Revisiting the archetypical wind accretor Vela X-1 in depth. Case study of a well-known X-ray binary and the limits of our knowledge

[Context]: The Vela X-1 system is one of the best-studied X-ray binaries because it was detected early, has persistent X-ray emission, and a rich phenomenology at many wavelengths. The system is frequently quoted as the archetype of wind-accreting high-mass X-ray binaries, and its parameters are referred to as typical examples. Specific values for these parameters have frequently been used in subsequent studies, however, without full consideration of alternatives in the literature, even more so when results from one field of astronomy (e.g., stellar wind parameters) are used in another (e.g., X-ray astronomy). The issues and considerations discussed here for this specific, very well-known example will apply to various other X-ray binaries and to the study of their physics.[Aims]: We provide a robust compilation and synthesis of the accumulated knowledge about Vela X-1 as a solid baseline for future studies, adding new information where available. Because this overview is targeted at a broader readership, we include more background information on the physics of the system and on methods than is usually done. We also attempt to identify specific avenues of future research that could help to clarify open questions or determine certain parameters better than is currently possible.[Methods]: We explore the vast literature for Vela X-1 and on modeling efforts based on this system or close analogs. We describe the evolution of our knowledge of the system over the decades and provide overview information on the essential parameters. We also add information derived from public data or catalogs to the data taken from the literature, especially data from the Gaia EDR3 release.[Results]: We derive an updated distance to Vela X-1 and update the spectral classification for HD 77518. At least around periastron, the supergiant star may be very close to filling its Roche lobe. Constraints on the clumpiness of the stellar wind from the supergiant star have improved, but discrepancies persist. The orbit is in general very well determined, but a slight difference exists between the latest ephemerides. The orbital inclination remains the least certain factor and contributes significantly to the uncertainty in the neutron star mass. Estimates for the stellar wind terminal velocity and acceleration law have evolved strongly toward lower velocities over the years. Recent results with wind velocities at the orbital distance in the range of or lower than the orbital velocity of the neutron star support the idea of transient wind-captured disks around the neutron star magnetosphere, for which observational and theoretical indications have emerged. Hydrodynamic models and observations are consistent with an accretion wake trailing the neutron star.[Conclusions]: With its extremely rich multiwavelength observational data and wealth of related theoretical studies, Vela X-1 is an excellent laboratory for exploring the physics of accreting X-ray binaries, especially in high-mass systems. Nevertheless, much room remains to improve the accumulated knowledge. On the observational side, well-coordinated multiwavelength observations and observing campaigns addressing the intrinsic variability are required. New opportunities will arise through new instrumentation, from optical and near-infrared interferometry to the upcoming X-ray calorimeters and X-ray polarimeters. Improved models of the stellar wind and flow of matter should account for the non-negligible effect of the orbital eccentricity and the nonspherical shape of HD 77581. There is a need for realistic multidimensional models of radiative transfer in the UV and X-rays in order to better understand the wind acceleration and effect of ionization, but these models remain very challenging. Improved magnetohydrodynamic models covering a wide range of scales are required to improve our understanding of the plasma-magnetosphere coupling, and they are thus a key factor for understanding the variability of the X-ray flux and the torques applied to the neutron star. A full characterization of the X-ray emission from the accretion column remains another so far unsolved challenge.IEM has received funding from the Research Foundation Flanders (FWO), from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 665501 and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 863412). SMN acknowledges funding by the Spanish Ministry MCIU under project RTI2018-096686-B-C21 (MCIU/AEI/FEDER, UE), co-funded by FEDER funds and by the Unidad de Excelencia María de Maeztu, ref. MDM-2017-0765. VG is supported through the Margarete von Wrangell fellowship by the ESF and the Ministry of Science, Research and the Arts Baden-Württemberg. IEM and PK acknowledge support from the ESA/ESAC Faculty Visiting Scientist Programme. JMA acknowledges support from the Spanish Government Ministerio de Ciencia through grant PGC2018-095049-B-C22. FJE acknowledges support from ESCAPE – The European Science Cluster of Astronomy & Particle Physics ESFRI Research Infrastructures, which received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement no. 824064. JvdE is supported by a Lee Hysan Junior Research Fellowship awarded by St. Hilda’s College.Peer reviewe