SSC/Stability/n1PolytropeLAWE/Pt4 - Revision history

Joel2: /* TERM2 */

2025-09-07T18:00:41Z

TERM2

← Older revision		Revision as of 14:00, 7 September 2025
Line 2,050:		Line 2,050:
	</table>		</table>
	where, we have completed the integral with the assistance of the [https://wolframalpha.com WolframAlpha] online integrator:		where, we have completed the integral with the assistance of the [https://wolframalpha.com WolframAlpha] online integrator:
	[[File:~~Mathematica05~~.png\|500px\|center\|Mathematica Integral]]		[[File:Mathematica06.png\|500px\|center\|Mathematica Integral]]

	=See Also=		=See Also=

2025-09-07T17:58:49Z

TERM2

← Older revision		Revision as of 13:58, 7 September 2025
Line 2,041:		Line 2,041:
	<math>		<math>
	- (3\gamma - 4) \biggl[ \frac{2^3 \cdot 3^8 }{5^4\pi} \biggr]^{1 / 2}		- (3\gamma - 4) \biggl[ \frac{2^3 \cdot 3^8 }{5^4\pi} \biggr]^{1 / 2}
	\cdot \frac{1}{8}\biggl\{		\cdot \frac{1}{4}\biggl\{
	\frac{\chi_\mathrm{surf}(8\chi^4_\mathrm{surf} + 25 \chi^2_\mathrm{surf}		\frac{\chi_\mathrm{surf}(4\chi^6_\mathrm{surf} + 53\chi^4_\mathrm{surf} + 40 \chi^2_\mathrm{surf}
	+ 15)}{(1+\chi^2_\mathrm{surf})^2} - 15\tan^{-1}(\chi_\mathrm{surf})		+ 15)}{(1+\chi^2_\mathrm{surf})^3} - 15\tan^{-1}(\chi_\mathrm{surf})
	\biggr\}		\biggr\}
	</math>		</math>

2025-09-07T17:54:16Z

TERM2

← Older revision		Revision as of 13:54, 7 September 2025
Line 2,031:		Line 2,031:
	\biggr\}		\biggr\}

			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			- (3\gamma - 4) \biggl[ \frac{2^3 \cdot 3^8 }{5^4\pi} \biggr]^{1 / 2}
			\cdot \frac{1}{8}\biggl\{
			\frac{\chi_\mathrm{surf}(8\chi^4_\mathrm{surf} + 25 \chi^2_\mathrm{surf}
			+ 15)}{(1+\chi^2_\mathrm{surf})^2} - 15\tan^{-1}(\chi_\mathrm{surf})
			\biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			where, we have completed the integral with the assistance of the [https://wolframalpha.com WolframAlpha] online integrator:
			[[File:Mathematica05.png\|500px\|center\|Mathematica Integral]]

	=See Also=		=See Also=

2025-09-07T17:41:04Z

TERM2

← Older revision		Revision as of 13:41, 7 September 2025
Line 2,025:		Line 2,025:
	<td align="left">		<td align="left">
	<math>		<math>
	- (3\gamma - 4) \biggl[ \frac{2^3 }{3\cdot 5^4\pi} \biggr]^{1 / 2}		- (3\gamma - 4) \biggl[ \frac{2^3 }{3\cdot 5^4\pi} \biggr]^{1 / 2} \cdot 3^{9 / 2}
	\int_0^{\chi_\mathrm{surf}}		\int_0^{\chi_\mathrm{surf}}
	\biggl\{~~3^2 \cdot 3^2 \cdot 3^{1 / 2}~~		\biggl\{
	\biggl[(5 - \chi^2)^2 \chi^4 \biggl(1 + \chi^2\biggr)^{-4} \biggr] d\chi		\biggl[(5 - \chi^2)^2 \chi^4 \biggl(1 + \chi^2\biggr)^{-4} \biggr] d\chi
	\biggr\}		\biggr\}

2025-09-07T17:38:34Z

TERM2

@@ Line 1,983: / Line 1,983: @@
 \biggl[(15 - \xi^2)^2 \xi^4 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-4}  \biggr] d\xi
 \biggr\}
 </math>
    </td>

2025-09-07T17:27:13Z

TERM2

@@ Line 1,966: / Line 1,966: @@
 <math>
 \frac{15 - \xi^2}{15} \, ,
 </math>
    </td>

2025-09-07T17:19:52Z

TERM2

@@ Line 1,952: / Line 1,952: @@
 \biggl[ \frac{2^3 \cdot 3^3}{\pi} \biggr]^{1 / 2}
 \biggl[\xi^4 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-4}  \biggr] d\xi\biggr\}
 </math>
    </td>

2025-09-07T17:15:44Z

Stability Analysis

← Older revision		Revision as of 13:15, 7 September 2025
Line 1,931:		Line 1,931:
	</tr>		</tr>
	</table>		</table>
			we can write,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 1,938:		Line 1,939:
	<td align="left">		<td align="left">
	<math>		<math>
	dr		[K_c^{-5 / 2} G^{3 / 2}]\biggl[ (3\gamma - 4) x^2 \biggl( - \frac{GM_r }{r}\biggr)4\pi \rho r^2 \biggr] dr
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			- (3\gamma - 4) x_P^2 \biggl\{
			\biggl[ \frac{2^3 \cdot 3^3}{\pi} \biggr]^{1 / 2}
			\biggl[\xi^4 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-4} \biggr] d\xi\biggr\}
	</math>		</math>
	</td>		</td>

2025-09-07T17:02:54Z

TERM2

@@ Line 1,927: / Line 1,927: @@
 \biggl[\xi^4 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-4}  \biggr] d\xi
 \, ,
 </math>
    </td>

2025-09-07T16:59:25Z

TERM2

@@ Line 1,913: / Line 1,913: @@
 ==TERM2==
-Given that,
+Given that (from above),
 =See Also=