Physical VariablesΒΆ

Once the Lane-Emden equation (12) has been solved, the density in each region can be evaluated by

\[\rho_{i} = \rho_{1,0} \, t_{i} \, \theta_{i}^{n_{i}}.\]

The pressure then follows from the equation-of-state (11) as

\[P_{i} = P_{1,0} \, \frac{n_{1}+1}{n_{i}+1} \, \frac{t_{i}^{2}}{B_{i}} \, \theta_{i}^{n_{i}+1}.\]

The interior mass \(m\) is evaluated by introducing the auxiliary quantity \(\mu\), which is defined in the first region by

\[\mu_{1}(z) = - z^{2} \theta'_{1} (z),\]

and in subsequent regions by

\[\mu_{i}(z) = \mu_{i-1}(z_{i-1/2}) - \frac{t_{i}}{B_{i}} \left[ z^{2} \theta'_{i} (z) - z_{i-1/2}^{2} \theta'_{i} (z_{i-1/2}) \right].\]

The interior mass then follows as

\[M_{r} = M \frac{\mu_{i}}{\mu_{\rm s}},\]

where \(\mu_{\rm s} \equiv \mu_{\nreg}(z_{\rm s})\).