Numerical models for stationary superfluid neutron stars in general relativity with realistic equations of state

We present a numerical model for uniformly rotating superfluid neutron stars in a fully general relativistic framework with, for the first time, realistic microphysics including entrainment. We compute stationary and axisymmetric configurations of neutron stars composed of two fluids, namely superfluid neutrons and charged particles (protons and electrons), rotating with different rates around a common axis. Both fluids are coupled by entrainment, a nondissipative interaction which in the case of a nonvanishing relative velocity between the fluids causes the fluid momenta to be not aligned with the respective fluid velocities. We extend the formalism put forth by Comer and Joynt in order to calculate the equation of state (EOS) and entrainment parameters for an arbitrary relative velocity as far as superfluidity is maintained. The resulting entrainment matrix fulfills all necessary sum rules, and in the limit of small relative velocity our results agree with Fermi liquid theory ones derived to lowest order in the velocity. This formalism is applied to two new nuclear equations of state which are implemented in the numerical model, which enables us to obtain precise equilibrium configurations. The resulting density profiles and moments of inertia are discussed employing both EOSs, showing the impact of entrainment and the dependence on the EOS.

Figure 1

Entrainment coefficients $Y_{XY}$ as functions of total baryon density $n_B$. Solid (dashed) lines refer to the $\mathrm{DDH}(\delta )$ EOS. Following the prescription given by Gusakov et al. [31], these coefficients are normalized to the constant $Y=3n_0/{\mu }^{\mathrm{n}}(3n_0)$, where $n_0=0.16{\mathrm{fm}}^{-3}$ stands for the saturation density. For $\mathrm{DDH}(\delta )$, $Y=2.55\times {10}^{41}{\mathrm{erg}}^{-1}{\mathrm{cm}}^{-3}$ ($Y=2.47\times {10}^{41}{\mathrm{erg}}^{-1}{\mathrm{cm}}^{-3}$).

Figure 2

Left: Influence of the interaction on the entrainment parameters ${ϵ}_X^0$. Solid (dashed) lines refer to $\mathrm{DDH}(\delta )$ EOS. For better clarity, only quantities defined in the zero-velocity frames are plotted. Right: Comparison between entrainment parameters defined in the zero-velocity frames (${ϵ}_X^0$, solid lines) and in the zero-momentum frames (${ϵ}_X^{\#}$, dashed lines), for the DDH EOS.

Figure 3

Density profiles $n_{\mathrm{n}}$ and $n_{\mathrm{p}}$ plotted with respect to the radial coordinate $r$ for a star with $M_G=1.4{\mathrm{M}}_{\odot }$ spinning at ${\mathrm{\Omega }}_{\mathrm{n}}/2\pi ={\mathrm{\Omega }}_{\mathrm{p}}/2\pi =716\mathrm{Hz}$, obtained with DDH and $\mathrm{DDH}\delta$ EOSs. Neutrons (protons) particle density $n_{\mathrm{n}}$ ($n_{\mathrm{p}}$) is plotted in red (green). Dashed and solid lines refer to profiles obtained in the polar and equatorial planes. Blue and orange vertical lines represent the coordinates of vanishing density of protons and neutrons. Some insets of the area surrounding the surfaces are also presented.

Figure 4

Moments of inertia plotted with respect to the angular velocity of the pulsar, taking both angular velocities to be equal and assuming $\beta$-equilibrium, for a (same) fixed total baryon mass $M^B=1.542{\mathrm{M}}_{\odot }$ corresponding to $M_G\simeq 1.4{\mathrm{M}}_{\odot }$. Results from the DDH ($\mathrm{DDH}\delta$) EOS are represented with solid (dashed) lines. Total quantities are shown in blue whereas neutron (proton) ones are plotted in red (green).

Figure 5

Neutron and proton angular momenta as functions of the neutron angular velocity ${\mathrm{\Omega }}_{\mathrm{n}}$, assuming the proton fluid to be at rest with respect to a static observer at spatial infinity (${\mathrm{\Omega }}_{\mathrm{p}}/2\pi =0\mathrm{Hz}$). Results from the DDH ($\mathrm{DDH}\delta$) EOS are shown with solid (dashed) lines. Quantities are plotted for a fixed total baryon mass (equal for the two EOSs), assuming $\beta$-equilibrium at the center.

Figure 6

3D interpolation scheme on a parallelepipedic grid (red crosses), in a point corresponding to a given value of $({\mathrm{\Delta }}^2,{\mu }^{\mathrm{n}},{\mu }^{\mathrm{p}})$ (green cross). On each plan where ${\mathrm{\Delta }}^2$ is constant, quantities are interpolated with a 2D thermodynamically consistent method on the chemical potentials (blue crosses). From these two values, a linear interpolation is used in ${\mathrm{\Delta }}^2$ in order to obtain the values of the quantities needed at the interesting point.

Figure 7

Single-fluid (red crosses and green triangles) and two-fluid (blue dots) areas in the plane $({\mu }^{\mathrm{n}},{\mu }^{\mathrm{p}})$, for ${\mathrm{\Delta }}^2=0$, using the $\mathrm{DDH}\delta$ EOS. The two-fluid zone is delimited by the limit lines $n_{\mathrm{n}}=0$ (orange) and $n_{\mathrm{p}}=0$ (purple), beyond which one fluid disappears. For chemical potentials below the rest masses $m_{\mathrm{n}}=939.6\mathrm{MeV}$ and $m_{\mathrm{p}}=938.8\mathrm{MeV}$ no fluid is present, as can be seen on the inset shown on the right. Note that $m_{\mathrm{p}}$ denotes here the sum of the rest mass of protons and that of electrons since the subscript “p” stands here for the fluid of charged particles. For better clarity, only a small proportion of the data contained in the table is plotted. In order to describe the area at low densities, where rapid variations occur, one uses a refined mesh. As realistic configurations are expected to be close to $\beta$-equilibrium and corotation, only the data around the ${\mu }^{\mathrm{n}}={\mu }^{\mathrm{p}}$ line will be used.