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

Tohline: /* Numerical Integration from the Center, Outward */ Correct expression for x_2 using newly derived power-series expression

2017-02-28T20:50:37Z

Numerical Integration from the Center, Outward: Correct expression for x_2 using newly derived power-series expression

← Older revision		Revision as of 20:50, 28 February 2017
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	</div>		</div>

	In order to kick-start the integration~~, at the center of the configuration <math>~(\xi_1 = 0)</math>~~, we will set the ~~eigenfunction~~ value to <math>~x_1 = 1</math>~~; and we will use~~ the ~~assumed symmetry condition, <math>~x_1' = 0 ~~\Rightarrow ~~ x_- = x_+</math>, in order to determine the value~~ of the ~~eigenfunction at the first grid location off center~~ <math>~(\~~xi_2~~ = ~~\Delta~~)~~</math>. That is, for <math>~i = 1~~</math>, the ~~discretized LAWE gives,~~		In order to kick-start the integration, we will set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then we will draw on the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \Delta_\xi</math>, away from the center. Specifically, we will 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_i \biggl\{ 1~~
	~~-\frac{\Delta_\xi^2 (n+1)}{2\theta_i}\biggl[ \frac{\sigma_c^2}{6\gamma_g} -~~
	~~\frac{\alpha}{\xi_i } (- \theta^')_i\biggr] \biggr\} \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	~~Given that an~~ [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#~~Polytropic_Lane-Emden_Function~~\|~~approximate~~ power-series ~~representation~~ of the ~~polytropic Lane-Emden~~ function~~]] is~~,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	<math>~\~~theta</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align~~=~~"left">~~
	~~<math>~~~
	~~1 -~~ \~~frac{\xi^2}{6} + \frac{n}{120} \xi^4 - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \xi^6 + \biggl[ \frac{n(122n^2 -183n + 70)}{3265920} \biggr]~~ \xi~~^8 + \cdots~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	~~we know that in this individual case <math>~(i=1)~~</math> ~~— that is~~, ~~for <math>~\xi = 0</math> — <math>~\theta = 1</math> and <math>~(-\theta')/\xi = \tfrac{1}{3}</math> for all values of~~ the ~~polytropic index~~. ~~Hence, for <math>~i=1</math>~~, we ~~can use the expression~~,
	<div align="center">		<div align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	x_1 [ 1 - \~~tfrac~~{1}{12} \Delta_\xi^2 ~~(n+1) \mathfrak~~{F} ] \, ,</math>		x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F} \Delta_\xi^2}{60} \biggr] \, ,</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Numerical Integration from the Center, Outward */

2017-02-25T22:10:08Z

Numerical Integration from the Center, Outward

← Older revision		Revision as of 22:10, 25 February 2017
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	</div>		</div>

	Given that ~~the~~ power-series representation of the Lane-Emden function is,		Given that an [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Polytropic_Lane-Emden_Function\|approximate power-series representation of the polytropic Lane-Emden function]] is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Adiabatic (Polytropic) Wave Equation */

2017-02-22T16:37:50Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 16:37, 22 February 2017
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	It is also the same as the radial pulsation equation for polytropic configurations that appears as equation (56) in the detailed discussion of "The Oscillations of Gas Spheres" published by [http://adsabs.harvard.edu/abs/1966ApJ...143..535H H M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535)]; hereafter, we will refer to this paper as HRW66. The relevant set of equations from HRW66 has been extracted as a single digital image and reprinted, here, as a boxed-in image.		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline: As is shown in the subsection on "Boundary Conditions," below, it appears as though the term on the right-hand-side of HRW66's equation (58) is incorrect, as published; it should be preceded with a negative sign.]]It is also the same as the radial pulsation equation for polytropic configurations that appears as equation (56) in the detailed discussion of "The Oscillations of Gas Spheres" published by [http://adsabs.harvard.edu/abs/1966ApJ...143..535H H M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535)]; hereafter, we will refer to this paper as HRW66. The relevant set of equations from HRW66 has been extracted as a single digital image and reprinted, here, as a boxed-in image.


	<div align="center" id="HRW66excerpt">		<div align="center" id="HRW66excerpt">
	<table border="2" cellpadding="10">		<table border="2" cellpadding="10">

Tohline: /* Adiabatic (Polytropic) Wave Equation */

2017-02-22T16:30:58Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 16:30, 22 February 2017
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	</div>		</div>

	As can be seen in the following framed image, this is the form of the ''polytropic'' wave equation published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astron. Soc. Australia, 6, 222)], at the beginning of their discussion of "Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models." (NOTE: There appears to be a sign error in the numerator of the second term of their published expression; there also appears to be an error in the definition of the coefficient, <math>~\alpha^*</math>, as given in the text of their paper.)		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline: There appears to be a sign error in the numerator of the second term of the polytropic wave equation published by Murphy & Fiedler; there also appears to be an error in the definition of the coefficient, α*, as given in the text of their paper.]]As can be seen in the following framed image, this is the form of the ''polytropic'' wave equation published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astron. Soc. Australia, 6, 222)], at the beginning of their discussion of "Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models."

	<div align="center">		<div align="center">

Tohline: /* Groundwork */

2017-02-21T17:32:45Z

Groundwork

← Older revision		Revision as of 17:32, 21 February 2017
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	</div>		</div>

	The correspondence with our derived expression is complete, assuming that,		<span id="HRW66frequency">The correspondence with our derived expression is complete, assuming that,</span>
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* n = 3 Polytrope */

2017-02-17T17:21:11Z

n = 3 Polytrope

← Older revision		Revision as of 17:21, 17 February 2017
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	=~~n = 3 Polytrope~~=		=Numerical Integration from the Center, Outward=
	Here we ~~perform~~ a ~~numerical integration of~~ the governing LAWE ~~for <math>~n=3</math> polytropes. We can directly compare our results with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S Schwarzschild's (1941)] published work on "Overtone Pulsations for~~ the ~~Standard [Stellar] Model~~."		Here we show how a relatively simple, finite-difference algorithm can be developed to numerically integrate the governing LAWE from the center of a polytropic configuration, outward to its surface.

	Drawing from our [[#Groundwork\|above discussion]], the LAWE for any polytrope of index, <math>~n</math>, may be written as,		Drawing from our [[#Groundwork\|above discussion]], the LAWE for any polytrope of index, <math>~n</math>, may be written as,
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	</div>		</div>

	It can be shown straightforwardly that this matches the LAWE used by [[User:Tohline/SSC/Perturbations#Schwarzschild_.281941.29\|Schwarzschild (1941)]], if <math>~n</math> is set to 3. But let's postpone making this substitution until we formulate a general approach to integrating this equation from the center of the configuration, outward. Following a [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Integrating_Outward_Through_the_Core\|parallel discussion]], we begin by multiplying the LAWE through by <math>~~~(\xi~~\theta)</math>, obtaining a 2<sup>nd</sup>-order ODE that is relevant at every individual coordinate location, <math>~\xi_i</math>, namely,		Following a [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Integrating_Outward_Through_the_Core\|parallel discussion]], we begin by multiplying the LAWE through by <math>~\theta</math>, obtaining a 2<sup>nd</sup>-order ODE that is relevant at every individual coordinate location, <math>~\xi_i</math>, namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Related Discussions */

2017-02-16T21:01:49Z

Related Discussions

← Older revision		Revision as of 21:01, 16 February 2017
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	=Related Discussions=		=Related Discussions=
			* Radial Oscillations of [[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres\|Uniform-density sphere]]
			* Radial Oscillations of Isolated Polytropes
			** [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|Setup]]
			** n = 1:  [[User:Tohline/SSC/Stability/n1PolytropeLAWE\|Attempt at Formulating an Analytic Solution]]
			** 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]]

	* In an accompanying [[User:Tohline/Appendix/Ramblings/SphericalWaveEquation#Playing_With_Spherical_Wave_Equation\|Chapter within our "Ramblings" Appendix]], we have played with the adiabatic wave equation for polytropes, examining its form when the primary perturbation variable is an enthalpy-like quantity, rather than the radial displacement of a spherical mass shell. This was done in an effort to mimic the approach that has been taken in studies of the [[User:Tohline/Apps/ImamuraHadleyCollaboration#Papaloizou-Pringle_Tori\|stability of Papaloizou-Pringle tori]].		* In an accompanying [[User:Tohline/Appendix/Ramblings/SphericalWaveEquation#Playing_With_Spherical_Wave_Equation\|Chapter within our "Ramblings" Appendix]], we have played with the adiabatic wave equation for polytropes, examining its form when the primary perturbation variable is an enthalpy-like quantity, rather than the radial displacement of a spherical mass shell. This was done in an effort to mimic the approach that has been taken in studies of the [[User:Tohline/Apps/ImamuraHadleyCollaboration#Papaloizou-Pringle_Tori\|stability of Papaloizou-Pringle tori]].

Tohline: Delete subsection discussing Yabushita's (1991) work on truncated isothermal spheres, as it has been moved to a separate chapter on this topic

2017-02-16T21:00:49Z

Delete subsection discussing Yabushita's (1991) work on truncated isothermal spheres, as it has been moved to a separate chapter on this topic

← Older revision		Revision as of 21:00, 16 February 2017
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	</div>		</div>
	With the exception of the leading negative sign on the right-hand side, this expression is identical to the outer boundary condition identified by equation (58) of HRW66 — see the [[User:Tohline/SSC/Stability/Polytropes#HRW66excerpt\|excerpt reproduced above]].		With the exception of the leading negative sign on the right-hand side, this expression is identical to the outer boundary condition identified by equation (58) of HRW66 — see the [[User:Tohline/SSC/Stability/Polytropes#HRW66excerpt\|excerpt reproduced above]].

	~~==Yabushita's (1992) Analysis==~~

	In the portion (§5) of his analysis that is focused on the stability of pressure-truncated polytropic spheres, [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y S. Yabushita (1992)] examined the eigenvalue problem governed by the following wave equation:
	~~<div align="center" id="HRW66excerpt">~~
	~~<table border="2" cellpadding="10">~~
	~~<tr>~~
	~~<th align="center">~~
	~~Radial Pulsation Equation Extracted<sup>&dagger;</sup> from p. 182 of [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y S. Yabushita (1992)]<p></p>~~
	~~"''Similarity Between the Structure and Stability of Isothermal and Polytropic Gas Spheres''"<p></p>~~
	~~Astrophysics and Space Science, vol. 193, pp. 173-183 © [http://www.springer.com/astronomy/astrophysics+and+astroparticles/journal/10509 Springer]~~
	~~</th>~~
	~~<tr>~~
	~~<td>~~
	~~[[File:Yabushita1992WaveEquation2.png\|650px\|center\|Yabushita (1992)]]~~
	~~</td>~~
	~~</tr>~~
	~~<tr><td align="left">~~
	~~<sup>&dagger;</sup>Equations and text displayed here exactly as it appears in the original publication.~~
	~~</td></tr>~~
	~~</table>~~
	~~</div>~~
	Let's examine the overlap between this pair of governing relations and the ones employed by HRW66. If we replace the variable <math>~X</math> with <math>~h</math>, set <math>~\gamma = (n+1)/n</math>, and set the dimensionless eigenfrequency, <math>~s</math>, to zero in the [[#HRW66excerpt\|radial pulsation equation employed by HRW66]], 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^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (n+1)\biggl[ 3 - \frac{4n}{(n+1)} \biggr] \biggl[ \frac{\theta^' h}{\theta x} \biggr]~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (3-n) \biggl[ \frac{\theta^' h}{\theta x} \biggr] \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	This matches equation (5.3) of [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y Yabushita (1992)] — see the above boxed-in image — except the <math>~(4/x)</math> term appears as <math>~(2/x)</math> in Yabushita's article; giving the benefit of the doubt, <font color="red">this is most likely a typographical error</font> in [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y Yabushita (1992)]. According to HRW66, the corresponding central boundary condition is,
	~~<div align="center">~~
	~~<math>\frac{dh}{dx} = 0</math>         at         <math>x=0 \, .</math>~~
	~~</div>~~
	While — after changing the sign on the right-hand side of HRW66's equation (58) as argued in our [[User:Tohline/SSC/Perturbations#ChristyCox\|accompanying discussion]] in order to align with the separate derivations presented by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)] — the corresponding boundary condition at the surface is,
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\frac{dh}{dx}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~- \frac{h}{x} \biggr[ 3 - \frac{4}{\gamma} + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math>~~
	~~</td>~~
	~~<td align="left" colspan="2">~~
	~~ ~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~        at        ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~x = x_0 \, .</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	This surface boundary condition, which has been used by the astrophysics community in the context of ''isolated'' polytropic configurations, is different from the one displayed as equation (5.4) of [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y Yabushita (1992)]. The surface boundary condition chosen by Yabushita — effectively,
	~~<div align="center">~~
	~~<math>~\frac{d \ln h}{d\ln x} = -3 \, ,</math>~~
	~~</div>~~
	— does seem to be more appropriate in the context of a study of the stability of ''pressure-truncated'' polytropes because, as argued by [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] and as reviewed in our [[User:Tohline/SSC/Perturbations#Set_the_Surface_Pressure_Fluctuation_to_Zero\|accompanying discussion]], it ensures that the pressure fluctuation ''at the surface'' is zero. It is worth noting that Yabushita's surface boundary condition matches the surface boundary condition chosen by [http://adsabs.harvard.edu/abs/1974MNRAS.168..427T Taff & Van Horn (1974)] in their study of pressure-truncated ''isothermal'' spheres; in their words (see p. 428 of their article):   [Setting the surface logarithmic derivative to negative 3] <font color="green">expresses the condition that the pressure at the perturbed surface always remain[s] equal to the confining pressure exerted by the external medium in which the [pressure-truncated] sphere must be embedded</font>.

	==Overview==		==Overview==

Tohline: Delete entire subsection on n=5 polytropic spheres because this material has been moved to a new chapter

2017-02-16T20:54:25Z

Delete entire subsection on n=5 polytropic spheres because this material has been moved to a new chapter

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Tohline: Delete entire subsection on n=1 polytropes and move that material to a new, separate chapter

2017-02-16T20:46:39Z

Delete entire subsection on n=1 polytropes and move that material to a new, separate chapter

Show changes