A comprehensive study of orbital evolution of LMC X-4: existence of a second derivative of the orbital period

We report here results from pulse arrival time delay analysis of the eclipsing high-mass X-ray binary (HMXB) pulsar LMC X-4 using observations made with the Rossi X-ray Timing Explorer, XMM-Newton, NuSTAR (Nuclear Spectroscopic Telescope ARray), and AstroSat. Combining the orbital parameters determined from these observations with the historical measurements dating back to 1998, we have extended the T_π/2 epoch history of LMC X-4 by about 4600 binary orbits spanning about 18 yr. We also report mid-eclipse time measurements (T_ecl) using data obtained from wide-field X-ray monitors of MAXI-GSC (Monitor of All-sky X-ray Image - Gas Slit Camera) and Swift-BAT (Burst Alert Telescope). Combining the new T_π/2 and T_ecl estimates with all the previously reported values, we have significantly improved the orbital evolution measurement, which indicates that the orbital period is evolving at a time-scale ($P_{\rm orb}/\dot{P}_{\rm orb}$) of about 0.8 Myr. For the first time in an accreting X-ray pulsar system, we confirm the existence of a second derivative of the orbital period, having an evolution time-scale ($\dot{P}_{\mathrm{ orb}}/\ddot{P}_{\mathrm{ orb}}$) of about 55 yr. Detection of a second derivative of the orbital period in LMC X-4 makes its orbital evolution time-scale more uncertain, which may also be true for other HMXBs. Independent solutions for the orbital evolution measurement using the mid-eclipse data and the pulse timing data are consistent with each other, and help us put an upper limit of 0.009 on the eccentricity of the binary system.