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 19 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

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


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

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

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

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

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