Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope - Revision history

Jet53man: /* Eureka! */

2023-08-06T18:07:07Z

Eureka!

← Older revision		Revision as of 14:07, 6 August 2023
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	</table>		</table>

	=Eureka!=		=Eureka (NOT!)=

	Here is an approach that has been mapped out with some success today (23 July 2023).		Here is an approach that has been mapped out with some success today (23 July 2023).

Jet53man: /* Modified Approach for n = 1 */

2023-07-25T22:11:59Z

Modified Approach for n = 1

← Older revision		Revision as of 18:11, 25 July 2023
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	<td align="right"><math>M_r</math></td>		<td align="right"><math>M_r</math></td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>\~~int_0~~^r 4\pi \rho r^2 dr = 4\pi \rho_c a_n^3 \~~int_C~~^{\eta-C} \phi (\eta - C)^2 d\eta</math></td>		<td align="left">
			<math>
			\int_{r=0}^{r=R} 4\pi \rho r^2 dr
			=
			4\pi \rho_c a_n^3 \int_{\eta=C}^{\eta=C+R/a_n} \phi (\eta - C)^2 d\eta
			</math>
			</td>
	</tr>		</tr>

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	<td align="right"> </td>		<td align="right"> </td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>- 4\pi \rho_c a_n^3 \~~int_C~~^{\eta-C} \biggl\{ \frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr] \biggr\} (\eta - C)^2 d\eta</math></td>		<td align="left"><math>- 4\pi \rho_c a_n^3 \int_{\eta=C}^{\eta=C+R/a_n} \biggl\{ \frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr] \biggr\} (\eta - C)^2 d\eta</math></td>
	</tr>		</tr>

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	<td align="right"> </td>		<td align="right"> </td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>\biggl[ - 4\pi \rho_c a_n^3 (\eta - C)^2 \frac{d\phi}{d\eta} \biggr]_C^{\eta-C} \, .</math></td>		<td align="left"><math>\biggl[ - 4\pi \rho_c a_n^3 (\eta - C)^2 \frac{d\phi}{d\eta} \biggr]_{\eta=C}^{\eta=C+R/a_n} \, .</math></td>
	</tr>		</tr>
	</table>		</table>
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	<td align="left">		<td align="left">
	<math>		<math>
	A \cdot \frac{\sin(\eta - B)}{(\eta - ~~m\pi~~)} \, ,		A \cdot \frac{\sin(\eta - B)}{(\eta - C)} \, ,
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{\cos(\eta - B)}{(\eta - ~~m\pi~~)}		\frac{\cos(\eta - B)}{(\eta - C)}
	-		-
	\frac{\sin(\eta - B)}{(\eta - ~~m\pi~~)^2} \, ;		\frac{\sin(\eta - B)}{(\eta - C)^2} \, ;
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	- \frac{\sin(\eta - B)}{(\eta - ~~m\pi~~)} - \frac{2\cos(\eta - B)}{(\eta - ~~m\pi~~)^2}		- \frac{\sin(\eta - B)}{(\eta - C)} - \frac{2\cos(\eta - B)}{(\eta - C)^2}
	+ \frac{2\sin(\eta - B)}{(\eta - ~~m\pi~~)^3} \, .		+ \frac{2\sin(\eta - B)}{(\eta - C)^3} \, .
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">RHS</td>		<td align="right">RHS</td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>
			\frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr]
			=
			\frac{d^2\phi}{d\eta^2} + \frac{2}{(\eta - C)} \cdot \frac{d\phi}{d\eta}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>		<math>
	A \biggl[		A \biggl[
	- \frac{\sin(\eta - B)}{(\eta - ~~m\pi~~)} - \frac{2\cos(\eta - B)}{(\eta - ~~m\pi~~)^2}		- \frac{\sin(\eta - B)}{(\eta - C)} - \frac{2\cos(\eta - B)}{(\eta - C)^2}
	+ \frac{2\sin(\eta - B)}{(\eta - ~~m\pi~~)^3}\biggr]		+ \frac{2\sin(\eta - B)}{(\eta - C)^3}\biggr]
	+		+
	A\biggl[\frac{2\cos(\eta - B)}{(\eta - ~~m\pi~~)^2}		A\biggl[\frac{2\cos(\eta - B)}{(\eta - C)^2}
	-		-
	\frac{2\sin(\eta - B)}{(\eta - ~~m\pi~~)^3} \biggr]		\frac{2\sin(\eta - B)}{(\eta - C)^3} \biggr]
	</math>		</math>
	</td>		</td>
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	<math>		<math>
	A \biggl[		A \biggl[
	- \frac{\sin(\eta - B)}{(\eta - ~~m\pi~~)} \biggr] = -\phi \, ,		- \frac{\sin(\eta - B)}{(\eta - C)} \biggr] = -\phi \, ,
	</math>		</math>
	</td>		</td>

Jet53man: /* Eureka! */

2023-07-25T21:43:03Z

Eureka!

@@ Line 2,446: / Line 2,446: @@
    <td align="left">
 <math>
-\frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr]
+\frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr] \, .
 </math>
    </td>
@@ Line 2,466: / Line 2,454: @@
 ===Modified Expression for Integrated Mass===
 The modified expression for the integrated mass in our <math>n=1</math> envelope is,
 <table border="0" align="center" cellpadding="5">
@@ Line 2,494: / Line 2,460: @@
    <td align="right"><math>M_r</math></td>
    <td align="center"><math>=</math></td>
-   <td align="left"><math>\int_0^r 4\pi \rho r^2 dr = 4\pi \rho_c a_n^3 \int_0^\eta \phi (\eta - m\pi)^2 d\eta</math></td>
+   <td align="left"><math>\int_0^r 4\pi \rho r^2 dr = 4\pi \rho_c a_n^3 \int_C^{\eta-C} \phi (\eta - C)^2 d\eta</math></td>
 </tr>
@@ Line 2,500: / Line 2,466: @@
    <td align="right">&nbsp;</td>
    <td align="center"><math>=</math></td>
-   <td align="left"><math>- 4\pi \rho_c a_n^3 \int_0^\xi \biggl\{ \frac{1}{(\eta - m\pi)^2} \frac{d}{d\eta}\biggl[(\eta - m\pi)^2 \frac{d\phi}{d\eta}\biggr] \biggr\}  (\eta - m\pi)^2 d\eta</math></td>
+   <td align="left"><math>- 4\pi \rho_c a_n^3 \int_C^{\eta-C} \biggl\{ \frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr] \biggr\}  (\eta - C)^2 d\eta</math></td>
 </tr>
@@ Line 2,506: / Line 2,472: @@
    <td align="right">&nbsp;</td>
    <td align="center"><math>=</math></td>
-   <td align="left"><math>- 4\pi \rho_c a_n^3  (\eta - m\pi)^2 \frac{d\phi}{d\eta} \, .</math></td>
+   <td align="left"><math>\biggl[ - 4\pi \rho_c a_n^3  (\eta - C)^2 \frac{d\phi}{d\eta} \biggr]_C^{\eta-C} \, .</math></td>
 </tr>
 </table>

Jet53man: /* Modified Expression for Integrated Mass */

2023-07-25T21:11:49Z

Modified Expression for Integrated Mass

@@ Line 2,466: / Line 2,466: @@
 ===Modified Expression for Integrated Mass===
 The modified expression for the integrated mass in our <math>n=1</math> envelope is,
 <table border="0" align="center" cellpadding="5">

Jet53man: /* Modified Approach for n = 1 */

2023-07-25T21:09:48Z

Modified Approach for n = 1

← Older revision		Revision as of 17:09, 25 July 2023
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	<table border="0" align="center" cellpadding="8">		<table border="0" align="center" cellpadding="8">
	<tr>		<tr>
	<td align="right"><math>\eta - ~~m \pi~~</math></td>		<td align="right"><math>\eta - C</math></td>
	<td align="center"><math>\equiv</math></td>		<td align="center"><math>\equiv</math></td>
	<td align="left"><math>\frac{r}{a_n}</math></td>		<td align="left"><math>\frac{r}{a_n}</math></td>
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	<td align="left">		<td align="left">
	<math>		<math>
	\frac{1}{(\eta - ~~m\pi~~)^2} \frac{d}{d\eta}\biggl[(\eta - ~~m\pi~~)^2 \frac{d\phi}{d\eta}\biggr]		\frac{1}{(\eta - C)^2} \frac{d}{d\eta}\biggl[(\eta - C)^2 \frac{d\phi}{d\eta}\biggr]
	</math>		</math>
	</td>		</td>
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	\frac{d^2\phi}{d\eta^2}		\frac{d^2\phi}{d\eta^2}
	+		+
	\frac{2}{(\eta - ~~m\pi~~)} \cdot \frac{d\phi}{d\eta} \, .		\frac{2}{(\eta - C)} \cdot \frac{d\phi}{d\eta} \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* Lane-Emden Equation */

2023-07-25T20:58:01Z

Lane-Emden Equation

← Older revision		Revision as of 16:58, 25 July 2023
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	<font color="red">ASIDE:</font>  Following along the lines of ~~§~~IV~~.5.b~~ of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], the expression for the enclosed mass is,		<font color="red">ASIDE:</font>  Following along the lines of Chapter IV of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>] — see also our [[SSC/Structure/Polytropes#IntegralMass\|accompanying derivation]] — the expression for the enclosed mass is,
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

Jet53man: /* Eureka! */

2023-07-24T18:10:48Z

Eureka!

@@ Line 2,429: / Line 2,429: @@
 ==Modified Approach for n = 1==
 Here, we deviate from tradition and adopt the following expression for the dimensionless radius,
 <table border="0" align="center" cellpadding="8">
@@ Line 2,463: / Line 2,464: @@
 </table>
 Let's ''guess'' a solution of the form,
 <table border="0" align="center" cellpadding="8">

Jet53man: /* Lane-Emden Equation */

2023-07-24T17:57:51Z

Lane-Emden Equation

← Older revision		Revision as of 13:57, 24 July 2023
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	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left"><math>- 4\pi \rho_c a_n^3 \biggl(\xi^2 \frac{d\Theta_H}{d\xi}\biggr) \, .</math></td>		<td align="left"><math>- 4\pi \rho_c a_n^3 \biggl(\xi^2 \frac{d\Theta_H}{d\xi}\biggr) \, .</math></td>
			</tr>

			<tr>
			<td align="center" colspan="3">
			§IV.5.b of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], p. 97, Eq. (67)

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

Jet53man: /* Lane-Emden Equation */

2023-07-24T17:50:55Z

Lane-Emden Equation

@@ Line 2,396: / Line 2,396: @@
 </div>
 Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> &#8212; and, from it in turn, determine the radial distribution of various dimensional physical variables &#8212; for various values of the polytropic index, {{Math/MP_PolytropicIndex}}.
 ==Modified Approach for n = 1==

Jet53man: /* Lane-Emden Equation */

2023-07-24T17:04:48Z

Lane-Emden Equation

← Older revision		Revision as of 13:04, 24 July 2023
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	</math>		</math>
	</div>		</div>
	where the mathematical relationship between <math>~H/H_c</math> and <math>~\rho/\rho_c</math> comes from the adopted barotropic (polytropic) relation ~~identified above~~. To accomplish this, we replace {{Math/VAR_Enthalpy01}} with <math>~H_c \Theta_H</math> on the left-hand-side of the governing differential equation and we replace {{Math/VAR_Density01}} with <math>~\rho_c \Theta_H^n</math> on the right-hand-side, then gather the constant coefficients together on the left. The resulting ODE is,		where the mathematical relationship between <math>~H/H_c</math> and <math>~\rho/\rho_c</math> comes from the adopted [[SR#Barotropic_Structure\|barotropic (polytropic) relation]]. To accomplish this, we replace {{Math/VAR_Enthalpy01}} with <math>~H_c \Theta_H</math> on the left-hand-side of the governing differential equation and we replace {{Math/VAR_Density01}} with <math>~\rho_c \Theta_H^n</math> on the right-hand-side, then gather the constant coefficients together on the left. The resulting ODE is,

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

	If we follow tradition and define a normalized radius, <math>\xi \equiv r/a_n</math>, then our governing ODE becomes what is referred to in the ~~astronomical~~ literature as the,		If we follow tradition and define a normalized radius, <math>\xi \equiv r/a_n</math>, then our governing ODE becomes what is referred to in the astrophysics literature as the,
	<div align="center">		<div align="center">
	<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>		<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>