Astron. Astrophys. 321, 207–212 (1997) ASTRONOMY AND ASTROPHYSICS The formation of black-holes in low-mass X-ray binaries Simon F. Portegies Zwart1, Frank Verbunt1, and Ene Ergma2 1 Astronomical Institute, Utrecht, Postbus 80000, 3508 TA Utrecht, The Netherlands 2 Tartu University, Physics Department, Ulikooli˜ 18, EE2400 Tartu, Estonia Received 24 April 1996 / Accepted 12 September 1996 Abstract. We calculate the formation rates of low-mass X-ray these persist in a more refined treatment (Sect. 3). We quantify binaries with a black hole. Both a semi-analytic and a more de- this in Sect. 4 and in Sect. 5 the results are discussed. tailed model predict formation rates two orders of magnitude lower than derived from the observations. Solution of this co- nundrum requires either that stars with masses less than 20 M 2. Semi-analytic approach can evolve into a black hole, or that stellar wind from a member In the standard scenario the formation of a low-mass X-ray bi- of a binary is accompanied by a much larger loss of angular nary starts with a high-mass primary and a low mass secondary momentum than hitherto assumed. star (mo ∼ 1M ) in a detached binary. For simplicity we as- sume that the initial orbit is circular. As the primary evolves Key words: binaries: close – stars: evolution – stars: neutron – and expands to giant dimensions it fills its Roche lobe and starts X-rays: stars 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 pri- mary’s envelope is fully ejected before the secondary star coa- lesces with the compact core of the primary. The reduction in 1. Introduction the semi-major axis during the spiral-in can be computed by comparing the binding energy of the primary’s envelope with Six low-mass X-ray binaries currently are known to have a mass the orbital binding-energy of the binary (Webbink 1984). The function that indicates a mass of the compact object larger than helium core continues its evolution and finally collapses into ∼ 3M , and these are therefore believed to have a black hole a neutron star or a black hole. The binary becomes an X-ray accretor. All these systems are soft X-ray transients and the total binary once the secondary star fills its Roche lobe. number in the galaxy of such systems is estimated to be between We illustrate this scenario for primary stars of initially = a few hundred and a few thousand; at any moment in time only Mo 20 and 60 which reach a maximum radius of 1000 (see a small fraction is X-ray active (for reviews, see Cowley 1992, M R Romani 1992). The scenario requires that the primary fills its Tanaka & Lewin 1995, Tanaka & Shibazaki 1996). Roche lobe during its evolution, i.e. that its Roche lobe has a The most popular formation scenario for low-mass X-ray bi- radius less than 1000 R . The corresponding semi-major axis naries proposes a relatively wide binary with an extreme mass of the binary orbit may be calculated using the equation for the ratio as the progenitor system (van den Heuvel 1983). The mas- radius R of the Roche lobe given by Eggleton (1983): sive star evolves to fill its Roche lobe and engulfs its low-mass L companion. Due to the extreme mass ratio the low-mass star R 0.49 spirals into the envelope of the high-mass star. A close binary L ≡ R ( )= (1) L q 2/3 −1/3 . remains if the spiral-in ceases before the low-mass companion a 0.6+q ln(1 + q ) coalesces with the compact helium core of the primary. The he- ≡ lium core continues its evolution and may turn into a neutron Here a is the semi-major axis of the binary and q m/M star or a black hole. Only a few studies apply this evolution- is the mass ratio and RL is the Roche-lobe radius for the star ary scenario specifically to the formation of low-mass X-ray with mass M. For the 20 and 60 M primaries we thus find a binaries with an accreting black hole (see Romani 1992). semi-major axis of about 1590 and 1440 R , respectively. In this paper we discuss some problems of this standard The scenario further requires that the binary survives the formation scenario in its simplest form (Sect. 2) and show that spiral-in that follows upon the first Roche-lobe contact, i.e. that both the helium core and the 1 M companion star are smaller Send offprint requests to: Simon Portegies Zwart than their Roche lobes. The mass Mc and radius Rc of the helium 208 S.F. Portegies Zwart et al.: The formation of black-holes in low-mass X-ray binaries core of the primary can be computed with (see Iben & Tutukov 1985 and de Loore & Doom 1992, respectively) 1.42 Mc =0.073Mo , (2) and 2 log Rc = −1.13+2.26 log Mc − 0.78(log Mc) . (3) 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 M primaries we thus find a semi-major axis after the spiral- in which should exceed 4.0 and 6.3 R , respectively. In both Fig. 1. Lower limit to the initial semi-major axis at which the binary cases the most stringent limit is set by the main-sequence star. survives the spiral-in, as function of the mass of the primary. The upper The ratio of the semi-major axes before and after the spiral-in (lower) dashed line gives the limit determined from the condition that can be computed by comparing the binding energy of the pri- the secondary star (helium core) is smaller than its Roche-lobe, The mary’s envelope with the binding energy released by the shrink- solid line gives the upper limit at which the primary reaches its Roche ing binary (see Webbink 1984): lobe at its maximum radius of Rmax = 1000R . The secondary is assumed to be a 1 M star − a M 2a M 1 f = c 1+ i e . (4) a M αλR m i 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 Here ai and af 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 inside its Roche lobe. We see that even in this case low-mass X- at the onset of spiral-in. We set αλ =0.5. We assume that the ray binaries with a black hole cannot be formed in the standard primary did not lose any mass before it fills its Roche lobe: scenario. M = Mo and Me = Mo − Mc. For the 20 and 60 M primaries we thus find a semi-major axis before the spiral-in which should 3. Detailed model exceed 1470 and 3160 R , respectively. For the 20 M primary, the lower limit on the semi-major Since the stellar parameters used in the previous Sect. are rather axis for survival of the spiral-in is smaller than the upper limit rough, the same computation is performed with a more detailed for Roche-lobe contact and a low-mass X-ray binary can be model for population I (Z =0.02) stars. We use the models formed when the initial semi-major axis is between these two with moderate core overshooting computed by Schaller et al. limits. For the 60 M primary, the lower limit on the semi-major (1992), using the radius (calculated from effective temperature axis for survival of the spiral-in is larger than the upper limit for and luminosity) and the mass of the star in the tabulated points. Roche-lobe contact and no initial orbit leads to the formation of The mass of the star decreases as a function of time due to stellar a low-mass X-ray binary: Roche-lobe overflow always leads to wind. The mass loss in the stellar wind causes an increase of the a merger. Roche-lobe radius of the primary, (mainly) by increasing the Fig. 1 shows, as function of the mass of the primary, the semi-major axis and (to a lesser extent) by increasing the mass- lower and upper limits to the semi-major axis of the initial binary ratio m/M. The increase in the semi-major axis is described, at which the binary survives the spiral-in and the primary reaches assuming an isotropic wind with high velocity according to the the Roche lobe. Only for low mass primaries (M ∼< 22M ) does Jeans approximation, with (van den Heuvel 1983): the binary survive the spiral-in. Thus, if only stars with an initial a(M + m) = constant. (5) mass larger than 40 M (van den Heuvel & Habets 1984) form a black hole, the formation of a low-mass X-ray binary with The relation between the semi-major axis at which the pri- a black hole as a compact object is excluded in the standard ai mary fills its Roche lobe and the initial semi-major axis scenario.