User:Tohline/SSC/UniformDensity - Revision history

Tohline: /* Numerical Integration */

2017-02-15T21:07:33Z

Numerical Integration

← Older revision		Revision as of 21:07, 15 February 2017
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{[4a (1 - a^2) - 2\Delta^2 a \mathfrak{F} \xi_a		<math>~\frac{[4a (1 - a^2) - 2\Delta^2 a \mathfrak{F} ]\xi_a
	+ [ \Delta( 4 - 6a^2 ) - 2a (1 - a^2)] \xi_- }{[2a (1 - a^2) + \Delta( 4 - 6a^2 ) ] } \, .		+ [ \Delta( 4 - 6a^2 ) - 2a (1 - a^2)] \xi_- }{[2a (1 - a^2) + \Delta( 4 - 6a^2 ) ] } \, .
	</math>		</math>

Tohline: /* Numerical Integration */

2017-02-12T02:06:23Z

Numerical Integration

← Older revision		Revision as of 02:06, 12 February 2017
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0</math>		<math>~x(1-x^2) \xi_1^{ ' ' } </math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(1-x^2) \xi_1^{ ' ~~' } + \frac{1}{~~x~~}\biggl[4 - 6x^2 \biggr]\xi_1^' +~~ \mathfrak{F} \xi_1 \, ,</math>		<math>~- (4 - 6x^2 )\xi_1^' - x \mathfrak{F} \xi_1 \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<math>~\sigma_c^2 \equiv \frac{3n^2 }{2\pi G \rho_c} \, .</math>		<math>~\sigma_c^2 \equiv \frac{3n^2 }{2\pi G \rho_c} \, .</math>
	</div>		</div>

			Following precisely the same logic as has been laid out in our [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Integrating_Outward_Through_the_Core\|separate discussion]], if we set the central value of the eigenfunction to <math>~\xi_0</math>, then the eigenfunction's value at the first zone (distance <math>~\Delta</math>) away from the center will be,

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

			<tr>
			<td align="right">
			<math>~
			\xi_+
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ 1 - \frac{\Delta^2 \mathfrak{F}}{2} \biggr] \xi_0 \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			While, for each successive coordinate location, <math>~a = x</math>, in the range, <math>~0 < x < 1</math>, we will use the general expression, namely,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			\Rightarrow~~~\xi_+
			</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~\frac{[4a (1 - a^2) - 2\Delta^2 a \mathfrak{F} \xi_a
			+ [ \Delta( 4 - 6a^2 ) - 2a (1 - a^2)] \xi_- }{[2a (1 - a^2) + \Delta( 4 - 6a^2 ) ] } \, .
			</math>
			</td>
			</tr>
			</table>
			</div>





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Tohline: /* The Stability of Uniform-Density Spheres */Begin laying out numerical integration

2017-02-12T01:47:36Z

The Stability of Uniform-Density Spheres: Begin laying out numerical integration

← Older revision		Revision as of 01:47, 12 February 2017
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	</table>		</table>
	</div>		</div>

			==Numerical Integration==

			In order to gain a more complete understanding of this type of modal analysis, let's attempt to obtain various eigenvectors by numerically integrating the governing LAWE from the center of the system, outward to the surface. This will be done in a manner similar to our [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope\|numerical study of radial oscillations in zero-zero bipolytropes]]. Following our [[User:Tohline/SSC/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29\|above review of Sterne's presentation]], the relevant LAWE 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-x^2) \xi_1^{ ' ' } + \frac{1}{x}\biggl[4 - 6x^2 \biggr]\xi_1^' + \mathfrak{F} \xi_1 \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<math>~\mathfrak{F} \equiv \frac{\sigma_c^2}{\gamma} - 2 \alpha = \frac{\sigma_c^2 + 8}{\gamma} - 6\, ,</math>
			and,
			<math>~\sigma_c^2 \equiv \frac{3n^2 }{2\pi G \rho_c} \, .</math>
			</div>



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Tohline: /* Mode j = 2: */ Finished demonstrating validity of j=2 eigenfunction

2015-06-17T23:25:21Z

Mode j = 2:: Finished demonstrating validity of j=2 eigenfunction

← Older revision		Revision as of 23:25, 17 June 2015
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- \frac{2^2\cdot 3^2}{5} + \biggl[~~\frac{2^2 \cdot 3^3 \cdot 11}{~~5 \~~cdot 7}~~		\frac{2^2\cdot 3^2}{5}\biggl[ -1 -4 + 5\biggr] + \frac{2^2\cdot 3^2}{5\cdot 7}\biggl[33 + 7 + 44 + 42 - 2\cdot 3^2 \cdot 7\biggr] x^2
	+ \frac{2^2\cdot 3^2~~}{5} \biggr] x^2 - \frac{2^2 \cdot 3^3 \cdot 11~~}{5 \cdot 7} ~~x^4~~		+ \frac{2^2 \cdot 3^3 \cdot 11}{5 \cdot 7} \biggl[ 3 - 2 - 1\biggr] x^4 \, ,
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	- \~~frac{2^4\cdot 3^2}{5}~~ +~~\biggl[ \frac{~~2^4 \cdot 3^2 ~~\cdot 11}{5~~ \cdot 7}
	~~+ \frac{2^3\cdot 3^3}{5}~~ \biggr]x^2 - \frac{2^3 \cdot 3^3 \cdot 11}{5 \cdot 7} ~~x^4~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~+ 2^2~~ \~~cdot~~ 3^2 - \~~frac{2^3\cdot 3^4}{5}~~ x~~^2 + \frac{2^2 \cdot 3~~^4 \~~cdot 11}{5 \cdot 7} x^4~~
	</math>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>

	which properly sums to zero for all <math>~x</math>. Next, it is clear that the inner boundary condition is satisfied because,		which properly sums to zero for all <math>~x</math>. Next, it is clear that the inner boundary condition is satisfied because,
	<div align="center">		<div align="center">
	<math>~\xi_1^'\biggr\|_{x = 0} = - \frac{14}{5} (0)= 0 \, .</math>		<math>~\xi_1^'\biggr\|_{x = 0} = - \frac{2^2\cdot 3^2}{5} (0) + \frac{2^2 \cdot 3^2 \cdot 11}{5 \cdot 7} (0) = 0 \, .</math>
	</div>		</div>
	And the expression for the outer boundary condition gives,		And the expression for the outer boundary condition gives,
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ -\~~biggl(~~\frac{14}{5} ~~\biggr)~~ x - 7 \biggl( 1 - \frac{7}{5}x^2 \biggr) \biggr]_{x=1} </math>		<math>~\biggl[ - \frac{2^2\cdot 3^2}{5} x + \frac{2^2 \cdot 3^2 \cdot 11}{5 \cdot 7} x^3
			- 2\cdot 3^2 \biggl( 1 - \frac{2\cdot 3^2}{5} x^2 + \frac{3^2 \cdot 11}{5 \cdot 7} x^4 \biggr) \biggr]_{x=1} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~\biggl[ -\biggl(~~\frac{14}{5} \~~biggr)~~ - 7 \biggl( 1 - ~~\frac{7}{5}~~ \biggr) \biggr] \, ,</math>		<math>~\frac{2\cdot 3^2}{5\cdot 7}\biggl[ -14 + 22 - \biggl( 35 - 126 + 99 \biggr) \biggr] \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>
	which also properly sums to zero.		which also properly sums to zero.


	===Properties of Eigenfunction Solutions===		===Properties of Eigenfunction Solutions===

Tohline: /* The Stability of Uniform-Density Spheres */ Middle of demonstrating that mode j=2 is correct

2015-06-17T22:57:48Z

The Stability of Uniform-Density Spheres: Middle of demonstrating that mode j=2 is correct

Show changes

Tohline: /* Stability */ Add extracted paragraph from Ledoux (1946)

2015-06-17T22:15:59Z

Stability: Add extracted paragraph from Ledoux (1946)

← Older revision		Revision as of 22:15, 17 June 2015
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	The plot titled, "Critical Adiabatic Index," that is [[User:Tohline/SSC/UniformDensity#Properties_of_Eigenfunction_Solutions\| presented above]] shows graphically how <math>~\gamma_\mathrm{crit}</math> varies with mode number, <math>~0 \le j \le 11</math>. All modes are stable as long as <math>~\gamma > 4/3</math>. As the adiabatic index is decreased below this value, the lowest order mode, <math>~j = 0</math>, becomes unstable, first; then successively higher order modes become unstable at ~~lower~~ and ~~lower~~ values of the index.		The plot titled, "Critical Adiabatic Index," that is [[User:Tohline/SSC/UniformDensity#Properties_of_Eigenfunction_Solutions\| presented above]] shows graphically how <math>~\gamma_\mathrm{crit}</math> varies with mode number over the range of mode numbers, <math>~0 \le j \le 11</math>. All modes are stable as long as <math>~\gamma > 4/3</math>. As the adiabatic index is decreased below this value, the lowest order mode, <math>~j = 0</math>, becomes unstable, first; then successively higher order modes become unstable at smaller and smaller values of the index. A very similar explanation and enunciation of Sterne's derived results regarding the stability of uniform-density spheres appears on p. 338 of the paper by [http://adsabs.harvard.edu/abs/1946ApJ...104..333L P. Ledoux (1946)] titled, ''On the Dynamical Stability of Stars''. The relevant paragraph from Ledoux's paper is reprinted as a boxed-in digital image in the following table:

			<div align="center">
			<table border="2" cellpadding="5" width="80%">
			<tr>
			<th align="center" colspan="1">
			Paragraph extracted<sup>&dagger;</sup> from p. 338 of [http://adsabs.harvard.edu/abs/1946ApJ...104..333L P. Ledoux (1946)]<p></p>
			"''On the Dynamical Stability of Stars''"<p></p>
			The Astrophysical Journal, vol. 104, pp. 333 - 346 © [http://aas.org/about/policies/copyright American Astronomical Society]
			</th>
			</tr>
			<tr>
			<td colspan="1" rowspan="1">
			[[File:Ledoux1946OnSterne01.png\|700px\|center\|Ledoux (1946)]]
			</td>
			</tr>
			<tr>
			<td align="left">
			<sup>&dagger;</sup>Our function, <math>~\gamma_\mathrm{crit}(j)</math>, is effectively the expression to which Ledoux is referring when he says, "… directly verify from equation (19) …"
			</td>
			</tr>
			</table>
			</div>


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Tohline: /* Stability */ Explain plot of critical adiabatic index

2015-06-17T19:19:07Z

Stability: Explain plot of critical adiabatic index

← Older revision		Revision as of 19:19, 17 June 2015
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	</div>		</div>

			The plot titled, "Critical Adiabatic Index," that is [[User:Tohline/SSC/UniformDensity#Properties_of_Eigenfunction_Solutions\| presented above]] shows graphically how <math>~\gamma_\mathrm{crit}</math> varies with mode number, <math>~0 \le j \le 11</math>. All modes are stable as long as <math>~\gamma > 4/3</math>. As the adiabatic index is decreased below this value, the lowest order mode, <math>~j = 0</math>, becomes unstable, first; then successively higher order modes become unstable at lower and lower values of the index.


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Tohline: /* Properties of Eigenfunction Solutions */ Insert plot of gamma_crit(j)

2015-06-17T02:22:09Z

Properties of Eigenfunction Solutions: Insert plot of gamma_crit(j)

← Older revision		Revision as of 02:22, 17 June 2015
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	<tr>		<tr>
	<td colspan="1" align="center">		<td colspan="1" align="center">
	[[File:Sterne1937SolutionPlot1.png\|~~500px~~\|center\|Sterne (1937)]]		[[File:Sterne1937SolutionPlot1.png\|350px\|center\|Sterne (1937)]]
			</td>
			<td colspan="1" align="center">
			[[File:Sterne1937CritGamma1.png\|350px\|center\|Sterne (1937)]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Stability */ Develop expression for gamma_crit(j)

2015-06-17T01:56:55Z

Stability: Develop expression for gamma_crit(j)

← Older revision		Revision as of 01:56, 17 June 2015
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	This latter condition is the mark of a dynamically unstable system. It is in this manner that the solution to an eigenvalue problem can provide critical information regarding the relative stability of equilibrium configurations.		This latter condition is the mark of a dynamically unstable system. It is in this manner that the solution to an eigenvalue problem can provide critical information regarding the relative stability of equilibrium configurations.

			For any given mode number, <math>~j</math>, then, the critical configuration separating stable from unstable systems occurs when the dimensionless eigenfrequency is zero. Therefore, the critical state occurs when,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0 = \frac{n_\mathrm{crit}^2}{4\pi G \bar\rho}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)\gamma_\mathrm{crit} -\frac{4}{3} </math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~~ \gamma_\mathrm{crit}(j)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{4}{3}\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)^{-1} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{4}{3}\biggl[1 + \frac{2j(2j+5)}{6} \biggr]^{-1} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{4}{3 + j(2j+5)} \, . </math>
			</td>
			</tr>
			</table>
			</div>



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Tohline: /* Validity Check */ Finished demonstrating validity of j=1 mode solution

2015-06-16T23:45:47Z

Validity Check: Finished demonstrating validity of j=1 mode solution

← Older revision		Revision as of 23:45, 16 June 2015
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	</div>		</div>
	which properly sums to zero. Next, because <math>~\xi_1^' = 0</math> everywhere, we know that it is zero at the center of the configuration, which satisfies the inner boundary condition. ~~Finally~~, the ~~expression~~ that should be satisfied at the outer boundary condition is,		which properly sums to zero. Next, because <math>~\xi_1^' = 0</math> everywhere, we know that it is zero at the center of the configuration, which satisfies the inner boundary condition. But, via the outer boundary condition, this also means that the product, <math>~(\tfrac{1}{2} \mathfrak{F} \xi_1)</math> should be zero; which it is, because <math>~\mathfrak{F} = 0</math> for this mode.

			====Mode j = 1:====
			In this case, according to [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)],
			<div align="center">
			<math>~\xi_1 = 1 - \frac{7}{5} x^2 \, ,</math>
			and
			<math>~\mathfrak{F} = 14 \, .</math>
			</div>
			Hence,
			<div align="center">
			<math>~\xi_1^' = - \frac{14}{5} x \, ;</math>
			and
			<math>\xi_1^{' '} = - \frac{14}{5} \, .</math>
			</div>
			So, the adiabatic wave equation gives,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~(1-x^2) \xi_1^{ ' ' } + \frac{1}{x}\biggl[4 - 6x^2 \biggr]\xi_1^' + \mathfrak{F} \xi_1 </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{14}{5}(1-x^2) - \frac{14}{5}\biggl[4 - 6x^2 \biggr] + 14 \biggl( 1 - \frac{7}{5} x^2 \biggr) </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{14}{5}\biggl[- (1-x^2) - (4 - 6x^2 ) +5 - 7 x^2 \biggr] \, , </math>
			</td>
			</tr>
			</table>
			</div>
			which properly sums to zero for all <math>~x</math>. Next, it is clear that the inner boundary condition is satisfied because,
			<div align="center">
			<math>~\xi_1^'\biggr\|_{x = 0} = - \frac{14}{5} (0)= 0 \, .</math>
			</div>
			And the expression for the outer boundary condition gives,

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

			<tr>
			<td align="right">
			<math>~\biggl[ \xi_1^' - \frac{\mathfrak{F} \xi_1}{2} \biggr]_{x=1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ -\biggl(\frac{14}{5} \biggr) x - 7 \biggl( 1 - \frac{7}{5}x^2 \biggr) \biggr]_{x=1} </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ -\biggl(\frac{14}{5} \biggr) - 7 \biggl( 1 - \frac{7}{5} \biggr) \biggr] \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			which also properly sums to zero.

	===Properties of Eigenfunction Solutions===		===Properties of Eigenfunction Solutions===