# User:Tohline/SSC/Structure/PowerLawDensity

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

# Power-Law Density Distributions (structure)

Here we begin with the same second-order, one-dimensional ODE that governs the structure of polytropic spheres, namely, the

Lane-Emden Equation
$\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n$ ,

and examine whether or not this governing relation can be satisfied by a power-law enthalpy distribution of the form,

$\Theta_H = A \xi^{-\alpha} ,$

where $A$ and $\alpha$ are assumed to be constants. We note, up front, that such a solution will not satisfy the 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,

$-\alpha (1 -\alpha) A \xi^{-(2 +\alpha)} = - A^n \xi^{-\alpha n} . $

Hence, the power-law enthalpy distribution works as long as,

$\alpha = \frac{2}{n-1} ~~~~~~\mathrm{and}~~~~~~ A = [\alpha (1 -\alpha)]^{1/(n-1)} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{1/(n-1)}.$

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,

$\frac{\rho}{\rho_c} = \biggl[ \frac{2(n-3)}{(n-1)^2} \biggr]^{n/(n-1)} \xi^{- 2n/(n-1)}.$

(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 $n > 3$. For $n=4$, the relevant power-law density distribution is,

$\frac{\rho}{\rho_c} = \biggl[ \frac{2}{9} \biggr]^{4/3} \xi^{- 8/3}.$

For $n=(3+\epsilon)$ and $\epsilon \ll 1$,

$\frac{\rho}{\rho_c} \approx \biggl[ \frac{\epsilon}{2} \biggr] \xi^{- 3}.$

For $n \gg 1$,

$\frac{\rho}{\rho_c} \approx \biggl[ \frac{2}{n} \biggr] \xi^{- 2}.$

Hence, for polytropic indices in the range $\infty > n > 3$, the relevant power-law density distribution lies between $\rho \propto \xi^{-2}$ and $\rho \propto \xi^{-3}$.

# Related Wikipedia Discussions

 © 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation