Apps/MaclaurinSpheroidSequence - Revision history

Jet53man: /* The Perturbed Configuration */

2023-04-20T19:11:31Z

The Perturbed Configuration

← Older revision		Revision as of 15:11, 20 April 2023
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	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">
	<tr>		<tr>
	<td align="right"><math>\Psi</math></td>		<td align="right" width="40%"><math>\Psi</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left"><math>- ~\int \frac{h^2(\varpi)}{\varpi^3} d\varpi \, ,</math></td>		<td align="left"><math>- ~\int \frac{h^2(\varpi)}{\varpi^3} d\varpi \, ,</math></td>
	~~</math>~~
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" colspan="3">{{ MPT77 }}, §III (p. 590), eq. (3.4)</td>		<td align="center" colspan="3">
			{{ OM68 }}, §IIId (p. 1084), eq. (44)<br />
			{{ MPT77 }}, §III (p. 590), eq. (3.4)
			</td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* The Perturbed Configuration */

2023-04-20T18:52:15Z

The Perturbed Configuration

@@ Line 1,743: / Line 1,743: @@
 </div>
 In both of these expressions, <math>\Phi</math> is the gravitational potential.  In the algebraic expression, <math>C_0</math> is a constant throughout the volume, and on the surface, of the equilibrium configuration.  Here, we seek a uniform-density (incompressible) configuration, in which [[SR#Time-Independent_Problems|the enthalpy]], <math>H = P/\rho</math>, goes to zero at all points across the surface.  And the centrifugal potential, <math>\Psi</math>, [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|is given by the expression]]
-<div align="center">
-<math>
+<table border="0" align="center" cellpadding="5">
-\Psi \equiv - \int \frac{h^2(\varpi)}{\varpi^3} d\varpi \, ,
+<tr>
 </math>
-</div>
+</tr>
 where, the (cylindrical) radial distribution of the specific angular momentum,
 <div align="center">
@@ Line 1,768: / Line 1,775: @@
 ===Uniform Rotation===
-{{ HE83 }} sought to find bifurcation points along the Maclaurin spheroid sequence where the associated, deformed equilibrium configuration was uniformly rotating.  From the set of familiar [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]], therefore, they set
+{{ HE83 }} sought to find bifurcation points along the Maclaurin spheroid sequence where the associated, deformed equilibrium configuration is uniformly rotating.  From the set of familiar [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]], therefore, they set
 <div align="center">
 <math>
@@ Line 1,799: / Line 1,806: @@
 </table>
 which is the same as their Appendix (&sect;A.1) Eq. (6), except they chose a different sign when defining the constant, <math>C_0</math>.
 ==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==

Jet53man: /* The Perturbed Configuration */

2023-04-20T17:57:02Z

The Perturbed Configuration

← Older revision		Revision as of 13:57, 20 April 2023
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	<div align="center">		<div align="center">
	<math>		<math>
	\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi \, ,		\Psi \equiv - \int \frac{h^2(\varpi)}{\varpi^3} d\varpi \, ,
	</math>		</math>
	</div>		</div>
	where, the (cylindrical) radial distribution of the specific angular momentum,		where, the (cylindrical) radial distribution of the specific angular momentum,
	<div align="center">		<div align="center">
	<math>~~~j(\varpi,z) = j~~(\varpi) = \varpi^2 \dot\varphi(\varpi) \, ,</math>		<math>h(\varpi) = \varpi^2 \dot\varphi(\varpi) \, ,</math>
	</div>		</div>
	is to be specified according to the physical problem in hand — usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]." Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,		is to be specified according to the physical problem in hand — usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]." Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="right"><math>\Phi - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi</math></td>		<td align="right"><math>\Phi - \int \frac{h^2(\varpi)}{\varpi^3} d\varpi</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>C_0 \, .</math></td>		<td align="left"><math>C_0 \, .</math></td>

Jet53man: /* The Perturbed Configuration */

2023-04-20T17:47:22Z

The Perturbed Configuration

← Older revision		Revision as of 13:47, 20 April 2023
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	</div>		</div>
	is to be specified according to the physical problem in hand — usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]." Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,		is to be specified according to the physical problem in hand — usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]." Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,
	<~~div~~ align="center">		<table border="0" cellpadding="5" align="center">
	<math>\Phi - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = C_0</math> .		<tr>
	</~~div~~>		<td align="right"><math>\Phi - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>C_0 \, .</math></td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			{{ EH85 }}, §2.1 (p. 290), Eq. (5)
			</td>
			</tr>
			</table>

	===Uniform Rotation===		===Uniform Rotation===

Jet53man: /* The Perturbed Configuration */

2023-04-20T00:20:40Z

The Perturbed Configuration

@@ Line 1,756: / Line 1,756: @@
 <math>\Phi - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = C_0</math> .
 </div>
 ==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==

Jet53man: /* The Perturbed Configuration */

2023-04-19T23:49:04Z

The Perturbed Configuration

← Older revision		Revision as of 19:49, 19 April 2023
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	</math>		</math>
	</div>		</div>
	where,		where, the (cylindrical) radial distribution of the specific angular momentum,
	<div align="center">		<div align="center">
	<math>~j(\varpi,z) = j(\varpi) = \varpi^2 \dot\varphi(\varpi) \, .</math>		<math>~j(\varpi,z) = j(\varpi) = \varpi^2 \dot\varphi(\varpi) \, ,</math>
			</div>
			is to be specified according to the physical problem in hand — usually chosen from a familiar set of "[[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|simple rotation profiles]]." Therefore, across the surface of each equilibrium configuration, the algebraic expression for hydrostatic balance takes the form,
			<div align="center">
			<math>\Phi - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = C_0</math> .
	</div>		</div>

Jet53man: /* Bifurcation Points Along Maclaurin-Spheroid Sequence */

2023-04-19T23:26:21Z

Bifurcation Points Along Maclaurin-Spheroid Sequence

← Older revision		Revision as of 19:26, 19 April 2023
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	=Bifurcation Points Along Maclaurin-Spheroid Sequence=		=Bifurcation Points Along Maclaurin-Spheroid Sequence=

			==The Perturbed Configuration==

			Referencing the [[AxisymmetricConfigurations/HSCF#Constructing_Two-Dimensional,_Axisymmetric_Structures\|Hachisu Self-Consistent Field (HSCF) technique]], our objective is to solve an ''algebraic'' expression for hydrostatic balance,
			<div align="center">
			<math>~H + \Phi + \Psi = C_0</math> ,
			</div>
			in conjunction with the Poisson equation in a form that is appropriate for two-dimensional, axisymmetric systems — written in cylindrical coordinates, for example,
			<div align="center">
			<math>~
			\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
			</math>
			</div>
			In both of these expressions, <math>\Phi</math> is the gravitational potential. In the algebraic expression, <math>C_0</math> is a constant throughout the volume, and on the surface, of the equilibrium configuration. Here, we seek a uniform-density (incompressible) configuration, in which [[SR#Time-Independent_Problems\|the enthalpy]], <math>H = P/\rho</math>, goes to zero at all points across the surface. And the centrifugal potential, <math>\Psi</math>, [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|is given by the expression]]
			<div align="center">
			<math>
			\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi \, ,
			</math>
			</div>
			where,
			<div align="center">
			<math>~j(\varpi,z) = j(\varpi) = \varpi^2 \dot\varphi(\varpi) \, .</math>
			</div>

	==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==		==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==

Jet53man: /* Oblate Spheroidal Coordinates */

2023-04-19T22:17:49Z

Oblate Spheroidal Coordinates

← Older revision		Revision as of 18:17, 19 April 2023
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	{{ Bardeen71 }}, §IV, p. 429, Eq. (12)		{{ Bardeen71 }}, §IV, p. 429, Eq. (12)<br />
			{{ HE83 }}, §A.1, p. 587, Eq. (1)
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
			{{ HE83 }}, §A.2, p. 588, Eq. (10)<br />
	{{ HTE87 }}, p. 610, Eq. (A5)		{{ HTE87 }}, p. 610, Eq. (A5)
	</td>		</td>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
			{{ HE83 }}, §A.2, p. 588, Eq. (9)<br />
	{{ HTE87hereafter }}, p. 610, Eq. (A4)		{{ HTE87hereafter }}, p. 610, Eq. (A4)
	</td>		</td>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
			{{ HE83 }}, §A.2, p. 588, Eq. (11)<br />
	{{ HTE87hereafter }}, p. 610, Eq. (A6)		{{ HTE87hereafter }}, p. 610, Eq. (A6)
	</td>		</td>

Jet53man at 21:51, 19 April 2023

2023-04-19T21:51:15Z

← Older revision		Revision as of 17:51, 19 April 2023
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	</tr>		</tr>
	</table>		</table>

	~~=Bifurcation Points Along Maclaurin-Spheroid Sequence=~~

	==Oblate Spheroidal Coordinates==		==Oblate Spheroidal Coordinates==
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	</td></tr>		</td></tr>
	</table>		</table>

			=Bifurcation Points Along Maclaurin-Spheroid Sequence=

	==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==		==Particularly Interesting Models Along the Maclaurin Spheroid Sequence==

Jet53man: /* Particularly Interesting Models Along the Maclaurin Spheroid Sequence */

2023-04-11T17:49:38Z

Particularly Interesting Models Along the Maclaurin Spheroid Sequence

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