Spectral approach to the relativistic inverse stellar structure problem

A new method for solving the relativistic inverse stellar structure problem is presented. This method determines a spectral representation of the unknown high-density portion of the stellar equation of state from a knowledge of the total masses $M$ and radii $R$ of the stars. Spectral representations of the equation of state are very efficient, generally requiring only a few spectral parameters to achieve good accuracy. This new method is able, therefore, to determine the high-density equation of state quite accurately from only a few accurately measured $[M,R]$ data points. This method is tested here by determining the equations of state from mock $[M,R]$ data computed from tabulated realistic neutron-star equations of state. The spectral equations of state obtained from these mock data are shown to agree, on average, with the originals to within a few percent (over the entire high-density range of the neutron-star interior) using only two $[M,R]$ data points. Higher accuracies are achieved when more data are used. The accuracies of the equations of state determined in these examples are shown to be nearly optimal, in the sense that their errors are comparable to the errors of the best-fit spectral representations of these realistic equations of state.

Figure 1

Exact mass-radius curves for the three realistic neutron-star equations of state, PAL6, APR3, and BGN1H1. Large dots illustrate the data points for the spectral method tests that use $N_{\mathrm{stars}}=5$ stellar models.

Figure 2

Ratios between various approximate equations of state, $ϵ(h,{\gamma }_k)$, obtained by fitting $[M,R]$ data, and the exact PAL6 equation of state, $ϵ(h)$. Note that $\log [ϵ(h,{\gamma }_k)/ϵ(h)]\approx [ϵ(h,{\gamma }_k)-ϵ(h)]/ϵ(h)$ measures the fractional error.

Figure 3

Ratios between various approximate equations of state, $ϵ(h,{\gamma }_k)$, obtained by fitting $[M,R]$ data, and the exact APR3 equation of state, $ϵ(h)$.

Figure 4

Ratios between various approximate equations of state, $ϵ(h,{\gamma }_k)$, obtained by fitting $[M,R]$ data, and the exact BGN1N1 equation of state, $ϵ(h)$.

Figure 5

Ratios between the masses, $M(h_c,{\gamma }_k)$ computed from the various fits, and $M(h_c)$ computed from the exact PAL6 equation of state, for a range of values of the central enthalpy $h_c$ of those models.

Figure 6

Ratios between the masses, $M(h_c,{\gamma }_k)$ computed from the various fits, and $M(h_c)$ computed from the exact APR3 equation of state for a range of values of the central enthalpy $h_c$ of those models.

Figure 7

Ratios between the masses, $M(h_c,{\gamma }_k)$ computed from the various fits, and $M(h_c)$ computed from the exact BGN1H1 equation of state for a range of values of the central enthalpy $h_c$ of those models.