Rapidly rotating axisymmetric neutron stars with quark cores

We present a systematic study of the properties of pure hadronic and hybrid compact stars. The nuclear equation of state (EoS) for beta-equilibrated neutron star matter was obtained using density dependent effective nucleon-nucleon interaction which satisfies the constraints from the observed flow data from heavy-ion collisions. The energy density of quark matter is lower than that of this nuclear EoS at higher densities implying the possibility of transition to quark matter inside the core. We solve the Einstein’s equations for rotating stars using pure nuclear matter and quark core. The beta- equilibrated neutron star matter with a thin crust is able to describe highly massive compact stars but find that the nuclear to quark matter deconfinement transition inside neutron stars causes reduction in their masses. Recent observations of the binary millisecond pulsar J1614-2230 by P. B. Demorest et al. [1] suggest that the masses lie within 1.97\pm0.04 M\odot where M\odot is the solar mass. In conformity with recent observations, pure nucleonic EoS determines that the maximum mass of NS rotating with frequency below r-mode instability is ~1.95 M\odot with radius ~10 kilometers. Although compact stars with quark cores rotating with Kepler’s frequency have masses up to ~2 M\odot, but if the maximum frequency is limited by the r-mode instability, the maximum mass ~1.7 M\odot turns out to be lower than the observed mass of 1.97\pm0.04 M\odot, by far the highest yet measured with such certainty, implying exclusion of quark cores for such massive pulsars.

💡 Research Summary

The paper presents a comprehensive investigation of the structure and observable properties of rapidly rotating, axisymmetric neutron stars (NSs) both with purely nuclear matter cores and with hybrid cores that contain a quark matter phase. The authors first construct a nuclear equation of state (EoS) for β‑equilibrated neutron‑star matter using a density‑dependent effective nucleon–nucleon interaction (DDM3Y). This interaction is calibrated to reproduce the saturation energy and density of symmetric nuclear matter and is constrained by flow data from heavy‑ion collisions, ensuring that both “soft” and “stiff” variants of the nuclear EoS lie within the experimentally allowed pressure region. The nuclear EoS provides pressure as a function of density, the symmetry energy, and the proton fraction required for charge neutrality and β‑equilibrium.

To model the possible appearance of deconfined quark matter at high density, the authors adopt a perturbative QCD–inspired version of the MIT bag model, including two mass‑less quark flavors and one massive flavor, together with a running coupling constant. The bag constant B is treated as a free parameter; the authors select B^1/4 = 110 MeV because it yields a crossing of the nuclear and quark energy‑density curves at ρ≈0.405 fm⁻³, indicating the onset of a quark core. Smaller B values would push the crossing to much higher densities (≈1.2 fm⁻³), resulting in an insignificant quark core, while larger B would suppress the quark phase altogether.

The stellar structure equations are solved using the publicly available “rns” code, which integrates the Einstein field equations for a stationary, axisymmetric, perfect‑fluid configuration. The metric employed contains four potentials (γ, ρ, α, ω) that depend only on the radial and polar coordinates. The crust is modeled with standard low‑density EoSs (FMT, BPS, BBP) up to a baryon density of 0.0458 fm⁻³, after which the β‑equilibrated nuclear or hybrid EoS is applied. This construction yields a continuous pressure‑density relation across the whole star.

Key results for the pure‑nuclear‑matter stars are:

• Static maximum mass M_max ≈ 1.92 M⊙ with radius ≈ 9.7 km.
• At the Keplerian (mass‑shedding) limit, M_max ≈ 2.27 M⊙ with equatorial radius ≈ 13.1 km.
• When the rotation frequency is limited by the r‑mode instability (period ≈ 1.5–2.0 ms), the maximum mass drops to ≈ 1.94–1.95 M⊙ and the radius stays near 9.8 km.

These numbers are in excellent agreement with the precisely measured mass of the binary millisecond pulsar J1614‑2230 (1.97 ± 0.04 M⊙), indicating that a stiff nuclear EoS alone can account for the most massive observed pulsars, provided the star does not exceed the r‑mode stability limit.

For hybrid stars containing a quark core, the picture changes dramatically:

• Static maximum mass falls to ≈ 1.68 M⊙ with radius ≈ 10.4 km.
• At the Keplerian limit, the mass can reach ≈ 2.02 M⊙ with an equatorial radius ≈ 14.3 km.
• However, imposing the r‑mode frequency ceiling reduces the maximum mass to ≈ 1.70 M⊙, well below the 1.97 M⊙ benchmark.

The reduction originates from the lower pressure of the quark phase at a given density, which softens the overall EoS and limits the star’s ability to support large masses. The authors also note that the coexistence (mixed) region between nuclear and quark matter is extremely narrow, essentially a constant‑pressure Maxwell construction, implying a sharp transition with negligible mixed‑phase volume.

The paper further discusses recent claims of even heavier neutron stars (e.g., PSR J1748‑2021B with a reported mass of 2.74 M⊙). The present DDM3Y nuclear EoS and the chosen quark EoS cannot accommodate such extreme masses, suggesting that either the observational inference is uncertain or that additional physics—such as strongly interacting quark matter, color‑superconductivity, or exotic degrees of freedom—must be invoked.

In conclusion, the study demonstrates that a density‑dependent nuclear interaction calibrated to heavy‑ion data yields an EoS capable of reproducing the observed ∼2 M⊙ pulsars when rotational stability limits are respected. Conversely, the inclusion of a conventional MIT‑bag‑type quark core leads to a substantial mass reduction, effectively excluding quark cores in the most massive, precisely measured pulsars. The authors argue that future work must explore more sophisticated quark‑matter models or alternative stiffening mechanisms if hybrid stars are to remain viable candidates for the heaviest known neutron stars.