23:35:47Z

Equilibrium Sequences

@@ Line 1,966: / Line 1,966: @@
 ===Equilibrium Sequences===
 <table border="1" align="center" cellpadding="5"><tr><td align="left">
@@ Line 2,393: / Line 2,395: @@
 </table>
 </div>
 ==Additional, Numerically Constructed Polytropic Configurations==

Jet53man: /* Equilibrium Sequences */

2022-10-05T23:47:44Z

Equilibrium Sequences

← Older revision		Revision as of 19:47, 5 October 2022
Line 1,970:		Line 1,970:
	Pulling from, and setting n = 5 in, our [[#Chieze's_Presentation\|above discussion of Chieze's presentation]], we find the following …		Pulling from, and setting n = 5 in, our [[#Chieze's_Presentation\|above discussion of Chieze's presentation]], we find the following …

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="8" align="center">

	<tr>		<tr>
Line 2,063:		Line 2,063:

	Hence,		Hence,


	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>		<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
	</td>		</td>
	<td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ ~~\biggl(1 +~~ \frac{~~\xi^2~~}{3} \biggr)^{5/2}~~\biggr]_~~\tilde\xi</math>		<td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[ \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Equilibrium Sequences */

2022-10-05T23:39:47Z

Equilibrium Sequences

@@ Line 2,340: / Line 2,340: @@
 </tr>
 </table>
 <div align="center" id="Fig1">
 <table border="1" align="center" cellpadding="8" width="1050px">
 <tr>
-   <td align="center" colspan="5">
+   <td align="center" colspan="6">
 <b>Figure 1:</b> &nbsp; Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
    </td>
@@ Line 2,385: / Line 2,354: @@
    <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
    <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
 </tr>
 <tr>
@@ Line 2,396: / Line 2,366: @@
    <td align="center" colspan="1" rowspan="4">
 [[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
    </td>
 </tr>
@@ Line 2,416: / Line 2,389: @@
    <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
    </td>
-</tr>
+   <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
    </td>
 </tr>

Jet53man: /* Equilibrium Sequences */

2022-10-05T22:35:58Z

Equilibrium Sequences

← Older revision		Revision as of 18:35, 5 October 2022
Line 2,388:		Line 2,388:
	<tr>		<tr>
	<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>		<td align="center" colspan="1"><font color="yellow" size="+2">●</font></td> <td align="center" colspan="1">√3</td>
	<td align="center" colspan="1" rowspan="3">		<td align="center" colspan="1" rowspan="4">
	[[File:~~N5Sequence01~~.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence01B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="3">		<td align="center" colspan="1" rowspan="4">
	[[File:~~N5Sequence02~~.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence02B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	<td align="center" colspan="1" rowspan="3">		<td align="center" colspan="1" rowspan="4">
	[[File:~~N5Sequence03~~.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]		[[File:N5Sequence03B.png\|300px\|center\|Pressure-Truncated Isothermal Equilibrium Sequence]]
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>		<td align="center" colspan="1"><font color="darkgreen" size="+2">●</font></td> <td align="center" colspan="1">3</td>
			</tr>
			<tr>
			<td align="center" colspan="1"><font color="purple" size="+2">●</font></td> <td align="center" colspan="1">√15</td>
	</tr>		</tr>
	<tr>		<tr>