Cygnus X-3 and the problem of the missing Wolf-Rayet X-ray binaries

Cygnus X-3 is a strong X-ray source ($L_{\rm X} \approx 10^{38}$ erg s-1) which is thought to consist of a compact object accreting matter from a helium star. We analytically find that the estimated ranges of mass-loss rate and orbital-period derivative for Cyg X-3 are consistent with two models: i) the system is detached and the mass loss from the system comes from the stellar wind of a massive helium star, of which only a fraction that allows for the observed X-ray luminosity is accreted, or ii) the system is semidetached and a Roche-lobe-overflowing low- or moderate-mass helium donor transfers mass to the compact object, followed by ejection of its excess over the Eddington rate from the system. These analytical results appear to be consistent with evolutionary calculations. By means of population synthesis we find that currently in the Galaxy there may exist ~1 X-ray binary with a black hole that accretes from a $\ga$$7~M_\odot$ Wolf-Rayet star and ~1 X-ray binary in which a neutron star accretes matter from a Roche-lobe-overflowing helium star with mass $\la$$1.5~M_\odot$. Cyg X-3 is probably one of these systems.