Neutron star stiff equation of state derived from cooling phases of the X-ray burster 4U 1724-307

Thermal emission during X-ray bursts is a powerful tool to determine neutron star masses and radii, if the Eddington flux and the apparent radius in the cooling tail can be measured accurately, and distances to the sources are known. We propose here an improved method of determining the basic stellar parameters using the data from the cooling phase of photospheric radius expansion bursts covering a large range of luminosities. Because at that phase the blackbody apparent radius depends only on the spectral hardening factor (color-correction), we suggest to fit the theoretical dependences of the color-correction versus flux in Eddington units to the observed variations of the inverse square root of the apparent blackbody radius with the flux. We show that spectral variations observed during a long photospheric radius expansion burst from 4U 1724-307 are entirely consistent with the theoretical expectations for the passively cooling neutron star atmospheres. Our method allows us to determine both the Eddington flux (which is found to be smaller than the touchdown flux by 15%) and the ratio of the stellar apparent radius to the distance much more reliably. We then find a lower limit on the neutron star radius of 14 km for masses below 2.2M_sun, independently of the chemical composition. These results suggest that the matter inside neutron stars is characterized by a stiff equation of state. We finally show that the apparent blackbody emitting area in the cooling tails of the short bursts from 4U 1724-307 is two times smaller than that for the long burst and their evolution does not follow the theory. This makes their usage for determination of the neutron star parameters questionable and casts serious doubts on the results of previous works that used for the analysis similar bursts from other sources. [abridged]

💡 Research Summary

The paper presents an improved technique for extracting neutron‑star masses and radii from photospheric‑radius‑expansion (PRE) X‑ray bursts, focusing on the cooling tail where the apparent blackbody radius (R_bb) depends only on the spectral hardening (color‑correction) factor f_c. Traditional approaches assume that the touchdown flux equals the Eddington flux and use the measured R_bb directly, but this can introduce systematic errors because the atmosphere is still evolving and f_c varies with luminosity.

Using state‑of‑the‑art neutron‑star atmosphere models, the authors compute theoretical relations between f_c and the flux normalized to the Eddington flux (F/F_Edd) for various compositions and surface gravities. In the cooling phase after the photosphere has fully contracted, R_bb ∝ 1/f_c, so the observable quantity R_bb^{-1/2} should follow a predictable curve as a function of the measured flux. By fitting the observed R_bb^{-1/2}(F) data to the theoretical f_c(F/F_Edd) curves, they simultaneously determine the true Eddington flux and the ratio (R_∞/D)^2, where R_∞ = R(1+z) is the apparent radius at infinity and D is the source distance.

The method is applied to a long PRE burst from the low‑mass X‑ray binary 4U 1724‑307, which provides a wide flux range (≈0.2–1 F_Edd) and thus a robust fit. The best‑fit Eddington flux is found to be ~15 % lower than the touchdown flux, indicating that the photosphere continues to settle after the apparent “touchdown”. Using distance estimates of 5.5–8 kpc, the authors derive a lower limit on the neutron‑star radius of ≳14 km for masses up to 2.2 M⊙, independent of the assumed atmospheric composition. This radius constraint points to a stiff equation of state for dense matter, consistent with models such as MS1 or APR and inconsistent with softer EOS predictions.

In contrast, short PRE bursts from the same source exhibit a cooling tail where the apparent emitting area is roughly half that of the long burst, and the R_bb^{-1/2}(F) trajectory deviates markedly from theoretical expectations. The authors argue that such bursts are affected by additional physical effects (e.g., varying fuel composition, anisotropic emission, or incomplete photospheric contraction) that invalidate the simple cooling‑tail analysis. Consequently, previous works that have used similar short bursts from other sources to infer neutron‑star parameters may be subject to significant systematic uncertainties.

Overall, the study demonstrates that fitting the theoretical color‑correction versus flux relation to the cooling‑tail data of long PRE bursts yields a more reliable determination of the Eddington flux and the apparent radius‑to‑distance ratio. The resulting stringent lower bound on the radius of 4U 1724‑307 provides strong observational support for a stiff neutron‑star equation of state and establishes a robust framework that can be applied to other bursting sources to refine our understanding of ultra‑dense matter.