# Structure Equations

## Lane-Emden Equation

In the $$i$$’th region, a composite polytrope satisfies the equation of hydrostatic equilibrium

$-\frac{1}{\rho_{i}} \deriv{P_{i}}{r} = \deriv{\Phi_{i}}{r}$

Substituting in the polytropic equation-of-state (11) yields

$\frac{(n_{i}+1) P_{i,0}}{\rho_{i,0}^{1+1/n_{i}}} \deriv{}{r} \left( \rho_{i}^{1/n_{i}} \right) = - \deriv{\Phi_{i}}{r},$

which can then be integrated with respect to $$r$$ to give

$\frac{(n_{i}+1)P_{i,0}}{\Phi_{i,0} \, \rho_{i,0}} \left( \frac{\rho_{i}^{1/n_{i}}}{\rho_{i,0}^{1/n_{i}}} - 1 \right) = \left( 1 - \frac{\Phi_{i}}{\Phi_{i,0}} \right).$

Here, the constants of integration have been chosen so that $$\Phi_{i} = \Phi_{i,0}$$ when $$\rho_{i} = \rho_{i,0}$$. Rearranging, the density follows as

$\rho_{i} = \rho_{i,0} \, \theta_{i}^{n_{i}},$

where the polytropic dependent variable is introduced as

$\theta_{i} = \left[ \frac{\Phi_{i,0} \, \rho_{i,0}}{(n_{i} + 1) P_{i,0}} \left( 1 - \frac{\Phi_{i}}{\Phi_{i,0}} \right) + 1 \right].$

With these expressions, Poisson’s equation

$\frac{1}{r^{2}} \deriv{}{r} \left( r^{2} \deriv{P_{i}}{r} \right) = 4 \pi G \rho_{i}$

is recast as

$\frac{1}{r^{2}} \deriv{}{r} \left( r^{2} \deriv{\theta_{i}}{r} \right) = - \frac{1}{A_{i}} \theta_{i}^{n_{i}},$

where

$A_{i} \equiv \frac{(n_{i} + 1) P_{i,0}}{4 \pi G \rho_{i,0}^{2}}.$

A change of variables to the polytropic independent variable $$z \equiv A_{1}^{-1/2} r$$ results in the dimensionless form

(12)$\frac{1}{z^{2}} \deriv{}{z} \left( z^{2} \deriv{\theta_{i}}{z} \right) = - B_{i} \theta_{i}^{n_{i}},$

where $$B_{i} \equiv A_{1}/A_{i}$$. This can be regarded as a generalization of the usual Lane-Emden equation to composite polytropes.

## Continuity Relations

At the boundary between adjacent regions, the pressure and interior mass must be continuous. If $$z_{i-1/2}$$ denotes the coordinate of the boundary between the $$i-1$$ and $$i$$ regions, then these continuity relations are expressed as

$\begin{split}\left. \begin{gathered} B_{i} = \frac{n_{i-1} + 1}{n_{i} + 1} \frac{\theta_{i}^{n_{i}+1}}{\theta_{i-1}^{n_{i-1}+1}} \frac{\rho_{i,0}^{2}}{\rho_{i-1,0}^{2}} \, B_{i-1}, \\ \theta'_{i} = \frac{n_{i-1} + 1}{n_{i} + 1} \frac{\theta_{i-1}^{n_{i-1}+1}}{\theta_{i}^{n_{i}+1}} \frac{\rho_{i,0}}{\rho_{i-1,0}} \, \theta'_{i-1}, \end{gathered} \right\} \quad \text{at} \ z = z_{i-1/2}\end{split}$

respectively.