SSC/Stability/BiPolytrope00 - Revision history

Jet53man: /* Expectation */

2022-10-12T16:00:58Z

Expectation

← Older revision		Revision as of 12:00, 12 October 2022
Line 1,166:		Line 1,166:

	===Expectation===		===Expectation===
	As we have reviewed in [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Boundary_Condition\|an accompanying discussion]], one astrophysically reasonable surface boundary condition provides a mathematical relationship between the logarithmic derivative of the eigenfunction with respect to the radius, in terms of the eigenfrequency as follows:		As we have reviewed in [[SSC/Stability/BiPolytrope00Details#Boundary_Condition\|an accompanying discussion]], one astrophysically reasonable surface boundary condition provides a mathematical relationship between the logarithmic derivative of the eigenfunction with respect to the radius, in terms of the eigenfrequency as follows:

Jet53man: /* Duh! */

2022-10-11T23:27:32Z

Duh!

← Older revision		Revision as of 19:27, 11 October 2022
Line 1,491:		Line 1,491:
	===Duh!===		===Duh!===

	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This numerical determination of surface boundary conditions was carried out inside spreadsheet "FDflex22" of Excel file ''analyticeigenvectorcorrected.xlsx''.]]After also checking conformance with the expected boundary condition in [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Check_Surface_Boundary_Condition\|the case of analytic eigenfunctions having <math>~\ell = 3</math>]] and, separately (not shown), for numerically generated eigenfunctions having a wide range of oscillation frequencies, it dawned on us that the [[#DesiredBoundaryCondition\|"desired" surface boundary condition]] may actually be a natural outcome of the envelope's LAWE.		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 4 February 2017: This numerical determination of surface boundary conditions was carried out inside spreadsheet "FDflex22" of Excel file ''analyticeigenvectorcorrected.xlsx''.]]After also checking conformance with the expected boundary condition in [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Check_Surface_Boundary_Condition\|the case of analytic eigenfunctions having <math>\ell = 3</math>]] and, separately (not shown), for numerically generated eigenfunctions having a wide range of oscillation frequencies, it dawned on us that the [[#DesiredBoundaryCondition\|"desired" surface boundary condition]] may actually be a natural outcome of the envelope's LAWE.

	By constraining our discussion to models for which <math>~g^2 = \mathcal{B}</math> and <math>~\mathcal{D} = q^3</math>, the [[#The_Envelope.27s_LAWE\|envelope's LAWE]] is,		By constraining our discussion to models for which <math>~g^2 = \mathcal{B}</math> and <math>~\mathcal{D} = q^3</math>, the [[#The_Envelope.27s_LAWE\|envelope's LAWE]] is,
Line 1,556:		Line 1,556:
	</div>		</div>

	And this is ''precisely'' the condition that derives from the astrophysically reasonable boundary condition that we have [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Boundary_Condition\|discussed separately]] and that has been [[#Expectation\|reviewed, above]].		And this is ''precisely'' the condition that derives from the astrophysically reasonable boundary condition that we have [[SSC/Stability/BiPolytrope00Details#Boundary_Condition\|discussed separately]] and that has been [[#Expectation\|reviewed, above]].

	===Broader Analysis===		===Broader Analysis===

Jet53man: /* Five Mode Summary */

2022-10-11T22:50:21Z

Five Mode Summary

← Older revision		Revision as of 18:50, 11 October 2022
Line 1,992:		Line 1,992:
	<td align="right">		<td align="right">
	<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 37.08874</math></div>		<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 37.08874</math></div>
	<font size="-2">[[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration21\|more details …]]</font>		<font size="-2">[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration21\|more details …]]</font>
	</td>		</td>
	<td align="right">		<td align="right">
	<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 35.95210</math></div>		<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 35.95210</math></div>
	<font size="-2">[[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration22\|more details …]]</font>		<font size="-2">[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration22\|more details …]]</font>
	</td>		</td>
	<td align="center"> </td>		<td align="center"> </td>
Line 2,016:		Line 2,016:
	<td align="right">		<td align="right">
	<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 12.452545</math></div>		<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 12.452545</math></div>
	<font size="-2">[[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration31\|more details …]]</font>		<font size="-2">[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration31\|more details …]]</font>
	</td>		</td>
	<td align="right">		<td align="right">
	<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 35.05461</math></div>		<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 35.05461</math></div>
	<font size="-2">[[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration32\|more details …]]</font>		<font size="-2">[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration32\|more details …]]</font>
	</td>		</td>
	<td align="right">		<td align="right">
	<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 87.41594</math></div>		<div align="center"><math>~\frac{3\omega^2}{2\pi G\rho_c} = 87.41594</math></div>
	<font size="-2">[[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Illustration33\|more details …]]</font>		<font size="-2">[[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration33\|more details …]]</font>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Piecing Together */

2022-10-11T21:31:49Z

Piecing Together

Show changes

Jet53man: /* The Envelope's LAWE */

2022-10-11T21:21:06Z

The Envelope's LAWE

← Older revision		Revision as of 17:21, 11 October 2022
Line 306:		Line 306:
	</div>		</div>

	Next, [[SSC/Stability/BiPolytrope00Details#Eureka_Regarding_Prasad.27s_1948_Paper\|we noticed]] the strong similarities between the mathematical properties of this eigenvalue problem and the one that was studied by {{ Prasad48full }} in connection with, what we now recognize to be, a closely related problem. Drawing heavily from ~~Prasad~~'s analysis, we discovered an infinite number of eigenfunctions (each, a truncated polynomial expression) and associated eigenfrequencies that satisfy this governing envelope LAWE. The eigenvectors associated with the lowest few modes are tabulated, below.		Next, [[SSC/Stability/BiPolytrope00Details#Eureka_Regarding_Prasad.27s_1948_Paper\|we noticed]] the strong similarities between the mathematical properties of this eigenvalue problem and the one that was studied by {{ Prasad48full }} in connection with, what we now recognize to be, a closely related problem. Drawing heavily from {{ Prasad48hereafter }}'s analysis, we discovered an infinite number of eigenfunctions (each, a truncated polynomial expression) and associated eigenfrequencies that satisfy this governing envelope LAWE. The eigenvectors associated with the lowest few modes are tabulated, below.

	==Eigenvector==		==Eigenvector==

Jet53man: /* Related Discussions */

2022-10-11T21:08:39Z

Related Discussions

← Older revision		Revision as of 17:08, 11 October 2022
Line 2,358:		Line 2,358:

	=Related Discussions=		=Related Discussions=
	* [[~~User:Tohline/~~Appendix/Ramblings/~~Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes~~#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]		* [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]
	* [[~~User:Tohline/~~Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|Numerically Determined Eigenvectors]]		* [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|Numerically Determined Eigenvectors]]

	<!-- OLD STUFF ...		<!-- OLD STUFF ...

Jet53man: /* Old, Incorrect Solutions */

2022-10-11T21:05:06Z

Old, Incorrect Solutions

Show changes

Jet53man: /* Example Solutions */

2022-10-11T18:47:29Z

Example Solutions

← Older revision		Revision as of 14:47, 11 October 2022
Line 986:		Line 986:


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

	===Old, Incorrect Solutions===		===Old, Incorrect Solutions===

Jet53man: /* Example Solutions */

2022-10-11T18:45:37Z

Example Solutions

← Older revision		Revision as of 14:45, 11 October 2022
Line 983:		Line 983:
	</div>		</div>

	Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[~~User:Tohline/~~Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]]. Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>~(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ''analytically defined'' eigenfunction displayed as Model C21 in [[#Table3\|Table 3]], but it has been rescaled — and by necessity inverted — to provide a central value of unity.		Each of the six plots displayed in [[#Table4\|Table 4]] (click on a panel in order to view a larger image) was [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|generated numerically]] by integrating the LAWE for the core, outward from the center of the configuration to the core/envelope interface, then integrating the LAWE for the envelope, from the interface outward to the surface. At the interface:   the ''value'' of the envelope eigenfunction is set to the ''value'' of the eigenfunction of the core; and the ''slope'' of the envelope's eigenfunction (highlighted graphically in each plot by the green, dashed line segment) was based on the slope of the core's eigenfunction (highlighted graphically by the orange, dashed line segment) but shifted in a discontinuous fashion [[#Step4\|according to the above "Step 4" discussion]]. Each of the graphically illustrated Table 4 eigenfunctions has been scaled in such a way that the central value is unity; note that the panel labeled <math>(\ell=2; \sigma_c^2 = +16)</math> displays an eigenfunction that is identical to the ''analytically defined'' eigenfunction displayed as Model C21 in [[#Table3\|Table 3]], but it has been rescaled — and by necessity inverted — to provide a central value of unity.

Jet53man: /* Piecing Together */

2022-10-11T18:38:22Z

Piecing Together

← Older revision		Revision as of 14:38, 11 October 2022
Line 596:		Line 596:
	</table>		</table>
	</div>		</div>
	In the context of this interface-matching constraint (see their equation 62.1)~~, [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)] state the following~~:   <font color="darkgreen">~~<b>~~In the static~~</b>~~</font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen"~~><b~~>model~~</b>~~</font> … <font color="darkgreen"~~><b~~>discontinuities in <math>~\rho</math> or in <math>~\gamma</math> might occur at some [radius]~~</b>~~</font>. <font color="darkgreen"~~><b~~>In the first case~~</b>~~</font> — that is, a discontinuity only in density, while <math>~\gamma_e = \gamma_c</math> — the interface conditions <font color="darkgreen"~~><b~~>imply the continuity of <math>~\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius]. In the second case~~</b>~~</font> — that is, a discontinuity in the adiabatic exponent — <font color="darkgreen"~~><b~~>the dynamical condition may be written~~</b>~~</font> as above. <font color="darkgreen"~~><b~~>This implies a discontinuity of the first derivative at any discontinuity of <math>~\gamma</math>~~</b>~~</font>.
			In the context of this interface-matching constraint, {{ LW58full }} state the following (see their equation 62.1):   <font color="darkgreen">"In the static</font> (''i.e.,'' unperturbed equilibrium) <font color="darkgreen">model</font> … <font color="darkgreen">discontinuities in <math>\rho</math> or in <math>\gamma</math> might occur at some [radius]</font>. <font color="darkgreen">In the first case</font> — that is, a discontinuity only in density, while <math>\gamma_e = \gamma_c</math> — the interface conditions <font color="darkgreen">imply the continuity of <math>\tfrac{1}{x} \cdot \tfrac{dx}{d\xi}</math> at that [radius]. In the second case</font> — that is, a discontinuity in the adiabatic exponent — <font color="darkgreen">the dynamical condition may be written</font> as above. <font color="darkgreen">This implies a discontinuity of the first derivative at any discontinuity of <math>\gamma</math>."</font>

	When evaluated at the interface, the logarithmic derivatives of our pair of parameterized eigenfunction expressions are, respectively,		When evaluated at the interface, the logarithmic derivatives of our pair of parameterized eigenfunction expressions are, respectively,
Line 857:		Line 858:


	The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0Details~~#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which the values of the two eigenfunctions match at the interface, and for which the first derivatives of the two eigenfunctions are discontinuous by the properly prescribed amount at the interface.		The physically relevant (real) root of this quartic equation in <math>~\Chi</math> — see our [[SSC/Stability/BiPolytrope00Details#Quartic\|accompanying detailed presentation]] — gives us the ''specific'' value of the dimensionless interface location, <math>~q</math>, for which the values of the two eigenfunctions match at the interface, and for which the first derivatives of the two eigenfunctions are discontinuous by the properly prescribed amount at the interface.

	===Example Solutions===		===Example Solutions===