Tohline: /* ASIDE: Whitworth's Scaling */ Delete extraneous R_rf in table

2012-10-22T14:41:26Z

ASIDE: Whitworth's Scaling: Delete extraneous R_rf in table

← Older revision		Revision as of 14:41, 22 October 2012
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	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~R_\mathrm{rf} =~~ \frac{2^2GM}{3\cdot 5 K}</math>		<math>\frac{2^2GM}{3\cdot 5 K}</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* ASIDE: Whitworth's Scaling */ Discuss Whitworth's normalizations

2012-10-20T23:02:29Z

ASIDE: Whitworth's Scaling: Discuss Whitworth's normalizations

← Older revision		Revision as of 23:02, 20 October 2012
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	=ASIDE: Whitworth's Scaling=		=ASIDE: Whitworth's Scaling=
			In his study of the "global gravitational stability [of] one-dimensional polytropes," [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, <math>R</math>, and for the pressure exerted by an external bounding medium, <math>P_\mathrm{ex}</math>, to quantities he refers to as, respectively, <math>R_\mathrm{rf}</math> and <math>P_\mathrm{rf}</math>. The paragraph from his paper in which these two reference quantities are defined is shown here:
			<div align="center">
			<table border="2">
			<tr><td>
			[[File:WhitworthScalingText.jpg\|600px\|center\|Whitworth (1981, MNRAS, 195, 967)]]
			</td></tr>
			</table>
			</div>
			In order to map Whitworth's terminology to ours, we note, first, that he uses <math>M_0</math> to represent the spherical configuration's total mass, which we refer to simply as <math>M</math>; and his parameter <math>\eta</math> is related to our {{User:Tohline/Math/MP_PolytropicIndex}} via the relation,
			<div align="center">
			<math>\eta = 1 + \frac{1}{n} \, .</math>
			</div>
			Hence, Whitworth writes the polytropic equation of state as,
			<div align="center">
			<math>P = K_\eta \rho^\eta \, ,</math>
			</div>
			whereas, using our standard notation, this same key relation is written as,
			<div align="center">
			{{User:Tohline/Math/EQ_Polytrope01}} ;
			</div>
			and his parameter <math>K_\eta</math> is identical to our {{User:Tohline/Math/MP_PolytropicConstant}}.

	=~~==Bounded Adiabatic===~~		According to the second (bottom) expression identified by the red outlined box drawn above,
	~~For adiabatic configurations~~ <math>(\~~delta_~~{1\~~gamma_g} = 0~~)</math>~~, equilibrium states exist at radii given by~~ the ~~roots of~~ the ~~following expression:~~		<div align="center">
			<math>
			P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, ,
			</math>
			</div>
			and inverting the expression inside the green outlined box gives,
	<div align="center">		<div align="center">
	<math>		<math>
	3(\~~gamma_g-1~~) ~~B\chi~~^{3 -~~3\gamma_g~~} ~~~-~A~~\~~chi~~^{-1} ~~-~ 3D\chi^3 = 0~~ \, .		K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, .
	</math>		</math>
	</div>		</div>
	This is precisely the same condition that derives from setting equation (3) to zero in [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's] (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes." The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,		Hence,
	<div align="center">		<div align="center">
	<math>		<math>
	~~P_e~~ = ~~K \biggl(~~ \frac{~~3M}{~~4~~\pi R_\mathrm{eq}~~^3} ~~\biggr)~~^{\~~gamma_g~~} - \biggl( \frac{~~3GM~~^2}~~{20~~\~~pi R_~~\mathrm{eq}^4} \biggr) \, .		P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{1}{G^3 M^2} \biggr)\biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{4/\eta} \, ,
	</math>		</math>
	</div>		</div>
	~~This exactly matches equation (5)~~ of ~~[http://adsabs.harvard.edu/abs~~/~~1981MNRAS.195..967W Whitworth]~~, ~~which reads:~~		or, gathering all factors of <math>P_\mathrm{rf}</math> to the left-hand side,
	<div align="center">		<div align="center">
	<table border="2">		<math>
	<tr><td>		P_\mathrm{rf}^{(4-3\eta)} = 2^{-2(4+\eta)} \biggl( \frac{3^4 5^3}{\pi} \biggr)^\eta \biggl[ \frac{K_n^4}{G^{3\eta} M^{2\eta}} \biggr] \, .
	[[~~File:WhitworthScalingText.jpg\|500px\|center\|Whitworth~~ (~~1981, MNRAS, 195, 967~~)]]		</math>
	</td></tr>		</div>
			Analogously, according to the first (top) expression identified inside the red outlined box,
			<div align="center">
			<math>
			R_\mathrm{rf} = \frac{2^2 GM}{3\cdot 5 K_1} = 2^{2/\eta} \biggl( \frac{GM}{3\cdot 5}\biggr) K_n^{-1/\eta} P_\mathrm{rf}^{(1-\eta)/\eta}
			~~~~\Rightarrow~~~~ R_\mathrm{rf}^\eta = \frac{2^{2}}{K_n} \biggl( \frac{GM}{3\cdot 5}\biggr)^\eta P_\mathrm{rf}^{(1-\eta)} \, .
			</math>
			</div>

			<div align="center">
			<table border="1" width="90%">
			<tr>
			<td colspan="4" align="center">'''Examples'''</td>
			</tr>
			<tr>
			<td align="center">
			{{User:Tohline/Math/MP_PolytropicIndex}}
			</td>
			<td align="center">
			<math>\eta = 1+1/n</math>
			</td>
			<td align="center">
			<math>P_\mathrm{rf}</math>
			</td>
			<td align="center">
			<math>R_\mathrm{rf}</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			1
			</td>
			<td align="center">
			2
			</td>
			<td align="center">
			<math>\frac{2^{6}\pi}{3^4 5^3} \biggl[ \frac{G^{3} M^{2} }{K^2}\biggr]</math>
			</td>
			<td align="center">
			<math>\biggl[ \frac{3^2 5}{2^4 \pi} \biggl( \frac{K}{G} \biggr) \biggr]^{1/2}</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			5
			</td>
			<td align="center">
			6/5
			</td>
			<td align="center">
			<math>\frac{3^{12} 5^{9}}{2^{26} \pi^3} \biggl[ \frac{K^{10}}{G^9 M^6} \biggr]</math>
			</td>
			<td align="center">
			<math>\biggl[ \frac{2^{12} \pi}{3^6 5^5} \biggl( \frac{G^5 M^4}{K^5} \biggr) \biggr]^{1/2}</math>
			</td>
			</tr>

			<tr>
			<td align="center">
			<math>\infty</math>
			</td>
			<td align="center">
			1
			</td>
			<td align="center">
			<math> \frac{3^4 5^3}{2^{10}\pi} \biggl[ \frac{K^4}{G^{3} M^{2} }\biggr]</math>
			</td>
			<td align="center">
			<math>R_\mathrm{rf} = \frac{2^2GM}{3\cdot 5 K}</math>
			</td>
			</tr>

	</table>		</table>
	</div>		</div>
	Ideally we would like to invert this equation to obtain an analytic expression for the configuration's equilibrium radius in terms of the physical parameters, <math>M</math>, <math>K</math>, and <math>P_e</math>. However, this cannot be accomplished for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>.


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: Create the ASIDE to explain Whitworth's (1981) scaling of pressure-bounded polytropes

2012-10-20T20:39:41Z

Create the ASIDE to explain Whitworth's (1981) scaling of pressure-bounded polytropes

New page

__FORCETOC__
{{LSU_HBook_header}}

=ASIDE: Whitworth's Scaling=

===Bounded Adiabatic===
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, equilibrium states exist at radii given by the roots of the following expression:
<div align="center">
<math>
3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, .
</math>
</div>
This is precisely the same condition that derives from setting equation (3) to zero in [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's] (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes." The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,
<div align="center">
<math>
P_e = K \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} - \biggl( \frac{3GM^2}{20\pi R_\mathrm{eq}^4} \biggr) \, .
</math>
</div>
This exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
<div align="center">
<table border="2">
<tr><td>
[[File:WhitworthScalingText.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</div>
Ideally we would like to invert this equation to obtain an analytic expression for the configuration's equilibrium radius in terms of the physical parameters, <math>M</math>, <math>K</math>, and <math>P_e</math>. However, this cannot be accomplished for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>.

{{LSU_HBook_footer}}

User:Tohline/SSC/Structure/PolytropesASIDE1 - Revision history

Tohline: /* ASIDE: Whitworth's Scaling */ Delete extraneous R_rf in table

Tohline: /* ASIDE: Whitworth's Scaling */ Discuss Whitworth's normalizations

Tohline: Create the ASIDE to explain Whitworth's (1981) scaling of pressure-bounded polytropes