Revisiting the archetypical wind accretor Vela X-1 in depth

Context: Vela X-1 is one of the best studied X-ray binaries. Frequently though, specific values for its parameters have been used in subsequent studies without considering alternatives. Aims: We aim to provide a robust compilation and synthesis of the accumulated knowledge about Vela X-1 as a solid baseline for future studies and identify specific avenues of possible future research. Methods: We explore the literature for Vela X-1 and on modelling efforts, describing the evolution of the system knowledge. We also add information derived from public data, especially the Gaia EDR3 release. Results: We update the distance to Vela X-1, the spectral classification for HD 77518 and find that the supergiant may be very close to filling its Roche lobe. Constraints on the clumpiness of the stellar wind have improved. The orbit is very well determined, but the uncertain inclination limits information on the neutron star mass. Estimates for the stellar wind have evolved towards lower velocities, supporting the idea of transient wind-captured disks around the neutron star. Hydrodynamic models and observations are consistent with an accretion wake trailing the neutron star. Conclusions: Vela X-1 is an excellent laboratory, but a lot of room remains to improve. Well-coordinated multi-wavelength observations and campaigns addressing the intrinsic variability are required. New opportunities will arise through new instrumentation. Models of the stellar wind should account for the orbital eccentricity and the non-spherical shape of HD 77581. Realistic multi-dimensional models of radiative transfer in the UV and X-rays are needed, but remain very challenging. Improved MHD models covering a wide range of scales would be required to improve understanding of the plasma-magnetosphere coupling. A full characterization of the accretion column remains another open challenge. (Abbreviated for arXiv)