Improved analytic modeling of neutron star interiors

Studies of neutron stars are extremely timely, given the recent detection of gravitational waves from a binary neutron star merger GW170817 and the International Space Station payload NICER currently in operation that aims to determine radii of neutron stars to a precision better than 5%. In many cases, neutron star solutions are constructed numerically due to the complexity of the field equations with realistic equations of state. However, to relate observables like the neutron star mass and radius to interior quantities like central density and pressure, it would be useful to provide an accurate, analytic modeling of a neutron star interior. One such solution for static and isolated neutron stars is the Tolman VII solution characterized only by two parameters (e.g., mass and radius), though its agreement with numerical solutions is not perfect. We introduce here an improved analytic model based on the Tolman VII solution by introducing an additional parameter to make the analytic density profile agree better with the numerically obtained one. This additional parameter can be fitted in terms of the stellar mass, radius, and central density in an equation of state–insensitive way. In most cases, we find that the new model more accurately describes realistic profiles than the original Tolman VII solution by a factor of 2–5. Our results are first-step calculations towards constructing analytic interior solutions for more realistic neutron stars under rotation or tidal deformation.

Figure 1

Energy density (top) and pressure (bottom) profiles for the original Tolman solution with two different parametrizations. We choose $R=11.4\mathrm{km}$ and either ${\rho }_c=9.9\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ or $M=1.4M_{\odot }$. We also present the numerical solution with the AP4 EoS and ${\rho }_c=9.9\times {10}^{14}\mathrm{g}/{\mathrm{cm}}^3$ that corresponds to $M=1.4M_{\odot }$ and $R=11.4\mathrm{km}$.

Figure 2

(Top) $\alpha$ [characterizing the density profile in the improved Tolman model in Eq. (21)] as a function of $C^n/{\rho }_c{R}^2$ with $n=0.903$ for 11 realistic EoS. Different colors correspond to different classes of EoS in Table 1 (soft in red, intermediate in green, and stiff in blue). We also present the fit in a black solid curve given by Eq. (25) with the coefficients given in the last row of Table 2. (Bottom) Relative fractional errors between numerical results and the fit. Notice that the relations are nearly EoS independent, with an EoS variation of 10% at most.

Figure 3

(Top) Profiles for energy density $\rho$ using the universal fit for $\alpha$ (top left), interior mass $m$ (top right), $\nu$ (related to the gravitational potential) (bottom left), and pressure $p$ (bottom right) for the original Tolman VII solution in terms of $(R,{\rho }_c)$ or $(R,M)$ and the improved Tolman VII solution. The values of ${\rho }_c$, $M$, and $R$ are the same as those in Fig. 1. We also present the numerical result with the AP4 EoS and $M=1.4M_{\odot }$. (Bottom) Fractional errors from the profile obtained numerically. Observe that the new model works better than the original Tolman solution especially for the $\rho$ and $m$ profiles.

Figure 4

The relative RMSE [defined in Eq. (33)] of $\rho$ (top left), $m$ (top right), $\nu$ (bottom left), and $p$ (bottom right) for the original Tolman solution with the $(R,M)$ parametrization (top panel), the improved model with the universal $\alpha$ (middle panel), and the improved model with the EoS-specific $\alpha$ (bottom panel) as a function of the NS mass, using 11 EoS with different stiffness in different colors as in Fig. 2. Observe how the improved models more accurately describe the realistic profiles (by having smaller relative RMSEs), especially for soft EoS.

Figure 5

The normalized relative RMSE difference between the original Tolman solution and the improved model with the universal $\alpha$. We show the results for $\rho$ (top left), $m$ (top right), $\nu$ (bottom left), and $p$ (bottom right) using the 11 EoS with different stiffness. The new model has an improvement over the original one if the normalized relative RMSE difference is negative (orange shaded region). Observe how the improved model more accurately describes the realistic profiles than the original Tolman solution in most cases.

Figure 6

Similar to Fig. 5 but with the EoS-specific fit for $\alpha$. Observe how the new model is further improved from Fig. 5 by using such an EoS-specific fit over the universal one.