Solution Method

Specification

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.

Solution

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.