User:Tohline/SSC/Stability/n5PolytropeLAWE - Revision history

Tohline: /* Numerical Integration of LAWE */

2017-04-18T03:50:05Z

Numerical Integration of LAWE

← Older revision		Revision as of 03:50, 18 April 2017
Line 190:		Line 190:
	==Numerical Integration of LAWE==		==Numerical Integration of LAWE==

	By numerically integrating the above LAWE, we have examined the properties of the displacement function that describes radial modes of oscillation in pressure-truncated, n = 5, polytropic configurations. Our brief description, here, of these modes parallels the more detailed description of radial oscillation modes in truncated isothermal spheres that has been [[User:Tohline/SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions\|presented in a separate chapter]].		By numerically integrating the above LAWE using the algorithm outlined in a [[User:Tohline/SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|separate chapter]], we have examined the properties of the displacement function that describes radial modes of oscillation in pressure-truncated, n = 5, polytropic configurations. Our brief description, here, of these modes parallels the more detailed description of radial oscillation modes in truncated isothermal spheres that has been [[User:Tohline/SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions\|presented in a separate chapter]].

	The animation sequence that appears in the right panel of Composite Display 1 shows how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the n=5 polytropic sphere, out to <math>~\xi = 10</math> — for sixteen different values of the square of the eigenfrequency, <math>~\sigma_c^2</math>, as denoted at the top of each animation frame. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies: (a) the radius at which the logarithmic derivative of the displacement function, <math>~d\ln x/d\ln\xi </math>, is negative three; and (b) the radius, <math>~\tilde\xi</math>, at which the n = 5 polytropic configuration has been truncated. As displayed here, in every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.		The animation sequence that appears in the right panel of Composite Display 1 shows how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the n=5 polytropic sphere, out to <math>~\xi = 10</math> — for sixteen different values of the square of the eigenfrequency, <math>~\sigma_c^2</math>, as denoted at the top of each animation frame. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies: (a) the radius at which the logarithmic derivative of the displacement function, <math>~d\ln x/d\ln\xi </math>, is negative three; and (b) the radius, <math>~\tilde\xi</math>, at which the n = 5 polytropic configuration has been truncated. As displayed here, in every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.

Tohline: /* Numerical Integration of LAWE */

2017-04-17T23:46:26Z

Numerical Integration of LAWE

← Older revision		Revision as of 23:46, 17 April 2017
Line 215:		Line 215:
	</div>		</div>

	Each full loop through the left-panel animation sequence can be viewed as evolution along the equilibrium sequence from <math>~\tilde\xi = 0.75</math> to <math>~\tilde\xi = 5</math>, then back again. During this evolution, the "×" marker moves through both turning points along the sequence: the maximum radius configuration — at <math>~\tilde\xi= \sqrt{3}</math> — and the maximum mass configuration — at <math>~\tilde\xi= 3</math>. Notice that <math>~\sigma_c^2</math> is positive for all models having <math>~\tilde\xi < 3</math> while it is negative for all models having <math>~\tilde\xi > 3</math>. Hence, models having <math>~\tilde\xi < 3</math> are dynamically unstable and the transition from stable to unstable models — that is, the ''marginally'' unstable model — occurs at <math>~\tilde\xi = 3</math>. For emphasis, the "×" marker (left panel) and the numerical value recorded for <math>~\sigma_c^2</math> (right panel) have been colored red for models that are not stable.		Each full loop through the left-panel animation sequence can be viewed as evolution along the equilibrium sequence from <math>~\tilde\xi = 0.75</math> to <math>~\tilde\xi = 5</math>, then back again. During this evolution, the "×" marker moves through both turning points along the sequence: the maximum radius configuration — at <math>~\tilde\xi= \sqrt{3}</math> — and the maximum mass configuration — at <math>~\tilde\xi= 3</math>. Notice that <math>~\sigma_c^2</math> is positive for all models having <math>~\tilde\xi < 3</math> while it is negative for all models having <math>~\tilde\xi > 3</math>. Hence, models having <math>~\tilde\xi > 3</math> are dynamically unstable and, as best we have been able to determine via these numerical integrations, the transition from stable to unstable models — that is, the ''marginally'' unstable model — occurs at <math>~\tilde\xi = 3</math>. (Via an ''analytic'' analysis, we prove, below, that this association is precise.) For emphasis, the "×" marker (left panel) and the numerical value recorded for <math>~\sigma_c^2</math> (right panel) have been colored red for models that are not stable.

	=Search for Analytic Solutions to the LAWE=		=Search for Analytic Solutions to the LAWE=

Tohline: /* Numerical Integration of LAWE */

2017-04-17T23:39:52Z

Numerical Integration of LAWE

← Older revision		Revision as of 23:39, 17 April 2017
Line 190:		Line 190:
	==Numerical Integration of LAWE==		==Numerical Integration of LAWE==

	~~Via an animation sequence~~, ~~Figure 1 displays (right panel) fundamental-mode eigenvectors — [blue] eigenfunction along with numerical value~~ of the ~~squared eigenfrequency, <math>~\sigma_c^2</math> — for~~ pressure-truncated, n = 5 ~~polytropes at (left panel) various locations along~~ the ~~mass-radius equilibrium sequence~~.		By numerically integrating the above LAWE, we have examined the properties of the displacement function that describes radial modes of oscillation in pressure-truncated, n = 5, polytropic configurations. Our brief description, here, of these modes parallels the more detailed description of radial oscillation modes in truncated isothermal spheres that has been [[User:Tohline/SSC/Stability/Isothermal#Previously_Published_Eigenvalues_and_Eigenfunctions\|presented in a separate chapter]].

	The right~~-hand column~~ of Composite Display ~~2, below, presents a pair of animations that display~~ how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the ~~isothermal~~ sphere, out to <math>~\xi = 9</math> — for ~~a variety of~~ values of the square of the eigenfrequency, <math>~\~~lambda_0~~^2</math>; each frame ~~of both animations is tagged by the relevant value of <math>~\lambda_0^2</math>~~. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies ~~the truncation radius of the relevant isothermal sphere, and~~ the radius at which the ~~boundary condition~~, <math>~d\ln x/d\ln\xi ~~= -3~~</math>, has been ~~enforced~~. In every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.		The animation sequence that appears in the right panel of Composite Display 1 shows how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the n=5 polytropic sphere, out to <math>~\xi = 10</math> — for sixteen different values of the square of the eigenfrequency, <math>~\sigma_c^2</math>, as denoted at the top of each animation frame. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies: (a) the radius at which the logarithmic derivative of the displacement function, <math>~d\ln x/d\ln\xi </math>, is negative three; and (b) the radius, <math>~\tilde\xi</math>, at which the n = 5 polytropic configuration has been truncated. As displayed here, in every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.

			The left panel of Composite Display 1 is also animated and has been provided in support of the animation on the right. Specifically, the number written at the top of each left-panel frame quantitatively identifies the radial location, <math>~\tilde\xi</math>, of the surface of the relevant truncated polytropic configuration; and, on each frame, "×" marks the location of that truncated configuration on the mass-radius equilibrium sequence.

	<div align="center" id="n5TruncatedMovie">		<div align="center" id="n5TruncatedMovie">
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
	<th align="center" colspan="2">~~Figure~~ 1:   Numerically Generated Eigenvectors</th>		<th align="center" colspan="2">Composite Display 1:   Numerically Generated Fundamental-Mode Eigenvectors</th>
	</tr>		</tr>
	<tr>		<tr>
Line 213:		Line 214:
	</table>		</table>
	</div>		</div>

			Each full loop through the left-panel animation sequence can be viewed as evolution along the equilibrium sequence from <math>~\tilde\xi = 0.75</math> to <math>~\tilde\xi = 5</math>, then back again. During this evolution, the "×" marker moves through both turning points along the sequence: the maximum radius configuration — at <math>~\tilde\xi= \sqrt{3}</math> — and the maximum mass configuration — at <math>~\tilde\xi= 3</math>. Notice that <math>~\sigma_c^2</math> is positive for all models having <math>~\tilde\xi < 3</math> while it is negative for all models having <math>~\tilde\xi > 3</math>. Hence, models having <math>~\tilde\xi < 3</math> are dynamically unstable and the transition from stable to unstable models — that is, the ''marginally'' unstable model — occurs at <math>~\tilde\xi = 3</math>. For emphasis, the "×" marker (left panel) and the numerical value recorded for <math>~\sigma_c^2</math> (right panel) have been colored red for models that are not stable.

	=Search for Analytic Solutions to the LAWE=		=Search for Analytic Solutions to the LAWE=

Tohline: /* Numerical Integration of LAWE */

2017-04-17T21:48:41Z

Numerical Integration of LAWE

← Older revision		Revision as of 21:48, 17 April 2017
Line 191:		Line 191:

	Via an animation sequence, Figure 1 displays (right panel) fundamental-mode eigenvectors — [blue] eigenfunction along with numerical value of the squared eigenfrequency, <math>~\sigma_c^2</math> — for pressure-truncated, n = 5 polytropes at (left panel) various locations along the mass-radius equilibrium sequence.		Via an animation sequence, Figure 1 displays (right panel) fundamental-mode eigenvectors — [blue] eigenfunction along with numerical value of the squared eigenfrequency, <math>~\sigma_c^2</math> — for pressure-truncated, n = 5 polytropes at (left panel) various locations along the mass-radius equilibrium sequence.

			The right-hand column of Composite Display 2, below, presents a pair of animations that display how our numerically derived displacement function, <math>~x(\xi)</math>, varies with radius — from the center of the isothermal sphere, out to <math>~\xi = 9</math> — for a variety of values of the square of the eigenfrequency, <math>~\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>~\lambda_0^2</math>. The segment of the <math>~x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency. In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>~d\ln x/d\ln\xi = -3</math>, has been enforced. In every frame, the <math>~x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.


	<div align="center" id="n5TruncatedMovie">		<div align="center" id="n5TruncatedMovie">
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">

Tohline: /* Numerical Integration of LAWE */

2017-04-17T21:22:37Z

Numerical Integration of LAWE

← Older revision		Revision as of 21:22, 17 April 2017
Line 190:		Line 190:
	==Numerical Integration of LAWE==		==Numerical Integration of LAWE==

			Via an animation sequence, Figure 1 displays (right panel) fundamental-mode eigenvectors — [blue] eigenfunction along with numerical value of the squared eigenfrequency, <math>~\sigma_c^2</math> — for pressure-truncated, n = 5 polytropes at (left panel) various locations along the mass-radius equilibrium sequence.
	<div align="center" id="n5TruncatedMovie">		<div align="center" id="n5TruncatedMovie">
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
			<th align="center" colspan="2">Figure 1:   Numerically Generated Eigenvectors</th>
			</tr>
			<tr>
			<td align="center">
			[[File:N5Truncated2.gif\|600px\|n5 Truncated movie]]
			</td>
	<td align="center">		<td align="center">
	[[File:~~N5Truncated2~~.gif~~\|n5 Truncated movie~~]]		Excel File:<br />
			[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n5Eigenvectors/n5TruncatedSphere.xlsx --- worksheet = OursPt1]]
			Movie File:<br />
			[[File:DataFileButton02.png\|right\|60px\|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/n5movie/ --- worksheet = n5Truncated2.gif]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Eureka Moment */

2017-03-30T23:10:11Z

Eureka Moment

← Older revision		Revision as of 23:10, 30 March 2017
Line 613:		Line 613:
	</tr>		</tr>
	</table>		</table>
			</div>

Tohline: /* Eureka Moment */

2017-03-10T21:57:23Z

Eureka Moment

← Older revision		Revision as of 21:57, 10 March 2017
Line 457:		Line 457:
	Hence, a parabolic eigenfunction does not work for arbitrary <math>~\alpha</math>.		Hence, a parabolic eigenfunction does not work for arbitrary <math>~\alpha</math>.
	END MORE GENERIC CASE -->		END MORE GENERIC CASE -->

			Let's express the parabolic displacement function, <math>~x</math>, as a function of the Lagrangian mass coordinate, instead of as a function of <math>~\xi</math>. Drawing upon our [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation#Exploration\|accompanying discussion]] where we have used <math>~\tilde\xi</math> to denote the truncation edge, we know that,

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

			<tr>
			<td align="right">
			<math>~r_\xi(\xi)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\xi \biggl\{
			\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
			\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			and that,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~
			r_\xi (m_\xi)
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\tilde{r}_\mathrm{edge}
			\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			<span id="DefineTildeC">where,</span>
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\tilde{C}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr) \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\tilde{r}_\mathrm{edge}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
			\, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			By equating <math>~r_\xi(\xi)</math> with <math>~r_\xi(m_\xi)</math>, we find,

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

			<tr>
			<td align="right">
			<math>~\xi \biggl\{
			\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
			\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\}
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
			\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \xi
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>
			This means that,
			<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_0 \biggl\{1 -
			\frac{1}{15} \biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]
			\biggr\} \, ;
			</math>
			</td>
			</tr>
			</table>
			</div>
			and, specifically for the critical case of <math>~\tilde\xi = 3</math>, in which case, <math>~\tilde{C} = 4</math>,
			<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_0 \biggl\{1 -
			\frac{1}{15} \biggl[\frac{3^2m_\xi^{2/3}}{4 - 3 m_\xi^{2/3}}\biggr]
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>



	{{ LSU_WorkInProgress }}		{{ LSU_WorkInProgress }}


	==Setup Using Lagrangian Radial Coordinate==		==Setup Using Lagrangian Radial Coordinate==

Tohline at 20:09, 6 March 2017

2017-03-06T20:09:31Z

← Older revision		Revision as of 20:09, 6 March 2017
Line 200:		Line 200:
	</div>		</div>

	<font color="red">Note from J. E. Tohline on 3/6/2017:</font>  Yesterday evening, after I finished putting together the above animation sequence using an Excel workbook, I noticed that the eigenfunction of the fundamental mode for the marginally unstable model <math>~(\sigma_c^2 = 0)</math> resembles a parabola. In an effort to see how well a parabola fits at least the central portion of this eigenfunction, I returned to my Excel spreedsheet and, in a brute-force manner, began to search for the pair of coefficients that would provide a best fit. What I discovered was that a parabola with the following formula fits perfectly!		=Search for Analytic Solutions to the LAWE=

			==Eureka Moment==

			<font color="red">Note from J. E. Tohline on 3/6/2017:</font>  Yesterday evening, after I finished putting together the [[#n5TruncatedMovie\|above animation sequence]] using an Excel workbook, I noticed that the eigenfunction of the fundamental mode for the marginally unstable model <math>~(\sigma_c^2 = 0)</math> resembles a parabola. In an effort to see how well a parabola fits at least the central portion of this eigenfunction, I returned to my Excel spreedsheet and, in a brute-force manner, began to search for the pair of coefficients that would provide a best fit. What I discovered was that a parabola with the following formula fits perfectly!
	<div align="center" i="AnalyticSoln">		<div align="center" i="AnalyticSoln">
	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">
	<tr>		<tr>
	<th align="center" colspan="1">Fundamental Mode Eigenfunction <~~p><~~/p>when <math>~\sigma_c^2 = 0</math> and <math>~\gamma = ~~\frac{~~6}{5} ~\Rightarrow~~\alpha=- ~~\frac{~~1}{3}</math></th>		<th align="center" colspan="1">Fundamental Mode Eigenfunction <br />
			when <math>~\sigma_c^2 = 0</math> and <math>~\gamma = 6/5 ~\Rightarrow~\alpha=- 1/3</math></th>
	</tr>		</tr>

Line 453:		Line 458:
	END MORE GENERIC CASE -->		END MORE GENERIC CASE -->

	~~=Search for Analytic Solutions to the LAWE=~~		{{ LSU_WorkInProgress }}


	==Setup Using Lagrangian Radial Coordinate==		==Setup Using Lagrangian Radial Coordinate==

Tohline: /* Numerical Integration of LAWE */

2017-03-06T20:02:11Z

Numerical Integration of LAWE

← Older revision		Revision as of 20:02, 6 March 2017
Line 285:		Line 285:
	Q. E. D.  I don't think that anyone has previously appreciated that the LAWE in this case admits to an analytic eigenvector solution.		Q. E. D.  I don't think that anyone has previously appreciated that the LAWE in this case admits to an analytic eigenvector solution.

	Now, let's see how the boundary condition comes into play. We see that the logarithmic derivative is,		Now, let's see how the boundary condition comes into play. We see that the logarithmic derivative of the parabolic eigenfunction is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 368:		Line 368:
	</div>		</div>
	Hence, although the parabolic eigenfunction provides an accurate solution to the <math>~n=5</math> LAWE throughout the entire configuration — that is, for all <math>~\xi</math> — the desired surface boundary condition will only be satisfied if the polytrope is truncated at <math>~\xi_\mathrm{surf} = 3</math>. The parabolic eigenfunction is therefore only physically relevant to the model that sits at the point along the equilibrium sequence that is associated with the <math>~P_\mathrm{max}</math> turning point.		Hence, although the parabolic eigenfunction provides an accurate solution to the <math>~n=5</math> LAWE throughout the entire configuration — that is, for all <math>~\xi</math> — the desired surface boundary condition will only be satisfied if the polytrope is truncated at <math>~\xi_\mathrm{surf} = 3</math>. The parabolic eigenfunction is therefore only physically relevant to the model that sits at the point along the equilibrium sequence that is associated with the <math>~P_\mathrm{max}</math> turning point.

			<!-- TRY MORE GENERIC CASE ...
			Finally, let's see how this plays out for arbitrary values of <math>~\alpha</math>. Guess a more general eigenfunction of the form,
			<div align="center" i="AnalyticSoln">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center">
			<math>~x = 1 - \frac{\xi^2}{A} </math>
			</td>
			</tr>
			</table>
			</div>
			In this more general case, we have,

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

			<tr>
			<td align="right">
			<math>~\frac{dx}{d\xi} = -\frac{2\xi}{A} \, ;</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~\frac{d^2x}{d\xi^2} = -\frac{2}{A} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			The LAWE then gives,

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

			<tr>
			<td align="right">
			<math>~
			(3+\xi^2) \frac{d^2x}{d\xi^2} + \biggl[12 - 2\xi^2 \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} +
			\biggl[ \cancelto{0}{\frac{\sigma_c^2}{3^{1/2} \gamma_g }} \cdot (3+\xi^2)^{3/2} -6\alpha \biggr] x
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\frac{2}{A}(3+\xi^2) -\frac{2}{A} \biggl[12 - 2\xi^2 \biggr] -6\alpha \biggl[1 - \frac{\xi^2}{A}\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-\frac{1}{A}\biggl\{
			2(3+\xi^2) + 2 \biggl[12 - 2\xi^2 \biggr] + 6\alpha \biggl[A - \xi^2\biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~-
			\frac{1}{A}\biggl[ \biggl(6 + 24 + 6\alpha A \biggr) + \biggl(2 -4-6\alpha \biggr)\xi^2\biggr]
			</math>
			</td>
			</tr>
			</table>
			</div>
			Hence, a parabolic eigenfunction does not work for arbitrary <math>~\alpha</math>.
			END MORE GENERIC CASE -->

	=Search for Analytic Solutions to the LAWE=		=Search for Analytic Solutions to the LAWE=

Tohline: /* Numerical Integration of LAWE */

2017-03-06T19:29:38Z

Numerical Integration of LAWE

← Older revision		Revision as of 19:29, 6 March 2017
Line 351:		Line 351:
	<td align="left">		<td align="left">
	<math>~3(15 - \xi^2)</math>		<math>~3(15 - \xi^2)</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~\xi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
			Hence, although the parabolic eigenfunction provides an accurate solution to the <math>~n=5</math> LAWE throughout the entire configuration — that is, for all <math>~\xi</math> — the desired surface boundary condition will only be satisfied if the polytrope is truncated at <math>~\xi_\mathrm{surf} = 3</math>. The parabolic eigenfunction is therefore only physically relevant to the model that sits at the point along the equilibrium sequence that is associated with the <math>~P_\mathrm{max}</math> turning point.

	=Search for Analytic Solutions to the LAWE=		=Search for Analytic Solutions to the LAWE=