Effective No-Hair Relations for Neutron Stars and Quark Stars: Relativistic Results

TL;DR

The paper tests EOS-insensitive no-hair–like relations for NSs and QSs by computing multipole moments up to hexadecapole order using slow-rotation expansions and two independent rapid-rotation codes. It demonstrates approximate universality in several moment relations (I–M2, S3–M2, M4–M2, M4/M2–S3) with typical variations of 10–20%, and shows the Newtonian limit closely tracks the GR results as compactness decreases. Quartic-in-spin terms are quantified, revealing potentially significant corrections for very fast rotators and implications for NICER/LOFT–style X-ray measurements, especially in eccentricity and quadrupole contributions. The findings provide a practical framework to reduce parameter degeneracies in neutron-star observations and offer insights into constructing analytic exterior spacetimes that respect these universal relations, while acknowledging limitations for QSs and nonuniformly rotating or magnetized stars.

Abstract

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. We 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 $\sim 10%$, 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 expansion. The quartic in spin, slowly-rotating solutions found here allow us to estimate the systematic errors in the measurement of the neutron star's mass and radius with future X-ray observations, such as NICER and 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.

Figures (14)

• Figure 1: (Color online) (Top) The (reduced dimensionless) hexadecapole $(\bar{M}_4)$--quadrupole $(\bar{M}_2)$ moments relation with various NS and QS EoSs and spins, together with the fit given by Eq. \ref{['eq:fit']} and the Newtonian relation found in Stein:2013ofa. Observe the relation approaches the Newtonian one as one increases $\bar{M}_2$. The Newtonian relation for an $n=0.5$ polytrope agrees with the relativistic fit for various realistic EoSs within 10% accuracy above the critical $\bar{M}_2$ 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: (Color online) (Left) $\bar{I}$--$C$ (top left), $\bar{M}_2$--$C$ (top right), $\bar{S}_3$--$C$ (bottom left) and $\bar{M}_4$--$C$ (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 $\bar{I}$--$C$ 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 $\bar{M}_4$ 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) $\bar{M}_4$ depends on $C$ for any given $\chi$ and (ii) $\bar{M}_4$ computed with the LORENE and RNS codes contains spin corrections higher than $\mathcal{O}(\chi^2)$.
• Figure 3: (Color online) (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: (Color online) (Top) $\bar{I}$--$\bar{M}_2$ 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: (Color online) (Top) $\bar{I}$--$\bar{M}_2$ 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 ${\cal{O}}(\chi^{2})$ corrections) of I-Love-Q-ScienceI-Love-Q-PRD and the fit to RNS data valid for all spins of Pappas:2013naa. (Bottom) Fractional difference between the data and the fits in Pappas:2013naa.