User:Tohline/SSC/Stability/BiPolytrope0 0Details - Revision history

Tohline: /* More General Solution */ Add span tag "EnvelopeLAWE"

2017-01-15T20:29:40Z

More General Solution: Add span tag "EnvelopeLAWE"

← Older revision		Revision as of 20:29, 15 January 2017
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	</table>		</table>
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	Hence, the wave equation becomes,		<span id="EnvelopeLAWE">Hence, the wave equation becomes,</span>

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Tohline: /* Radial Oscillations of a Zero-Zero Bipolytrope */ Insert initial warning statement regarding improper expression for g^2

2017-01-10T18:10:17Z

Radial Oscillations of a Zero-Zero Bipolytrope: Insert initial warning statement regarding improper expression for g^2

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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Radial Oscillations of a Zero-Zero Bipolytrope=		=Radial Oscillations of a Zero-Zero Bipolytrope=

			<font color="red" size="+1"><b>IMPORTANT NOTE!</b></font>  While the overall development in this chapter is correct and, in particular, the chronology of discovery is properly reflected, the final quantitative results — for example, the recorded root(s) of the governing quartic equation — are incorrect because, beginning in the [[#Example21\|Example21 subsection, below]], we started using an incorrect expression for the function <math>~g^2</math>. In an [[User:Tohline/SSC/Stability/BiPolytrope0_0#Five_Mode_Summary\|accompanying summary presentation]], we have corrected this mistake.


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	====Example21====		====Example21====
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	Let's define both <math>~a_0</math> and <math>~b_0</math> such that the ''values'' of both eigenfunctions is unity at the interface <math>~(\xi=1)</math>. This means that,		Let's define both <math>~a_0</math> and <math>~b_0</math> such that the ''values'' of both eigenfunctions is unity at the interface <math>~(\xi=1)</math>. This means that,

Tohline: /* Roots of Quartic Equation */ Correct expression for g^2 in terms of q^3

2017-01-08T02:18:35Z

Roots of Quartic Equation: Correct expression for g^2 in terms of q^3

← Older revision		Revision as of 02:18, 8 January 2017
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	</math>		</math>

			</div>

			<div align="center">
			<table border="1" cellpadding="8" align="center" width="70%">
			<tr><td align="left">
			<font color="red">'''WRONG!'''</font>   The first of these three expressions should read,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1+8q^3}{(1+2q^3)^2} = \frac{1+8\mathcal{D} }{(1+2\mathcal{D})^2}</math>
			</td>
			</tr>
			</table>
			</div>
			</td></tr>
			</table>
	</div>		</div>

Tohline: /* Setup */ Correct expression that gives g^2 as a function of math cal{D}

2017-01-08T02:15:54Z

Setup: Correct expression that gives g^2 as a function of math cal{D}

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	</table>		</table>
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			<div align="center">
			<table border="1" cellpadding="8" align="center" width="70%">
			<tr><td align="left">
			<font color="red">'''WRONG!'''</font>   This relation should read,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1+8q^3}{(1+2q^3)^2} = \frac{1+8\mathcal{D} }{(1+2\mathcal{D})^2}</math>
			</td>
			</tr>
			</table>
			</div>
			</td></tr>
			</table>
			</div>


	we can view this expression as a conditional relationship between <math>~q</math> and <math>~\gamma_e</math>. Once a <math>~(q,\gamma_e)</math> pair has been found that satisfies this condition, we can use the matching-frequency condition to give the corresponding, required value of <math>~\gamma_c</math>; specifically, for <math>~(\ell,j) = (2,1)</math>,		we can view this expression as a conditional relationship between <math>~q</math> and <math>~\gamma_e</math>. Once a <math>~(q,\gamma_e)</math> pair has been found that satisfies this condition, we can use the matching-frequency condition to give the corresponding, required value of <math>~\gamma_c</math>; specifically, for <math>~(\ell,j) = (2,1)</math>,
	<div align="center">		<div align="center">

Tohline: /* Example21 */ Correct expression relating g^2 to mathcal{D}

2017-01-08T02:11:47Z

Example21: Correct expression relating g^2 to mathcal{D}

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	<math>\frac{1}{g} = (1+2\mathcal{D}) \, .</math>		<math>\frac{1}{g} = (1+2\mathcal{D}) \, .</math>
	</div>		</div>

			<div align="center">
			<table border="1" cellpadding="8" align="center" width="70%">
			<tr><td align="left">
			<font color="red">'''WRONG!'''</font>   This relation should read,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~g^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{1+8q^3}{(1+2q^3)^2} = \frac{1+8\mathcal{D} }{(1+2\mathcal{D})^2}</math>
			</td>
			</tr>
			</table>
			</div>
			</td></tr>
			</table>
			</div>


	Hence,		Hence,

Tohline: /* Envelope Solution Outline */ Give alternate expression for sigma_env^2

2017-01-01T22:50:03Z

Envelope Solution Outline: Give alternate expression for sigma_env^2

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(c_0 + 3j)(c_0 + 3j+5) + 2\~~alpha~~ \, ,</math>		<math>~(c_0 + 3j)(c_0 + 3j+5) + 2\alpha_e </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3[\alpha_e + c_0(2j+1) + j(3j+5)] \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	where,		where, the last expression results from recognizing that <math>~\alpha_e = c_0^2 + 2c_0 \, ,</math> and we have adopted the notation,
	<div align="center">		<div align="center">
	<math>~		<math>~

Tohline: /* Allow Different Adiabatic Exponents */

2016-12-24T22:12:35Z

Allow Different Adiabatic Exponents

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	<math>~		<math>~
	\frac{ 2\alpha_e + (c_0 + 3\ell)(c_0 + 3\ell +5) }{ 2\alpha_c + 2j(2j+5) } \, .		\frac{ 2\alpha_e + (c_0 + 3\ell)(c_0 + 3\ell +5) }{ 2\alpha_c + 2j(2j+5) }
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~ \Rightarrow ~~~ \gamma_c [ 6 + 2j(2j+5)] </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~8 +
			\gamma_e [2\alpha_e + (c_0 + 3\ell)(c_0 + 3\ell +5) ]\frac{\rho_e}{\rho_c} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Related Discussions */

2016-12-19T01:23:15Z

Related Discussions

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	=Related Discussions=		=Related Discussions=
			* [[User:Tohline/SSC/Stability/BiPolytrope0_0#Radial_Oscillations_of_a_Zero-Zero_Bipolytrope\|Summary of Above, Detailed Derivation]]
			* [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes\|Searching for Additional, Analytically Specified Eigenvectors of Zero-Zero Bipolytropes]]


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Tohline: /* Roots of Quartic Equation */

2016-12-17T18:29:58Z

Roots of Quartic Equation

← Older revision		Revision as of 18:29, 17 December 2016
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	====Roots of Quartic Equation====		====Roots of Quartic Equation====
			<!--
	TO BE DONE:		TO BE DONE:
	* In the ''specific'' case of <math>~\gamma_e = 4/3</math>, the expression governing the relevant value of <math>~q</math> can be readily viewed as a quartic equation in <math>~\Chi \equiv q^3</math>. One of the real roots of this quartic equation should provide an analytic expression for this value of <math>~q</math>. Indeed, it seems likely that the more ''general'' expression for <math>~q(\gamma_e)</math> can also be viewed as a quartic equation in <math>~\Chi</math> whose root is expressible analytically.		* In the ''specific'' case of <math>~\gamma_e = 4/3</math>, the expression governing the relevant value of <math>~q</math> can be readily viewed as a quartic equation in <math>~\Chi \equiv q^3</math>. One of the real roots of this quartic equation should provide an analytic expression for this value of <math>~q</math>. Indeed, it seems likely that the more ''general'' expression for <math>~q(\gamma_e)</math> can also be viewed as a quartic equation in <math>~\Chi</math> whose root is expressible analytically.
	* Need to also generate a table of solutions for the case of <math>~c_0</math> (minus).		* Need to also generate a table of solutions for the case of <math>~c_0</math> (minus).
			-->

	Let,		Let,
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	To solve this equation analytically, we follow the [http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method Summary of Ferrari's method] that is presented in Wikipedia's discussion of the Quartic Function to identify the roots of an arbitrary quartic equation.		To solve this equation analytically, we follow the [http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method Summary of Ferrari's method] that is presented in Wikipedia's discussion of the Quartic Function to identify the roots of an arbitrary quartic equation.

	<div align="center">		<div align="center" id="Quartic">
	<table border="1" cellpadding="8" width="85%">		<table border="1" cellpadding="8" width="85%">
	<tr><td align="left">		<tr><td align="left">

Tohline: Rename file containing these detailed derivations

2016-12-15T01:32:46Z

Rename file containing these detailed derivations

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