SSC/Structure/BiPolytropes/Analytic51 - Revision history

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-05T16:44:39Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 12:44, 5 August 2023
Line 512:		Line 512:
	An examination of their equations (A3) reveals that {{ EFC98hereafter }} continue to use <math>\theta</math> to represent the polytropic function throughout the envelope — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. Henceforth we will assume that these functions are interchangeable, that is, <math>\theta_\mathrm{efc} \leftrightarrow \phi</math>, and examine whether or not their various physical parameter expressions match ours.		An examination of their equations (A3) reveals that {{ EFC98hereafter }} continue to use <math>\theta</math> to represent the polytropic function throughout the envelope — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. Henceforth we will assume that these functions are interchangeable, that is, <math>\theta_\mathrm{efc} \leftrightarrow \phi</math>, and examine whether or not their various physical parameter expressions match ours.

	In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline: As detailed in the text, there appears to be a type-setting error in both of these expressions; as published by EFC98, the exponent on the coefficient of theta_i should be 6 and 5, respectively, whereas it appears as 4.]]In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,

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

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T22:13:19Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 18:13, 4 August 2023
Line 513:		Line 513:

	In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,		In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,

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

			<tr>
			<td align="right"><math>P</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>p_0 \theta_c^4 \theta_\mathrm{efc}^2 = K_c \rho_0^{6 / 5} \theta_i^4 \phi^2</math></td>
			<td align="center">        and,</td>
			<td align="right"><math>\rho</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>\frac{\rho_0}{\alpha} \cdot \theta_c^4 \theta_\mathrm{efc}
			= \rho_0\biggl(\frac{\mu_e}{\mu_c}\biggr) \theta_i^4 \phi \, ,</math></td>
			</tr>
			</table>

			where, in both expressions, we have replaced their label for the value of the polytropic function at the core-envelope interface, <math>\theta_c</math>, with our label, <math>\theta_i</math>. Both of their expressions match ours <font color="red"><b>EXCEPT … NOTE:</b></font>   in both of their expressions, <math>\theta_i</math> is raised to the 4<sup>th</sup> power whereas, according to our derivation, this interface value should be raised to the 6<sup>th</sup> power in the expression for pressure and it should be raised to the 5<sup>th</sup> power in the expression for the density. We are confident that our expressions are the correct ones and therefore presume that type-setting errors are present in both of the {{ EFC98hereafter }} expressions.

	----		----

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T21:33:16Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 17:33, 4 August 2023
Line 510:		Line 510:
	<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">		<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">

	An examination of their equations (A3) reveals that~~, throughout the envelope,~~ {{ EFC98hereafter }} continue to use <math>\theta</math> to represent the polytropic function — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. Henceforth we will assume that these functions are interchangeable, that is, <math>\theta_\mathrm{efc} \leftrightarrow \phi</math>.		An examination of their equations (A3) reveals that {{ EFC98hereafter }} continue to use <math>\theta</math> to represent the polytropic function throughout the envelope — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. Henceforth we will assume that these functions are interchangeable, that is, <math>\theta_\mathrm{efc} \leftrightarrow \phi</math>, and examine whether or not their various physical parameter expressions match ours.

	In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,		In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,

Jet53man: /* Step 4: Throughout the core (0 ≤ ξ ≤ ξi) */

2023-08-04T21:27:29Z

Step 4: Throughout the core (0 ≤ ξ ≤ ξi)

← Older revision		Revision as of 17:27, 4 August 2023
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	</tr>		</tr>
	</table>		</table>
			Hence,

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

			<tr>
			<td align="right"><math>\theta</math></td>
			<td align="center"><math>=</math></td>
			<td align="left"><math>\biggl[ 1 + \frac{\xi^2}{3}\biggr]^{-1 / 2} \, ,</math></td>
			</tr>
			</table>
			which matches our expression for the core's polytrope function, <math>\theta</math>.

			----

			Now, look at the {{ EFC98hereafter }} expression for the core's integrated mass.
	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

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	</tr>		</tr>
	</table>		</table>
			This expression matches ours.
	</td></tr></table>		</td></tr></table>

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T19:06:11Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 15:06, 4 August 2023
Line 495:		Line 495:
	<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">		<table border="1" align="center" cellpadding="5" width="80%"><tr><td align="left">

	~~Examining~~ equations (A3)~~, we see~~ that, throughout the envelope, {{ ~~EFC98~~ }} continue to use <math>\theta</math> to represent the polytropic function — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. We state that it has the form,		An examination of their equations (A3) reveals that, throughout the envelope, {{ EFC98hereafter }} continue to use <math>\theta</math> to represent the polytropic function — for clarity, we will attach the subscript <math>\theta_\mathrm{efc} </math> — whereas we use <math>\phi</math>. Henceforth we will assume that these functions are interchangeable, that is, <math>\theta_\mathrm{efc} \leftrightarrow \phi</math>.

			In their Eqs. (A3), {{ EFC98hereafter }} state that, throughout the envelope,

			----

			We state that the envelope's polytropic function has the form,

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

Jet53man: /* Step 4: Throughout the core (0 ≤ ξ ≤ ξi) */

2023-08-04T18:47:21Z

Step 4: Throughout the core (0 ≤ ξ ≤ ξi)

@@ Line 125: / Line 125: @@
    <td align="center"><math>=</math></td>
    <td align="left"><math>p_0 \rho_0^{-6/5} \, ;</math></td>
 </tr>

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T03:08:14Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 23:08, 3 August 2023
Line 642:		Line 642:
	<math>		<math>
	-A \cdot		-A \cdot
	\frac{4\pi \~~rho_0 }~~{9}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}		\biggl(\frac{2 }{\pi}\biggr)^{1 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta^{-2}_i
	\biggl[		\biggl[
	\sin(B-\eta) + \eta \cos(B-\eta)		\sin(B-\eta) + \eta \cos(B-\eta)
	\biggr]		\biggr]
	~~\biggl\{~~ \biggl[ \frac{G \rho_0^{4/5}~~}{K_c~~} \biggr]^{1/2~~} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i (2\pi)^{1 / 2} \biggr\}^{-1}~~		\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2}
	~~\biggl\{ 3 \biggl[ \frac{K_c }{G \rho_0^{4/5}} \cdot \biggl( \frac{3}{2\pi}\biggr) \biggr] \biggr\~~}
	</math>		</math>
	</td>		</td>

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T02:55:08Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

@@ Line 631: / Line 631: @@
 \biggr]
 \biggl\{ \biggl[ \frac{G \rho_0^{4/5}}{K_c} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i (2\pi)^{1 / 2}  \biggr\}^{-1}
-\biggl\{ a_\mathrm{efc}^2  \biggr\}
+\biggl\{ 3 \biggl[ \frac{K_c }{G \rho_0^{4/5}} \cdot \biggl( \frac{3}{2\pi}\biggr) \biggr] \biggr\}
 </math>
    </td>

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T02:50:48Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

@@ Line 578: / Line 578: @@
    <td align="center"><math>=</math></td>
    <td align="left">
-<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{-\eta^2 \frac{d\phi}{d\eta} \biggr\}</math>
+<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{- A \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr]\biggr\}</math>
    </td>
 </tr>
@@ Line 586: / Line 586: @@
    <td align="center"><math>=</math></td>
    <td align="left">
-<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{A \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr]\biggr\}</math>
+<math>-A \biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl[\sin(B - \eta) + \eta\cos(B - \eta) \biggr] \, .</math>
    </td>
 </tr>
 </table>
 According to {{ EFC98hereafter }},
@@ Line 606: / Line 603: @@
 \sin[\beta(r_s - r)] + \beta r \cos[\beta(r_s - r)]
 \biggr\}
 </math>
    </td>

Jet53man: /* Step 8: Throughout the envelope (ηi ≤ η ≤ ηs) */

2023-08-04T02:34:09Z

Step 8: Throughout the envelope (ηi ≤ η ≤ ηs)

← Older revision		Revision as of 22:34, 3 August 2023
Line 579:		Line 579:
	<td align="left">		<td align="left">
	<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{-\eta^2 \frac{d\phi}{d\eta} \biggr\}</math>		<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{-\eta^2 \frac{d\phi}{d\eta} \biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5}} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl\{A \biggl[ \eta\cos(\eta-B) - \sin(\eta-B) \biggr]\biggr\}</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>