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Two different types of stellar BH formation exist: (1) direct core collapse of a massive star and (2) AIC of a NS.
A BH of the first type presumably will have a mass comparable to that of the collapsing star and can be born both from a single star and in a binary system. The AIC BH can be formed only in binary systems. Their masses must be much lower, about the Oppenheimer-Volkoff limit, , and the number of such AIC BH will be sensitive to this limit. AIC BH can be formed at the stage of supercritical accretion in a binary system at high accretion rates yr. An effective means of formation of such BH can be spiral-in of a NS in the extended atmosphere of the giant companion at the common envelope (CE) stage (Chevalier, 1993), when neutrino losses can effectively remove the gravitational energy released, and a Bondi-Hoyle accretion may be established. The characteristic time of mass growth of the NS at this stage is only five to seven times less than the time-scale of angular momentum drag due to spiral-in (Chevalier, 1993), so AIC BH with masses close to the OV-limit in circular orbits around the remaining star core (a Wolf-Rayet star (WR) or a white dwarf (WD)) will form, unless the spiral-in ends by coalescence of the compact star with the stellar core and formation of a Thorne-Zytkow object. Some BH binaries thus formed can be disrupted during a second supernova explosion. The remaining WR+BH binaries will yield PSR+BH systems. Firm theoretical estimation of the number of AIC BH is very difficult, however one can evaluate their possible number from the observational data.
At present about a dozen single-line WR stars with suspected binary companions are known (Cherepashchuk, 1991), some of which can be considered as progenitors of these binary pulsars (namely, those which produce notable X-rays). We can make a simple estimation:
Here yr and -3 yr are typical pulsar and WR lifetimes, respectively. N(WR+compact) relates mainly to WR+NS that produce no X-rays due to fast rotation of the magnetized NS (see Lipunov, 1992, p. 293). In reality, we know only one WR-star with a compact X-ray companion, Cyg X-3 (van Kerkwijk, 1993), so we must reduce the upper limit by 10 times . Taking the total galactic number of pulsars 3 , we obtain the expected fraction of binary PSR+AIC BH ( 0.1 per 1000 isolated pulsars). Although we believe that the real number of PSR+AIC BH may be much smaller, we note nevertheless that the distribution of such binary pulsars on orbital periods will significantly differ from that of PSR+BH formed from massive stars, in having shorter orbital periods of several hours to several days.