Solution Method


The structure of a composite polytrope is specified completely by

  • a set of \(\nreg\) polytropic indices \(n_{i}\)

  • a set of \(\nreg-1\) boundary coordinates \(z_{i-1/2}\)

  • a set of \(\nreg\) density jumps \(\Delta_{i-1/2} \equiv \ln [\rho_{i}(z_{i-1/2})/\rho_{i-1}(z_{i-1/2}]\)

Although the normalizing densities \(\rho_{i,0}\) have so far been left unspecified, it’s convenient to choose them as the density at the beginning of their respective regions.


The structure equations may be solved as an initial value problem. In the first region (\(i=1\)) this IVP involves integrating the Lane-Emden equation (12) from the center \(z=0\) to the first boundary \(z=z_{3/2}\), with the initial conditions

\[\begin{split}\left. \begin{gathered} \theta_{i} = 1, \\ \theta'_{i} = 0, \\ B_{1} = 1, \\ t_{1} = 1 \end{gathered} \right\} \quad \text{at}\ z=0\end{split}\]

(here, \(t_{i} \equiv \rho_{i,0}/\rho_{1,0}\)).

The IVP in the intermediate regions (\(i = 2,\ldots,\nreg-1\)) involves integrating from \(z=z_{i-1/2}\) to \(z=z_{i+1/2}\), with initial conditions established from the preceding region via

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

The IVP in the final region (\(i=\nreg\)) involves integrating from \(z_{\nreg-1/2}\) until \(\theta_{\nreg} = 0\). This point defines the stellar surface, \(z=z_{\rm s}\). For some choices of \(n_{i}\), \(z_{i-1/2}\) and/or \(\Delta_{i-1/2}\), the point \(\theta=0\) can arise in an earlier region \(i = \nreg_{\rm t} < \nreg\); in such cases, the model specification must be truncated to \(\nreg_{\rm t}\) regions.