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.
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