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Classical Supergiant X-ray Binaries host a neutron star orbiting a supergiantOB star and display persistent X-ray luminosities of 10$^{35}$ to 10$^{37}$erg/s. The stellar wind from the massive companion is believed to be the mainsource of matter accreted by the compact object. With this first paper, weintroduce a ballistic model to characterize the structure of the wind at theorbital scale as it accelerates, from the stellar surface to the vicinity ofthe accretor. Thanks to the parametrization we retained and the numericalpipeline we designed, we can investigate the supersonic flow and the subsequentobservables as a function of a reduced set of characteristic numbers andscales. We show that the shape of the permanent flow is entirely determined bythe mass ratio, the filling factor, the Eddington factor and the $\alpha$-forcemultiplier which drives the stellar wind acceleration. Provided scales such asthe orbital period are known, we can trace back the observables to evaluate themass accretion rates, the accretion mechanism, the shearing of the inflow andthe stellar parameters. We confront our model to three persistent SupergiantX-ray Binaries (Vela X-1, IGR J18027-2016, XTE J1855-026) and discuss thelikelihood of wind-formed accretion discs around the accretors in each case,further investigated in a following paper.

A numerical investigation of wind accretion in persistent Supergiant X-ray Binaries I - Structure of the flow at the orbital scale