Neutron star modelling with polytrope

Question

I want to model the density profile of a neutron star with 1.4 $M_\odot$ with a polytrope $p=\kappa\rho^{\gamma}$ with $\gamma=1+\frac{1}{n}$. I started with $n=\frac{1}{2}$ and solved the Lane-Emden equation for the Newtonian case with Mathematica. Then I put the resulting numerical solution for $\rho(r)$ into the structure equations for stars (with TOV equation for $\frac{\text{d}p}{\text{d}r}$) and solved them numerically.

For the boundary conditions I used $R=13\,$km and the conditions from the Lane-Emden equation: $\kappa=\rho_c^{1-\frac{1}{n}}\frac{4\pi G}{n+1}\frac{1}{a^2}$, $a=\frac{\xi_0}{R}$, $\rho_c=\left(-\frac{3}{\xi_0}\partial_z\theta(\xi_0)\right)^{-1}\rho_{\text{avg}}$, $\rho_{\text{avg}}=\frac{3M}{4\pi R^3}$. Where $\xi_0$ is the Root of $\theta(z)$.

Now when I put $M=1.4\,M_{\odot}$ the resulting mass after solving the TOV equation is $\approx 0.9\,M_{\odot}$. Only if I go up to around $3.9\,M_{\odot}$ I get the desired mass. Also changing the polytrope index $n$ doesn't really help.

Somehow I think, I have the wrong approach by mixing the Newtonian and full GR (TOV equation) case. Is there a way to get a realistic density profile with a polytrope and $M=1.4\,M_{\odot}$ by using the TOV equation?

Answer 1

Yeah, mixing the Newtonian and GR approach is definitely troublesome. In fact, the TOV and Lane-Emden equations are essentially doing the same thing (albeit in different forms).

There's a general-relativistic generalization of the Lane-Emden equation that should give you more consistent results. You can find it in Section II of this paper, for instance: https://articles.adsabs.harvard.edu/pdf/1973ApJ...183..637B

The equation to look at is ($2.8$), which reduces to the regular (Newtonian) Lane-Emden form when the parameter $\sigma \equiv \dfrac{P}{\rho c^2}$ goes to zero.