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

[Context] 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.[Aims] 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.[Methods] 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.[Results] Optical pulses lag the X-ray pulses by ∼150 μs 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, δϕ ∈ (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 (∼100 km) from the pulsar, where its striped wind encounters the accretion disk inflow.G.I. is supported by the AASS Ph.D. joint research program between the University of Rome “Sapienza” and the University of Rome “Tor Vergata”, with the collaboration of the National Institute of Astrophysics (INAF). F. A., G. I., A.P., L.S., D.d.M., and L.Z. acknowledge financial support from the Italian Space Agency (ASI) and National Institute for Astrophysics (INAF) under agreements ASI-INAF I/037/12/0 and ASI-INAF n.2017-14-H.0, from INAF ‘Sostegno alla ricerca scientifica main streams dell’INAF’, Presidential Decree 43/2018. F. A., G. I., A.P., and L.S. also acknowledge funding from the Italian Ministry of University and Research (MUR), PRIN 2020 (prot. 2020BRP57Z). A.P. and D.d.M. acknowledge financial contribution from INAF ‘SKA/CTA projects’, Presidential Decree 70/2016. A.M.Z. is supported by PRIN-MIUR 2017 UnIAM (Unifying Isolated and Accreting Magnetars, PI S. Mereghetti). F.C.Z. is supported by a Juan de la Cierva fellowship. The work of F.C.Z and D.F.T. is also partially supported by the program Unidad de Excelencia María de Maeztu (CEX2020-001058-M). D.F.T. work has been supported by the grants PID2021-124581OB-I00 and acknowledges as well USTC and the Chinese Academy of Sciences Presidential Fellowship Initiative 2021VMA0001. M.T. acknowledges funding from the ERC under the EU’s Horizon 2020 research and innovation programme (consolidator grant agreement no. 101002352).Peer reviewe