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

5. Discussion

We briefly discuss the assumptions and model uncertainties that affect our computations most.

In our calculations the effect of convective overshooting on the mass of the helium core is taken into account in a relatively simple manner, since the tabulated stellar evolution tracks do not provide masses for the stellar helium core. The importance of convective overshooting is uncertain, and therefore we investigate small variations in its effects.

If we increase the mass of the core by increasing the multiplication factor for Eq.  2 discussed in paragraph 2 of Sect. 3 from 1.125 to 1.250, the lower limit to the semi-major axis for which a binary survives the spiral-in is slightly reduced, and consequently the birthrate of neutron star binaries is enhanced by , and the birthrate of black hole binaries increases with a factor . An increase of the exponent in Eq.  2 to 1.57, which is not unrealistic according to Iben & Tutukov 1985, reduces the discrepancy between the birthrates for black holes and neutron stars in low-mass X-ray binaries to a factor 15. Thus neither of these changes in Eq.  2 leads to a sufficiently high birthrate of black-hole binaries relative to neutron-star binaries.

The efficiency at which the common-envelope is expelled upon the spiral-in is highly uncertain. The effect of varying on the galactic formation rate for binaries that reach Roche-lobe contact and survive the spiral-in is demonstrated in Fig. 4. Decreasing results, as expected, in a strong reduction in the formation rate of low-mass X-ray binaries with a neutron star as well as with a black hole. By increasing to a value larger than about unity, the formation rate of low-mass X-ray binaries approaches a constant rate, limited by the number of primordial binaries formed in the galaxy. Note that might have different values for neutron star- and black-hole progenitors.

Fig. 4. Birthrate (per year in the galaxy) of low-mass X-ray binaries with a neutron star (upper lines) and a black hole companion (lower lines) as function of the common-envelope efficiency-parameter . The full lines are from the stellar evolution models from Schaller et al. 1992, the dotted lines are from the low metallicity models with from Charbonnel et al. 1993 and the dashed lines are computed with from Schaerer et al. 1993

To investigate the effect of the stellar metallicity on the formation rate of low-mass X-ray binaries we compute two more models using the evolutionary tracks calculated by Charbonnel et al. 1993 for and Schaerer et al. 1993 for (see Fig. 4). Apart from the enhanced mass-loss rates in the model with high metallicity which results in a larger formation rate of low-mass X-ray binaries, these model variations have little influence on our conclusions. The models with a lower metallicity lose less mass in the stellar wind and the fraction of binaries that survive the spiral-in decreases accordingly.

The limit of , which we use throughout the calculations, was originally derived from the observation of the high-mass X-ray binary LMC X-3 (van den Heuvel & Habets 1984). Recently the analysis of the data of Wray 977, the BIa hypergiant companion of the X-ray pulsar GX301-2, has increased the empirical lower mass limit for the formation of a black-hole in a binary to (Kaper et al. 1995). This only exacerbates the problem of too low a formation rate of black-hole binaries in our calculations. Maeder 1992 argues on the basis of the galactic ratio of the abundances of metals and of helium that the lower limit for the formation of a black hole is 20 - 25  . As can be seen in Fig. 3, such decrease of the lower mass-limit has little effect on our computations, because in the range of 25-40  .

The stellar evolutionary tracks of Schaller et al. 1992 follow the single stars beyond the point of carbon burning. The large amount of mass and angular-momentum lost in the stellar wind during earlier phases of the evolution of the more massive primaries cause the binary orbit to widen and prohibits mass transfer after the onset of core helium burning (see Fig. 2). For the amount of angular momentum lost per unit mass in the stellar wind we used the standard description of isotropic mass loss from the donor (see Eq.  5). Mass transfer at an evolutionary stage well after the onset of helium fusion (say, beyond point 25 in Fig. 2) can only be an effective channel for the formation of low-mass X-ray binaries with a neutron star as the compact object, if the loss of angular momentum is high enough for the orbit to shrink. For a more massive progenitor, larger than , which leaves a black hole instead of a neutron star after the supernova, such late mass transfer is not likely to become important even for such high loss of angular momentum: the star has evolved into a naked helium core and a merger will be unavoidable upon Roche-lobe contact.

The evolution of the helium star and the effect of the supernova on the binary orbit are neglected in our calculations: both effects tend to lower the formation rate of low-mass X-ray binaries (see e.g. Portegies Zwart & Verbunt 1996). Mass transfer from the helium star to its companion can result in a merger and the asymmetry of a supernova can dissociate the binary. Both effects, however, tend to reduce the formation-rate of neutron stars as well as black holes in low-mass X-ray binaries and do consequently not solve the birth-rate problem discussed in this paper.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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