Effective no-hair relations for neutron stars and quark stars: Relativistic results

Astrophysical charge-free black holes are known to satisfy no-hair relations through which all multipole moments can be specified in terms of just their mass and spin angular momentum. We here investigate the possible existence of no-hair-like relations among multipole moments for neutron stars and quark stars that are independent of their equation of state. We calculate the multipole moments of these stars up to hexadecapole order by constructing uniformly rotating and unmagnetized stellar solutions to the Einstein equations. For slowly rotating stars, we construct stellar solutions to quartic order in spin in a slow-rotation expansion, while for rapidly rotating stars, we solve the Einstein equations numerically with the LORENE and RNS codes. We find that the multipole moments extracted from these numerical solutions are consistent with each other and agree with the quartic-order slow-rotation approximation for spin frequencies below roughly 500 Hz. We also confirm that the current dipole is related to the mass quadrupole in an approximately equation-of-state-independent fashion, which does not break for rapidly rotating neutron stars or quark stars. We further find that the current-octupole and the mass-hexadecapole moments are related to the mass quadrupole in an approximately equation-of-state-independent way to roughly $\mathcal{O}(10\%)$, worsening in the hexadecapole case. All of our findings are in good agreement with previous work that considered stellar solutions to leading order in a weak-field, Newtonian expansion. In fact, the hexadecapole-quadrupole relation agrees with the Newtonian one quite well even in moderately relativistic regimes. The quartic in spin, slowly rotating solutions found here allows us to estimate the systematic errors in the measurement of the neutron star’s mass and radius with future x-ray observations, such as Neutron star Interior Composition ExploreR (NICER) and Large Observatory for X-ray Timing (LOFT). We find that the effect of these quartic-in-spin terms on the quadrupole and hexadecapole moments and stellar eccentricity may dominate the error budget for very rapidly rotating neutron stars. The new universal relations found here should help to reduce such systematic errors.

Figure 1

(Top) The (reduced dimensionless) hexadecapole $({\overline{M}}_4)$-quadrupole $({\overline{M}}_2)$ moments relation with various NS and QS EoSs and spins, together with the fit given by Eq. (90) and the Newtonian relation found in Ref. [26]. Observe the relation approaches the Newtonian one as one increases ${\overline{M}}_2$. The Newtonian relation for an $n=0.5$ polytrope agrees with the relativistic fit for various realistic EoSs within 10% accuracy above the critical ${\overline{M}}_2$ denoted by the dotted-dashed, vertical line. (Bottom) Fractional difference between the data and the fit. Observe the relation is universal to roughly 20%. This means that the hexadecapole moment can be approximately expressed in terms of just the stellar mass, spin, and quadrupole moment.

Figure 2

(Left) $\overline{I}$–$C$ (top left), ${\overline{M}}_2$–$C$ (top right), ${\overline{S}}_3$–$C$ (bottom left) and ${\overline{M}}_4$–$C$ (bottom right) relations for an $n=0.5$ polytropic EoS, where $C$ is the stellar compactness. The black circles are calculated to leading-order in slow-rotation, while the green plusses and red crosses are computed with the LORENE and RNS codes respectively for a sequence of spins (increasing from top to bottom). Observe that the $\overline{I}$–$C$ relation is almost insensitive to spin, while the other three relations depend clearly on spins. (Right) Fractional difference, as a function of dimensionless spin, of ${\overline{M}}_4$ computed with the LORENE or RNS codes and in the slow-rotation approximation. Observe that the fractional difference scales as ${\chi }^2$ as expected. The scattering is because (i) ${\overline{M}}_4$ depends on $C$ for any given $\chi$ and (ii) ${\overline{M}}_4$ computed with the LORENE and RNS codes contains spin corrections higher than $\mathcal{O}({\chi }^2)$.

Figure 3

(Top) Critical spin parameter ${\chi }_{\mathrm{cr}}$ against stellar compactness, below which the difference between the $\mathcal{O}({\chi }^0)$ and $\mathcal{O}({\chi }^2)$ parts of the stellar mass (red cross), moment of inertia (green plus), and quadrupole moment (blue circle) with an APR EoS in the slow-rotation approximation is less than 1%. The black dashed line roughly corresponds to ${\chi }_{\mathrm{cr}}$, below which the difference between the $\mathcal{O}({\chi }^0)$ and $\mathcal{O}({\chi }^4)$ parts is less than 1%, assuming that the latter is expressed as the leading-order term times ${\chi }^4$. (Bottom) Same as the top panel, except for the critical spin frequency $f_{\mathrm{cr}}$.

Figure 4

(Top) $\overline{I}$–${\overline{M}}_2$ relation for an APR EoS in the slow-rotation limit, including quadratic-order spin corrections and full-order spin corrections using LORENE for various spin frequencies. (Bottom) Spin dependence of the fractional difference. Observe the scaling of ${\chi }^4$ as expected.

Figure 5

(Top) $\overline{I}$-${\overline{M}}_2$ relations computed in the slow-rotation approximation [including $\mathcal{O}({\chi }^2)$ corrections] for various realistic NS and QS EoSs and with $\chi =0.3$ and 0.5. Observe that the QS relation is almost the same as the NS one. We also show the fit to the slow-rotation results [without $\mathcal{O}({\chi }^2)$ corrections] of Refs. [19, 20] and the fit to RNS data valid for all spins of Ref. [24]. (Bottom) Fractional difference between the data and the fits in Ref. [24].

Figure 6

(Top) $\overline{I}$–${\overline{M}}_2$ relation as a function of $\chi$. The blue plane shows the relation for NSs, consistent with that found in Ref. [24], while red points show the relation for QSs. Observe that the QS points lie on the NS plane, showing that the QS relation is almost identical to the NS one. (Bottom) Same as the left panel but from a different viewpoint.

Figure 7

(Top) ${\overline{S}}_3$–${\overline{M}}_2$ relation with various realistic NS and QS EoSs and spins, together with the fit in Eq. (90) and the Newtonian relation for the $n=0.5$ polytropes in Ref. [26]. The meaning of the dotted-dashed vertical line is the same as in Fig. 1. Observe that the QS relation is almost the same as the NS one. (Bottom) Fractional difference between the data and the fit.

Figure 8

Fractional difference between the ${\overline{M}}_4$–${\overline{M}}_2$ relation for rapidly rotating stars with RNS and the one in the slow-rotation limit for an APR EoS with various spin parameters.

Figure 9

(Top) ${\overline{M}}_4/{\overline{S}}_3$–${\overline{M}}_2$ relation with various EoSs and spins, together with the fit in Eq. (90) and the EoS-independent Newtonian relation. The meaning of the dotted-dashed line is the same as in Fig. 1. (Bottom) Fractional difference from the fit.

Figure 10

(Top left) ${\overline{M}}_2$ against $M_0$ for various EoSs and spins, together with the fractional difference from the fit created for each EoS. Observe that the relation for a given EoS depends very weakly on the spin. (Top right) Same as the top panel, but for ${\overline{S}}_3$ against $M_0$. (Bottom) Same as the top panel, but for ${\overline{M}}_4$ against $M_0$.

Figure 11

Fractional difference between the numerical values and the Newtonian ones for the ${\overline{S}}_3$–${\overline{M}}_2$, ${\overline{M}}_4$–${\overline{M}}_2$, and ${\overline{M}}_4/{\overline{S}}_3$–${\overline{M}}_2$ relations with the $n=0.5$ polytropes.

Figure 12

(Top) Stellar eccentricity-compactness relation in the slow-rotation limit for various realistic EoSs and the fit for the NS relation given by Eq. (109). (Bottom) Fractional difference between the numerical values and the NS fit. Observe that the QS relation is different from the NS one by $\mathcal{O}(10\%)$.

Figure 13

(Top) Compactness dependence on the spin correction to the stellar eccentricity for various realistic EoSs, together with the fit given in Eq. (90) for the NS sequence. (Bottom) Fractional difference from the fit.

Figure 14

Estimated relative error in the calculation of $q_4$ for the SLy4 EoS, for a different number of omitted points and different rotation rates (various curves of the same color). The different colors correspond to the relative differences, $(q_4^{4\#}-q_4^{n\#})/q_4^{4\#}$, with $n\#$ points omitted. The gold lines correspond to the comparison of $q_4$ between four points and three points being omitted. The figure shows that at the high- compactness/low-${\overline{M}}_2$ end of the plot the truncation error of $q_4^{4\#}$ dominates, while at the low-compactness/high-${\overline{M}}_2$ end of the plot, the singularity error of $q_4^{3\#}$ dominates: