SSC/Stability/Polytropes - Revision history

Jet53man: /* Numerical Integration from the Center, Outward */

2022-05-28T17:27:18Z

Numerical Integration from the Center, Outward

← Older revision		Revision as of 13:27, 28 May 2022
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	</div>		</div>

	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 [[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,		<span id="KickStart">In order to kick-start the integration</span>, 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 [[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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-27T15:57:03Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 11:57, 27 October 2021
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	</div>		</div>

	[[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 {{ MF85bfull }}, at the beginning of their discussion.		As can be seen in the following set of retyped expressions, this is the form of the ''polytropic'' wave equation published by {{ MF85bfull }}, at the beginning of their discussion.

	<div align="center">		<div align="center">
	<table border="2" cellpadding="10" width="75%">		<table border="2" cellpadding="10" width="80%">
	<tr>		<tr>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td>		<td>
	<!-- [[File:MurphyFiedler1985b.png\|500px\|center\|Murphy & Fiedler (1985b)]] -->		[[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 as published by Murphy & Fiedler and as retyped here; there also appears to be an error in the definition of the coefficient, α*, as given in the text of their paper.]]<!-- [[File:MurphyFiedler1985b.png\|500px\|center\|Murphy & Fiedler (1985b)]] -->
	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	<tr>		<tr>
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	[[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 {{ HRW66hereafter }}'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 {{ HRW66full }}; hereafter, {{ HRW66hereafter }}. The relevant set of equations from {{ HRW66hereafter }} ~~has been extracted as a single digital image and reprinted,~~ here~~, as a boxed-in image~~.		It is also the same as the radial pulsation equation for polytropic configurations that appears as equation (56) in {{ HRW66full }}; hereafter, {{ HRW66hereafter }}. The relevant set of equations from {{ HRW66hereafter }} is retyped here.

	<div align="center" id="HRW66excerpt">		<div align="center" id="HRW66excerpt">
	<table border="2" cellpadding="10">		<table border="2" cellpadding="10" width="80%">
	<tr>		<tr>
	<td align="center">		<td align="center">
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	</table>		</table>

	and		[[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 {{ HRW66hereafter }}'s equation (58) is incorrect, as published and as retyped here; it should be preceded with a negative sign.]]and

	<table border="0" align="center" cellpadding="8" width="~~100~~%">		<table border="0" align="center" cellpadding="8" width="80%">
	<tr>		<tr>
	<td align="right">		<td align="right">
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	<div align="center">		<div align="center">
	<table border="1" align="center" width="75%" cellpadding="5">		<table border="1" align="center" width="80%" cellpadding="5">
	<tr><td align="left">		<tr><td align="left">
	<font color="maroon">'''ASIDE:'''</font> In their equation (46), {{ HRW66hereafter }} convert the eigenfrequency, <math>~s</math> — which has units of inverse time — to a dimensionless eigenfrequency, <math>~s^'</math>, via the relation,		<font color="maroon">'''ASIDE:'''</font> In their equation (46), {{ HRW66hereafter }} convert the eigenfrequency, <math>~s</math> — which has units of inverse time — to a dimensionless eigenfrequency, <math>~s^'</math>, via the relation,

Jet53man: /* Tables */

2021-10-26T21:39:31Z

Tables

← Older revision		Revision as of 17:39, 26 October 2021
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	Excerpts from Table 1 of		Excerpts from Table 1 of

	~~[http://adsabs.harvard.edu/abs/1966ApJ...143..535H Hurley, Roberts, & Wright (1966)]~~		{{ HRW66 }}

	<math>~s^2 (n+1)/(4\pi G\rho_c)</math>		<math>~s^2 (n+1)/(4\pi G\rho_c)</math>

Jet53man: /* Overview */

2021-10-26T21:38:43Z

Overview

← Older revision		Revision as of 17:38, 26 October 2021
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	* P. LeDoux & Th. Walraven (1958, Handbuch der Physik, 51, 353) —		* P. LeDoux & Th. Walraven (1958, Handbuch der Physik, 51, 353) —
	* [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353)] — ''Pulsation Theory''		* [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353)] — ''Pulsation Theory''
	* ~~[http://adsabs.harvard.edu/abs/1966ApJ...143..535H M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535)]~~ — ''The Oscillations of Gas Spheres''		* {{ HRW66full }} — ''The Oscillations of Gas Spheres''
	* [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] — ''Pulsating Stars''		* [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] — ''Pulsating Stars''

Jet53man: /* Boundary Conditions */

2021-10-26T21:37:43Z

Boundary Conditions

← Older revision		Revision as of 17:37, 26 October 2021
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	<math>~\frac{dx}{d\xi} = 0</math>        at         <math>~\xi = 0 \, .</math>		<math>~\frac{dx}{d\xi} = 0</math>        at         <math>~\xi = 0 \, .</math>
	</div>		</div>
	This is precisely the inner boundary condition specified by ~~HRW66~~ — see their equation (57), which has been reproduced in the above excerpt from HWR66.		This is precisely the inner boundary condition specified by {{ HRW66hereafter }} — see their equation (57), which has been reproduced in the above excerpt from HWR66.


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	</div>		</div>

	Adopting notation used by ~~HRW66~~, specifically, as demonstrated above,		Adopting notation used by {{ HRW66hereafter }}, specifically, as demonstrated above,
	<div align="center">		<div align="center">
	<math>~-\omega^2 \biggl( \frac{1+n}{4\pi G \rho_c } \biggr) \rightarrow (s^')^2 \, , </math>		<math>~-\omega^2 \biggl( \frac{1+n}{4\pi G \rho_c } \biggr) \rightarrow (s^')^2 \, , </math>
	</div>		</div>
	and, from equation (50) of ~~HRW66~~,		and, from equation (50) of {{ HRW66hereafter }},
	<div align="center">		<div align="center">
	<math>~-\theta^' \rightarrow q </math>		<math>~-\theta^' \rightarrow q </math>
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	</table>		</table>
	</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 [[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 {{ HRW66hereafter }} — see the [[SSC/Stability/Polytropes#HRW66excerpt\|excerpt reproduced above]].

	==Overview==		==Overview==

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-26T21:36:21Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 17:36, 26 October 2021
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	[[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 {{ HRW66full }}; hereafter, {{ HRW66hereafter }}. The relevant set of equations from {{ HRW66hereafter }} 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 {{ HRW66hereafter }}'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 {{ HRW66full }}; hereafter, {{ HRW66hereafter }}. The relevant set of equations from {{ HRW66hereafter }} 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">
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	</div>		</div>

	In order to make clearer the correspondence between our derived expression and the one published by ~~HRW66~~, we will rewrite the ~~HRW66~~ radial pulsation equation: (1) Gathering all terms on the same side of the equation; (2) making the substitution,		In order to make clearer the correspondence between our derived expression and the one published by {{ HRW66hereafter }}, we will rewrite the {{ HRW66hereafter }} radial pulsation equation: (1) Gathering all terms on the same side of the equation; (2) making the substitution,
	<div align="center">		<div align="center">
	<math>\theta^' \rightarrow -\frac{\theta V}{x} \, ;</math>		<math>\theta^' \rightarrow -\frac{\theta V}{x} \, ;</math>
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	<table border="1" align="center" width="75%" cellpadding="5">		<table border="1" align="center" width="75%" cellpadding="5">
	<tr><td align="left">		<tr><td align="left">
	<font color="maroon">'''ASIDE:'''</font> In their equation (46), ~~HRW66~~ convert the eigenfrequency, <math>~s</math> — which has units of inverse time — to a dimensionless eigenfrequency, <math>~s^'</math>, via the relation,		<font color="maroon">'''ASIDE:'''</font> In their equation (46), {{ HRW66hereafter }} convert the eigenfrequency, <math>~s</math> — which has units of inverse time — to a dimensionless eigenfrequency, <math>~s^'</math>, via the relation,
	<div align="center">		<div align="center">
	<math>~s = \biggl( \frac{4\pi G \rho_c}{1+n} \biggr)^{1/2} s^' ~~~~~~~\cdots\cdots~~~~~~~(46)</math>		<math>~s = \biggl( \frac{4\pi G \rho_c}{1+n} \biggr)^{1/2} s^' ~~~~~~~\cdots\cdots~~~~~~~(46)</math>
	</div>		</div>
	Then, immediately following equation (46), they state that they will "omit the prime on <math>~s</math> henceforward." As a result, the ''dimensionless'' eigenfrequency that appears in their equations (56) and (58) is unprimed. This is unfortunate as it somewhat muddies our efforts, here, to demonstrate the correspondence between the ~~HRW66~~ ''polytropic'' radial pulsation equation and ours. In our subsequent manipulation of equation (56) from ~~HRW66~~ we reattach a prime to the quantity, <math>~s</math>, to emphasize that it is a ''dimensionless'' frequency. But this prime on <math>~s</math> should not be confused with the prime on <math>~\theta</math> (~~HRW66~~ equation 56) or with the prime on <math>~X</math> (~~HRW66~~ equation 57), both of which denote differentiation with respect to the radial coordinate.		Then, immediately following equation (46), they state that they will "omit the prime on <math>~s</math> henceforward." As a result, the ''dimensionless'' eigenfrequency that appears in their equations (56) and (58) is unprimed. This is unfortunate as it somewhat muddies our efforts, here, to demonstrate the correspondence between the {{ HRW66hereafter }} ''polytropic'' radial pulsation equation and ours. In our subsequent manipulation of equation (56) from {{ HRW66hereafter }} we reattach a prime to the quantity, <math>s</math>, to emphasize that it is a ''dimensionless'' frequency. But this prime on <math>~s</math> should not be confused with the prime on <math>\theta</math> ({{ HRW66hereafter }} equation 56) or with the prime on <math>~X</math> ({{ HRW66hereafter }} equation 57), both of which denote differentiation with respect to the radial coordinate.
	</td></tr>		</td></tr>
	</table>		</table>
	</div>		</div>

	With these modifications, the ~~HRW66~~ radial pulsation equation becomes,		With these modifications, the {{ HRW66hereafter }} radial pulsation equation becomes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	As has been explained in the above "<font color="maroon">'''ASIDE'''</font>," this is exactly the factor that ~~HRW66~~ use to normalize their eigenfrequency, <math>~s</math>, and make it dimensionless <math>~(s^')</math>. It is clear, as well, that ~~HRW66~~ have adopted a sign convention for the square of their eigenfrequency that is the opposite of the sign convention that we have adopted for <math>~\omega^2</math>. That is, it is clear that,		As has been explained in the above "<font color="maroon">'''ASIDE'''</font>," this is exactly the factor that {{ HRW66hereafter }} use to normalize their eigenfrequency, <math>~s</math>, and make it dimensionless <math>(s^')</math>. It is clear, as well, that {{ HRW66hereafter }} have adopted a sign convention for the square of their eigenfrequency that is the opposite of the sign convention that we have adopted for <math>~\omega^2</math>. That is, it is clear that,
	<div align="center">		<div align="center">
	<math>~s^2 ~~\leftrightarrow~~ - \omega^2 \, .</math>		<math>~s^2 ~~\leftrightarrow~~ - \omega^2 \, .</math>

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-26T21:30:39Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 17:30, 26 October 2021
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	[[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.		[[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 {{ HRW66full }}; hereafter, {{ HRW66hereafter }}. The relevant set of equations from {{ HRW66hereafter }} 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">
	<tr>		<tr>
	<th align="center">		<td align="center">
	Radial Pulsation Equation as Presented<sup>&dagger;</sup> by ~~[http://adsabs.harvard.edu/abs/1966ApJ...143..535H M. Hurley, P. H. Roberts, & K. Wright (1966)]<p>~~</p>		Radial Pulsation Equation as Presented<sup>&dagger;</sup> by<br />
	~~"''The Oscillations of Gas Spheres''"<p>~~</p>		{{ HRW66figure }}<br />
	~~The Astrophysical Journal, vol. 143, pp. 535 - 551~~ © American Astronomical Society		© American Astronomical Society
	</th>		</td>
	<tr>		<tr>
	<td align="left">		<td align="left">

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-26T21:09:55Z

Adiabatic (Polytropic) Wave Equation

@@ Line 240: / Line 240: @@
    </th>
 <tr>
-   <td>
+   <td align="left">
-[[File:HRW66_PolytropicWaveEquation.png|600px|center|Hurley, Roberts &amp; Wright (1966)]]
+<!-- [[File:HRW66_PolytropicWaveEquation.png|600px|center|Hurley, Roberts &amp; Wright (1966)]] -->
    </td>
 </tr>

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-26T17:58:31Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 13:58, 26 October 2021
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	<table border="2" cellpadding="10" width="75%">		<table border="2" cellpadding="10" width="75%">
	<tr>		<tr>
	<th align="center">		<td align="center">
	''Polytropic'' Wave Equation extracted<sup>&dagger;</sup> from ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b)]<p>~~</p>		''Polytropic'' Wave Equation extracted<sup>&dagger;</sup> from<br />
	~~"''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models''"<p>~~</p>		{{ MF85bfigure }}<br />
	~~Proceeding of the Astronomical Society of Australia, vol. 6, pp. 222 - 226~~ © Astronomical Society of Australia		© Astronomical Society of Australia
	</th>		</td>
	<tr>		<tr>
	<td>		<td>
	[[File:MurphyFiedler1985b.png\|500px\|center\|Murphy & Fiedler (1985b)]]		<!-- [[File:MurphyFiedler1985b.png\|500px\|center\|Murphy & Fiedler (1985b)]] -->
			<table border="0" align="center" cellpadding="8">
			<tr>
			<td align="right">
			<math>
			\frac{d^2\eta}{d\zeta^2}
			+ \biggl( \frac{4+(n + 1)V}{\zeta}\biggr) \frac{d\eta}{d\zeta}
			+ \biggl( \frac{\omega_k^2 \theta_c}{\theta} - \frac{\alpha^*(n+1)V}{\zeta^2}\biggr)\eta
			</math>
			</td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>0 \, ,</math>
			</td>
			</tr>
			</table>
			<div align="center">where:    <math>\omega_k^2 = \frac{\sigma_k^2 \alpha_n^2 \rho_c}{\gamma P_c} \, .</math></div>
	</td>		</td>
	</tr>		</tr>
	<tr><td align="left"><sup>&dagger;</sup>~~Equations~~ displayed here, ~~as a single digital image, with~~ layout modified from the original publication.</td></tr>		<tr><td align="left"><sup>&dagger;</sup>As displayed here, the layout of the equations has been modified from the original publication.</td></tr>
	</table>		</table>
	</div>		</div>

Jet53man: /* Adiabatic (Polytropic) Wave Equation */

2021-10-26T17:41:22Z

Adiabatic (Polytropic) Wave Equation

← Older revision		Revision as of 13:41, 26 October 2021
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	</div>		</div>

	[[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~~."		[[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 {{ MF85bfull }}, at the beginning of their discussion.

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