User:Tohline/Appendix/Ramblings/BordeauxSequences - Revision history

Tohline at 00:00, 17 December 2020

2020-12-17T00:00:59Z

← Older revision		Revision as of 00:00, 17 December 2020
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			=See Also=
			<ul>
			<li>Université de Bordeaux (Part 1):  [[User:Tohline/Appendix/Ramblings/Bordeaux\|External Gravitational Potential of Toroids]]</li>
			<li>Université de Bordeaux (Part 3):  [[User:Tohline/Appendix/Ramblings/BordeauxPostDefense\|Discussions Following Dissertation Defense]]</li>
			</ul>


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Tohline: /* Fixed Mass Ratio */

2020-12-03T23:24:00Z

Fixed Mass Ratio

← Older revision		Revision as of 23:24, 3 December 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{e^2} \, .		\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{2e^2} \, .
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>~		<math>~
	\frac{1}{2}(3 - 2e^2)(1 - e^2)^{3 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)^2 }{e^2}		\frac{1}{2}(3 - 2e^2)(1 - e^2)^{3 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)^2 }{2e^2}
	</math>		</math>
	</td>		</td>

Tohline: /* Fixed Ω */

2020-12-03T22:58:17Z

Fixed Ω

← Older revision		Revision as of 22:58, 3 December 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{e^2} \, ,		\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{2e^2} \, ,
	</math>		</math>
	</td>		</td>

Tohline: /* Fixed Mass Ratio */

2020-12-03T18:44:50Z

Fixed Mass Ratio

← Older revision		Revision as of 18:44, 3 December 2020
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	<td align="left">		<td align="left">
	<math>~\frac{2q}{3\pi} </math>		<math>~\frac{2q}{3\pi} </math>
			</td>
			</tr>
			</table>
			where [[#Surrounding_Torus\|again, as above]]
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\alpha_t \equiv \frac{a}{R_\mathrm{eq}} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[\frac{1}{3\Omega^2} (1 - e^2)^{1 / 2} \biggr]^{1 / 3} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Fixed Mass Ratio */

2020-12-03T18:41:37Z

Fixed Mass Ratio

← Older revision		Revision as of 18:41, 3 December 2020
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	Note that, for the limiting value, <math>~\beta_\mathrm{max} = (\alpha_t - 1)</math>, the ~~relevan~~ relation becomes,		Note that, for the limiting value, <math>~\beta_\mathrm{max} = (\alpha_t - 1)</math>, the relevant relation becomes,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="right">		<td align="right">
	<math>~		<math>~
	\alpha_t (1 - e^2)^{- 1 / 2}		(\alpha_t - 1)^2\alpha_t (1 - e^2)^{- 1 / 2}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\biggl(~~ \frac{2}{3\pi} ~~\biggr) \frac{q}{ \beta_t^2 }~~ </math>		<math>~\frac{2q}{3\pi} </math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Fixed Mass Ratio */

2020-12-03T18:37:06Z

Fixed Mass Ratio

← Older revision		Revision as of 18:37, 3 December 2020
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	</table>		</table>

	Note that, for the limiting value, <math>~\beta_\mathrm{max}</math>, ~~this~~ relation becomes,		Note that, for the limiting value, <math>~\beta_\mathrm{max} = (\alpha_t - 1)</math>, the relevan relation becomes,

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

			<tr>
			<td align="right">
			<math>~
			\alpha_t (1 - e^2)^{- 1 / 2}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } </math>
			</td>
			</tr>
			</table>


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Tohline: /* Fixed Mass Ratio */

2020-12-03T18:34:52Z

Fixed Mass Ratio

← Older revision		Revision as of 18:34, 3 December 2020
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	<td align="right">		<td align="right">
	<math>~\Rightarrow~~~		<math>~\Rightarrow~~~
	3\Omega^2 (1 - e^2)		\Omega^2 (1 - e^2)
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ \biggl( \frac{3\pi}{2} \biggr) \frac{ \beta_t^2 }{q} \biggr]^3 \, .</math>		<math>~\frac{1}{3}\biggl[ \biggl( \frac{3\pi}{2} \biggr) \frac{ \beta_t^2 }{q} \biggr]^3 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			But, for central models along the MLS, we also must satisfy the relation [[#Model_Sequences\|given above]], namely,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Omega^2 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2}(3 - 2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)}{e^2} \, .
			</math>
			</td>
			</tr>
			</table>
			Hence, for each <math>~(q, \beta_t)</math> parameter pair, the relevant central-object eccentricity is given by the root of the relation,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\frac{1}{2}(3 - 2e^2)(1 - e^2)^{3 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{3(1-e^2)^2 }{e^2}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{3}\biggl[ \biggl( \frac{3\pi}{2} \biggr) \frac{ \beta_t^2 }{q} \biggr]^3 \, .</math>
			</td>
			</tr>
			</table>

			Note that, for the limiting value, <math>~\beta_\mathrm{max}</math>, this relation becomes,


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Tohline: /* Fixed Mass Ratio */

2020-12-03T18:16:59Z

Fixed Mass Ratio

← Older revision		Revision as of 18:16, 3 December 2020
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \, .</math>		<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \, ,</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			that is, from the [[#Surrounding_Torus\|above expression for <math>~\alpha_t</math>]],
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>~
	\~~Rightarrow ~~~~~\~~alpha_t~~ (1 - e^2)^{- 1 / 2}		\biggl[\frac{1}{3\Omega^2} (1 - e^2)^{1 / 2} \biggr] (1 - e^2)^{- 3 / 2}
	</math>		</math>
	</td>		</td>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \, .</math>		<math>~\biggl[ \biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \biggr]^3</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			3\Omega^2 (1 - e^2)
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \biggl( \frac{3\pi}{2} \biggr) \frac{ \beta_t^2 }{q} \biggr]^3 \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Fixed Mass Ratio */

2020-12-03T18:10:23Z

Fixed Mass Ratio

← Older revision		Revision as of 18:10, 3 December 2020
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	</tr>		</tr>
	</table>		</table>

			The sequence is constructed by choosing <math>~q</math> (held fixed), and varying the value of <math>~0 < \beta_t \le \beta_\mathrm{max}</math>; then, for each chosen parameter pair, recognize that,

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

			<tr>
			<td align="right">
			<math>~
			\alpha_t (1 - e^2)^{- 1 / 2}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \, .</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~
			\Rightarrow ~~~\alpha_t (1 - e^2)^{- 1 / 2}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{2}{3\pi} \biggr) \frac{q}{ \beta_t^2 } \, .</math>
			</td>
			</tr>
			</table>


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Tohline: /* Spheroid-Ring Systems */

2020-12-03T17:58:09Z

Spheroid-Ring Systems

← Older revision		Revision as of 17:58, 3 December 2020
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			===Model Sequences===

			====Fixed Ω====
	<span id="jSquared">The Maclaurin sequence (MLS) is defined by the relations:</span>		<span id="jSquared">The Maclaurin sequence (MLS) is defined by the relations:</span>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	This matches the expression for the isolated Maclaurin spheroid [[#jSquared\|derived above]].		This matches the expression for the isolated Maclaurin spheroid [[#jSquared\|derived above]].

			====Fixed Mass Ratio====

			Let's define the mass ratio,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~q</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{M_t}{M_c} \, ,
			</math>
			</td>
			</tr>
			</table>

			and build a sequence along which this ratio is held constant. For the problem being considered here, the relevant expression for <math>~q</math> is,

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

			<tr>
			<td align="right">
			<math>~q</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ 2\pi^2 R_\mathrm{eq}^3 \rho \alpha_t \beta_t^2 \biggr]
			\biggl[ \frac{4\pi}{3} \rho R_\mathrm{eq}^3 (1 - e^2)^{1 / 2} \biggr]^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{3\pi}{2} \biggr) \alpha_t \beta_t^2 (1 - e^2)^{- 1 / 2}\, .
			</math>
			</td>
			</tr>
			</table>

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