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

Tohline: /* Unconstrained LAWEs */

2017-02-03T22:59:27Z

Unconstrained LAWEs

← Older revision		Revision as of 22:59, 3 February 2017
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{ ~~- 2~~\Delta [ ~~\Delta~~( 3 +4\mathcal{H} - 6 \mathcal{D} )		\frac{ \Delta [ 2( 1 + \mathcal{H} - \mathcal{D} )- \Delta( 3 +4\mathcal{H} - 6 \mathcal{D} ) ] x_q'
	- 2( 1 + \mathcal{H} - \mathcal{D} )] x_q'		+ x_a [ 2( 1 + \mathcal{H} - \mathcal{D} )
	+ x_a [ 4( 1 + \mathcal{H} - \mathcal{D} )		- \Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} -\alpha_e ) ] }{ 2 ( 1 + \mathcal{H} - \mathcal{D} ) } \, .
	- 2\Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} -\alpha_e ) ] }{ 4 ( 1 + \mathcal{H} - \mathcal{D} ) } \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Unconstrained LAWEs */

2017-02-03T22:55:33Z

Unconstrained LAWEs

← Older revision		Revision as of 22:55, 3 February 2017
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	<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e = \frac{1}{\gamma_e}\biggl[\frac{\sigma_c^2}{(\rho_e/\rho_c)} + 8\biggr] - 6		<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e = \frac{1}{\gamma_e}\biggl[\frac{\sigma_c^2}{(\rho_e/\rho_c)} + 8\biggr] - 6
	\, .		\, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			After defining the new parameter,
			<div align="center">
			<math>~\mathcal{H} \equiv \frac{g^2 - \mathcal{B}}{\mathcal{A}} \, ,</math>
			</div>

			this LAWE for the envelope may be written as,

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

			<tr>
			<td align="right">
			<math>~a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 ) x_a'' </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- ( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a x_a'
			-
			[ \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ]x_a \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Inserting the same finite-difference expressions for the first and second derivatives, we therefore have,

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

			<tr>
			<td align="right">
			<math>~a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )\biggl[ \frac{x_+ - 2x_a + x_-}{\Delta^2} \biggr] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- ( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a \biggl[ \frac{x_+ - x_-}{2 \Delta} \biggr]
			- [ \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ]x_a
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )\biggl[ \frac{x_+ }{\Delta^2} \biggr]
			+ ( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a \biggl[ \frac{x_+ }{2 \Delta} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a \biggl[ \frac{x_-}{2 \Delta} \biggr]
			- a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )\biggl[ \frac{x_-}{\Delta^2} \biggr]
			- [ \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ]x_a + a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )\biggl[ \frac{2x_a }{\Delta^2} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~x_+ [2 a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )
			+ \Delta ( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a ]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			x_- [ \Delta( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a
			- 2a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )]
			+ x_a [ 4a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )
			- 2\Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ) ] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			Now, at the interface (only), we need to relate <math>~x_-</math> to <math>~x_+</math> in such a way that the slope gives the proper value at the interface. Specifically, we need to set,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~x_-</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~x_+ - 2\Delta (x_q') \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where, <math>~x_q'</math> takes the [[#InterfaceSlope\|value that was determined for the core]]. Hence, ''at'' the interface <math>~(a = 1)</math>, the first step into the envelope is special and demands that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~x_+ [2 a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )
			+ \Delta ( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a ]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			[x_+ - 2\Delta (x_q')][ \Delta( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a
			- 2a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )]
			+ x_a [ 4a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )
			- 2\Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ) ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ x_+ [4 a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 ) ]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- 2\Delta [ \Delta( 3 +4\mathcal{H}a - 6 \mathcal{D} a^3 ) a
			- 2a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )] x_q'
			+ x_a [ 4a^2( 1 + \mathcal{H}a - \mathcal{D} a^3 )
			- 2\Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} a^3 -\alpha_e ) ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			and, setting,       <math>~a = 1 ~~~~\Rightarrow ~~~
			x_+
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{ - 2\Delta [ \Delta( 3 +4\mathcal{H} - 6 \mathcal{D} )
			- 2( 1 + \mathcal{H} - \mathcal{D} )] x_q'
			+ x_a [ 4( 1 + \mathcal{H} - \mathcal{D} )
			- 2\Delta^2( \mathcal{D} \mathfrak{F}_\mathrm{env} -\alpha_e ) ] }{ 4 ( 1 + \mathcal{H} - \mathcal{D} ) } \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Unconstrained LAWEs */

2017-02-03T22:14:24Z

Unconstrained LAWEs

← Older revision		Revision as of 22:14, 3 February 2017
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	(g^2 - \xi^2)\frac{d^2x}{d\xi^2} +		(g^2 - \xi^2)\frac{d^2x}{d\xi^2} +
	( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}		( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}
	+ \mathfrak{F}_\mathrm{core} x \, .		+ \mathfrak{F}_\mathrm{core} x \, ,
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>
			and we should be able to numerically integrate from the center, outward through the core, exactly as [[#Integrating_Outward_Through_the_Core\|described above]].

	Separately, the [[User:Tohline/SSC/Stability/BiPolytrope0_0#The_Envelope.27s_LAWE\|LAWE for the envelope]] is,		Separately, the [[User:Tohline/SSC/Stability/BiPolytrope0_0#The_Envelope.27s_LAWE\|LAWE for the envelope]] is,

Tohline: /* Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope */ Begin subsection using the "unconstrained LAWEs"

2017-02-03T22:02:36Z

Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope: Begin subsection using the "unconstrained LAWEs"

← Older revision		Revision as of 22:02, 3 February 2017
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	[[File:NumericanModel31.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration31]]		[[File:NumericanModel31.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration31]]
	</td></tr>		</td></tr>
			</table>
			</div>

			==Unconstrained LAWEs==
			Here we use the most general expressions for the pair of governing LAWEs; that is, we will ''not'' force <math>~g^2 = \mathcal{B}</math>. Drawing from our most general summary discussion, the [[User:Tohline/SSC/Stability/BiPolytrope0_0#The_Core.27s_LAWE\|LAWE for the core]] is,

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

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(1 - \eta^2)\frac{d^2x}{d\eta^2} +
			( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta}
			+ \mathfrak{F}_\mathrm{core} x \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			where,
			<div align="center">
			<math>~\eta \equiv \frac{\xi}{g} \, ,</math>
			and
			<math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c = \frac{(\sigma_c^2 +8)}{\gamma_c} -6 \, ,</math>
			</div>
			and,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g^2</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			1 + \biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) +
			\frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Hence, we may also write the core's LAWE as,

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

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(g^2 - \xi^2)\frac{d^2x}{d\xi^2} +
			( 4g^2 - 6\xi^2 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}
			+ \mathfrak{F}_\mathrm{core} x \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


			Separately, the [[User:Tohline/SSC/Stability/BiPolytrope0_0#The_Envelope.27s_LAWE\|LAWE for the envelope]] is,

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

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ 1 + \frac{(g^2-\mathcal{B}) \xi}{\mathcal{A}} - \mathcal{D} \xi^3\biggr] \frac{d^2x}{d\xi^2}
			+ \biggl\{ 3 + \frac{4(g^2-\mathcal{B}) \xi}{\mathcal{A}} - 6\mathcal{D} \xi^3 \biggr\}
			\frac{1}{\xi} \cdot \frac{dx}{d\xi}
			+ \biggl[
			\mathcal{D} \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e
			\biggr]\frac{x}{\xi^2} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mathcal{A}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~2\biggl(\frac{\rho_e}{\rho_c}\biggr) \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathcal{B}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~1 + 2\biggl(\frac{\rho_e}{\rho_c}\biggr) - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathcal{D}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{1}{\mathcal{A}}\biggl( \frac{\rho_e}{\rho_c}\biggr)^2 = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr]
			\, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\mathfrak{F}_\mathrm{env}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e = \frac{1}{\gamma_e}\biggl[\frac{\sigma_c^2}{(\rho_e/\rho_c)} + 8\biggr] - 6
			\, .
			</math>
			</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Results */ Insert sub subsection titled, "Motivated by Analytic31"

2017-01-27T23:45:54Z

Results: Insert sub subsection titled, "Motivated by Analytic31"

← Older revision		Revision as of 23:45, 27 January 2017
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	<tr><td align="center">		<tr><td align="center">
	[[File:NumericalModel22.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]		[[File:NumericalModel22.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]
			</td></tr>
			</table>
			</div>


			====Motivated by Analytic31====

			We have also numerically constructed an eigenfunction that matches our [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Illustration31\|accompanying analytic Illustration31]]. In Figure 4, the numerically derived solution has been plotted on top of the analytically derived solution.
			<div align="center" id="Figure4">
			<table border="1" align="center" cellpadding="5">
			<tr><th align="center">
			<font size="+1">Figure 4:</font><br />
			<br />
			<math>~(q,\gamma_e,\gamma_c) = ( 0.4059596, 1.180462, 1.008887)</math>
			</th></tr>
			<tr><td align="center">
			[[File:NumericanModel31.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration31]]
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Varying the Oscillation Frequency */

2017-01-23T22:30:39Z

Varying the Oscillation Frequency

← Older revision		Revision as of 22:30, 23 January 2017
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\mathfrak{F}_\mathrm{~~core~~}</math>		<math>~\mathfrak{F}_\mathrm{env}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 2,035:		Line 2,035:

	====Motivated by Analytic21====		====Motivated by Analytic21====
	~~Using~~ the above-described numerical algorithm, ~~we have constructed~~ 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>. The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession — in order of ''decreasing'' values of <math>~\sigma_c^2</math> — in the form of a looped animation sequence. Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.		Continuing with our analysis of the equilibrium model that is defined by the parameters, <math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>, we have used the above-described numerical algorithm, to construct 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>. The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession — in order of ''decreasing'' values of <math>~\sigma_c^2</math> — in the form of a looped animation sequence. Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.

	<div align="center" id="Figure2">		<div align="center" id="Figure2">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr><th align="center">Figure 2</th></tr>		<tr><th align="center">
			<font size="+1">Figure 2:</font><br />
			<br />
			<math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>
			</th></tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:ImageTrio.png\|500px\|center\|Three movie frames]]		[[File:ImageTrio.png\|500px\|center\|Three movie frames]]
Line 2,056:		Line 2,060:
	<div align="center" id="Figure3">		<div align="center" id="Figure3">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr><th align="center">Figure 3</th></tr>		<tr><th align="center">
			<font size="+1">Figure 3:</font><br />
			<br />
			<math>~(q,\gamma_e,\gamma_c) = ( 0.886575, 1.798817, 1.021798)</math>
			</th></tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:NumericalModel22.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]		[[File:NumericalModel22.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]

Tohline: /* Results */

2017-01-23T21:33:36Z

Results

← Older revision		Revision as of 21:33, 23 January 2017
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	===Results===		===Results===

			====Motivated by Analytic21====
	Using the above-described numerical algorithm, we have constructed 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>. The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession — in order of ''decreasing'' values of <math>~\sigma_c^2</math> — in the form of a looped animation sequence. Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.		Using the above-described numerical algorithm, we have constructed 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>. The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession — in order of ''decreasing'' values of <math>~\sigma_c^2</math> — in the form of a looped animation sequence. Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.

Line 2,049:		Line 2,050:

	Three frames from the animation sequence have been displayed side-by-side in the top panel of Figure 2. This image montage is presented, in part, to illustrate the degree to which our numerically generated eigenfunction matches the analytically generated eigenfunction in the ''specific'' case <math>~(\sigma_c^2 = 28.9)</math> for which we have been able to obtain an analytic solution to the combined/matched, core/envelope LAWEs.		Three frames from the animation sequence have been displayed side-by-side in the top panel of Figure 2. This image montage is presented, in part, to illustrate the degree to which our numerically generated eigenfunction matches the analytically generated eigenfunction in the ''specific'' case <math>~(\sigma_c^2 = 28.9)</math> for which we have been able to obtain an analytic solution to the combined/matched, core/envelope LAWEs.

			====Motivated by Analytic22====

			We have also numerically constructed an eigenfunction that matches our [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Illustration22\|accompanying analytic Illustration22]]. In Figure 3, the numerically derived solution has been plotted on top of the analytically derived solution.
			<div align="center" id="Figure3">
			<table border="1" align="center" cellpadding="5">
			<tr><th align="center">Figure 3</th></tr>
			<tr><td align="center">
			[[File:NumericalModel22.png\|350px\|center\|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]
			</td></tr>
			</table>
			</div>

	=Discussion=		=Discussion=

Tohline: /* Additional Possibilities */

2017-01-22T22:53:11Z

Additional Possibilities

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	* Tie this new (numerical) technique into the [[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.		* Tie this new (numerical) technique into the [[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes\|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.

			* The Figure 2 animation displays numerically determined solutions, <math>~x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>~\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations. We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Discussion */ Clean up text

2017-01-22T22:46:44Z

Discussion: Clean up text

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	=Discussion=		=Discussion=
	==Conclusions==		==Conclusions==
	Initially, I was surprised to find that, by employing our above-described numerical algorithm, we were able to solve the combined/matched LAWEs for a continuum set — rather than a discrete set — of dimensionless oscillation frequencies, <math>~\sigma_c^2</math>. After all, I have been taught to believe that radial oscillation modes are obtained by solving an eigenvalue problem. After a bit of thought, I recognized that the continuum set of solutions has been obtained in the absence of a specified ''surface'' boundary condition. I suspect that the continuum of solutions can only be relevant to a real ''astrophysical'' problem after a physically meaningful surface boundary condition has been imposed; for example, a specification of the ''slope'' of the eigenfunction. This should naturally reduce the continuum set to a discrete set.		Initially, I was surprised to find that, by employing our above-described numerical algorithm, we were able to solve the combined/matched LAWEs for a continuum set — rather than a discrete set — of dimensionless oscillation frequencies, <math>~\sigma_c^2</math>. After all, I have been taught to believe that radial oscillation modes are obtained by solving an eigenvalue problem. After a bit of thought, I recognized that the continuum set of solutions has been obtained in the absence of a specified ''surface'' boundary condition. I suspect that the continuum of solutions can only be relevant to a real ''astrophysical'' problem after a physically meaningful surface boundary condition has been imposed; for example, a specification of the ''slope'' of the eigenfunction at the equilibrium configuration's surface. This should naturally reduce the continuum set to a discrete set of eigenvectors.

	Watching the animation sequence reveals, for example, that as the value of <math>~\sigma_c^2</math> is reduced, the number of nodes inside the configuration is reduced, in a predictable, quantized fashion. At the same time — between each drop in the integer number of nodes — the slope of the eigenfunction varies between large positive, and large negative values. Hence, we should be able to find a matched solution whose slope at the surface ''also'' matches any reasonably specified boundary condition.		Watching the animation sequence reveals, for example, that as the value of <math>~\sigma_c^2</math> is reduced, the number of nodes inside the configuration is reduced, in a predictable, quantized fashion. At the same time — between each drop in the integer number of nodes — the slope of the eigenfunction at the surface <math>~(r/R = 1)</math> varies between large positive, and large negative values. Hence, we should be able to find a matched solution whose slope at the surface ''also'' matches any reasonably specified boundary condition.

	==Additional Possibilities==		==Additional Possibilities==

Tohline: /* Approach */ Finished filling out, then explaining, Table 1

2017-01-22T22:40:54Z

Approach: Finished filling out, then explaining, Table 1

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	<td align="right">		<td align="right">
	<math>~		<math>~
	~~x_1~~		x_{k=1}
	</math>		</math>
	</td>		</td>
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	</div>		</div>

	<font color="red"><b>TEST:</b></font>   We tested this finite-difference algorithm on a grid of resolution, <math>~\mathrm{N}_\mathrm{core} = \mathrm{N}_\mathrm{core} = 50</math>, by first setting <math>~\sigma_c^2 = 28.91158</math>. The resulting, numerically constructed eigenfunction matched to high accuracy the analytically generated eigenfunction shown, above, as [[#Figure1\|Figure 1]]; see also, the middle image in the top panel of [[#Figure2\|Figure 2]], ~~immediately below~~.		<font color="red"><b>TEST:</b></font>   We tested this finite-difference algorithm on a grid of resolution, <math>~\mathrm{N}_\mathrm{core} = \mathrm{N}_\mathrm{core} = 50</math>, by first setting <math>~\sigma_c^2 = 28.91158</math>. The resulting, numerically constructed eigenfunction matched to high accuracy the analytically generated eigenfunction shown, above, as [[#Figure1\|Figure 1]]; see also, the middle image in the top panel of [[#Figure2\|Figure 2]]. Representative values of the numerically determined eigenfunction, <math>~x(\xi)</math> at various discrete grid locations are provided in Table 1, along with the numerically determined value of the slope at the interface, <math>~x_q'</math>. At each grid location, the associated value of the dimensionless radius, <math>~r/R</math>, may be obtained by simply multiplying each tabulated value of <math>~\xi</math> by the parameter, <math>~q</math>.

	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
			<tr>
			<th align="center" colspan="7">
			<font size="+1">Table 1:</font><br />
			Example Numerical Determination of Eigenfunction
			</th>
			</tr>
			<tr>
			<td align="center" colspan="7">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(q,\gamma_e,\gamma_c)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(0.684012, 1.194030, 1.845579)</math>
			</td>
			</tr>
			<tr><td align="center" colspan="3">and</td></tr>
			<tr>
			<td align="center" colspan="3">
			<math>~\sigma_c^2 = 28.91158</math>
			</td>
			</tr>
			</table>
			</td>
			</tr>
	<tr>		<tr>
	<th align="center" colspan="3">Core</th>		<th align="center" colspan="3">Core</th>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~g^2~~</math>		<math>~\alpha_e </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~1.~~323609~~</math>		<math>~-0.35</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\~~Delta_c~~</math>		<math>~\Delta_e</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="center">2</td>		<td align="center">2</td>
	<td align="center">1.0184785</td>		<td align="center">1.0184785</td>
	<td align="center">---</td>		<td align="center">-0.095792</td>
	</tr>		</tr>
	<tr>		<tr>