SSC/Stability/UniformDensity - Revision history

Jet53man: /* Sterne's Presentation */

2022-10-11T17:35:57Z

Sterne's Presentation

← Older revision		Revision as of 13:35, 11 October 2022
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	</div>		</div>

	The first few solutions are displayed in the following boxed-in image that has been extracted directly from §2 (p. 587) of {{ Sterne37 }}; to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of solutions presented by {{ Sterne37hereafter }}.		The first few solutions are displayed in the following boxed-in image that has been extracted directly from §2 (p. 587) of {{ Sterne37 }}; to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of the solutions presented by {{ Sterne37hereafter }}.

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

Jet53man: /* Stability */

2022-10-11T17:33:09Z

Stability

← Older revision		Revision as of 13:33, 11 October 2022
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	</div>		</div>

	The plot titled, "Critical Adiabatic Index," that is [[#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 {{ Sterne37hereafter}}'s derived results regarding the stability of uniform-density spheres appears at the bottom of p. 338 of {{ Ledoux46full }}~~; the~~ relevant paragraph from ~~Ledoux's paper is reprinted as a boxed-in digital image in the following table~~:		The plot titled, "Critical Adiabatic Index," that is [[#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 {{ Sterne37hereafter}}'s derived results regarding the stability of uniform-density spheres appears at the bottom of p. 338 of {{ Ledoux46full }}. The relevant paragraph from {{ Ledoux46 }} follows:

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

Jet53man: /* Stability */

2022-10-11T17:28:54Z

Stability

← Older revision		Revision as of 13:28, 11 October 2022
Line 958:		Line 958:
	</div>		</div>

	The plot titled, "Critical Adiabatic Index," that is [[#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 {{ Sterne37hereafter}}'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:		The plot titled, "Critical Adiabatic Index," that is [[#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 {{ Sterne37hereafter}}'s derived results regarding the stability of uniform-density spheres appears at the bottom of p. 338 of {{ Ledoux46full }}; the relevant paragraph from Ledoux's paper is reprinted as a boxed-in digital image in the following table:

	<div align="center">		<div align="center">
	<table border="2" cellpadding="5" width="80%">		<table border="2" cellpadding="5" width="80%">
	<tr>		<tr>
	<th align="center" colspan="1">		<td 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>~~		Paragraph extracted<sup>&dagger;</sup> from the bottom of p. 338 of …<br />
	~~"''On the Dynamical Stability of Stars''"<p>~~</p>		{{ Ledoux46figure }}
	~~The Astrophysical Journal, vol. 104, pp. 333 - 346 © [http://aas.org/about/policies/copyright American Astronomical Society]~~		</td>
	</th>
	</tr>		</tr>
	<tr>		<tr>
	<td colspan="1" rowspan="1">		<td colspan="1" rowspan="1">
	[[File:Ledoux1946OnSterne01.png\|700px\|center\|Ledoux (1946)]]		<!-- [[File:Ledoux1946OnSterne01.png\|700px\|center\|Ledoux (1946)]] -->
			<font color="darkgreen">
			"Another point brought out clearly by Sterne's analysis is that, while the fundamental mode becomes unstable for <math>\Gamma < \tfrac{4}{3}\, ,</math> the higher harmonics continue to be stable. Indeed, as we may directly verify from equation (19), the first harmonic becomes unstable only if <math>\Gamma < \tfrac{2}{5}</math> and the second one if <math>\Gamma < \tfrac{4}{21} \, .</math>"
			</font>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Stability */

2022-10-10T22:28:20Z

Stability

← Older revision		Revision as of 18:28, 10 October 2022
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	==Stability==		==Stability==

	The [[SSC/Perturbations#2ndOrderODE\|Adiabatic Wave Equation]] that defines this [[SSC/Perturbations#The_Eigenvalue_Problem\|eigenvalue problem]] has been derived from the fundamental set of nonlinear [[PGE#Principal_Governing_Equations\|Principal Governing Equations]] by assuming that, for example, the radial position, <math>~r(m,t)</math>, at any time, <math>~t</math>, and of each mass shell throughout our spherical configuration can be described by the expression,		The [[SSC/Perturbations#2ndOrderODE\|Adiabatic Wave Equation]] that defines this [[SSC/Perturbations#The_Eigenvalue_Problem\|eigenvalue problem]] has been derived from the fundamental set of nonlinear [[PGE#Principal_Governing_Equations\|Principal Governing Equations]] by assuming that, for example, the radial position, <math>r(m,t)</math>, at any time, <math>t</math>, and of each mass shell throughout our spherical configuration can be described by the expression,
	<div align="center">		<div align="center">
	<math>~r(m,t) = r_0(m) [ 1 + x(m) e^{i\omega t} ] \, ,</math>		<math>r(m,t) = r_0(m) [ 1 + x(m) e^{i\omega t} ] \, ,</math>
	</div>		</div>
	where, the fractional displacement, <math>~\|x\| \ll 1</math>. Switching to ~~Sterne~~'s variable notation, this should be written as,		where, the fractional displacement, <math>\|x\| \ll 1</math>. Switching to {{ Sterne37hereafter }}'s variable notation, this should be written as,
	<div align="center">		<div align="center">
	<math>~\xi(x,t) = \xi_0(x) [ 1 + A\xi_1(x) e^{i n t} ] \, ,</math>		<math>\xi(x,t) = \xi_0(x) [ 1 + A\xi_1(x) e^{i n t} ] \, ,</math>
	</div>		</div>
	with the presumption that the coefficient, <math>~\|A\| \ll 1</math>, and the understanding that, in ~~Sterne's presentation~~, the variable, <math>~x</math>, is used to identify individual mass shells. Specifically, given <math>~R</math> and <math>~\bar\rho</math>,		with the presumption that the coefficient, <math>\|A\| \ll 1</math>, and the understanding that, in {{ Sterne37hereafter}}, the variable, <math>x</math>, is used to identify individual mass shells. Specifically, given <math>R</math> and <math>\bar\rho</math>,
	<div align="center">		<div align="center">
	<math>~m \equiv M_r = \frac{4}{3}\pi \xi_0^3 \bar\rho = \frac{4}{3}\pi (R x)^3 \bar\rho </math>		<math>m \equiv M_r = \frac{4}{3}\pi \xi_0^3 \bar\rho = \frac{4}{3}\pi (R x)^3 \bar\rho </math>
	<math>~\Rightarrow</math>		<math>\Rightarrow</math>
	<math>~x = \biggl( \frac{3m}{4\pi R^3 \bar\rho} \biggr)^{1/3} \, .</math>		<math>x = \biggl( \frac{3m}{4\pi R^3 \bar\rho} \biggr)^{1/3} \, .</math>
	</div>		</div>

	~~Sterne's~~ [[#Sterne.27s_General_Solution\|general solution of this eigenvalue problem]] describes mathematically how a self-gravitating, uniform-density configuration will vibrate if perturbed away from its equilibrium state; the oscillatory behavior associated with each pure radial mode, <math>~j</math> — among an infinite number of possible modes — is fully defined by the polynomial expression for the eigenvector, <math>~\xi_1(x)</math>, and the corresponding value of the square of the eigenfrequency, <math>~n^2</math>.		The [[#Sterne.27s_General_Solution\|general solution of this eigenvalue problem]] presented by {{ Sterne37 }} describes mathematically how a self-gravitating, uniform-density configuration will vibrate if perturbed away from its equilibrium state; the oscillatory behavior associated with each pure radial mode, <math>j</math> — among an infinite number of possible modes — is fully defined by the polynomial expression for the eigenvector, <math>\xi_1(x)</math>, and the corresponding value of the square of the eigenfrequency, <math>n^2</math>.


	If, for any mode, <math>~j</math>, the ''square'' of the derived eigenfrequency, <math>~n^2</math>, is positive, then the eigenfrequency itself will be a real number — specifically,		If, for any mode, <math>j</math>, the ''square'' of the derived eigenfrequency, <math>n^2</math>, is positive, then the eigenfrequency itself will be a real number — specifically,
	<div align="center">		<div align="center">
	<math>~n = \pm \sqrt{\|n^2\|} \, .</math>		<math>n = \pm \sqrt{\|n^2\|} \, .</math>
	</div>		</div>
	As a result, the radial location of every mass shell will vary sinusoidally in time according to the expression,		As a result, the radial location of every mass shell will vary sinusoidally in time according to the expression,
	<div align="center">		<div align="center">
	<math>~\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm i \sqrt{\|n^2\|} t} \, .</math>		<math>\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm i \sqrt{\|n^2\|} t} \, .</math>
	</div>		</div>
	If, on the other hand, <math>~n^2</math>, is negative, then the eigenfrequency will be an imaginary number — specifically,		If, on the other hand, <math>n^2</math>, is negative, then the eigenfrequency will be an imaginary number — specifically,
	<div align="center">		<div align="center">
	<math>~n = \pm i \sqrt{\|n^2\|} \, .</math>		<math>n = \pm i \sqrt{\|n^2\|} \, .</math>
	</div>		</div>
	As a result, the radial location of every mass shell will grow (or damp) exponentially in time according to the expression,		As a result, the radial location of every mass shell will grow (or damp) exponentially in time according to the expression,
	<div align="center">		<div align="center">
	<math>~\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm \sqrt{\|n^2\|} t} \, .</math>		<math>\frac{\xi(x,t)}{\xi_0} - 1 \propto e^{\pm \sqrt{\|n^2\|} t} \, .</math>
	</div>		</div>
	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,		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">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~0 = \frac{n_\mathrm{crit}^2}{4\pi G \bar\rho}</math>		<math>0 = \frac{n_\mathrm{crit}^2}{4\pi G \bar\rho}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)\gamma_\mathrm{crit} -\frac{4}{3} </math>		<math>\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)\gamma_\mathrm{crit} -\frac{4}{3} </math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~~~ \gamma_\mathrm{crit}(j)</math>		<math>\Rightarrow ~~~~~ \gamma_\mathrm{crit}(j)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{4}{3}\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)^{-1} </math>		<math>\frac{4}{3}\biggl(1 + \frac{\mathfrak{F}}{6} \biggr)^{-1} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{4}{3}\biggl[1 + \frac{2j(2j+5)}{6} \biggr]^{-1} </math>		<math>\frac{4}{3}\biggl[1 + \frac{2j(2j+5)}{6} \biggr]^{-1} </math>
	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{4}{3 + j(2j+5)} \, . </math>		<math>\frac{4}{3 + j(2j+5)} \, . </math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

	The plot titled, "Critical Adiabatic Index," that is [[#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:		The plot titled, "Critical Adiabatic Index," that is [[#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 {{ Sterne37hereafter}}'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">		<div align="center">

Jet53man: /* Validity Check */

2022-10-10T22:11:01Z

Validity Check

Show changes

Jet53man: /* Sterne's Presentation */

2022-10-10T22:04:11Z

Sterne's Presentation

← Older revision		Revision as of 18:04, 10 October 2022
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	<div align="center">		<div align="center">
	<table border="2" cellpadding="5">		<table border="2" cellpadding="5" width="70%">
	<tr>		<tr>
	<th align="center" colspan="1">		<td align="center" colspan="1">
	Table extracted from ~~[http://adsabs.harvard~~.~~edu/abs/1937MNRAS..97..582S T. E. Sterne (1937~~)~~]<p></p>~~		Table of exact eigenvector expressions extracted from §2 (p. 587) of …<br />
	~~"''Modes~~ of ~~Radial Oscillation''"<p>~~</p>		{{ Sterne37figure }}
	~~Monthly Notices of the Royal Astronomical Society, vol. 97, pp. 582 - © [https://www.ras.org.uk/ Royal Astronomical Society]~~		</td>
	</th>		<td align="center" colspan="1">
	<th align="center" colspan="1">		<math>\frac{n^2}{4\pi G \bar\rho}</math>
	<math>~\frac{n^2}{4\pi G \bar\rho}</math>		</td>
	</th>		</tr>
	<tr>		<tr>
	<td colspan="1" rowspan="4">		<td colspan="1" rowspan="1">
	[[File:Sterne1937SolutionTable1.png\|600px\|center\|Sterne (1937)]]		<!-- [[File:Sterne1937SolutionTable1.png\|600px\|center\|Sterne (1937)]] -->
			<table border="0" align="left">
			<tr>
			<td align="right"><math>j=0 \, ;</math>     </td>
			<td align="right"><math>\mathfrak{F}=0 \, ;</math>     </td>
			<td align="right"> <math>\xi_1 = 1</math></td>
			</tr>
			</table>
	</td>		</td>
	<td align="center"><math>~\gamma - 4/3</math></td>		<td align="center"><math>\gamma - 4/3</math></td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center"><math>~2(5\gamma - 2)/3</math></td>		<td colspan="1" rowspan="1">
			<table border="0" align="left">
			<tr>
			<td align="right"><math>j=1 \, ;</math>     </td>
			<td align="right"><math>\mathfrak{F}= 14 \, ;</math>     </td>
			<td align="right"><math>\xi_1 = 1 - (7/5)x^2</math></td>
			</tr>
			</table>
			</td>
			<td align="center"><math>2(5\gamma - 2)/3</math></td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center"><math>~7\gamma - 4/3</math></td>		<td colspan="1" rowspan="1">
			<table border="0" align="left">
			<tr>
			<td align="right"><math>j=2 \, ;</math>     </td>
			<td align="right"><math>\mathfrak{F}= 36 \, ;</math>     </td>
			<td align="right"><math>\xi_1 = 1 - (18/5)x^2 + (99/35)x^4</math></td>
			</tr>
			</table>
			</td>
			<td align="center"><math>7\gamma - 4/3</math></td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="center"><math>~12\gamma - 4/3</math></td>		<td colspan="1" rowspan="1">
			<table border="0" align="left">
			<tr>
			<td align="right"><math>j=3 \, ;</math>     </td>
			<td align="right"><math>\mathfrak{F}=66 \, ;</math>     </td>
			<td align="right"><math>\xi_1 = 1 - (33/5)x^2 + (429/35)x^4 - (143/21)x^6</math></td>
			</tr>
			</table>
			</td>
			<td align="center"><math>12\gamma - 4/3</math></td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* The Approach Taken by Sterne (1937) */

2022-10-10T21:38:07Z

The Approach Taken by Sterne (1937)

← Older revision		Revision as of 17:38, 10 October 2022
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	<tr>		<tr>
	<td align="right" >		<td align="right" >
	<math>~ r_0 \frac{d\ln x}{dr_0}</math>		<math>r_0 \frac{d\ln x}{dr_0}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>        at         <math>~r_0 = R \, ,</math>		<math>\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>        at         <math>~r_0 = R \, ,</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Sterne's Presentation */

2022-10-10T21:36:44Z

Sterne's Presentation

← Older revision		Revision as of 17:36, 10 October 2022
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	===Sterne's Presentation===		===Sterne's Presentation===
	In what follows, as before, text presented in a green font has been taken verbatim from ~~the paper by [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)]~~. ~~Sterne~~ begins by writing the unknown eigenfunction as a power series expanded about the origin, specifically,		In what follows, as before, text presented in a green font has been taken verbatim from {{ Sterne37hereafter }}. He begins by writing the unknown eigenfunction as a power series expanded about the origin, specifically,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\xi_1</math>		<math>\xi_1</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\sum\limits_{0}^{\infty} a_k x^k \, ,</math>		<math>\sum\limits_{0}^{\infty} a_k x^k \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	with, <math>~a_0 = 1</math>. <font color="green">It is found by substitution that the terms in odd powers of <math>~x</math> vanish, and that the coefficients of the even terms satisfy the recurrence formula</font>,		with, <math>a_0 = 1</math>. <font color="green">It is found by substitution that the terms in odd powers of <math>x</math> vanish, and that the coefficients of the even terms satisfy the recurrence formula</font>,

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~a_{k+2}</math>		<math>a_{k+2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~a_k \cdot \frac{k^2 + 5k - \mathfrak{F}}{(k+2)(k+5)} \, .</math>		<math>a_k \cdot \frac{k^2 + 5k - \mathfrak{F}}{(k+2)(k+5)} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	</div>		</div>

	The wave equation and attending boundary conditions will all <font color="green">be satisfied if we choose <math>\mathfrak{F}</math> so as to make the series solution terminate with some term, say the <math>~2 j^\mathrm{th}</math> where <math>~j</math> is zero or any positive integer. This it will do</font> [via the above recurrence relation] if,		The wave equation and attending boundary conditions will all <font color="green">be satisfied if we choose <math>\mathfrak{F}</math> so as to make the series solution terminate with some term, say the <math>2 j^\mathrm{th}</math> where <math>j</math> is zero or any positive integer. This it will do</font> [via the above recurrence relation] if,
	<div align="center">		<div align="center">
	<math>~\mathfrak{F} = 2j(2j+5) \, .</math>		<math>\mathfrak{F} = 2j(2j+5) \, .</math>
	</div>		</div>

	The first few solutions are displayed in the following boxed-in image that has been extracted directly from §2 (p. 587) of ~~[http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)]~~; to the right of ~~Sterne's~~ table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of ~~Sterne's~~ solutions.		The first few solutions are displayed in the following boxed-in image that has been extracted directly from §2 (p. 587) of {{ Sterne37 }}; to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of solutions presented by {{ Sterne37hereafter }}.

	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>


	===Validity Check===		===Validity Check===

Jet53man: /* Setup as Presented by Sterne (1937) */

2022-10-10T21:30:19Z

Setup as Presented by Sterne (1937)

← Older revision		Revision as of 17:30, 10 October 2022
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	===Setup as Presented by Sterne (1937)===		===Setup as Presented by Sterne (1937)===

	In §2 of his paper, ~~Sterne (1937)~~ details the structural properties of an equilibrium, uniform-density sphere as follows. (Text taken verbatim from ~~Sterne's paper~~ are presented here in green.) Given that <font color="green">the undisturbed density is constant and equal to the mean density</font>, <math>~\bar\rho</math>, the <font color="green">mass within any radius is</font>,		In §2 of his paper, {{ Sterne37 }} details the structural properties of an equilibrium, uniform-density sphere as follows. (Text taken verbatim from {{ Sterne37hereafter }} are presented here in green.) Given that <font color="green">the undisturbed density is constant and equal to the mean density</font>, <math>~\bar\rho</math>, the <font color="green">mass within any radius is</font>,
	<div align="center">		<div align="center">
	<math>M_r = \biggl( \frac{4\pi}{3} \biggr) \bar\rho \xi_0^3 \, ;</math>		<math>M_r = \biggl( \frac{4\pi}{3} \biggr) \bar\rho \xi_0^3 \, ;</math>
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	</div>		</div>

	Hence, for this particular equilibrium model, ~~Sterne's derived~~ wave equation — his equation (1.91), as displayed above — becomes,		Hence, for this particular equilibrium model, the wave equation derived by {{ Sterne37hereafter }} — his equation (1.91), as displayed above — becomes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* The Approach Taken by Sterne (1937) */

2022-10-10T21:26:23Z

The Approach Taken by Sterne (1937)

← Older revision		Revision as of 17:26, 10 October 2022
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	<tr>		<tr>
	<td align="center" colspan="1">		<td align="center" colspan="1">
	Reprint of a paragraph from …<br />		Reprint of the last paragraph of §1 (p. 585) from …<br />
	{{ Sterne37figure }}		{{ Sterne37figure }}
	</td>		</td>