  We examine the influence of the gravity of the companion (the secondary) to
the massive primary star Eta Carinae on the winds blown by the primary and the
secondary. The two winds collide with each other after passing through two
respective shock waves, and escape the system while strongly emitting in the
X-ray band. While during most of the 5.5 years orbital period, the companion's
gravity has a negligible effect on the winds, we find that near periastron, the
companion's gravity may significantly influence the flow and the companion
might accrete from the primary's wind under certain circumstances. Near
periastron passage, the collision region of the two winds may collapse onto the
secondary star, a process that could substantially reduce the X-ray luminosity.
We suggest that such an accretion process produces the long, almost flat, X-ray
minimum in Eta Carinae.
