SSC/Stability/InstabilityOnsetOverview - Revision history

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-26T13:35:48Z

Pressure and Density Displacement Functions

← Older revision		Revision as of 09:35, 26 September 2023
Line 771:		Line 771:
	</table>		</table>

	For later use (immediately below) we note as well that,		For later use ([[#Derivative\|immediately below]]) we note as well that,

	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
Line 841:		Line 841:
	</table>		</table>

	-~~---~~		=====Cross-Check=====

	~~<font color="red"><b>Double Check:</b></font>  ~~ Can you determine <math>x_P</math>, given this expression for <math>p_P</math>?		Can you determine <math>x_P</math>, given this expression for <math>p_P</math>?

	When the square of the radial-oscillation frequency is set to zero, the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations\|linearized Euler + Poisson equations]] states,		When the square of the radial-oscillation frequency is set to zero, the [[SSC/Perturbations#Summary_Set_of_Linearized_Equations\|linearized Euler + Poisson equations]] states,
Line 1,021:		Line 1,021:
	</table>		</table>

	As a result, we find that,		<span id="Derivative">As a result</span>, we find that,

	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">

Jet53man: /* Properties of the Marginally Unstable, Fundamental-Mode Eigenfunctions */

2023-09-26T13:31:37Z

Properties of the Marginally Unstable, Fundamental-Mode Eigenfunctions

← Older revision		Revision as of 09:31, 26 September 2023
Line 1,143:		Line 1,143:
	</table>		</table>

	~~Compare with~~ …		Q. E. D.

			=====Displacement Functions Summary=====

			<table border="1" align="center" width="60%" cellpadding="8"><tr><td align="left">

			<div align="center"><b>Summary …</b></div>

			<table border="0" align="center" cellpadding="8">

	~~<table border="0" cellpadding="5" align="center">~~
	<tr>		<tr>
	<td align="~~center~~" ~~colspan~~="1"><~~font color~~="~~maroon~~">~~<b>Exact Solution to the~~ <math>~(3 \le n < \~~infty~~)</math> ~~Polytropic LAWE</b></font>~~</td>		<td align="right"><math>x_P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr)
			\frac{d\theta}{d\xi}\biggr] \, ,</math>
			</td>
	</tr>		</tr>

	<tr>		<tr>
	<td align="left">		<td align="right"><math>d_P</math></td>
	<math>~~x_P~~ \~~equiv~~ \~~frac~~{~~3(n~~-~~1)}{2n}\biggl[1 + \biggl(~~\frac{n-3}{~~n-1~~}~~\biggr)~~ \biggl( \frac{1}{~~\xi~~ \theta^{n}}\~~biggr)~~ \frac{d\theta}{d\xi}\biggr] \, ,</math>		<td align="center"><math>=</math></td>
	</td>		<td align="left"><math>
			-3 \biggl\{1
			-
			\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
			\, ,
			</math></td>
			</tr>

			<tr>
			<td align="right"><math>p_P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>
			\biggl( \frac{n+1}{n} \biggr) d_P
			\, .
			</math></td>
	</tr>		</tr>
	</table>		</table>

			</td></tr></table>

	====Related Thoughts====		====Related Thoughts====

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T22:27:55Z

Pressure and Density Displacement Functions

@@ Line 1,108: / Line 1,108: @@
 \biggr]
 + \frac{6(n-1)}{n}
 </math>
    </td>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T22:21:37Z

Pressure and Density Displacement Functions

@@ Line 1,096: / Line 1,096: @@
 + \frac{3(n+1)}{n}
 </math>
    </td>
 </tr>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T22:10:32Z

Pressure and Density Displacement Functions

← Older revision		Revision as of 18:10, 25 September 2023
Line 1,028:		Line 1,028:
	<td align="right"><math>\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}</math></td>		<td align="right"><math>\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left"><math>
	<math>		\biggl\{
	-\frac{1}{(n+1)}		\frac{1}{(n+1)}
	\cdot \xi \theta \biggl(~~-\xi^2~~ \frac{d\theta}{d\xi}\biggr)^{-1}		\cdot \frac{\theta}{\xi} \biggl(\frac{d\theta}{d\xi}\biggr)^{-1}
	\frac{3(n+1)(n-3)}{2n}		\biggr\}
	\biggl[		\frac{3(n+1)(n-3)}{n}
	(n+1)\theta^{-(n+1)}\biggl(\frac{d\theta}{d\xi} \biggr)^2		\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
	+		+
	2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} + 1		2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
	\biggr]\frac{\xi}{\theta}		\biggr\} \biggl(\frac{d\theta}{d\xi}\biggr) \frac{\xi}{\theta}
	</math>		</math></td>
	</td>
	</tr>		</tr>

Line 1,045:		Line 1,044:
	<td align="right"> </td>		<td align="right"> </td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left"><math>
	<math>		\frac{3(n-3)}{n}
	\frac{3(n-3)}{2n}		\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
	\biggl[		+
	(n+1)\theta^{-(n+1)}\biggl(\frac{d\theta}{d\xi} \biggr)		2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
	+		\biggr\} \, ;
	2\xi^{-1}\theta^{-n} + \biggl(\frac{d\theta}{d\xi}\biggr)~~^{-~~1}		</math></td>
	\biggr] \, ;
	</math>
	</td>
	</tr>		</tr>
	</table>		</table>
Line 1,069:		Line 1,065:
	-		-
	p_P		p_P
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{3(n-3)}{n}
			\biggl\{ \frac{(n+1)}{2\theta^{n+1}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
			+
			2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
			\biggr\}
			+ \frac{3(n+1)}{n} \biggl\{1
			-
			\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{3(n-3)}{n}\biggl\{
			2\xi^{-1}\theta^{-n}\biggl(\frac{d\theta}{d\xi}\biggr) + 1
			\biggr\}
			+ \frac{3(n+1)}{n}
	</math>		</math>
	</td>		</td>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T21:57:50Z

Pressure and Density Displacement Functions

@@ Line 825: / Line 825: @@
 \frac{2}{\theta} \biggl[-2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} -1\biggr]
 \biggr\} \biggl(\frac{d\theta}{d\xi}\biggr)
 </math></td>
 </tr>
@@ Line 988: / Line 1,000: @@
 \biggl[\frac{1}{(n+1)}  \biggr]
 \cdot \xi \theta \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)^{-1}
 </math>
    </td>
 </tr>
 <tr>
-   <td align="right"><math>\biggl[ \frac{P_0}{\rho_0 r_0 g_0} \biggr] \frac{d(p_P)}{d\ln \xi}</math></td>
+   <td align="right">
-   <td align="center"><math>=</math></td>
+&nbsp;
-   <td align="left"><math>
+  </td>
--~ \biggl[\frac{1}{(n+1)}
+   <td align="center">
-\cdot \xi \theta \biggl(-\xi^2 \frac{d\theta}{d\xi}\biggr)^{-1}
+<math>=</math>
-\biggr]
+  </td>
-\frac{3(n+1)(n-3)}{2n} \cdot
+   <td align="left">
-\biggl\{
+<math>
-\frac{(n+1)}{\theta^{n+1}}\biggl(\frac{d\theta}{d\xi}\biggr)^2
+-\frac{1}{(n+1)}
-- \frac{1}{\theta^{n}}
+\cdot \frac{\theta}{\xi} \biggl(\frac{d\theta}{d\xi}\biggr)^{-1}
-\biggl[ \frac{d^2\theta}{d\xi^2} \biggr]
+\, .
-\biggr\}\frac{\xi}{\theta}
+</math>
-</math></td>
+  </td>
 </tr>
 </table>
 As a result, we find that,

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T21:45:58Z

Pressure and Density Displacement Functions

← Older revision		Revision as of 17:45, 25 September 2023
Line 763:		Line 763:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>		<td align="left"><math>
	-		-\frac{3(n+1)}{n} \biggl\{1
	\frac{3(n+1)}{n}\biggl\{ 1		-
	+		\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
	\frac{(n-3)}{2n}\biggl[ \frac{d(\theta^{-n})}{d\xi} \biggr]
	\biggr\}
	\, .		\, .
	</math></td>		</math></td>
Line 781:		Line 779:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>		<td align="left"><math>
	-		\frac{3(n+1)(n-3)}{2n} \frac{d}{d\xi}
	\frac{3(n+1)(n-3)}{2n^2}		\biggl[ \theta^{-(n+1)} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr]
	\frac{d}{d\xi}\biggl[ ~~\frac{d(~~\theta^{-n})}{d\xi} \biggr]
	</math></td>		</math></td>
	</tr>		</tr>
Line 791:		Line 788:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>		<td align="left"><math>
	-		\frac{3(n+1)(n-3)}{2n}
	\frac{3(n+1)(n-3)}{2n^2}		\biggl[ -\frac{(n+1)}{\theta^{n+2}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
	~~\frac{d}{d\xi}~~\biggl[ - ~~n\theta^{-(n+1)}~~\frac{~~d\theta}{d\xi} \biggr]~~		+
	~~</math></td>~~		2\theta^{-(n+1)} \frac{d^2\theta}{d\xi^2}
	~~</tr>~~		\biggr] \biggl(\frac{d\theta}{d\xi}\biggr)

	~~<tr>~~
	~~<td align="right"> </td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left"><math>~~
	-
	~~\frac{3~~(n+1~~)(n-3~~)}{~~2n^2}~~
	~~\biggl[~~
	~~n(n+1)~~\theta^{-(n+2)}\biggl(\frac{d\theta}{d\xi} \biggr)^2
	~~- n~~\theta^{-(n+1)}\frac{d^2\theta}{d\xi^2}
	\biggr]
	~~</math></td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right"> </td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left"><math>~~
	~~\frac{3(n+1)(n-3)}{2n}~~
	~~\biggl[~~
	~~\theta^{-n}\frac{d^2\theta}{d\xi^2}~~
	-
	~~(n+1)\theta^{-(n+1)}~~\biggl(\frac{d\theta}{d\xi} \biggr)^2
	~~\biggr]\frac{1}{\theta} \, .~~
	</math></td>		</math></td>
	</tr>		</tr>
Line 847:		Line 820:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>		<td align="left"><math>
	-\frac{3(n+1)(n-3)}{2n}		\frac{3(n+1)(n-3)}{2n}
	\biggl[		\biggl\{ -\frac{(n+1)}{\theta^{n+2}}\biggl( \frac{d\theta}{d\xi}\biggr)^2
	(n+1)\theta^{-(n+1)}\biggl(\frac{d\theta}{d\xi} \biggr)^2		+
	+		\frac{2}{\theta} \biggl[-2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} -1\biggr]
	2\xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} + 1		\biggr\} \biggl(\frac{d\theta}{d\xi}\biggr)
	\biggr]\frac{1}{\~~theta~~} \~~, .~~
	</math></td>		</math></td>
	</tr>		</tr>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T21:23:55Z

Pressure and Density Displacement Functions

← Older revision		Revision as of 17:23, 25 September 2023
Line 618:		Line 618:
	-\xi^{-2}\theta^{-n} \frac{d\theta}{d\xi}		-\xi^{-2}\theta^{-n} \frac{d\theta}{d\xi}
	-		-
	n\xi^{-1}\theta^{-n-1} \frac{d\theta}{d\xi}		n\xi^{-1}\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2
	+		+
	\xi^{-1}\theta^{-n} \frac{d^2\theta}{d\xi^2}		\xi^{-1}\theta^{-n} \frac{d^2\theta}{d\xi^2}
Line 634:		Line 634:
	-\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}		-\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}
	-		-
	n\theta^{-n-1} \frac{d\theta}{d\xi}		n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2
	+		+
	\theta^{-n} \frac{d^2\theta}{d\xi^2}		\theta^{-n} \frac{d^2\theta}{d\xi^2}
Line 704:		Line 704:
	-3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}		-3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi}
	-		-
	n\theta^{-n-1} \frac{d\theta}{d\xi}		n\theta^{-n-1} \biggl( \frac{d\theta}{d\xi} \biggr)^2
	-1		-1
	\biggr]\frac{1}{\xi} \, ;		\biggr]\frac{1}{\xi} \, ;
Line 720:		Line 720:
	<td align="left"><math>-3\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \biggr\}		<td align="left"><math>-3\biggl\{ \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \biggr\}
	-		-
	\biggl\{ \frac{3(n-3)}{2n}\biggl[ -3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} - n\theta^{-n-1} \frac{d\theta}{d\xi} - 1		\biggl\{ \frac{3(n-3)}{2n}\biggl[ -3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} - n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2 - 1
	\biggr]\biggr\} </math></td>		\biggr]\biggr\} </math></td>
	</tr>		</tr>
Line 730:		Line 730:
	+ \frac{9(n-3)}{2n} \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr]		+ \frac{9(n-3)}{2n} \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr]
	+		+
	\biggl\{ \frac{3(n-3)}{2n}\biggl[ 3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} + n\theta^{-n-1} \frac{d\theta}{d\xi} + 1		\biggl\{ \frac{3(n-3)}{2n}\biggl[ 3\xi^{-1}\theta^{-n} \frac{d\theta}{d\xi} + n\theta^{-n-1} \biggl(\frac{d\theta}{d\xi}\biggr)^2 + 1
	\biggr]\biggr\} </math></td>		\biggr]\biggr\} </math></td>
	</tr>		</tr>
Line 740:		Line 740:
	\biggr]		\biggr]
	+		+
	\frac{3(n-3)}{2n}\biggl[ \frac{n}{\theta^{n+1}} \frac{d\theta}{d\xi} + 1		\frac{3(n-3)}{2n}\biggl[ \frac{n}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2 + 1
	\biggr] </math></td>		\biggr] </math></td>
	</tr>		</tr>
Line 748:		Line 748:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>		<td align="left"><math>
	-3		-3 \biggl\{1
	+		-
	\frac{3(n-3)}{2n}\biggl[ \frac{n}{\theta^{n+1}} \frac{d\theta}{d\xi} \biggr]		\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl( \frac{d\theta}{d\xi}\biggr)^2 \biggr] \biggr\}
	~~</math></td>~~		\, .
	~~</tr>~~

	~~<tr>~~
	~~<td align="right"> </td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left"><math>~~
	~~-3\biggl\{ 1~~
	+
	~~\frac{(n-3~~)~~}{2n}\biggl[ \frac{d(\theta~~^~~{-n})}{d\xi}~~ \biggr]
	\biggr\} \, .
	</math></td>		</math></td>
	</tr>		</tr>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T14:41:15Z

Pressure and Density Displacement Functions

@@ Line 1,106: / Line 1,106: @@
 \xi^{-1}\theta^{-n}\frac{d\theta}{d\xi} + 1
 \biggr]\frac{\xi}{\theta}
 </math>
    </td>

Jet53man: /* Pressure and Density Displacement Functions */

2023-09-25T14:35:08Z

Pressure and Density Displacement Functions

@@ Line 957: / Line 957: @@
 <tr>
    <td align="right">
-<math>~a_n</math>
+<math>a_n</math>
    </td>
    <td align="center">
-<math>~=</math>
+<math>=</math>
    </td>
    <td align="left">
-<math>~\biggl[\frac{(n+1)K}{4\pi G} \cdot \rho_c^{(1-n)/n} \biggr]^{1/2} \, .</math>
+<math>\biggl[\frac{4\pi G a_n^2}{(n+1)} \cdot \rho_c^{(n-1)/n} \biggr] \, .</math>
    </td>
 </tr>
@@ Line 1,021: / Line 1,031: @@
 </tr>
 </table>
-Also,
+As a result, we find that,
 <table border="0" align="center" cellpadding="8">
@@ Line 1,125: / Line 1,084: @@
 </math></td>
 </tr>
 </table>