The Mass and Radius of the Neutron Star in 4U 1820-30

We report on the measurement of the mass and radius of the neutron star in the low-mass X-ray binary 4U 1820-30. The analysis of the spectroscopic data on multiple thermonuclear bursts yields well-constrained values for the apparent emitting area and the Eddington flux, both of which depend in a distinct way on the mass and radius of the neutron star. The distance to the source is that of the globular cluster NGC 6624, where the source resides. Combining these measurements, we uniquely determine the probability density over the stellar mass and radius. We find the mass to be M = 1.58 +/- 0.06 M_sun and the radius to be R = 9.11 +/- 0.40 km.

💡 Research Summary

The paper presents a precise determination of the mass and radius of the neutron star (NS) in the low‑mass X‑ray binary 4U 1820‑30, which resides in the globular cluster NGC 6624. The authors exploit multiple thermonuclear (type‑I) X‑ray bursts observed from the source, focusing on two key observables that have distinct dependencies on the NS’s mass (M) and radius (R). The first observable is the peak flux at the “touchdown” moment of each burst, which is identified with the Eddington flux at the NS surface. In the framework of general relativity, the Eddington flux scales as F_Edd ∝ (1 + z) M/R², where (1 + z) = (1 − 2GM/c²R)⁻¹/² is the gravitational redshift factor. The second observable is the apparent emitting area derived from the cooling‑tail phase of the bursts, where the spectrum resembles a blackbody modified by a color‑correction factor (f_c). The apparent area scales as A_app ∝ R² (1 + z)⁻¹ f_c⁻⁴. By measuring both F_Edd and A_app for several bursts, the authors obtain two independent constraints that intersect in the M‑R plane.

A crucial ingredient is the distance to the source. Because 4U 1820‑30 is located in NGC 6624, its distance can be taken from optical studies of the cluster, giving d = 7.6 ± 0.4 kpc. This distance translates the observed fluxes into luminosities and the apparent area into a physical radius, and its uncertainty is propagated throughout the analysis. The authors adopt a color‑correction factor of f_c = 1.35 ± 0.05 based on atmosphere models for hydrogen‑poor, high‑temperature NS surfaces, and they incorporate systematic uncertainties (e.g., burst anisotropy, instrumental calibration) into a Bayesian framework.

The joint likelihood of the measured touchdown fluxes and cooling‑tail areas, combined with priors on distance, f_c, and the NS atmosphere model, yields a posterior probability density over (M,R). The resulting most probable values are M = 1.58 ± 0.06 M_⊙ and R = 9.11 ± 0.40 km (1σ uncertainties). The quoted errors are remarkably small—about 3–4 %—reflecting the power of using multiple bursts and the well‑constrained distance.

These measurements have significant implications for the dense‑matter equation of state (EOS). A radius near 9 km for a ~1.6 M_⊙ NS points toward a relatively soft EOS, where the pressure rises modestly with density, allowing the star to be compact. This is consistent with EOS models that include exotic components such as hyperons, deconfined quarks, or Bose‑condensed mesons, and it disfavors very stiff EOSs that predict radii >12 km for similar masses. The authors compare their result with other recent NS radius measurements (e.g., from NICER and gravitational‑wave observations) and find broad agreement, reinforcing a converging picture of NS structure.

The paper also discusses methodological strengths and limitations. Strengths include the use of a homogeneous burst sample, the simultaneous exploitation of two independent observables, and the incorporation of all known sources of statistical and systematic error in a coherent Bayesian analysis. Limitations arise mainly from uncertainties in the atmosphere model (particularly the exact value of f_c) and the assumption of isotropic burst emission. The authors suggest that future observations with higher spectral resolution (e.g., NICER, Athena) and independent distance determinations (e.g., Gaia parallaxes for the host cluster) could further tighten the constraints.

In summary, by combining precise spectroscopic measurements of multiple thermonuclear bursts with a well‑determined cluster distance, the authors uniquely determine the mass–radius probability distribution for the neutron star in 4U 1820‑30, obtaining M = 1.58 ± 0.06 M_⊙ and R = 9.11 ± 0.40 km. This result provides a stringent empirical benchmark for dense‑matter physics and demonstrates the power of burst spectroscopy as a tool for neutron‑star astrophysics.