Formation of recollimation shocks in jets of high-mass X-ray binaries

We study conditions for formation of recollimation shocks in jets interacting with stellar winds in high-mass X-ray binaries. We show the existence of a critical jet power, dependent on the wind rate and velocity and the jet velocity, above which a recollimation shock is not formed. For jet powers below critical, we derive the location of the shock. We show that surface shocks may still exist above the critical power, but only occupy a small volume of the jet and do not significantly alter the jet opening angle. We test these prediction by 3D numerical simulations, which confirm the existence and the value of the critical power. We apply our results to Cyg X-1 and Cyg X-3.