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

Tohline: /* Derivation */

2016-11-19T23:41:56Z

Derivation

← Older revision		Revision as of 23:41, 19 November 2016
Line 39:		Line 39:
	</math>		</math>
	</div>		</div>
	(Note that, in this case, the subscript ''c'' should not represent the central conditions but, rather, conditions at some characteristic radial position within the configuration.)		(Note that, in this case, the subscript ''c'' should not represent the central conditions but, rather, conditions at some characteristic radial position within the configuration.) As has been mentioned in our [[User:Tohline/SSC/Structure/Polytropes#Other_.28All.29_Solutions\|accompanying discussion of analytically defined structures of isolated polytropes]], this expression for the power-law density profile can be found in the introductory section of the article by [http://adsabs.harvard.edu/abs/2012JMP....53f2503M Patryk Mach (2012, J. Math. Phys., 53, 062503)].

	==Examples==		==Examples==

Tohline: /* Power-Law Density Distributions (structure) */ Modify opening title

2012-11-05T00:18:26Z

Power-Law Density Distributions (structure): Modify opening title

← Older revision		Revision as of 00:18, 5 November 2012
Line 1:		Line 1:
	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Power-Law Density Distributions ~~(structure)~~=		=Power-Law Density Distributions=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	[[Image:LSU_Structure_still.gif\|74px\|left]]		[[Image:LSU_Structure_still.gif\|74px\|left]]

Tohline: Move title image

2012-11-05T00:17:44Z

Move title image

← Older revision		Revision as of 00:17, 5 November 2012
Line 1:		Line 1:
	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
			=Power-Law Density Distributions (structure)=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Power-Law Density Distributions (structure)=~~
	[[Image:LSU_Structure_still.gif\|74px\|left]]		[[Image:LSU_Structure_still.gif\|74px\|left]]
	Here we begin with the same second-order, one-dimensional ODE that governs the structure of [[User:Tohline/SSC/Structure/Polytropes#Governing_Relations\|polytropic spheres]], namely, the		Here we begin with the same second-order, one-dimensional ODE that governs the structure of [[User:Tohline/SSC/Structure/Polytropes#Governing_Relations\|polytropic spheres]], namely, the
Line 108:		Line 107:
	* [http://en.wikipedia.org/wiki/Lane-Emden_equation Lane-Emden equation]		* [http://en.wikipedia.org/wiki/Lane-Emden_equation Lane-Emden equation]
	* [http://en.wikipedia.org/wiki/Polytrope Polytrope]		* [http://en.wikipedia.org/wiki/Polytrope Polytrope]


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Isothermal Equation of State */ Finish isothermal EOS

2010-04-02T20:52:32Z

Isothermal Equation of State: Finish isothermal EOS

← Older revision		Revision as of 20:52, 2 April 2010
Line 88:		Line 88:
	<div align="center">		<div align="center">
	<math>		<math>
	~~- \beta c_s^2 r_0~~\frac{\~~rho_0~~}{\~~rho}~~ \biggl( \frac{r}{~~r_0~~} \biggr)~~^{-\beta-1} = -\frac{d\Phi}{dr}~~ .		\frac{d\Phi}{dr} = \beta \biggl(\frac{c_s^2}{r}\biggr) .
	</math>		</math>
	</div>		</div>
			Therefore, the Poisson equation gives,
			<div align="center">
			<math>
			\frac{1}{r^2}\frac{d}{dr}\biggl[r^2 \frac{d\Phi}{dr}\biggr] = \beta \biggl(\frac{c_s^2}{r^2}\biggr)= 4\pi G \rho_0 \biggl( \frac{r}{r_0} \biggr)^{-\beta} .
			</math>
			</div>
			This relation can be satisfied only if,
			<div align="center">
			<math>
			\beta = 2 ~~~~~\mathrm{and}~~~~~ \rho_0 = \frac{c_s^2}{2\pi G r_0^2} .
			</math>
			</div>
			Hence, hydrostatic balance can be achieved for an isothermal gas with a power-law density distribution of the form, <math>\rho \propto r^{-2}</math>. Because an isothermal <math>P(\rho)</math> equation of state is obtained by setting <math>n = \infty</math> in the more general polytropic equation of state, the result just derived is consistent with the above, more general analysis which showed that, for values of the polytropic index <math>n \gg 1</math>, the equilibrium power-law density distribution tends toward a <math>\rho \propto r^{-2}</math> distribution.

	=Related Wikipedia Discussions=		=Related Wikipedia Discussions=

Tohline: Isothermal gas

2010-04-02T13:21:39Z

Isothermal gas

← Older revision		Revision as of 13:21, 2 April 2010
Line 65:		Line 65:
	Hence, for polytropic indices in the range <math>\infty > n > 3</math>, the relevant power-law density distribution lies between <math> \rho \propto \xi^{-2}</math> and <math> \rho \propto \xi^{-3}</math>.		Hence, for polytropic indices in the range <math>\infty > n > 3</math>, the relevant power-law density distribution lies between <math> \rho \propto \xi^{-2}</math> and <math> \rho \propto \xi^{-3}</math>.

			==Isothermal Equation of State==

			Suppose the gas is isothermal so that the relevant equation of state is,
			<div align="center">
			<math>
			P = c_s^2 \rho ,
			</math>
			</div>
			where <math>c_s</math> is the sound speed. To determine what power-law density distribution will satisfy hydrostatic equilibrium in this case, it is better to return to the original statement of [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#HydrostaticBalance\|hydrostatic balance for spherically symmetric configurations]],
			<div align="center">
			<math>
			\frac{1}{\rho} \frac{dP}{dr} = -\frac{d\Phi}{dr} .
			</math>
			</div>
			Plugging in the isothermal equation of state and assuming a radial density distribution of the form,
			<div align="center">
			<math>
			\rho(r) = \rho_0 \biggl( \frac{r}{r_0} \biggr)^{-\beta} ,
			</math>
			</div>
			we obtain,
			<div align="center">
			<math>
			- \beta c_s^2 r_0\frac{\rho_0}{\rho} \biggl( \frac{r}{r_0} \biggr)^{-\beta-1} = -\frac{d\Phi}{dr} .
			</math>
			</div>

Tohline: Begin discussion of power-law density distributions

2010-04-02T02:26:19Z

Begin discussion of power-law density distributions

New page

__FORCETOC__

{{LSU_HBook_header}}

=Power-Law Density Distributions (structure)=
[[Image:LSU_Structure_still.gif|74px|left]]
Here we begin with the same second-order, one-dimensional ODE that governs the structure of [[User:Tohline/SSC/Structure/Polytropes#Governing_Relations|polytropic spheres]], namely, the
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
<math>\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n</math> ,
</div>
and examine whether or not this governing relation can be satisfied by a power-law enthalpy distribution of the form,
<div align="center">
<math>
\Theta_H = A \xi^{-\alpha} ,
</math>
</div>
where <math>A</math> and <math>\alpha</math> are assumed to be constants. We note, up front, that such a solution will not satisfy the [[User:Tohline/SSC/Structure/Polytropes#Boundary_Conditions|boundary conditions that are imposed on polytropic spheres]]. But the simplistic form of a power-law solution can nevertheless sometimes be instructive.

==Derivation==

Plugging the power-law expression for the dimensionless enthalpy into both sides of the Lane-Emden equation gives,
<div align="center">
<math>
-\alpha (1 -\alpha) A \xi^{-(2 +\alpha)} = - A^n \xi^{-\alpha n} .
</math>
</div>
Hence, the power-law enthalpy distribution works as long as,
<div align="center">
<math>
\alpha = \frac{2}{n-1} ~~~~~~\mathrm{and}~~~~~~ A = [\alpha (1 -\alpha)]^{1/(n-1)} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{1/(n-1)}.
</math>
</div>

This means that hydrostatic balance can be established at all radial positions within a spherically symmetric configuration for power-law density distributions of the form,
<div align="center">
<math>
\frac{\rho}{\rho_c} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{n/(n-1)} \xi^{- 2n/(n-1)}.
</math>
</div>
(Note that, in this case, the subscript ''c'' should not represent the central conditions but, rather, conditions at some characteristic radial position within the configuration.)

==Examples==

It looks like the derived solution makes some physical sense only for polytropic indices <math> n > 3</math>.
For <math>n=4</math>, the relevant power-law density distribution is,
<div align="center">
<math>
\frac{\rho}{\rho_c} = \biggl[ \frac{2}{9} \biggr]^{4/3} \xi^{- 8/3}.
</math>
</div>
For <math>n=(3+\epsilon)</math> and <math>\epsilon \ll 1</math>,
<div align="center">
<math>
\frac{\rho}{\rho_c} \approx \biggl[ \frac{\epsilon}{2} \biggr] \xi^{- 3}.
</math>
</div>
For <math>n \gg 1</math>,
<div align="center">
<math>
\frac{\rho}{\rho_c} \approx \biggl[ \frac{2}{n} \biggr] \xi^{- 2}.
</math>
</div>
Hence, for polytropic indices in the range <math>\infty > n > 3</math>, the relevant power-law density distribution lies between <math> \rho \propto \xi^{-2}</math> and <math> \rho \propto \xi^{-3}</math>.

=Related Wikipedia Discussions=
* [http://en.wikipedia.org/wiki/Lane-Emden_equation Lane-Emden equation]
* [http://en.wikipedia.org/wiki/Polytrope Polytrope]

{{LSU_HBook_footer}}