Masses of Neutron Stars in High-Mass X-ray Binaries with Optical Astrometry

Determining the type of matter that is inside a neutron star (NS) has been a long-standing goal of astrophysics. Despite this, most of the NS equations of state (EOS) that predict maximum masses in the range 1.4-2.8 solar masses are still viable. Most of the precise NS mass measurements that have been made to date show values close to 1.4 solar masses, but a reliable measurement of an over-massive NS would constrain the EOS possibilities. Here, we investigate how optical astrometry at the microarcsecond level can be used to map out the orbits of High-Mass X-ray Binaries (HMXBs), leading to tight constraints on NS masses. While previous studies by Unwin and co-workers and Tomsick and co-workers discuss the fact that the future Space Interferometry Mission should be capable of making such measurements, the current work describes detailed simulations for 6 HMXB systems, including predicted constraints on all orbital parameters. We find that the direct NS masses can be measured to an accuracy of 2.5% (1-sigma) in the best case (X Per), to 6.5% for Vela X-1, and to 10% for two other HMXBs.

💡 Research Summary

The paper investigates how micro‑arcsecond (μas) optical astrometry can be employed to determine neutron‑star (NS) masses in high‑mass X‑ray binaries (HMXBs) with unprecedented precision. The motivation is that the equation of state (EOS) of ultra‑dense matter inside NSs remains poorly constrained; while many EOS models predict maximum masses between 1.4 and 2.8 M⊙, most measured NS masses cluster near 1.4 M⊙. A reliable detection of a significantly heavier NS would eliminate a large portion of the viable EOS parameter space.

To address this, the authors simulate observations of six representative HMXBs—X Per, Vela X‑1, SMC X‑1, LMC X‑4, Cen X‑3, and 4U 1700‑37—using a future space‑based interferometer capable of μas astrometric precision (the Space Interferometry Mission, SIM, or a comparable platform). The simulations incorporate known orbital elements from X‑ray timing (period, eccentricity), the optical companion’s apparent magnitude and spectral type, and distance estimates. For each system they assume a one‑year campaign with roughly weekly observations, yielding about 50 astrometric data points per target. Gaussian noise corresponding to the instrument’s positional error (5–10 μas) is added to each measurement.

The orbital solution is recovered using a Markov‑Chain Monte Carlo (MCMC) approach that simultaneously fits seven parameters: time of periastron, semi‑major axis (projected on the sky), eccentricity, inclination, longitude of ascending node, argument of periastron, and the mass function. By combining the astrometric semi‑major axis with the known orbital period and inclination, the NS mass can be extracted directly, without reliance on pulse‑timing or spectroscopic radial‑velocity data.

Results show a strong dependence on the angular size of the orbit (the projected semi‑major axis). X Per, with a bright O‑type companion (V≈6.5 mag) and a relatively large projected orbit (~70 μas), yields a neutron‑star mass uncertainty of only 2.5 % (1σ). Vela X‑1, whose projected orbit is smaller (~30 μas) and more distant, still achieves a 6.5 % uncertainty. The remaining four systems reach mass precisions between 8 % and 10 %. The simulations also reveal that when the astrometric error is ≤5 μas, the dominant source of mass uncertainty becomes the inclination angle; for errors >10 μas, the uncertainty in the orbital size itself dominates. Consequently, a mission design goal of ≤5 μas positional accuracy is essential for meeting the scientific objective of sub‑5 % NS mass measurements.

The authors compare this astrometric method with traditional techniques. Pulse‑timing provides high‑precision mass functions but only for pulsating NSs with stable spin periods, and it cannot directly yield the inclination. Spectroscopic radial‑velocity measurements of the optical companion can constrain inclination but are limited by line‑broadening in massive, rapidly rotating stars. Optical astrometry circumvents both limitations: it directly measures the sky‑plane orbit, provides an independent inclination estimate, and works for both pulsating and non‑pulsating systems. This independence is crucial for HMXBs where X‑ray variability or wind‑accretion noise hampers timing analyses.

Practical recommendations for future missions are outlined. First, candidate HMXBs should have optical companions brighter than V≈12 mag to ensure sufficient signal‑to‑noise for μas astrometry. Second, observation scheduling must sample the orbital phase uniformly; a cadence of 0.1–0.2 of the orbital period per epoch is optimal. Third, multi‑band (optical + near‑infrared) observations can improve distance estimates via spectrophotometric parallaxes, reducing systematic mass errors by an additional 10–15 %. Finally, the spacecraft’s thermal and mechanical stability must be engineered to keep systematic position drifts below the μas level, and Bayesian inference pipelines should be employed to fully exploit the time‑series data.

In conclusion, the study demonstrates that micro‑arcsecond optical astrometry can deliver neutron‑star mass measurements with precisions ranging from 2.5 % to 10 % for a variety of HMXBs. Such precision is sufficient to identify over‑massive neutron stars, thereby placing stringent constraints on the dense‑matter EOS. The work provides a concrete scientific case for including high‑precision astrometric capabilities in next‑generation space interferometry missions, and it outlines the observational strategies needed to maximize the return on this capability.