SSC/Perturbations - Revision history

Jet53man: /* Ledoux and Pekeris (1941) */

2022-12-25T22:00:43Z

Ledoux and Pekeris (1941)

@@ Line 1,329: / Line 1,329: @@
 Historically, by the 1940s, the [[#2ndOrderODE|expression just derived]] was a relatively familiar one to astrophysicists.  For example, the opening paragraph of {{ LP41full }} reads:
 <div align="center">
-<table border="1" cellpadding="5">
+<table border="1" cellpadding="5" width="80%">
 <tr><td align="center">
-Paragraph extracted from <div align="center">{{ LP41figure }}</div> &copy; American Astronomical Society
+Paragraph extracted from &sect;1 of<br />{{ LP41figure }}<br /> &copy; American Astronomical Society
 </td></tr>
 <tr>
 <td>
-[[File:LedouxPekeris1941.jpg|600px|center|Ledoux &amp; Pekeris (1941, ApJ, 94, 124)]]
+<!--[[File:LedouxPekeris1941.jpg|600px|center|Ledoux &amp; Pekeris (1941, ApJ, 94, 124)]]-->
 </td>
 </tr>

Jet53man: /* Eddington (1926) */

2022-12-25T21:37:29Z

Eddington (1926)

← Older revision		Revision as of 17:37, 25 December 2022
Line 1,261:		Line 1,261:
	===Eddington (1926)===		===Eddington (1926)===

	[[File:Eddington1930Cover.png\|thumb\|right\|200px\|Eddington (1926)]]To our knowledge, a derivation of [[#2ndOrderODE\|this governing 2nd-order ODE]] was first presented by A. S. Eddington (1926) in a book titled, ''The Internal Constitution of the Stars''. (This entire book has been digitally scanned and is now [https://~~archive~~.~~org~~/~~details~~/~~TheInternalConstitutionOfTheStars~~ available online].) The derived expression, which appears on p. 188 as equation (127.6) of Eddington's book, is presented in the following, framed image.		[[File:Eddington1930Cover.png\|thumb\|right\|200px\|Eddington (1926)]]To our knowledge, a derivation of [[#2ndOrderODE\|this governing 2nd-order ODE]] was first presented by A. S. Eddington (1926) in a book titled, ''The Internal Constitution of the Stars''. (This entire book has been digitally scanned and is now [https://www.google.com/books/edition/The_Internal_Constitution_of_the_Stars/N6UNAQAAIAAJ?hl=en&gbpv=1&printsec=frontcover available online].) The derived expression, which appears on p. 188 as equation (127.6) of Eddington's book, is presented in the following, framed image.


	<div align="center">		<div align="center">
	<table border="1" cellpadding="5" width="75%">		<table border="1" cellpadding="5" width="65%">
	<tr>		<tr>
	<th align="center">		<td align="center">
	''Pulsation equation'' extracted~~<sup>~~&~~dagger~~;~~</sup> from~~ p. 188 of [https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)]<~~p><~~/p>		Pulsation equation extracted from §127 (p. 188) of<br />[https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)]<br />
	"''The Internal Constitution of the Stars''~~"<p>~~</p>(Cambridge: Cambridge University Press)		''The Internal Constitution of the Stars''<br />Cambridge: Cambridge University Press
	</th>		</td>
	<tr>		<tr>
	<td>		<td>
	[[File:Eddington1930.png\|600px\|center\|Eddington (1926)]]		<!--[[File:Eddington1930.png\|600px\|center\|Eddington (1926)]]-->
			<table border="0" width="100%" cellpadding="10" align="center">
			<tr>
			<td align="right" width="60%"><math>
			\frac{d^2\xi_1}{d\xi_0^2} + \frac{4-\mu}{\xi_0} \frac{d\xi_1}{d\xi_0} + \biggl\{ \frac{n^2\rho_0}{\gamma P_0} - \biggl(3 - \frac{4}{\gamma}\biggr)
			\frac{\mu}{\xi_0^2} \biggr\}\xi_1
			</math></td>
			<td align="center" width="3%"><math>=</math></td>
			<td align="left"><math>0</math></td>
			<td align="right" width="10%">(127.6)</td>
			</tr>

			<tr>
			<td align="left" colspan="4">where            <math>\mu = g_0\rho_0\xi_0/P_0 \, .</math></td>
			</tr>
			</table>
	</td>		</td>
	</tr>		</tr>
	~~<tr><td align="left"><sup>&dagger;</sup>Mathematical expressions have been displayed here, as a single digital image, with a layout that is modified from the original publication.</td></tr>~~
	</table>		</table>
	</div>		</div>


	The similarity between Eddington's expression and the [[#2ndOrderODE\|governing 2nd-order ODE that we have derived]] is immediately apparent. Specifically, simply after inserting Eddington's definition of his composite variable, <math>~\mu</math>, and making the substitutions,		The similarity between Eddington's expression and the [[#2ndOrderODE\|governing 2nd-order ODE that we have derived]] is immediately apparent. Specifically, simply after inserting Eddington's definition of his composite variable, <math>\mu</math>, and making the substitutions,
	<div align="center">		<div align="center">
	<math>~\xi_1 \rightarrow x \, ,</math>		<math>\xi_1 \rightarrow x \, ,</math>

	<math>\xi_0 \rightarrow r_0 \, ,</math>		<math>\xi_0 \rightarrow r_0 \, ,</math>
Line 1,295:		Line 1,309:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~		<math>
	\frac{d^2 x}{dr_0^2} + \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}		\frac{d^2 x}{dr_0^2} + \biggl[ \frac{4}{r_0} - \frac{g_0 \rho_0}{P_0} \biggr] \frac{dx}{dr_0}
	+ \frac{\rho_0}{\gamma P_0} \biggl\{\omega^2 + \biggl(4 - 3\gamma\biggr) \frac{g_0 }{r_0} \biggr\}x		+ \frac{\rho_0}{\gamma P_0} \biggl\{\omega^2 + \biggl(4 - 3\gamma\biggr) \frac{g_0 }{r_0} \biggr\}x
Line 1,301:		Line 1,315:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~0 \, ,</math>		<math>0 \, ,</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Eddington (1926) */

2022-12-25T21:07:08Z

Eddington (1926)

← Older revision		Revision as of 17:07, 25 December 2022
Line 1,261:		Line 1,261:
	===Eddington (1926)===		===Eddington (1926)===

	[[File:Eddington1930Cover.png\|thumb\|right\|Eddington (1926)]]To our knowledge, a derivation of [[#2ndOrderODE\|this governing 2nd-order ODE]] was first presented by A. S. Eddington (1926) in a book titled, ''The Internal Constitution of the Stars''. (This entire book has been digitally scanned and is now [https://archive.org/details/TheInternalConstitutionOfTheStars available online].) The derived expression, which appears on p. 188 as equation (127.6) of Eddington's book, is presented in the following, framed image.		[[File:Eddington1930Cover.png\|thumb\|right\|200px\|Eddington (1926)]]To our knowledge, a derivation of [[#2ndOrderODE\|this governing 2nd-order ODE]] was first presented by A. S. Eddington (1926) in a book titled, ''The Internal Constitution of the Stars''. (This entire book has been digitally scanned and is now [https://archive.org/details/TheInternalConstitutionOfTheStars available online].) The derived expression, which appears on p. 188 as equation (127.6) of Eddington's book, is presented in the following, framed image.

Jet53man: /* Continuity Equation */

2022-09-09T16:56:27Z

Continuity Equation

← Older revision		Revision as of 12:56, 9 September 2022
Line 547:		Line 547:
	</tr>		</tr>
	</table>		</table>
			<table border="1" cellpadding="8"><tr><td align="center">
			NOTE:   An alternate approach to deriving this expression can be found in an [[SSC/Structure/BiPolytropes/51RenormaizePart2#STEP4\|accompanying <font color="red">ASIDE</font>]].
			</td></tr></table>
	</div>		</div>

Jet53man: /* Entropy Conservation */

2022-05-24T22:45:01Z

Entropy Conservation

← Older revision		Revision as of 18:45, 24 May 2022
Line 411:		Line 411:
	</tr>		</tr>
	</table>		</table>

			This is identical to the algebraic relationship between <math>p(m)</math> and <math>d(m)</math> that has been derived, [[#Adiabatic_form_of_the_First_Law_of_Thermodynamics\|immediately above]].

	====Continuity Equation====		====Continuity Equation====

Jet53man: /* Entropy Conservation */

2022-05-24T22:38:50Z

Entropy Conservation

@@ Line 300: / Line 300: @@
 <math>P_0(m) \rho_0(m)^{-\gamma_g}
 \biggl[1 + p(m) e^{i\omega t}  \biggr]
 \biggl[1 + d(m) e^{i\omega t}  \biggr]^{-\gamma_g}
 </math>
    </td>

Jet53man: /* Entropy Conservation */

2022-05-24T22:07:19Z

Entropy Conservation

@@ Line 284: / Line 284: @@
    <td align="left">
 <math>
-\biggl\{ P \biggr\}\biggl\{\rho \biggr\}^{-\gamma_g}
+\biggl\{ P_0(m) \biggl[1 + p(m) e^{i\omega t}  \biggr] \biggr\}
 </math>
    </td>

Jet53man: /* Entropy Conservation */

2022-05-24T22:00:15Z

Entropy Conservation

← Older revision		Revision as of 18:00, 24 May 2022
Line 271:		Line 271:

	Plugging the perturbed expressions for <math>P(m,t)</math> and <math>\rho(m,t)</math> into this expression gives,		Plugging the perturbed expressions for <math>P(m,t)</math> and <math>\rho(m,t)</math> into this expression gives,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>
			\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]
			</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl\{ P \biggr\}\biggl\{\rho \biggr\}^{-\gamma_g}
			</math>
			</td>
			</tr>
			</table>

	====Continuity Equation====		====Continuity Equation====

Jet53man: /* Entropy Conservation */

2022-05-24T21:55:42Z

Entropy Conservation

← Older revision		Revision as of 17:55, 24 May 2022
Line 269:		Line 269:
	</tr>		</tr>
	</table>		</table>

			Plugging the perturbed expressions for <math>P(m,t)</math> and <math>\rho(m,t)</math> into this expression gives,

	====Continuity Equation====		====Continuity Equation====

Jet53man: /* Entropy Conservation */

2022-05-24T21:52:15Z

Entropy Conservation

@@ Line 251: / Line 251: @@
 ====Entropy Conservation====
 If, instead, we simply demand that the specific entropy remain constant in time, then the expression that relates <math>P</math> to <math>\rho</math> is,
 ====Continuity Equation====