Investigating the origin of optical and X-ray pulsations of the transitional millisecond pulsar PSR J1023+0038

PSR J1023+0038 is the first millisecond pulsar that was ever observed as an optical and UV pulsar. So far, it is the only optical transitional millisecond pulsar. The rotation- and accretion-powered emission mechanisms hardly individually explain the observed characteristics of optical pulsations. A synergistic model, combining these standard emission processes, was proposed to explain the origin of the X-ray/UV/optical pulsations. We study the phase lag between the pulses in the optical and X-ray bands to gain insight into the physical mechanisms that cause it. We performed a detailed timing analysis of simultaneous or quasi-simultaneous observations in the X-ray band, acquired with the XMM-Newton and NICER satellites, and in the optical band, with the fast photometers SiFAP2 (mounted at the 3.6 m Telescopio Nazionale Galileo) and Aqueye+ (mounted at the 1.8 m Copernicus Telescope). We estimated the time lag of the optical pulsation with respect to that in the X-rays by modeling the folded pulse profiles with two harmonic components. Optical pulses lag the X-ray pulses by $\sim$ 150 $\mu$s in observations acquired with instruments (NICER and Aqueye+) whose absolute timing uncertainty is much smaller than the measured lag. We also show that the phase lag between optical and X-ray pulsations lies in a limited range of values, $\delta \phi \in$ (0 $-$ 0.15), which is maintained over timescales of about five years. This indicates that both pulsations originate from the same region, and it supports the hypothesis of a common emission mechanism. Our results are interpreted in the shock-driven mini pulsar nebula scenario. This scenario suggests that optical and X-ray pulses are produced by synchrotron emission from the shock that formed within a few light cylinder radii away ($\sim$ 100 km) from the pulsar, where its striped wind encounters the accretion disk inflow.