Superorbital Phase Evolution and a Soft-Hard X-ray Phase Shift in LMC X-4

Abstract

The superorbital period of LMC X-4 is among the most stable known in Roche-lobe overflow, high-mass X-ray binaries. We analyzed 33 years of monitoring data from the Compton Gamma Ray Observatory Burst and Transient Source Experiment (CGRO BATSE), the Rossi X-ray Timing Explorer All-Sky Monitor (RXTE ASM), the Neil Gehrels Swift Burst Alert Telescope (Swift BAT), the Monitor of All-sky X-ray Image Gas Slit Camera (MAXI GSC), and the Fermi Gamma-ray Burst Monitor (Fermi GBM). The measured phases show a smooth long-term trend with superposed systematic fluctuations. Fits with cubic, quartic, and sinusoidal models indicate that the quartic and sinusoidal forms provide significantly better descriptions, with the sinusoidal model yielding an $8900^{+210}_{-230}$-day modulation. Such a long timescale is unlikely to arise from orbital motion around a tertiary companion. The fluctuations resemble stochastic, glitch-like events on several-hundred-day timescales. Their rms period variation exceeds that of the smooth trend, yet the total rms period variation over 33 years remains only 0.55\%, demonstrating the exceptional stability of the superorbital period. During MJD 57000-60461, we detect a phase offset of 0.044$\pm$0.010 cycles between the soft and hard X-ray bands. This offset can be reproduced by including a higher-harmonic term in the azimuthal disk model, allowing a transition from antisymmetric to asymmetric structure. A contemporaneous decline in the hard X-ray flux suggests a partial obscuration of the emission region, similar to the anomalous low state in Her X-1. This evolving-disk scenario may also explain the superorbital phase shift previously reported in Her X-1.

Figures (7)

• Figure 1: Light curves collected by five instruments for analysis in this study. The bin size of these light curves is 30.34 days (about a superorbital cycle). The count rate units are cts s$^{-1}$ for RXTE ASM, $10^{-3}$ cts cm$^{-2}$ s$^{-1}$ for Swift BAT and $10^{-3}$ ph cm$^{-2}$ s$^{-1}$ for MAXI GSC.
• Figure 2: Top: A typical superorbital modulation profile of a data segment, generated from the MAXI GSC light curve obtained between MJD 56351.44 and 56472.71, folded by the linear ephemeris in Eq.\ref{['feph']}. The dashed red line represents the baseline level for MAXI GSC (0.005 ph cm$^{-2}$ s$^{-1}$) and the dot black line represents the zero count rate. Bottom: Modulation profile obtained by folding the entire MAXI GSC light curve with the optimal sinusoidal ephemeris (Eq. \ref{['sinueph']}).
• Figure 3: Evolution of superorbital phases fitted by the cubic (top), quartic (middle), and sinusoidal (bottom) models from 1991 to 2024. Soft X-ray phases from RXTE ASM and MAXI GSC are plotted in red, while hard X-ray phases from CGRO BATSE, Swift BAT, and Fermi GBM are plotted in black. The solid lines represent the corresponding best-fit models describing the long-term phase evolution trend. The residuals obtained after subtracting the corresponding trend are shown below. The vertical dotted lines mark MJD 57000 after which phase shifts between soft and hard X-ray bands become apparent (Section \ref{['ps']}).
• Figure 4: Superorbital modulation profiles of the light curves from five instruments folded by the ephemerides derived from the cubic (top), quartic (middle), and sinusoidal (bottom) models.
• Figure 5: Comparison of superorbital phase distributions between soft (red) and hard (black) X-ray bands between MJD 53500 to 57000 (Part 1, top panel) and MJD 57000 to 60300 (Part 2, bottom panel), where the phases are evaluated by utilizing the ephemerides derived from cubic (top), quartic (middle), and sinusoidal (bottom) models. The dashed lines and horizontal error bars indicate the corresponding mean values and their 1$\sigma$ uncertainties, respectively.