Polytropic fits of modern and unified equations of state

Background: Equations of state for a cold neutron star's interior are presented in three-column tables that relate the baryonic density, the energy density, and the pressure. A few analytical expressions for those tables have been established these past two decades as a convenient way to present a large number of nuclear models for neutron-star matter. Some of those analytical representations are based on nonunified equations of state, in the sense that the high- and the low-density part of the star are not computed with the same nuclear model.

Purpose: Fits of equations of state based on a piecewise polytropic representation are revised by using unified tables of equations of state, that is to say, models which have been calculated consistently for the core and the crust.

Methods: A set of 52 unified equations of state is chosen. Each one is divided in seven polytropes via an adaptive segmentation, and two parameters per polytrope are fit to the tabulated equation of state. The total mass, radius, tidal deformability, and moment of inertia of neutron stars are modeled from the fits and compared with the quantities calculated from the original tables to ensure the accuracy of the fits on macroscopic parameters.

Results: We provide the polytropes parameters for 15 nucleonic relativistic mean-field models, seven hyperonic relativistic mean-field models, five hybrid relativistic mean-field models, 24 nucleonic Skyrme models, and one ab initio model.

Conclusions: The fit error on the macroscopic parameters of neutron stars is small and well within the estimated measurement accuracy from current and next-generation telescopes.

Figure 1

Mass-radius sequence for unified Skyrme EoSs and the ab initio model BCPM. Mass measurement of $\mathrm{J}0740+6620$ and $\mathrm{J}1614-2230$ within $1\sigma$ precision are presented, as well as NICER mass-radius contours of $\mathrm{J}0740+6620$ and $\mathrm{J}0030+0451$ within $1\sigma$ (see text for details), and mass-radius constraint on GW170817. Exclusion regions for finite pressure and subluminal EoSs are represented respectively in yellow and green.

Figure 2

Mass-radius (MR) sequence for unified nucleonic and hyperonic relativistic mean field EoSs used in this paper. Mass measurements of $\mathrm{J}0740+6620$ and $\mathrm{J}1614-2230$ within $1\sigma$ precision are presented, as well as NICER mass-radius contours of $\mathrm{J}0740+6620$ and $\mathrm{J}0030+0451$ within $1\sigma$ (see text for details) and the mass-radius constraint on GW170817. Exclusion regions for finite pressure and subluminal EoSs are represented respectively in yellow and green.

Figure 3

MR sequence for the unified hybrid relativistic mean field EoSs used in this paper. Mass measurements of $\mathrm{J}0740+6620$ and $\mathrm{J}1614-2230$ with $1\sigma$ precision are presented as well as NICER mass-radius contours of $\mathrm{J}0740+6620$ and $\mathrm{J}0030+0451$ within $1\sigma$ (for details see text) and the mass-radius constraint on GW170817. NL3 models are in blue and DD2 models are in green. The pressure bag constant $B$ is either set to zero (plain lines) or takes a nonzero value (dotted lines). The exclusion region for subluminal EoSs is represented in green.

Figure 4

Symmetry energy and its slope at saturation density respectively denoted $J$ and $L$ for Skyrme models used in this paper. Experimental data constraints are presented: in blue is the compiled constraint presented in Ref. [42], and in red that of PREX-II. The names of the EoSs in green refer to nucleonic models that do not permit the direct Urca process.

Figure 5

Symmetry energy and its slope at saturation density respectively denoted $J$ and $L$ for the relativistic mean-field models used in this paper. Experimental data constraints are presented: in blue is the compiled constraint presented in Ref. [42], and in red is that of PREX-II. The names of the EoSs in green refer to nucleonic models that do not permit the direct Urca process.

Figure 6

Equation of state (relation between the pressure $P$ and the density $\rho$) for unified EoS H3 compared with Read's piecewise polytropic fit of H3 (PPFRead). Transition between polytropes of PPFRead are presented with blue points.

Figure 7

Radius $R$, tidal deformability $\mathrm{\Lambda }$, and moment of inertia $I$ as a function of the mass $M$ for unified tables H3, H4, and DH, as well as Read's piecewise polytropic fits (PPFRead). In the right panel, the relative difference at given mass $M$ between unified tables and PPFRead for each macroscopic parameters in the three cases of EoS is presented. For EoSs H3 and H4, the relative uncertainty is shown up to 98% of the maximum mass in plain lines, and the last two percent in dotted lines, see Sec. 4a for details.

Figure 8

Adiabatic index $\mathrm{\Gamma }$ as a function of the baryonic density for the nucleonic, hyperonic, and hybrid EoS DD2.

Figure 9

Relative difference at given mass $M$ in percent between unified tables and our fits for the radius $\mathrm{\Delta }R/R$, the tidal deformability $\mathrm{\Delta }\mathrm{\Lambda }/\mathrm{\Lambda }$, and the moment of inertia $\mathrm{\Delta }I/I$ as a function of the mass $M$ for EoS H3 and EoS H4. The relative difference between PPFRead and unified EoSs H3 and H4 is presented in gray. For our fits, the relative difference is presented up to 98% of the maximum mass in plain lines, and the last two percent in dotted lines, see text for details.

Figure 10

Relative difference $\mathrm{\Delta }C$ as a function of the tidal deformability $\mathrm{\Lambda }$ for EoS NL3 on the left and EoS DD2 on the right.

Figure 11

Relative uncertainty $\mathrm{\Delta }C$ as a function of the tidal deformability $\mathrm{\Lambda }$ for EOS H3 on the left and EOS H4 on the right.