User:Tohline/SSC/Structure/StahlerMassRadius - Revision history

Tohline: Insert "first finding" and "second finding"

2014-10-02T01:35:34Z

Insert "first finding" and "second finding"

Tohline: /* Stahler's Mass-Radius Relationship for Embedded Polytropes */ Begin experimentation in an effort show more general consistency between our virial and Stahler's analysis

2014-09-30T21:04:27Z

Stahler's Mass-Radius Relationship for Embedded Polytropes: Begin experimentation in an effort show more general consistency between our virial and Stahler's analysis

← Older revision		Revision as of 21:04, 30 September 2014
Line 447:		Line 447:
	On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors. The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side. We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)^{-1}</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>. But this is precisely the case for which Stahler provides an analytic mass-radius relationship. What happens to the virial expression if we multiply through by <math>~(5-n)</math>?		On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors. The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side. We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)^{-1}</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>. But this is precisely the case for which Stahler provides an analytic mass-radius relationship. What happens to the virial expression if we multiply through by <math>~(5-n)</math>?

			==Experimentation==
			Let's break the form factor, <math>~\mathfrak{f}_A</math>, into two distinct pieces, namely,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\tilde\mathfrak{f}_{A1} </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\tilde\mathfrak{f}_{A2} </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>
			\tilde\Theta^{n+1} \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			then multiply the virial equilibrium expression through by <math>~(5-n)</math>.

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

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl[ \frac{(5-n)\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr]
			\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
			~-~[(5-n)\mathfrak{f}_{A1} + (5-n)\mathfrak{f}_{A2}]
			\biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}
			\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
			\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n}
			~+~ (5-n)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{3}{4\pi} \biggl( \frac{n+1}{n} \biggr) \biggl\{
			\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
			~-~\frac{n}{(4\pi)^{1/n}} (\tilde\theta')^2 \biggl[ \frac{3}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}
			\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
			\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			~-~(5-n)\biggl[ \tilde\theta^{n+1}
			\biggl( \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr)^{(n+1)/n}
			\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
			\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n}
			~+~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggr]
			</math>
			</td>
			</tr>
			</table>
			</div>

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Tohline: /* Comparison */

2014-09-30T19:09:27Z

Comparison

← Older revision		Revision as of 19:09, 30 September 2014
Line 445:		Line 445:
	</div>		</div>

	On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors. The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side. We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>. But this is precisely the case for which Stahler provides an analytic mass-radius relationship. What happens to the virial expression if we multiply through by <math>~(5-n)</math>?		On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors. The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side. We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)^{-1}</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>. But this is precisely the case for which Stahler provides an analytic mass-radius relationship. What happens to the virial expression if we multiply through by <math>~(5-n)</math>?


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Tohline: /* Comparison */ Add sentence indicating that I've shown that the last virial expression is satisfied for all polytropic indexes

2014-09-30T19:08:15Z

Comparison: Add sentence indicating that I've shown that the last virial expression is satisfied for all polytropic indexes

← Older revision		Revision as of 19:08, 30 September 2014
Line 444:		Line 444:
	</table>		</table>
	</div>		</div>

			On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors. The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side. We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>. But this is precisely the case for which Stahler provides an analytic mass-radius relationship. What happens to the virial expression if we multiply through by <math>~(5-n)</math>?


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Tohline: /* Stahler's Mass-Radius Relationship for Embedded Polytropes */ Significant derivation, trying to compare with Stahler's mass-radius relation

2014-09-30T00:01:29Z

Stahler's Mass-Radius Relationship for Embedded Polytropes: Significant derivation, trying to compare with Stahler's mass-radius relation

Show changes

Tohline: Begin separate chapter discussion Staher's (1983) mass-radius relationship for n=5 embedded polytropes

2014-09-29T21:27:52Z

Begin separate chapter discussion Staher's (1983) mass-radius relationship for n=5 embedded polytropes

New page

__FORCETOC__

=Stahler's Mass-Radius Relationship for Embedded Polytropes=
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In [[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|an accompanying chapter that discusses detailed force-balanced models of embedded (and pressure-truncated) polytropes]], we review [http://adsabs.harvard.edu/abs/1983ApJ...268..165S S. W. Stahler's (1983)] pair of parametric [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|relations for the equilibrium mass and equilibrium radius]] for such systems, namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~M
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
M_\mathrm{SWS} \biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \biggl\{ \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \biggr\}_{\xi_e}
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~R_\mathrm{eq}
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
R_\mathrm{SWS} \biggl( \frac{n}{4\pi} \biggr)^{1/2} \biggl\{ \xi \theta_n^{(n-1)/2} \biggr\}_{\xi_e}
</math>
</td>
</tr>
</table>
</div>
where,
<div align="center">
<math>M_\mathrm{SWS} =
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>
</div>
<div align="center">
<math>
R_\mathrm{SWS} = \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,
</math>
</div>

and point out that [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)] (see his equation B13) explicitly states that [[User:Tohline/SSC/Structure/PolytropesEmbedded#Overlap_with_Stahler.27s_Presentation_2|the relevant mass-radius relationship]] for <math>~n = 5</math> embedded polytropes is,
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 - 5 \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)
+ \frac{20\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4
</math>
</td>
<td align="center">
<math>~=~</math>
</td>
<td align="left">
<math>
~0 \, .
</math>
</td>
</tr>
</table>
</div>

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