Astron. Astrophys. 321, 207-212 (1997)

2. Semi-analytic approach

In the standard scenario the formation of a low-mass X-ray binary starts with a high-mass primary and a low mass secondary star () in a detached binary. For simplicity we assume that the initial orbit is circular. As the primary evolves and expands to giant dimensions it fills its Roche lobe and starts to transfer mass onto its companion. Due to the small mass ratio mass transfer is highly unstable and results in a common-envelope phase. A close binary remains provided that the primary's envelope is fully ejected before the secondary star coalesces with the compact core of the primary. The reduction in the semi-major axis during the spiral-in can be computed by comparing the binding energy of the primary's envelope with the orbital binding-energy of the binary (Webbink 1984). The helium core continues its evolution and finally collapses into a neutron star or a black hole. The binary becomes an X-ray binary once the secondary star fills its Roche lobe.

We illustrate this scenario for primary stars of initially and 60  which reach a maximum radius of 1000  (see Romani 1992). The scenario requires that the primary fills its Roche lobe during its evolution, i.e. that its Roche lobe has a radius less than 1000  . The corresponding semi-major axis of the binary orbit may be calculated using the equation for the radius of the Roche lobe given by Eggleton 1983:

Here a is the semi-major axis of the binary and is the mass ratio and is the Roche-lobe radius for the star with mass M. For the 20 and 60  primaries we thus find a semi-major axis of about 1590 and 1440  , respectively.

The scenario further requires that the binary survives the spiral-in that follows upon the first Roche-lobe contact, i.e. that both the helium core and the 1  companion star are smaller than their Roche lobes. The mass and radius of the helium core of the primary can be computed with (see Iben & Tutukov 1985 and de Loore & Doom 1992, respectively)

and

The mass and radius of the secondary star are not affected by the spiral-in. The mass ratio after the spiral-in gives the sizes of the Roche lobes of the helium core and the secondary in units of the semi-major axis. From the radii of the helium star and its companion we can thus derive the minimum semi-major axis for which both stars fit inside their Roche lobes. For the 20 and 60  primaries we thus find a semi-major axis after the spiral-in which should exceed 4.0 and 6.3  , respectively. In both cases the most stringent limit is set by the main-sequence star.

The ratio of the semi-major axes before and after the spiral-in can be computed by comparing the binding energy of the primary's envelope with the binding energy released by the shrinking binary (see Webbink 1984):

Here and are the semi-major axes at the onset and end of the spiral-in, and M and R are the mass and radius of the primary at the onset of spiral-in. We set . We assume that the primary did not lose any mass before it fills its Roche lobe: and . For the 20 and 60  primaries we thus find a semi-major axis before the spiral-in which should exceed 1470 and 3160  , respectively.

For the 20  primary, the lower limit on the semi-major axis for survival of the spiral-in is smaller than the upper limit for Roche-lobe contact and a low-mass X-ray binary can be formed when the initial semi-major axis is between these two limits. For the 60  primary, the lower limit on the semi-major axis for survival of the spiral-in is larger than the upper limit for Roche-lobe contact and no initial orbit leads to the formation of a low-mass X-ray binary: Roche-lobe overflow always leads to a merger.

Fig. 1 shows, as function of the mass of the primary, the lower and upper limits to the semi-major axis of the initial binary at which the binary survives the spiral-in and the primary reaches the Roche lobe. Only for low mass primaries () does the binary survive the spiral-in. Thus, if only stars with an initial mass larger than 40  (van den Heuvel & Habets 1984) form a black hole, the formation of a low-mass X-ray binary with a black hole as a compact object is excluded in the standard scenario.

Fig. 1. Lower limit to the initial semi-major axis at which the binary survives the spiral-in, as function of the mass of the primary. The upper (lower) dashed line gives the limit determined from the condition that the secondary star (helium core) is smaller than its Roche-lobe, The solid line gives the upper limit at which the primary reaches its Roche lobe at its maximum radius of . The secondary is assumed to be a 1  star

One may argue that the binary survives the spiral-in even if the main-sequence star is larger than its Roche lobe after spiral-in, as long as the helium star fits inside its Roche lobe . The main-sequence star can shrink within its Roche lobe by transferring mass to the helium core. This mass transfer is stable because the main-sequence star is less massive than the helium core. In Fig. 1 we also show the lower limit to the semi-major axis obtained from the condition that only the helium star fits inside its Roche lobe. We see that even in this case low-mass X-ray binaries with a black hole cannot be formed in the standard scenario.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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