Apps/MaclaurinToroid - Revision history

Jet53man: /* Overview */

2023-04-07T16:46:24Z

Overview

← Older revision		Revision as of 12:46, 7 April 2023
Line 3:		Line 3:
	=Overview=		=Overview=

	~~The following except has been taken from (p. 598 of) {{ HTE87full }}:~~

			<table border="0" align="center" width="80%" cellpadding="3"><tr><td align="left">
	<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/\|W\| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/\|W\| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>		<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/\|W\| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/\|W\| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>

	<font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/\|W\|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point. We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/\|W\| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>		<font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/\|W\|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point. We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/\|W\| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>
			</td></tr>
			<tr><td align="right">
			— Drawn from (p. 598 of) {{ HTE87full }}
			</td></tr>
			</table>

	<table border="1" align="center" cellpadding="5" width="80%">
			<table border="1" align="center" cellpadding="5" width="90%">
	<tr>		<tr>
	<td align="center>		<td align="center>

Jet53man: /* Overview */

2023-04-06T22:28:58Z

Overview

@@ Line 50: / Line 50: @@
    </li>
    <li>
    </li>
 </ol>

Jet53man: /* Maclaurin Toroid (MPT77) */

2023-04-06T22:07:56Z

Maclaurin Toroid (MPT77)

← Older revision		Revision as of 18:07, 6 April 2023
Line 90:		Line 90:
	{{ BO73 }}, Eq. (3)<br />		{{ BO73 }}, Eq. (3)<br />
	{{ EH85 }}, Eq. (1)<br />		{{ EH85 }}, Eq. (1)<br />
	{{ HTE87 }}, Eq. (1)		{{ HTE87 }}, Eq. (6)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Maclaurin Toroid (MPT77) */

2023-04-06T22:07:11Z

Maclaurin Toroid (MPT77)

← Older revision		Revision as of 18:07, 6 April 2023
Line 89:		Line 89:
	{{ BO70 }}, Eq. (12)<br />		{{ BO70 }}, Eq. (12)<br />
	{{ BO73 }}, Eq. (3)<br />		{{ BO73 }}, Eq. (3)<br />
	{{ EH85 }}, Eq. (1)		{{ EH85 }}, Eq. (1)<br />
			{{ HTE87 }}, Eq. (1)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Overview */

2023-04-06T22:06:15Z

Overview

@@ Line 9: / Line 9: @@
 <font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/|W|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point.  We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/|W| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>
-<table border="1" align="center" cellpadding="0">
+<table border="1" align="center" cellpadding="5" width="80%">
 <tr>
    <td align="center>
@@ Line 19: / Line 19: @@
 [[File:HTE78Fig10.png|700px|HTE78Fig10]]
    </td>
 </tr>
 </table>

Jet53man: /* Maclaurin Toroid (MPT77) */

2023-04-06T21:46:52Z

Maclaurin Toroid (MPT77)

← Older revision		Revision as of 17:46, 6 April 2023
Line 31:		Line 31:

	Here, we summarize the work of {{ MPT77full }} — hereafter, {{ MPT77hereafter }} — who constructed a sequence of toroidal-shaped, self-gravitating, incompressible configurations that are not uniformly rotating but, rather, have a distribution of angular momentum that is identical to the distribution found in a uniformly rotating, uniform-density sphere. As we have pointed out in our [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_(n'_=_0)\|associated overview of "simple rotation curves"]], this chosen (cylindrical) radial distribution of specific angular momentum is given by the expression,		Here, we summarize the work of {{ MPT77full }} — hereafter, {{ MPT77hereafter }} — who constructed a sequence of toroidal-shaped, self-gravitating, incompressible configurations that are not uniformly rotating but, rather, have a distribution of angular momentum that is identical to the distribution found in a uniformly rotating, uniform-density sphere. As we have pointed out in our [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_(n'_=_0)\|associated overview of "simple rotation curves"]], this chosen (cylindrical) radial distribution of specific angular momentum is given by the expression,
	~~<div align="center">~~
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 57:		Line 57:
	</tr>		</tr>
	</table>		</table>
	~~</div>~~
	where, <math>L</math> is the total angular momentum, <math>M</math> is the total mass, the mass fraction,		where, <math>L</math> is the total angular momentum, <math>M</math> is the total mass, the mass fraction,
	<div align="center">		<div align="center">

Jet53man: /* Overview */

2023-04-06T21:16:16Z

Overview

← Older revision		Revision as of 17:16, 6 April 2023
Line 6:		Line 6:

	<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/\|W\| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/\|W\| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>		<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/\|W\| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/\|W\| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>

			<font color="darkgreen">"{{ Bardeen71 }}, however, did not examine the behavior of axisymmetric modes higher than <math>P_4.</math> {{ EH85 }} have tabulated bifurcation points along the Maclaurin spheroidal sequence for a number of axisymmetric modes higher than <math>P_4</math> and have found the point that occurs at the lowest value of <math>T/\|W\|</math> on the sequence to be the <math>P_6(\eta)</math> and not at the <math>P_4(\eta)</math> bifurcation point. We suspect, therefore that when a general linear perturbation analysis is performed, the first dynamical axisymmetric (ring) mode instability will be found to set in at <math>T/\|W\| = 0.4512</math> and that the unstable mode will be of a <math>P_6(\eta)</math> geometric form."</font>

	<table border="1" align="center" cellpadding="0">		<table border="1" align="center" cellpadding="0">

Jet53man: /* Overview */

2023-04-06T21:08:45Z

Overview

← Older revision		Revision as of 17:08, 6 April 2023
Line 5:		Line 5:
	The following except has been taken from (p. 598 of) {{ HTE87full }}:		The following except has been taken from (p. 598 of) {{ HTE87full }}:

	<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ ~~EH85~~ }}, …</font>		<font color="darkgreen">"It has been previously stated in the literature that the Maclaurin toroid sequence bifurcates from the Maclaurin spheroidal sequence at the <math>P_4(\eta)</math> bifurcation point ({{ EH85full }}, … Since the <math>P_4(\eta)</math> bifurcation point occurs along the spheroidal sequence at <math>T/\|W\| = 0.4574</math> … this is synonymous with saying that Maclaurin spheroids that have <math>T/\|W\| < 0.4574</math> are dynamically stable against linear, axisymmetric perturbations. This conclusion is based, in large part, on the fact that</font> the linear perturbation analysis performed by {{ Bardeen71full }}… <font color="darkgreen">found that the ring mode instability against a <math>P_4(\eta)</math> mode perturbation sets in at exactly the <math>P_4(\eta)</math> bifurcation point."</font>

	<table border="1" align="center" cellpadding="0">		<table border="1" align="center" cellpadding="0">

Jet53man at 21:00, 6 April 2023

2023-04-06T21:00:17Z

@@ Line 1: / Line 1: @@
 __FORCETOC__ <!-- will force the creation of a Table of Contents -->
 <!-- __NOTOC__ will force TOC off -->
 =Maclaurin Toroid (MPT77)=
 {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"

Jet53man: /* Relationship with the Dyson-Wong One-Ring Sequence */

2023-04-03T22:47:57Z

Relationship with the Dyson-Wong One-Ring Sequence

← Older revision		Revision as of 18:47, 3 April 2023
Line 808:		Line 808:
	=Relationship with the Dyson-Wong One-Ring Sequence=		=Relationship with the Dyson-Wong One-Ring Sequence=

			{{ CKST95bfull }} discuss how to interpret the physical meaning of the Maclaurin Toroid sequence, especially in the context of its relationship to the Maclaurin spheroid sequence and to the Dyson-Wong one-ring sequence. In this discussion, reference is made to the work of {{ HTE87full }}.

	=See Also=		=See Also=