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

3. Detailed model

Since the stellar parameters used in the previous Sect. are rather rough, the same computation is performed with a more detailed model for population I () stars. We use the models with moderate core overshooting computed by Schaller et al. 1992, using the radius (calculated from effective temperature and luminosity) and the mass of the star in the tabulated points. The mass of the star decreases as a function of time due to stellar wind. The mass loss in the stellar wind causes an increase of the Roche-lobe radius of the primary, (mainly) by increasing the semi-major axis and (to a lesser extent) by increasing the mass-ratio . The increase in the semi-major axis is described, assuming an isotropic wind with high velocity according to the Jeans approximation, with (van den Heuvel 1983):

The relation between the semi-major axis at which the primary fills its Roche lobe and the initial semi-major axis is thus given by:

Here and M are the zero-age mass of the primary and its mass at the moment it fills its Roche lobe. For each tabulated point of the evolutionary track we equate the radius R of the primary to the Roche-lobe radius in Eq.  6 to calculate the corresponding maximum initial semi-major axis . The values for and R for a 20 and a 60  star accompanied by a 1  companion are shown in Fig. 2. A value for smaller than reached at an earlier stage of the evolution implies that Roche lobe overflow would have occurred at that earlier moment.

Fig. 2a and b. a and indicate radii for subsequent evolutionary stages of the 20  primary as tabulated in Schaller et al. 1992. From the mass of the primary at the tabulated point one may calculate the semi-major axis of a binary with a 1  secondary in which the primary fills its Roche lobe, and from this the semi-major axis , shown as a solid line, of the binary at the beginning of its evolution. For each tabulated evolutionary stage we calculate the minimum semi-major axis at which the core of the primary survives the spiral-in of a 1  companion, and from this the minimum semi-major axis , indicated with the dashed line, of the binary at the beginning of its evolution. Primaries at the evolutionary stages marked with a cannot fill their Roche lobe for the first time at that stage but will reach their Roche lobe at an earlier point in their evolution. A low mass X-ray binary is formed when for evolutionary stages indicated with . b as a for a 60  primary

For each tabulated point of the evolutionary track we calculate the mass of the envelope by subtracting the core mass from the total mass. Because the stellar evolution models incorporate overshooting, which tends to increase the core mass, we calculate the core mass by multiplying the value found from Eq.  2 with 1.125 (Maeder & Meynet 1989). We also know the minimum separation after spiral-in for a detached binary. From this we calculate the minimum separation at the onset of spiral-in with Eq.  4, and the minimum separation of the initial binary with Eq.  6. These minimum separations are also shown in Fig. 2.

The value of of the first evolutionary point at which , which we denote as , corresponds to the minimum initial separation that the binary must have to survive the spiral-in. The maximum of the values of for all evolutionary stages, indicated with , corresponds to the maximum initial separation of the binary at which the primary can reach its Roche lobe. Only binaries with an initial separation in the range - can evolve into low-mass X-ray binaries. For a 20  primary with a 1  secondary this range is 1000 - 1590 .

The Roche lobe can only be reached for the first time in those evolutionary stages for which is larger than the 's at all earlier evolutionary stages. Those stages are marked in Fig. 2 with . Note that core hydrogen burning ends in tabulated point 13, and helium core burning begins in point 21. Helium core burning ends in point 43, and carbon burning starts in point 46. For a star of , the radius of the star expands following the end of core hydrogen burning, and mass transfer during this first ascent of the giant branch is called case B. At the onset of core helium burning, the star shrinks. It expands once more after the end of core helium burning, and mass transfer during this second ascent of the giant branch is called case C. For the star shown in Fig. 2, however, the radius does not shrink at the onset of each new phase of core fusion, but continues its expansion throughout its evolution, once the Hertzsprung gap is passed. The shrinks at the onset of helium fusion in the core, mainly due to extensive mass loss.

As shown by Fig. 2 a 60  primary with a 1  secondary can, according to the same reasoning, only evolve into a low-mass X-ray binary if its semi-major axis is in the very small range of 980 - 1100 .

We determine the mass and size of primaries in a range of masses at the moment they fill their Roche lobes at for each tabulated stellar evolution track. The masses and radii at of the stars that are not tabulated by Schaller et al. 1992 are determined by a linear interpolation between the tabulated models. The resulting values for are shown as a solid line in Fig. 3.

Fig. 3. Lower limit to the initial semi-major axis at which the binary survives the spiral-in, as function of the initial mass of the primary, calculated with use of the evolutionary sequences by Schaller et al. (1992). 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. The dotted line indicates the initial semi-major axis of a binary in which a 1  secondary can fill its Roche lobe after the primary has lost its entire envelope without ever having filled its Roche lobe

If Roche-lobe overflow for all initial semi-major axes smaller than leads to a merger, then is not properly defined. A lower limit can be obtained by computing with the stellar parameters that correspond to the point where is reached.

Mass loss from stars more massive than is so copious that these stars lose their entire hydrogen envelope before they expand on the giant branch, i.e. before Roche-lobe contact is achieved. The common-envelope phase hardly leads to a spiral-in, whereas further attrition of the core to a small final mass (   according to Schaller et al. 1992) causes the binary orbit to expand. As a result the final orbit is so wide that the 1  secondary never reaches its Roche lobe.

Fig. 3 shows, as a function of the zero-age mass of the primary, the lower limits to the semi-major axis of the initial binary at which the binary survives the spiral-in (or its conservative lower limit), and the upper limit for which the primary reaches its Roche lobe .

Comparison of Fig. 3 with Fig. 1 illustrates that the mass loss of massive stars and the concurrent widening of the binary reduces the maximum initial separation for which the binary reaches Roche-lobe contact. For stars with the maximum stellar radius is smaller than 1000  , which leads to a further reduction of . The wind mass-loss affects the minimum initial separation necessary to survive the spiral-in in two ways: the widening of the orbit reduces this separation, whereas the reduction of the envelope mass enlarges it.

Fig. 3 indicates that the formation rate of low-mass X-ray binaries with a neutron star greatly exceeds the formation-rate of low-mass X-ray binaries with a black hole, because the range of allowed initial separations is larger for neutron star progenitors.

Romani 1992 mentions the possibility that the secondary star might fill its Roche lobe as it evolves on the asymptotic giant-branch after the primary collapsed into a remnant without ever having reached Roche-lobe contact. If this happens the binary orbit also widens dramatically, according to Eq.  5. If the orbit widens too much, the secondary will not reach its Roche lobe. In Fig. 3 we show the maximum initial semi-major axis for which a 1  star can reach its Roche lobe after its companion has lost its envelope. It is seen that this semi-major axis is so small, that it invalidates the assumption that Roche lobe contact of the primary has been avoided.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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