Effect of symmetry energy on properties of rapidly rotating neutron stars and universal relations

TL;DR

Problem: how do the nuclear matter parameters $L_{\rm sym}$ and $Q_{\rm sat}$ shape rapidly rotating neutron stars at the Kepler limit, and do $I$-Love-$Q$ universal relations survive in this regime? Approach: the LS23 covariant density functionals provide a controlled EoS family spanning $L_{\rm sym}$ and $Q_{\rm sat}$; the authors compute static and Kepler-limit stellar properties with the public RNS code and perform fits for the $I^>(C)$, $Q(C)$, and $I^>-Q$ relations across the EoS family. Findings: Kepler-limit stars exhibit systematic but bounded dependence of $M_{\max}$ and radii on $L_{\rm sym}$ and $Q_{\rm sat}$, and the universal relations among $I^>$, $Q$, and $C$ persist with deviations typically from a few percent to about $10\%$; the slow-rotation limit yields even tighter ($\sim$1%) accuracy for the $I$-$Q$ universality. Significance: supports using $I$–Love–$Q$ universal relations to model maximally rotating neutron stars and their gravitational-wave emission in merger remnants, even when $L_{\rm sym}$ and $Q_{\rm sat}$ vary within current constraints.

Abstract

We investigated universal relations for compact stars rotating at the Keplerian (mass-shedding) limit, which is highly relevant for understanding the rapidly rotating objects formed in the aftermath of a neutron star-neutron star merger. Our analysis is based on a set of nucleonic equations of state (EoSs) featuring systematic variations in the symmetry energy slope parameter $L_{\rm sym}$ and the isoscalar skewness parameter $Q_{\rm sat}$, varied within ranges that are broadly consistent with current laboratory and astrophysical constraints. The global observable properties of isolated maximally rotating stars are examined, focusing on the mass-radius relation, moment of inertia, quadrupole moment, and the Keplerian (maximum) rotation frequency, as well as their variations in the $L_{\rm sym}$-$Q_{\rm sat}$ parameter space. Next, we demonstrate that, in the limit of Keplerian rotation, universal relations remain valid across the same set of EoSs characterized by varying $L_{\rm sym}$ and $\Qsat$. In particular, we present explicit results for the moment of inertia ($I$) and quadrupole moment ($Q$) as functions of compactness, as well as for the moment of inertia-quadrupole moment relation. All of these relations exhibit excellent universality, with deviations typically within a range from a few percent to 10\% across a wide range of parameters. Additionally, we verify for our set of EoSs that the universality of $I$-$Q$ holds to higher accuracy (at the level of 1\%) in the slow-rotation approximation compared with the Kepler limit, where the relative error increases up to $\lesssim 10\%$. Our findings support the applicability of $I$-Love-$Q$-type universal relations in observational modeling of maximally rotating compact stars and the gravitational wave emitted by them.

Figures (9)

• Figure 1: Left panel: mass-radius relations for nucleonic EoS models of nonrotating CSs with different pairs of values of $Q_{\text{sat }}$ and $L_{\text{sym }} =30$ (blue), 40 (orange), 50 (green), 60 (red), 70 (violet), 80 (brown), 90 (magenta), 100 (grey), and 110 (cyan) [MeV]. The color regions show the 90% confidence interval (CI) ellipses from each of the two NICER modeling groups for PSR J0030+0451 and J0740+6620 NICER:2021aNICER:2021b, the 90% CI regions for each of the two CSs that merged in the gravitational-wave event GW170817 Abbott_2019, and finally the 90% CI for the mass of the secondary component of GW190814 Abbott_2020. Right panel: same as in the left panel, but for CSs rotating at Keplerian frequency. The stellar radius corresponds to the equatorial radius.
• Figure 2: Left panel: dependence of the moment of inertia of Keplerian CSs on mass for varying values of $Q_{\text{sat }}$ and $L_{\text{sym }}$ (in units of MeV) as indicated in the plot. Right panel: dependence of the (dimensional) quadrupole moment of Keplerian CSs on mass for varying values of $Q_{\text{sat }}$ and $L_{\text{sym }}$ as indicated in the plot. The color convention is the same as in Fig. \ref{['fig:MR']}.
• Figure 3: Left panel: dependence of the Keplerian frequency on the mass of configuration for a pair of values of $L_{\text{sym }} = 30$ (violet) and 90 (orange) and $Q_{\text{sat }}$ (in units of MeV) as indicated in the plot. Right panel: same as in the left panel, but for the full set of EoSs. The color convention is the same as in Fig. \ref{['fig:MR']}.
• Figure 4: Left panel: dependence of Keplerian frequency $\Omega_{\mathrm{Kep}}$ on kinetic-to-gravitational energy ratio $T/W$ for a pair of values of $L_{\text{sym }} = 30$ (violet) and 90 (orange) and $Q_{\text{sat }}$ (in units of MeV) as indicated in the plot. Right panel: same as in the left panel, but for the full set of EoSs. The color convention is the same as in Fig. \ref{['fig:MR']}.
• Figure 5: Log-log plot of the dimensionless moment of inertia $I^{>}$ versus compactness $C$ for maximally fast rotating (Keplerian) configurations. The compactness is defined via the equatorial radius. It is limited from above by the maximum mass configuration (the curves are extended slightly in the unstable region where masses are below the maximum). The compactness is limited from below by the configuration with mass 0.1 $M_{\odot}$ for $L=30$ MeV and 0.5 $M_{\odot}$ for $L=110$ MeV. The bottom panel shows the relative residual errors $\delta I^{>}$, where $\delta I^{>}= \vert ({I}^{>}-{I}^>_{\rm fit})\vert/{I}_{\rm fit}^{>}$ with respect to the fit by Eq. \ref{['eq:bar_I']}. In each subpanel, the upper panel shows the data points by varying the parameter $L_{\text{sym }} = 30$ (red $\square$), 40 (green $\lozenge$), 50 (blue $\triangle$), 60 (violet $+$), 70 (orange $\times$), 80 (magneta $\ast$), 90 (brown $\square$), 100 (rose $\lozenge$) and 110 (grey $\triangle$) in MeV for fixed $Q_{\text{sat }}$ and the best-fit curve based on a third-order polynomial in $C$. The lower panel displays the relative deviation between the numerical data and the fit.