User:Tohline/Appendix/Ramblings/Hybrid Scheme Implications - Revision history

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2020-09-11T15:58:00Z

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← Older revision		Revision as of 15:58, 11 September 2020
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	2 a b \lambda \Omega_f (x^2 + y^2) (a^2 y^2 + b^2 x^2)		2 a b \lambda \Omega_f (x^2 + y^2) (a^2 y^2 + b^2 x^2)
	- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]		- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\biggl\{		\biggl\{
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	+		+
	2\biggl[\lambda^2 a^2 b^2 - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr]y^2x^2		2\biggl[\lambda^2 a^2 b^2 - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr]y^2x^2
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4		\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4
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	+		+
	\biggl[\lambda^2 b^4 - 2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 + 2ab^3 \lambda \Omega_f + \lambda^2(a^2-b^2)b^2\biggr] x^4		\biggl[\lambda^2 b^4 - 2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 + 2ab^3 \lambda \Omega_f + \lambda^2(a^2-b^2)b^2\biggr] x^4
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4		\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4

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2020-09-11T15:52:53Z

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← Older revision		Revision as of 15:52, 11 September 2020
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	c^2 A_3 \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) - \biggl(A_1 x^2 + A_2y^2 \biggr)		c^2 A_3 \biggl( \frac{x^2}{a^2} + \frac{y^2}{b^2}\biggr) - \biggl(A_1 x^2 + A_2y^2 \biggr)
	\biggr]		\biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\frac{1}{\varpi^3} \biggl\{		\frac{1}{\varpi^3} \biggl\{
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	\frac{2\pi G \rho a^2 \varpi^2}{(a^2 - b^2)}		\frac{2\pi G \rho a^2 \varpi^2}{(a^2 - b^2)}
	\biggl[ c^2 A_3 y^2 - A_2 b^2 y^2 \biggr]		\biggl[ c^2 A_3 y^2 - A_2 b^2 y^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+\biggl[\frac{a^2 b^2}{a^2 - b^2} \biggr]		+\biggl[\frac{a^2 b^2}{a^2 - b^2} \biggr]
	\biggl\{		\biggl\{
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	2\pi G \rho a^2 \varpi^2 y^2		2\pi G \rho a^2 \varpi^2 y^2
	\biggl[ c^2 A_3 - A_2 b^2 \biggr]		\biggl[ c^2 A_3 - A_2 b^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\biggl\{		\biggl\{
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	</tr>		</tr>
	</table>		</table>

	Also,		Also,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	Hence,		Hence,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
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	2\pi G \rho b^2 \varpi^2 x^2		2\pi G \rho b^2 \varpi^2 x^2
	\biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2} + \frac{a b \lambda \Omega_f }{\pi G \rho} - A_1 a^2 \biggr]		\biggl[ \frac{( A_1 - A_2 )a^2b^2}{b^2 - a^2} + \frac{a b \lambda \Omega_f }{\pi G \rho} - A_1 a^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	2\pi G \rho a^2 \varpi^2 y^2		2\pi G \rho a^2 \varpi^2 y^2
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	\frac{2\pi G \rho b^2 \varpi^2 x^2 }{(b^2 - a^2) }		\frac{2\pi G \rho b^2 \varpi^2 x^2 }{(b^2 - a^2) }
	\biggl[ ( A_1 - A_2 )a^2b^2 - A_1 a^2 (b^2 - a^2) \biggr]		\biggl[ ( A_1 - A_2 )a^2b^2 - A_1 a^2 (b^2 - a^2) \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	\frac{ 2\pi G \rho a^2 \varpi^2 y^2 }{ (b^2 - a^2)}		\frac{ 2\pi G \rho a^2 \varpi^2 y^2 }{ (b^2 - a^2)}
	\biggl[ ( A_1 - A_2 )a^2b^2 - A_2 b^2(b^2 - a^2) \biggr]		\biggl[ ( A_1 - A_2 )a^2b^2 - A_2 b^2(b^2 - a^2) \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
	+		+
	2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)		2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)

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2020-09-10T23:16:04Z

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← Older revision		Revision as of 23:16, 10 September 2020
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	<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>		<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (34)</font> ]</td></tr>
	</table>		</table>

	~~=====Junk=====~~

	~~=====Temporary=====~~

	<font color="red">'''Radial Component:'''</font>   In steady-state, this partial time-derivative also must be zero, so we require,		<font color="red">'''Radial Component:'''</font>   In steady-state, this partial time-derivative also must be zero, so we require,

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2020-09-10T23:15:12Z

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2020-09-10T23:14:50Z

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← Older revision		Revision as of 23:14, 10 September 2020
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	</table>		</table>

	~~----~~		=====Junk=====

	<font color="red">'''Radial Component:'''</font>   In steady-state, this partial time-derivative also must be zero, so we require,		<font color="red">'''Radial Component:'''</font>   In steady-state, this partial time-derivative also must be zero, so we require,

Tohline: /* Compare */

2020-09-10T23:12:12Z

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← Older revision		Revision as of 23:12, 10 September 2020
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	</td>		</td>
	</tr>		</tr>
			<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">Chapter 7, §48, Eq. (33)</font> ]</td></tr>
	</table>		</table>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

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2020-09-10T23:10:20Z

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← Older revision		Revision as of 23:10, 10 September 2020
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	+		+
	\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4		\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4
			</math>
			</td>
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			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2a^2 b^2 x^2 y^2 \biggl[ \lambda^2 + \Omega_f^2 \biggr]
			+
			\biggl[ \Omega_f^2 + \lambda^2 \biggr] a^2 b^2 x^4
			+
			\biggl[ \Omega_f^2 + \lambda^2 \biggr] a^2 b^2y^4
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \lambda^2 + \Omega_f^2\biggr]a^2 b^2 \biggl[ x^4 +2x^2y^2 + y^4 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			\frac{2\pi G \rho }{(a^2 - b^2) }
			\biggl[ A_1 a^2 - A_2 b^2 \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \lambda^2 + \Omega_f^2\biggr] \, .
	</math>		</math>
	</td>		</td>

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2020-09-10T22:55:25Z

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← Older revision		Revision as of 22:55, 10 September 2020
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	+		+
	\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4		\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4
			</math>
			</td>
			</tr>

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			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			x^2 y^2 \biggl\{ 2 \biggl[ \lambda^2 a^2 b^2 - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr] + 2ab\lambda \Omega_f (a^2 + b^2) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+
			\biggl[\lambda^2 b^4 - 2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 + 2ab^3 \lambda \Omega_f + \lambda^2(a^2-b^2)b^2\biggr] x^4
			+
			\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 + 2ab\lambda \Omega_f a^2 - \lambda^2(a^2-b^2)a^2\biggr] y^4
	</math>		</math>
	</td>		</td>

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2020-09-10T22:47:22Z

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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	2 a b \lambda \Omega_f ~~\varpi~~^2 (a^2 y^2 + b^2 x^2)		2 a b \lambda \Omega_f (x^2 + y^2) (a^2 y^2 + b^2 x^2)
	- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]		- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]
	+		+
	\biggl\{		\biggl\{
	\biggl[\lambda b^2 - a b \Omega_f \biggr]x^2		\biggl[\lambda b^2 - a b \Omega_f \biggr]^2 x^4
	+		+
	\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2		2\biggl[\lambda b^2 - a b \Omega_f \biggr]\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2x^2
	\biggr\}^2		+
			\biggl[ \lambda a^2 - a b \Omega_f\biggr]^2 y^4
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2 a b \lambda \Omega_f ( a^2 x^2 y^2 + b^2 x^4 + a^2 y^4 + b^2x^2y^2 )
			- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>~
			+
			\biggl[\lambda^2 b^4 - 2a b \Omega_f \lambda b^2 + a^2 b^2 \Omega_f^2 \biggr] x^4
			+
			2\biggl[\lambda^2 a^2 b^2 - \lambda a b^3 \Omega_f - \lambda a^3 b\Omega_f + a^2 b^2 \Omega_f^2 \biggr]y^2x^2
			+
			\biggl[ \lambda^2 a^4 - 2a b \Omega_f \lambda a^2 + a^2 b^2 \Omega_f^2 \biggr] y^4
	</math>		</math>
	</td>		</td>

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2020-09-10T22:33:26Z

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← Older revision		Revision as of 22:33, 10 September 2020
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	<td align="left">		<td align="left">
	<math>~		<math>~
	2\pi G \rho b^2 \varpi^2 x^2		\frac{2\pi G \rho b^2 \varpi^2 x^2 }{(b^2 - a^2) }
	\biggl[		\biggl[ ( A_1 - A_2 )a^2b^2 - A_1 a^2 (b^2 - a^2) \biggr]
	~~\frac{~~( A_1 - A_2 )a^2b^2~~}{b~~^2 ~~- a~~^2} - ~~A_1~~ a^2
	\biggr]
	+		+
	2\pi G \rho a^2 \varpi^2 y^2		\frac{ 2\pi G \rho a^2 \varpi^2 y^2 }{ (b^2 - a^2)}
	\biggl[ ~~\frac{~~( A_1 - A_2 )a^2b^2}{b^2 ~~- a~~^2} - ~~A_2 b~~^2 \biggr]		\biggl[ ( A_1 - A_2 )a^2b^2 - A_2 b^2(b^2 - a^2) \biggr]
	+		+
	2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)		2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2\pi G \rho a^2 b^2 \varpi^4 }{(b^2 - a^2) }
			\biggl[ A_1 a^2 - A_2 b^2 \biggr]
			+
			2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)
			</math>
			</td>
			</tr>
			</table>

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~
			\frac{2\pi G \rho }{(a^2 - b^2) }
			\biggl[ A_1 a^2 - A_2 b^2 \biggr]a^2 b^2 \varpi^4
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			2 a b \lambda \Omega_f \varpi^2 (a^2 y^2 + b^2 x^2)
			- \lambda^2 (a^2 - b^2) [ a^2 y^4 - b^2 x^4 ]
			+
			\biggl\{
			\biggl[\lambda b^2 - a b \Omega_f \biggr]x^2
			+
			\biggl[ \lambda a^2 - a b \Omega_f\biggr]y^2
			\biggr\}^2
	</math>		</math>
	</td>		</td>