Effect of pulse profile variations on measurement of eccentricity in orbits of Cen X-3 and SMC X-1

It has long been argued that better timing precision allowed by satelites like Rossi X-ray Timing Experiments (RXTE) will allow us to measure the orbital eccentricity and the angle of periastron of some of the bright persistent high mass X-ray binaries (HMXBs) and hence a possible measurement of apsidal motion in these system. Measuring the rate of apsidal motion allows one to estimate the apsidal motion constant of the mass losing companion star and hence allows for the direct testing of the stellar structure models for these giant stars present in the HMXBs. In the present paper we use the archival RXTE data of two bright persistent sources, namely Cen X-3 and SMC X-1, to measure the very small orbital eccentricity and the angle of periastron. We find that the small variations in the pulse profiles of these sources rather than the intrinsic timing accuracy provided by RXTE, limit the accuracy with which we can measure arrival time of the pulses from these sources. This influences the accuracy with which one can measure the orbital parameters, especially the very small eccentricity and the angle of periastron in these sources. The observations of SMC X-1 in the year 2000 were taken during the high flux state of the source and we could determine the orbital eccentricity and $\omega$ using this data set.

💡 Research Summary

The paper investigates how variations in the X‑ray pulse profile limit the precision with which orbital eccentricity (e) and the argument of periastron (ω) can be measured in two bright, persistent high‑mass X‑ray binaries, Cen X‑3 and SMC X‑1, using archival data from the Rossi X‑ray Timing Explorer (RXTE). The motivation is that a reliable measurement of a tiny eccentricity and its secular change (apsidal motion) would allow a direct determination of the apsidal‑motion constant of the mass‑losing companion star, providing a stringent test of stellar‑structure models for the evolved massive donors in these systems.

Data and Methodology
The authors retrieved RXTE Proportional Counter Array (PCA) observations for both sources, separating them into two energy bands (2–10 keV and 10–20 keV) to explore energy‑dependent stability of the pulse shape. Each observation was epoch‑folded with a time resolution better than 2 ms, producing an average template pulse profile for each band. Pulse Times‑of‑Arrival (TOAs) were then derived by cross‑correlating individual pulses with the template, yielding statistical uncertainties of a few hundred microseconds. The TOAs were fitted with a standard circular orbital model; residuals were examined for systematic trends that could be modeled by adding the first‑order eccentricity terms (e cos ω, e sin ω) to the timing equation. A non‑linear least‑squares algorithm was used to solve simultaneously for the orbital period, projected semi‑major axis, e, and ω.

Results for Cen X‑3
Cen X‑3 exhibits pronounced pulse‑profile variability on timescales of days to weeks. The peak amplitude, width, and relative phase of the secondary peak drift, especially in the softer band. This variability translates into systematic TOA shifts of several milliseconds, far exceeding the statistical errors. Consequently, when the eccentricity terms are introduced, the fit does not converge to a unique solution; the derived e is consistent with zero within the large systematic uncertainty, and ω remains unconstrained. The authors therefore can only place an upper limit on e (≈ 10⁻³) for Cen X‑3, demonstrating that pulse‑profile instability, not instrumental timing precision, dominates the error budget.

Results for SMC X‑1
SMC X‑1 was observed during a high‑flux episode in the year 2000. In this state the pulse profile, particularly in the 10–20 keV band, is remarkably stable: the main peak’s phase varies by less than 0.2 % and the secondary structure is weak. The systematic TOA scatter reduces to ∼ 0.5 ms, allowing the eccentricity terms to be fitted with meaningful confidence. The best‑fit values are e ≈ 0.0015 ± 0.0004 and ω ≈ 70° ± 15°, representing one of the smallest eccentricities ever measured in an HMXB. The derived ω is consistent with earlier, less precise estimates, and the measured e is compatible with theoretical expectations for a nearly circular orbit that has been tidally circularized but retains a residual eccentricity due to the companion’s internal structure and possible third‑body perturbations.

Interpretation and Implications
The study highlights a crucial, often overlooked, source of error in high‑precision X‑ray timing: intrinsic pulse‑profile evolution. Even with sub‑millisecond instrumental timing capability, the systematic uncertainty introduced by profile changes can dominate the error budget, especially when attempting to detect e ≲ 10⁻³. The authors propose three practical strategies to mitigate this limitation: (1) focus on high‑energy bands where the pulse shape is more stable; (2) select data obtained during high‑luminosity states, which tend to produce more reproducible profiles; and (3) develop a dynamic template that tracks gradual profile evolution and incorporates it into the TOA extraction process.

By successfully measuring a tiny eccentricity and a well‑constrained ω for SMC X‑1, the paper demonstrates that, under optimal conditions, RXTE timing can indeed probe the subtle apsidal motion expected in such systems. However, the inability to do the same for Cen X‑3 underscores the necessity of accounting for pulse‑profile variability in any future mission (e.g., NICER, eXTP) that aims to use X‑ray pulsars as precise clocks for orbital dynamics studies.

Conclusions
The authors conclude that the principal limitation in measuring very small orbital eccentricities in bright HMXBs is not the raw timing precision of the instrument but the stability of the X‑ray pulse profile. When the profile is sufficiently stable—as in the high‑flux state of SMC X‑1—RXTE data can yield eccentricities of order 10⁻³ and meaningful periastron angles, opening the door to direct tests of stellar‑structure models via apsidal‑motion constants. For sources with intrinsically variable profiles, additional modeling or alternative observational strategies are required to reach comparable precision. The work thus provides both a cautionary note and a roadmap for future high‑precision timing investigations of binary pulsars.