  We present in this paper a first step toward a semi-analytical treatment of
the accretion process in wind-fed neutron star supergiant X-ray binaries with
eccentric orbits. We consider the case of a spherically symmetric wind for the
supergiant star and a simplified model for the accretion onto the compact
object. A self-consistent calculation of the photoionization of the stellar
wind by the X-rays from the accreting neutron star is included. This effect is
convolved with the modulation of the mass accretion rate induced by the
eccentric orbit to obtain the expected X-ray luminosity of a system along the
orbit. As part of our results, we first show that the bi-modality of low and
high X-ray luminosity solutions for supergiant X-ray binaries reported in
previous papers is likely to result from the effect of the neutron star
approaching first and then moving away from the companion (without coexisting
simultaneously). We propose that episodes of strong wind photoionization can
give rise to off-states of the sources. Our calculations are applied to the
case of a few classical supergiant X-ray binary systems with known
eccentricities (Vela X-1, 4U 1907+09, GX 301-2) and to the case of the only
supergiant fast X-ray transient with a confirmed eccentric orbit, IGR
J08408-4503. The results are compared with observational findings on these
sources. We also discuss the next steps needed to expand the calculations
toward a more comprehensive treatment in future publications.
