Relativistic numerical models for stationary superfluid neutron stars

We have developed a theoretical model and a numerical code for stationary rotating superfluid neutron stars in full general relativity. The underlying two-fluid model is based on Carter’s covariant multifluid hydrodynamic formalism. The two fluids, representing the superfluid neutrons on one hand, and the protons and electrons on the other, are restricted to uniform rotation around a common axis, but are allowed to have different rotation rates. We have performed extensive tests of the numerical code, including quantitative comparisons to previous approximative results for these models. The results presented here are the first “exact” calculations of such models in the sense that no approximations (other than that inherent in a discretized numerical treatment) are used. Using this code we reconfirm the existence of prolate-oblate shaped configurations. We studied the dependency of the Kepler rotation limit and of the mass-density relation on the relative rotation rate. We further demonstrate how one can simulate a (albeit fluid) neutron-star crust by letting one fluid extend further outwards than the other, which results in interesting cases where the Kepler limit is actually determined by the outermost but slower fluid.

Figure 1

Relative differences $\Delta Q$ in the total baryon mass $M$, and the equatorial and polar radii $R_{\mathrm{e}\mathrm{q}}$ and $R_{\mathrm{p}\mathrm{o}\mathrm{l}}$. The first row is the Newtonian case, while the second row shows the relativistic results. In the first column, we used a fixed spherical inner domain, while in the second column the inner domain-grid is adapted to the stellar surface (which is the default in the single-fluid case).

Figure 2

Virial violations GRV2 and GRV3 for (a) 1 domain and (b) 2 domains covering the star. Only the Newtonian case is shown, as the relativistic results are very similar. ${\Omega }_{\mathrm{K}}$ denotes the Kepler rotation rate.

Figure 3

Relative difference $\diamond Q({\Omega }_{\mathrm{n}})$ between the numerical code in “Newtonian mode” and the slow-rotation analytic solution of Appendix , for the equatorial radii $R_{\mathrm{n}}$ and $R_{\mathrm{p}}$. In (a) we used the normal physical EOS-inversion, while (b) shows the results when using a “slow-rotation style” EOS-inversion.

Figure 4

Meridional cross section of an oblate-prolate two-fluid configuration. The dotted lines represent lines of constant “gravitational potential” $N$, while the thick lines are the respective surfaces of the neutron- and proton fluids.

Figure 5

Kepler limit ${\Omega }_{\mathrm{K}}$ as a function of relative rotation rate $\mathcal{R}$ for EOS-models I, II and III. The dashed line (NSR) represents the result from the analytic Newtonian slow-rotation solution (cf. Appendix )

Figure 6

Kepler-configurations for EOS I. In figure (a) the relative rotation rate is $\mathcal{R}=0.1$, while in (b) it is $\mathcal{R}=0.01$.

Figure 7

Gravitational mass $\mathcal{M}$ as a function of central baryon number density $n_c$ for EOS I with fixed central proton fraction of $x_{\mathrm{p}}=0.05$. The four curves correspond to the nonrotating case, the corotating Kepler configuration ($\mathcal{R}=0$), and two Kepler-configurations with relative rotation rates of $\mathcal{R}=0.5$ and $\mathcal{R}=-0.5$ respectively. The circle indicates a static configuration with central chemical potentials of ${\mu }_{\mathrm{n}}={\mu }_{\mathrm{p}}=0.3m_b{c}^2$. The box indicates the maximum-mass configuration in the static case. The dotted line represents the sequence of constant baryon mass connecting to the static maximum-mass configuration. Configurations above this line have no stable nonrotating counterpart and are called supramassive stars.

Figure 8

Configuration with protons rotating at the speed of the fastest known millisecond pulsar, ${\Omega }_{\mathrm{p}}/2\pi =641\mathrm{H}\mathrm{z}$, and ${\Omega }_{\mathrm{n}}/2\pi =645\mathrm{H}\mathrm{z}$. The protons are extending further outside than the neutrons. The physical parameters are ${\mu }_{\mathrm{n}}=0.228m_b{c}^2$ and ${\mu }_{\mathrm{p}}=0.220m_b{c}^2$, resulting in central baryon number density $n_c=0.561{\mathrm{f}\mathrm{m}}^{-3}$, proton fraction $x_{\mathrm{p}}=0.09$ and a gravitational mass of $\mathcal{M}=1.39M_{\odot }$.

Figure 9

Kepler-configuration with $\mathcal{R}=0.01$, ${\mu }_{\mathrm{n}}=0.28m_b{c}^2$ and ${\mu }_{\mathrm{p}}=0.3m_b{c}^2$, corresponding to a central baryon number density of $n_c=0.716{\mathrm{f}\mathrm{m}}^{-3}$, proton fraction $x_{\mathrm{p}}=0.09$ and a gravitational mass of $\mathcal{M}=1.57M_{\odot }$. The protons extend to the outer surface. The maximal rotation rates are found as ${\Omega }_{\mathrm{n}}/2\pi =924.5\mathrm{H}\mathrm{z}$ and ${\Omega }_{\mathrm{p}}/2\pi =915.3\mathrm{H}\mathrm{z}$.