User:Tohline/SSC/Stability/n3PolytropeLAWE - Revision history

Tohline: /* Related Discussions */

2017-04-11T16:39:25Z

Related Discussions

← Older revision		Revision as of 16:39, 11 April 2017
Line 1,518:		Line 1,518:
	** n = 3:  [[User:Tohline/SSC/Stability/n3PolytropeLAWE\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]		** n = 3:  [[User:Tohline/SSC/Stability/n3PolytropeLAWE\|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
	** n = 5:  [[User:Tohline/SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]		** n = 5:  [[User:Tohline/SSC/Stability/n5PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]


			{{LSU_HBook_footer}}

Tohline: /* Analytic Inquiry */

2017-04-10T02:04:08Z

Analytic Inquiry

← Older revision		Revision as of 02:04, 10 April 2017
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	=Analytic Inquiry=		=Analytic Inquiry=

			<font color="red"><b>NOTE</b> (from J. E. Tohline in April, 2017):</font>   The following subsections present some exploratory ideas that were pursued while I was searching for analytic solutions to the polytropic LAWE. For the most part this material has been superseded by a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Polytropic\|separate discussion]] in which we describe the desired analytic solution, which we discovered in March, 2017.

	==Fundamental-Mode, Homentropic Oscillations==		==Fundamental-Mode, Homentropic Oscillations==

Tohline: /* Behavior for Known n=5 Solution */

2017-03-11T21:42:55Z

Behavior for Known n=5 Solution

← Older revision		Revision as of 21:42, 11 March 2017
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	<math>~		<math>~
	(n-2) \frac{2\xi}{\theta_5}\biggl( \frac{d\theta_5}{d\xi} \biggr)^2		(n-2) \frac{2\xi}{\theta_5}\biggl( \frac{d\theta_5}{d\xi} \biggr)^2
	+ \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ (3-n~~) (n + 1~~)\theta_5^2 + 1 ~~+ n~~ \biggr]		+ (n+1)\biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ (3-n) \theta_5^2 + 1 \biggr]
	- 2\xi \theta_5^n		- 2\xi \theta_5^n \, .
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>
			Again, let's check to see if the case of n=5 works …

			<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( \frac{d\theta_5}{d\xi} \biggr)^2
			+ \frac{\theta_5}{\xi} \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ 1 - 2 \theta_5^2 \biggr]
			- \frac{\theta_5^{6}}{3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			3\xi^2(3+\xi^2)^{-3}
			-3(3+\xi^2)^{-2}\biggl[ 1 - 6(3+\xi^2)^{-1} \biggr]
			- 3^2(3+\xi^2)^{-3}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~(3+\xi^2)^{-3}\biggl\{
			3\xi^2 -3\biggl[ (3+\xi^2) -6 \biggr] - 3^2 \biggr\}
			</math>     Yes!
	</td>		</td>
	</tr>		</tr>

Tohline: /* Behavior for Known n=5 Solution */

2017-03-11T21:06:31Z

Behavior for Known n=5 Solution

← Older revision		Revision as of 21:06, 11 March 2017
Line 1,335:		Line 1,335:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~\Rightarrow~~~~~		<math>~0
	~~\frac{1}{\theta} \frac{d\theta }{d\xi} \biggl[ (3-n)(1-f_n) - (n+1)\xi \frac{df_n}{d\xi} \biggr]~~
	</math>		</math>
	</td>		</td>
Line 1,345:		Line 1,344:
	<math>~		<math>~
	\xi \cdot\frac{d^2f_n}{d\xi^2}		\xi \cdot\frac{d^2f_n}{d\xi^2}
	+ 4 \cdot \frac{~~df_n~~}{d\xi} \, .		- \frac{(3-n)}{\theta} \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ 1-f_n \biggr]
			+ \biggl[ 4 + \frac{(n+1)\xi}{\theta} \biggl( \frac{d\theta }{d\xi} \biggr) \biggr] \biggl[ \frac{df_n}{d\xi} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\xi \biggl\{ - \frac{2}{n\theta_5^3} \cdot \frac{d^2\theta_5}{d\xi^2} + \frac{6}{n\theta_5^4} \cdot \biggl( \frac{d\theta_5}{d\xi} \biggr)^2\biggr\}
			- \biggl[ 4 + \frac{(n+1)\xi}{\theta} \biggl( \frac{d\theta }{d\xi} \biggr) \biggr] \biggl[ \frac{2}{n\theta_5^3} \cdot \frac{d\theta_5}{d\xi}\biggr]
			- \frac{(3-n)}{\theta} \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ \frac{(n + 1)\theta_5^2 -1 }{n\theta_5^2} \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{n\theta_5^3}\biggl\{
			\biggl[ 2\xi \cdot \frac{d^2\theta_5}{d\xi^2} - \frac{6\xi}{\theta_5} \cdot \biggl( \frac{d\theta_5}{d\xi} \biggr)^2\biggr]
			+ 2\biggl[ 4 + \frac{(n+1)\xi}{\theta} \biggl( \frac{d\theta }{d\xi} \biggr) \biggr] \frac{d\theta_5}{d\xi}
			+ (3-n) \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ (n + 1)\theta_5^2 -1 \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1}{n\theta_5^3}\biggl\{
			2\xi \cdot \frac{d^2\theta_5}{d\xi^2}
			+ (n-2) \frac{2\xi}{\theta_5}\biggl( \frac{d\theta_5}{d\xi} \biggr)^2
			+ \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ (3-n) (n + 1)\theta_5^2 + 5+ n \biggr]
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Now, from the polytropic Lane-Emden equation, we also know that,
			<div align="center">
			{{ User:Tohline/Math/EQ_SSLaneEmden01 }}
			</div>
			That is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~2\xi \biggl(\frac{d\theta_5}{d\xi}\biggr) + \xi^2 \cdot \frac{d^2\theta_5}{d\xi^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \xi^2 \theta_5^n</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~ \frac{d^2\theta_5}{d\xi^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \theta_5^n - \frac{2}{\xi} \biggl(\frac{d\theta_5}{d\xi}\biggr)</math>
			</td>
			</tr>
			</table>
			</div>
			So, the LAWE becomes,

			<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>~
			(n-2) \frac{2\xi}{\theta_5}\biggl( \frac{d\theta_5}{d\xi} \biggr)^2
			+ \biggl( \frac{d\theta }{d\xi} \biggr) \biggl[ (3-n) (n + 1)\theta_5^2 + 1 + n \biggr]
			- 2\xi \theta_5^n
	</math>		</math>
	</td>		</td>

Tohline: /* Analytic Inquiry */

2017-03-11T19:02:04Z

Analytic Inquiry

Show changes

Tohline: /* Fundamental-Mode, Homentropic Oscillations */

2017-03-11T06:20:08Z

Fundamental-Mode, Homentropic Oscillations

← Older revision		Revision as of 06:20, 11 March 2017
Line 758:		Line 758:

	I'm not sure if anyone else has previously appreciated that the "fundamental mode" polytropic LAWE can be written in this form. I'm even less sure that this form sheds light on its solution.		I'm not sure if anyone else has previously appreciated that the "fundamental mode" polytropic LAWE can be written in this form. I'm even less sure that this form sheds light on its solution.


			Play a little more …   Start by letting, <math>~A \equiv [x \xi^3 \theta^{(n+1)}] \, ,</math> in which case we have,
			<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>~
			\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
			+ \frac{d\ln x}{d\ln \xi} \cdot \biggl\{ \frac{1}{A} \cdot \frac{dA}{d\ln \xi} \biggr\} + \frac{d\ln \theta^{(3-n) }}{d\ln\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{A} \cdot \frac{d}{d\ln\xi} \biggl[ A \cdot \frac{d\ln x}{d\ln \xi} \biggr] + \frac{d\ln \theta^{(3-n) }}{d\ln\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{d}{d\xi} \biggl[ A \cdot \frac{d\ln x}{d\ln \xi} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{A}{\theta^{(3-n)}} \cdot \frac{d\theta^{(3-n) }}{d\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{d}{d\xi} \biggl[ x \xi^3 \theta^{(n+1)} \cdot \frac{d\ln x}{d\ln \xi} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \frac{x \xi^3 \theta^{(n+1)}}{\theta^{(3-n)}} \cdot \frac{d\theta^{(3-n) }}{d\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{d}{d\xi} \biggl[ \xi^4 \theta^{(n+1)} \cdot \frac{dx}{d\xi} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- x \xi^3 \theta^{2(n-1)} \cdot \frac{d\theta^{(3-n) }}{d\xi} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			Now, what if, <math>~x \equiv [\xi^{-2}\theta^{-n}]</math>   ?

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

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\xi^2 \theta^{(n+1)} \cdot \frac{d(\xi^{-2}\theta^{-n})}{d\xi}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d\theta}{d\xi} - \frac{1}{\xi^{2}\theta^{n}} \cdot \frac{d(\xi^2 \theta^{(n+1)} )}{d\xi} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			in which case, the LAWE becomes,


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

			<tr>
			<td align="right">
			<math>~
			\frac{d}{d\xi} \biggl[ \xi^2 \frac{d\theta}{d\xi} - \frac{1}{\theta^{n}} \cdot \frac{d(\xi^2 \theta^{(n+1)} )}{d\xi}\biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- [\xi^{-2}\theta^{-n}] \xi^3 \theta^{2(n-1)} \cdot \frac{d\theta^{(3-n) }}{d\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			\frac{d}{d\xi} \biggl[ \xi^2 \frac{d\theta}{d\xi} \biggr] - \frac{d}{d\xi} \biggl[\frac{1}{\theta^{n}} \cdot \frac{d(\xi^2 \theta^{(n+1)} )}{d\xi}\biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \xi \theta^{n-2} \cdot \frac{d\theta^{(3-n) }}{d\xi}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow~~~
			-\xi^2\theta^n - \frac{d}{d\xi} \biggl[\frac{1}{\theta^{n}} \cdot \frac{d(\xi^2 \theta^{(n+1)} )}{d\xi}\biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \xi (3-n)\frac{d\theta }{d\xi}
			</math>
			</td>
			</tr>
			</table>
			</div>

	==Specific case of n = 3 Polytropes==		==Specific case of n = 3 Polytropes==

Tohline: /* Fundamental-Mode, Homentropic Oscillations */

2017-03-11T03:10:07Z

Fundamental-Mode, Homentropic Oscillations

← Older revision		Revision as of 03:10, 11 March 2017
Line 685:		Line 685:
	</div>		</div>

	~~Notice~~ that this equation may be ~~further~~ rewritten entirely as an expression that relates the logarithmic derivatives of ~~various key quantities~~. Multiplying through by <math>~\xi^2/x</math>, then drawing on a differential relation that has been [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Idea_Involving_Logarithmic_Derivatives\|derived in a separate context]], namely,		We have noticed that this equation may furthermore be rewritten entirely as an expression that relates the logarithmic derivatives of <math>~x, \xi,</math> and <math>~\theta</math>. Multiplying through by <math>~\xi^2/x</math>, then drawing on a differential relation that has been [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Idea_Involving_Logarithmic_Derivatives\|derived in a separate context]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<math>~		<math>~
	\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]		\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
	+ \frac{d\ln x}{d\ln \xi} \cdot \biggl[ \frac{d\ln x}{d\ln \xi} -1 + 4 - (n+1) Q \biggr] - (3-n) Q		+ \frac{d\ln x}{d\ln \xi} \cdot \biggl[ \frac{d\ln x}{d\ln \xi} + 3 - (n+1) Q \biggr] - (3-n) Q
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
			+ \frac{d\ln x}{d\ln \xi} \cdot \biggl\{ \frac{d\ln [x \xi^3 \theta^{(n+1)}] }{d\ln \xi} \biggr\} + \frac{d\ln \theta^{(3-n) }}{d\ln\xi} \, .
	</math>		</math>
	</td>		</td>
Line 741:		Line 756:
	</table>		</table>
	</div>		</div>

			I'm not sure if anyone else has previously appreciated that the "fundamental mode" polytropic LAWE can be written in this form. I'm even less sure that this form sheds light on its solution.

	==Specific case of n = 3 Polytropes==		==Specific case of n = 3 Polytropes==

Tohline: /* Fundamental-Mode, Homentropic Oscillations */

2017-03-11T02:52:37Z

Fundamental-Mode, Homentropic Oscillations

← Older revision		Revision as of 02:52, 11 March 2017
Line 655:		Line 655:
	==Fundamental-Mode, Homentropic Oscillations==		==Fundamental-Mode, Homentropic Oscillations==

	~~As has been stated~~, [[#Background\|above]], ~~the LAWE~~ that is relevant to polytropic spheres is,		The LAWE, [[#Background\|presented above]], that is relevant to polytropic spheres, may be rewritten as,
			<div align="center">
			{{ User:Tohline/Math/EQ_RadialPulsation02 }}
			</div>

			Now, if we assume that oscillations occur adiabatically with an adiabatic index that is consistent with the chosen polytropic index — that is to say,
			<div align="center">
			<math>~\gamma_g = \frac{n+1}{n}</math>      <math>~\Rightarrow</math>       <math>~\alpha = \frac{3-n}{n+1} \, ,</math>
			</div>
			in which case the configuration remains homentropic as it oscillates — and if we look only for a (marginally unstable) configuration that has <math>~\sigma_c^2 = 0</math>, then the relevant LAWE is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 667:		Line 676:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{d^2x}{d\xi^2} + \biggl[~~\frac{~~4~~}{\xi}~~ - ~~\frac{~~(n+1)}{\~~theta} \biggl(-~~ \frac{~~d\theta~~}{d\xi}\~~biggr)\biggr]~~ \frac{dx}{d\xi} +		<math>~
	~~\frac{~~(n+1)~~}{\theta}\biggl[~~\frac{~~\sigma_c^2}{6\gamma_g } -~~		\frac{d^2x}{d\xi^2} + \biggl[ 4 - (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}
	~~\frac{\alpha~~}{\xi} ~~\biggl(- \frac{d\theta}{d\xi}\biggr)\biggr] x~~ \, .</math>		- (3-n) Q \frac{x}{\xi^2} \, .
			</math>
	</td>		</td>
	</tr>		</tr>
Line 675:		Line 685:
	</div>		</div>

	~~Defining~~ the ~~function~~,		Notice that this equation may be further rewritten entirely as an expression that relates the logarithmic derivatives of various key quantities. Multiplying through by <math>~\xi^2/x</math>, then drawing on a differential relation that has been [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Idea_Involving_Logarithmic_Derivatives\|derived in a separate context]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 681:		Line 691:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~Q(\xi)</math>		<math>~\frac{\xi^2}{x} \cdot \frac{d^2x}{d\xi^2}
	</td>		</math>
			</td>
	<td align="center">		<td align="center">
	<math>~~~\equiv~~</math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\xi}{\~~theta~~} \~~cdot~~ \frac{d\~~theta~~}{d\xi} = \frac{d\ln~~\theta~~}{d\ln\xi} \, ,</math>		<math>~
	</td>		\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
			+ \biggl[ 1 \frac{d\ln x}{d\ln \xi} -1 \biggr]\cdot \frac{d\ln x}{d\ln \xi} \, ,
			</math>
			</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	this ~~general~~ LAWE becomes,		this polytropic LAWE becomes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 704:		Line 718:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{d~~^2x~~}{d\xi^2} + \biggl[ ~~4 + (n+1) Q \biggr]~~ \frac{~~1}{~~\~~xi} \cdot \frac{dx~~}{d\xi}		<math>~
	+ ~~(n+1)~~ \biggl[\frac{\~~sigma_c^2~~}{6\~~gamma_g~~ } \~~biggl(~~ \frac{\~~xi^2~~}{\~~theta~~}\~~biggr~~)		\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
	~~+ \alpha~~ Q\biggr] \frac{x}{\xi^2} ~~\, .~~</math>		+ \biggl[ \frac{d\ln x}{d\ln \xi} -1 \biggr]\cdot \frac{d\ln x}{d\ln \xi} + \biggl[ 4 - (n+1) Q \biggr] \frac{d\ln x}{d\ln\xi} - (3-n) Q
			</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~</div>~~
	~~Now, if we assume that oscillations occur adiabatically with an adiabatic index that is consistent with the chosen polytropic index — that is to say,~~
	~~<div align="center">~~
	~~<math>~\gamma_g = \frac{n+1}{n}</math>      <math>~\Rightarrow</math>       <math>~\alpha = \frac{3-n}{n+1} \, ,</math>~~
	~~</div>~~
	in which case the configuration remains homentropic as it oscillates — and we look only for a (marginally unstable) configuration that has <math>~\sigma_c^2 = 0</math>, then the relevant LAWE is,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~0 </math>~~
	</td>		</td>
	<td align="center">		<td align="center">
Line 728:		Line 734:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{d~~^2x~~}{d\xi^2} + \biggl[ ~~4 + (n+1) Q~~ \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi}		\frac{d}{d\ln\xi} \biggl[ \frac{d\ln x}{d\ln \xi} \biggr]
	+ (3-n) Q ~~\frac{x}{\xi^2} \, .~~		+ \frac{d\ln x}{d\ln \xi} \cdot \biggl[ \frac{d\ln x}{d\ln \xi} -1 + 4 - (n+1) Q \biggr] - (3-n) Q
	</math>		</math>
	</td>		</td>

Tohline: /* From the Core to the Surface */

2017-03-09T02:17:36Z

From the Core to the Surface

← Older revision		Revision as of 02:17, 9 March 2017
Line 408:		Line 408:
	Each frame of the Figure 2 animation also displays, as smooth solid curves, the radial eigenfunctions that Schwarzschild (1941) obtained for the fundamental mode (blue curve) and the first three overtone modes (green, purple, & orange curves, repectively) for his model with <math>~\alpha = 0.4</math>. These are the same curves that appear in the left-hand panel of Figure 1, but here the displacement amplitude has been renormalized such that <math>~x_0 = 1</math>, and, along the horizontal axis, the radial location is marked in terms of the fractional radius, <math>~r/R \equiv \xi/\xi_\mathrm{max}</math>. In our examination of this model, as we approached each specific value of a modal eigenfrequency identified by Schwarzschild — see the frequencies highlighted in pink in our Table 1 — we fine-tuned our choice of the eigenfrequency in order to find a displacement function whose surface ''amplitude'' matched, to a high level of precision, the surface amplitude associated with Schwarzschild's corresponding published eigenfunction. The column of our Table 2 whose heading is "Match Schwarzschild" identifies — to at least 10 digits precision — the frequency choice that was required in order for these surface ''amplitudes'' to match in each case.		Each frame of the Figure 2 animation also displays, as smooth solid curves, the radial eigenfunctions that Schwarzschild (1941) obtained for the fundamental mode (blue curve) and the first three overtone modes (green, purple, & orange curves, repectively) for his model with <math>~\alpha = 0.4</math>. These are the same curves that appear in the left-hand panel of Figure 1, but here the displacement amplitude has been renormalized such that <math>~x_0 = 1</math>, and, along the horizontal axis, the radial location is marked in terms of the fractional radius, <math>~r/R \equiv \xi/\xi_\mathrm{max}</math>. In our examination of this model, as we approached each specific value of a modal eigenfrequency identified by Schwarzschild — see the frequencies highlighted in pink in our Table 1 — we fine-tuned our choice of the eigenfrequency in order to find a displacement function whose surface ''amplitude'' matched, to a high level of precision, the surface amplitude associated with Schwarzschild's corresponding published eigenfunction. The column of our Table 2 whose heading is "Match Schwarzschild" identifies — to at least 10 digits precision — the frequency choice that was required in order for these surface ''amplitudes'' to match in each case.

	It is gratifying to see that our resulting frequencies match well the values published by Schwarzschild (as highlighted in pink, above). But this does not satisfactorily explain why, among the entire range of displacement functions displayed (in red) in the Figure 2 animation, Schwarzschild labeled these specific ones as the eigenmodes. As we shall now demonstrate, his eigenmode identifications resulted from the imposition of a specific, physically justified constraint on the ''slope'', rather than the value, of the displacement function at the surface of the configuration.		It is gratifying to see that our resulting frequencies match well the values published by Schwarzschild (as highlighted in pink, above). But this does not satisfactorily explain why, among the entire range of displacement functions displayed (in red) in the Figure 2 animation, Schwarzschild labeled these specific ones as the eigenmodes. As we shall now demonstrate, his eigenmode identifications resulted from the imposition of a specific, physically justified constraint on the ''slope'', rather than the value, of the displacement function at the surface of the configuration. (See also our separate brief answer to the question, "[[User:Tohline/SSC/Perturbations#What_Makes_This_an_Eigenvalue_Problem.3F\|What makes this an eigenvalue problem?]]".)

	===Surface Boundary Condition===		===Surface Boundary Condition===

Tohline: /* Another Potential Option */

2017-03-08T02:08:45Z

Another Potential Option

← Older revision		Revision as of 02:08, 8 March 2017
Line 878:		Line 878:
	</table>		</table>
	</div>		</div>

			Referring to the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden\|power-series expansion of the polytropic Lane-Emden function]], <math>~\theta(\xi)</math>, about the configuration's center, we see that the product, <math>~\xi\theta</math>, goes to zero as the first power of <math>~\xi</math>. This means that the right-hand side of this last differential equation blows up at the center. This, therefore, does not appear to provide a physically viable avenue by which to identify an alternative fundamental-mode eigenfunction.

	=Related Discussions=		=Related Discussions=