Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes - Revision history

Jet53man: /* Try Mode33 ~(\ell,j) = (3,3) */

2022-10-16T21:05:56Z

Try Mode33 ~(\ell,j) = (3,3)

← Older revision		Revision as of 17:05, 16 October 2022
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	The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.		The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.

	==Try Mode33 ~~<math>~~~(\ell,j) = (3,3)~~</math>~~==		==Try Mode33 (&ell;, j) = (3, 3)==

	===Setup33===		===Setup33===

Jet53man: /* Try Mode32 (\ell,j) = (3,2) */

2022-10-16T21:05:05Z

Try Mode32 (\ell,j) = (3,2)

← Older revision		Revision as of 17:05, 16 October 2022
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	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

	==Try Mode32 ~~<math>~~(\ell,j) = (3,2)~~</math>~~==		==Try Mode32 (&ell;, j) = (3, 2)==

	===Setup32===		===Setup32===

Jet53man: /* Try Mode31 ~(\ell,j) = (3,1) */

2022-10-16T21:03:07Z

Try Mode31 ~(\ell,j) = (3,1)

← Older revision		Revision as of 17:03, 16 October 2022
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	The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.		The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.

	==Try Mode31 ~~<math>~~~(\ell,j) = (3,1)~~</math>~~==		==Try Mode31 (&ell;, j) = (3, 1)==
	===Setup31===		===Setup31===

Jet53man: /* Try Mode22 ~(\ell,j) = (2,2) */

2022-10-16T21:01:34Z

Try Mode22 ~(\ell,j) = (2,2)

← Older revision		Revision as of 17:01, 16 October 2022
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	The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.		The black-dotted curve in the ''Bottom-Right Panel'' displays the behavior of this constraint; accordingly, the step function depicted by the solid black line segments must necessarily drop from unity to a point on this black-dotted curve for any equilibrium model — such as the example illustrated here — that has an analytically prescribable radial-oscillation eigenvector.

	==Try Mode22 ~~<math>~~~(\ell,j) = (2,2)~~</math>~~==		==Try Mode22 (&ell;, j) = (2, 2)==

	===Setup22===		===Setup22===

Jet53man: /* Seek Alternate Solution to Mode21 (\ell,j) = (2,1) */

2022-10-16T21:00:28Z

Seek Alternate Solution to Mode21 (\ell,j) = (2,1)

← Older revision		Revision as of 17:00, 16 October 2022
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	==Seek Alternate Solution to Mode21 ~~<math>~~(\ell,j) = (2,1)~~</math>~~==		==Seek Alternate Solution to Mode21 (&ell;, j) = (2, 1)==

	===Setup21===		===Setup21===

Jet53man: /* Summary Plots */

2022-04-12T18:08:13Z

Summary Plots

← Older revision		Revision as of 14:08, 12 April 2022
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	<table border="1" cellpadding="5">		<table border="1" cellpadding="5">
	<tr>		<tr>
	<td align="center" bgcolor="white"><math>~(\ell, j) = (2,1)</math></td>		<td align="center" bgcolor="white"><math>(\ell, j) = (2,1)</math></td>
	<td align="center" bgcolor="white"><math>~(\ell, j) = (2,2)</math></td>		<td align="center" bgcolor="white"><math>(\ell, j) = (2,2)</math></td>
	<td align="center" bgcolor="white"> </td>		<td align="center" bgcolor="white"> </td>
	</tr>		</tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center" bgcolor="white"><math>~(\ell, j) = (3,1)</math></td>		<td align="center" bgcolor="white"><math>(\ell, j) = (3,1)</math></td>
	<td align="center" bgcolor="white"><math>~(\ell, j) = (3,2)</math></td>		<td align="center" bgcolor="white"><math>(\ell, j) = (3,2)</math></td>
	<td align="center" bgcolor="white"><math>~(\ell, j) = (3,3)</math></td>		<td align="center" bgcolor="white"><math>(\ell, j) = (3,3)</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 12.452545</math></td>		<td align="center"><math>\frac{3\omega^2}{2\pi G\rho_c} = 12.452545</math></td>
	<td align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 35.05461</math></td>		<td align="center"><math>\frac{3\omega^2}{2\pi G\rho_c} = 35.05461</math></td>
	<td align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 87.41594</math></td>		<td align="center"><math>\frac{3\omega^2}{2\pi G\rho_c} = 87.41594</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Check Surface Boundary Condition */

2022-04-12T18:03:24Z

Check Surface Boundary Condition

← Older revision		Revision as of 14:03, 12 April 2022
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	</div>		</div>

	which precisely matches the desired surface boundary condition for <math>~\ell=3</math> that has been detailed in [[SSC/Stability/BiPolytrope00#DesiredBoundaryCondition\|an accompanying summary discussion]].		which precisely matches the desired surface boundary condition for <math>\ell=3</math> that has been detailed in [[SSC/Stability/BiPolytrope00#DesiredBoundaryCondition\|an accompanying summary discussion]].


	{{ ~~LSU_WorkInProgress~~ }}		{{ SGFworkInProgress }}

	==Try Mode32 <math>(\ell,j) = (3,2)</math>==		==Try Mode32 <math>(\ell,j) = (3,2)</math>==

Jet53man: /* Solve Cubic Equation */

2022-04-12T17:45:30Z

Solve Cubic Equation

← Older revision		Revision as of 13:45, 12 April 2022
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	===Solve Cubic Equation===		===Solve Cubic Equation===

	Here we draw from a [[Appendix/Ramblings/~~PPTori~~#Cubic_Equation_Solution\|separate discussion of solutions to a cubic equation]].		Here we draw from a [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution\|separate discussion of solutions to a cubic equation]].

	<div align="center">		<div align="center">
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	Upon evaluation, we have found that the expression inside of the square root is negative over the region of parameter space that is of most physical interest. Hence, we need to call upon [[Appendix/Ramblings/~~PPTori~~#Cubic_Equation_Solution\|a separate discussion]] in which the cube root of complex numbers was discussed.		Upon evaluation, we have found that the expression inside of the square root is negative over the region of parameter space that is of most physical interest. Hence, we need to call upon [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution\|a separate discussion]] in which the cube root of complex numbers was discussed.

Jet53man at 17:42, 12 April 2022

2022-04-12T17:42:56Z

Show changes

Jet53man: /* Searching for Additional Eigenvectors of Zero-Zero Bipolytropes */

2022-04-12T17:36:05Z

Searching for Additional Eigenvectors of Zero-Zero Bipolytropes

← Older revision		Revision as of 13:36, 12 April 2022
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Searching for Additional Eigenvectors of Zero-Zero Bipolytropes=		=Searching for Additional Eigenvectors of Zero-Zero Bipolytropes=
	This chapter is an extension of two accompanying discussions: [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|The original ''discovery'' and detailed derivation]]; and the [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|more readable, summary]].		This chapter is an extension of two accompanying discussions: [[SSC/Stability/BiPolytrope00Details#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|The original ''discovery'' and detailed derivation]]; and the [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|more readable, summary]].

	In our [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|accompanying summary]], we have demonstrated how analytically specified eigenvectors can be constructed for the mode labeled, <math>~(\ell, j) = (2,1)</math>. This was done by specifying <math>~\gamma_e</math>, then solving a quartic equation for <math>~q</math>. Shortly after completing this summary chapter, we noticed that an alternate approach may be to specify <math>~q</math>, then solve for <math>~\gamma_e</math>; and this path may be simpler because it may only involve solution of a quadratic equation. (Actually, we later have realized that the relevant equation is cubic, rather than quadratic. This is nevertheless simpler than the quartic equation.) If this proves to be the case, then it may also be possible to analytically construct eigenvectors of additional modes. Let's see.		In our [[SSC/Stability/BiPolytrope00#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|accompanying summary]], we have demonstrated how analytically specified eigenvectors can be constructed for the mode labeled, <math>(\ell, j) = (2,1)</math>. This was done by specifying <math>\gamma_e</math>, then solving a quartic equation for <math>q</math>. Shortly after completing this summary chapter, we noticed that an alternate approach may be to specify <math>q</math>, then solve for <math>\gamma_e</math>; and this path may be simpler because it may only involve solution of a quadratic equation. (Actually, we later have realized that the relevant equation is cubic, rather than quadratic. This is nevertheless simpler than the quartic equation.) If this proves to be the case, then it may also be possible to analytically construct eigenvectors of additional modes. Let's see.


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	<table border="1" cellpadding="8" align="center" width="70%">		<table border="1" cellpadding="8" align="center" width="70%">
	<tr><td align="left">		<tr><td align="left">
	<font color="red">'''CAUTION!'''</font>   All of the "core" eigenfunction expressions are suspect because (I think!) they have been developed herein under the assumption that <math>~g^{-2} = (1+2q^3)^2</math>, whereas the relation should read,		<font color="red">'''CAUTION!'''</font>   All of the "core" eigenfunction expressions are suspect because (I think!) they have been developed herein under the assumption that <math>g^{-2} = (1+2q^3)^2</math>, whereas the relation should read,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{1}{g^2}</math>		<math>\frac{1}{g^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{(1+2q^3)^2}{1+8q^3} = \frac{(1+2\Chi)^2}{1+8\Chi }\, .</math>		<math>\frac{(1+2q^3)^2}{1+8q^3} = \frac{(1+2\Chi)^2}{1+8\Chi }\, .</math>
	</td>		</td>
	</tr>		</tr>
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	==Seek Alternate Solution to Mode21 <math>~(\ell,j) = (2,1)</math>==		==Seek Alternate Solution to Mode21 <math>(\ell,j) = (2,1)</math>==


	===Setup21===		===Setup21===
	According to [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#STEP4\|STEP 4 in our accompanying summary discussion]], we need to solve the following "derivative matching" expression:		According to [[SSC/Stability/BiPolytrope00#STEP4\|STEP 4 in our accompanying summary discussion]], we need to solve the following "derivative matching" expression:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">