Ways to constrain neutron star equation of state models using relativistic disc lines

Relativistic spectral lines from the accretion disc of a neutron star low-mass X-ray binary can be modelled to infer the disc inner edge radius. A small value of this radius tentatively implies that the disc terminates either at the neutron star hard surface, or at the innermost stable circular orbit (ISCO). Therefore an inferred disc inner edge radius either provides the stellar radius, or can directly constrain stellar equation of state (EoS) models using the theoretically computed ISCO radius for the spacetime of a rapidly spinning neutron star. However, this procedure requires numerical computation of stellar and ISCO radii for various EoS models and neutron star configurations using an appropriate rapidly spinning stellar spacetime. We have fully general relativistically calculated about 16000 stable neutron star structures to explore and establish the above mentioned procedure, and to show that the Kerr spacetime is inadequate for this purpose. Our work systematically studies the methods to constrain EoS models using relativistic disc lines, and will motivate future X-ray astronomy instruments.

💡 Research Summary

The paper investigates how relativistically broadened iron Kα emission lines from the accretion disks of neutron‑star low‑mass X‑ray binaries (LMXBs) can be used to constrain neutron‑star equation‑of‑state (EoS) models. The basic premise is that the shape of the iron line, especially the extent of its red wing, encodes the radius of the inner edge of the disk (r_in) in units of GM/c². If r_in is small, the disk may terminate either at the neutron‑star surface (radius R) or at the innermost stable circular orbit (ISCO). Consequently, a measurement of r_in provides either a direct estimate of R or, if the disk ends at the ISCO, a way to compare the observed r_in with the theoretically computed ISCO radius for a given stellar spacetime, thereby testing the underlying EoS.

The authors point out that, unlike black holes whose exterior spacetime is described exactly by the Kerr metric (characterized only by mass and spin), rapidly rotating neutron stars have more complex spacetimes that depend on the EoS, the central density, and the rotation rate. Therefore, using Kerr‑based line models for neutron‑star systems can introduce systematic errors, especially when the stellar radius exceeds the ISCO radius. To address this, they perform fully general‑relativistic calculations of rotating neutron‑star structures using the Bardeen‑Cook‑Datta metric, solving Einstein’s equations for four representative EoS models:

• Model A – very stiff chiral‑sigma‑model EoS (M_max ≈ 2.59 M⊙)
• Model B – APR‑type stiff EoS with three‑body forces (M_max ≈ 2.20 M⊙)
• Model C – intermediate stiffness from Brueckner‑Bethe‑Goldstone theory (M_max ≈ 1.79 M⊙)
• Model D – very soft hyperon‑rich EoS (M_max ≈ 1.41 M⊙)

For each EoS they compute equilibrium sequences at 15 spin frequencies ranging from 0 to 750 Hz, generating roughly 16 000 stable neutron‑star models. For every model they calculate the equatorial radius R, the total angular momentum J, and the ISCO radius r_ISCO by locating the radius where the effective potential satisfies dV/dr = 0 and d²V/dr² = 0 for corotating circular orbits. They then compare the dimensionless spin parameter j = Jc/GM² with r_in c²/GM for both the stellar surface (R) and the ISCO.

Key findings include:

1. Deviation from Kerr: When R < r_ISCO the j–r_in curve lies close to the Kerr curve, but the deviation is non‑negligible and grows with higher spin, lower mass, and stiffer EoS. When R > r_ISCO (the disk is truncated by the surface) the deviation becomes large because the stellar equatorial radius expands with spin—a situation absent for black holes.

2. Mass scaling: Plotting j versus r_in for a fixed ratio M/M_max (instead of absolute M) reduces the spread among different EoS, indicating that the maximum mass of an EoS is a useful normalizing parameter.

3. Observational relevance: Existing measurements of r_in c²/GM in neutron‑star LMXBs typically fall in the range 6–15, with many best‑fit values pegged at the lower limit of 6. This suggests that for many sources r_in < 6 GM/c², i.e., the disk likely reaches the ISCO or the stellar surface, making the ISCO‑based method applicable.

4. Kerr inadequacy: Since many sources have r_in values near the Kerr lower bound (6 GM/c²), using Kerr models can give acceptable fits but will misinterpret whether the truncation is due to the ISCO or the surface, leading to biased EoS constraints.

5. Practical constraints: By fixing the spin frequency (which can be measured accurately from pulsations or burst oscillations) and treating R c²/GM and M as variables constrained by independent methods (thermonuclear burst spectroscopy, orbital dynamics), the authors produce a set of diagnostic plots (r_in versus ν_spin, R c²/GM, and M). These plots enable a direct translation of an observed r_in into allowed regions of the (M, R, ν_spin) parameter space for each EoS.

The paper concludes that a systematic, fully relativistic treatment of rotating neutron‑star spacetimes is essential for extracting reliable EoS constraints from relativistic disc lines. The extensive grid of models they have generated provides a ready‑to‑use reference for future high‑resolution X‑ray missions (e.g., XRISM, Athena) that will deliver higher‑quality iron‑line spectra. By combining precise spin measurements with independent mass and radius estimates, the method promises to narrow down the viable neutron‑star EoS and thereby probe the physics of ultra‑dense matter beyond the reach of terrestrial experiments.