A Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE). XXIII. NGC 4569 past orbit reconstruction with ORFEO

Extended ionised gas tails can preserve a dynamical record of a galaxy's recent interaction with the intracluster medium (ICM). Recovering that history from the observed tail requires solving an inverse dynamical problem tied to the physical conditions for gas removal. We present the first application of ORFEO to NGC 4569, using the sky positions and line-of-sight (LOS) velocities of the galaxy and 94 H$α$-emitting regions to reconstruct its recent orbit and tail chronology. The model combines data with gas surface-density constraints, a spherical Virgo Cluster A gravitational potential derived with MG-MAMPOSSt, and an X-ray-based ICM density profile. For an assumed LOS distance, it infers its orbit and ram-pressure history. At the fiducial distance $d_0=16.5$ Mpc, ORFEO recovers a coherent phase-space sequence with a dynamical tail age $T_{\rm tail}=250^{+37}_{-19}$ Myr. The ram-pressure maximum occurred at $t_{P_{\rm ram},{\rm peak}}=-163^{+11}_{-15}$ Myr, close to the nearest approach to the Virgo Cluster A centre at $305^{+19}_{-25}$ kpc, placing NGC 4569 after peak ram pressure. Scanning the LOS distance does not select a unique solution but separates comparably fitting reconstructions into a low-distance post-peak branch and a high-distance pre-peak branch. Comparison with an independent stripping threshold at the observed disc truncation radius favours the low-distance solutions and supports a post-stripping interpretation. On this branch, ORFEO recovers the mostly west-to-east projected motion inferred from the H$α$ morphology and is consistent with previous constraints on the interaction chronology. Requiring only $\sim10$--11 min of wall-clock time for NGC 4569, ORFEO provides a computationally inexpensive inverse framework for deriving orbital, tail, and ram-pressure histories that can guide tailored simulations and be extended to larger samples of stripped galaxies.