The need for hypercritical accretion in massive black hole binaries with large Kerr parameters

Enrique Moreno M endez 0 0 Argelander-Institut fur Astronomie, Bonn University , Auf dem Hugel 71, 53121 Bonn, Germany A B S T R A C T Recent measurements of the Kerr parameters of the black holes in M33 X-7 and LMC X-1 yield a = 0.84 0.05 and a = 0.90+00..0049, respectively. We study massive binary evolution scenarios that can reproduce such high values for the Kerr parameters. We first discuss a model with Case C mass transfer leading to a common envelope and tidal synchronization of the primary before it collapses into a black hole. We also study a Case M evolution model (which involves tidally locked, rotationally mixed, chemically homogeneous stars in a close binary). Our analysis suggests that, regardless of the specific scenario, the observed Kerr parameters for the black holes in M33 X-7 and LMC X-1 had to be obtained through hypercritical mass accretion. 1 I N T R O D U C T I O N When a star accretes mass on to its surface, it releases energy in the form of photons. These in turn regulate the accretion rate to what is known, in the spherically symmetric case (or Bondi accretion; Bondi 1951), as the Eddington limit: M Edd = , where LEdd = is the Eddington luminosity, with mp the mass of the proton, T the Thompson cross-section of the electron, R and M the radius and mass of the star and es the opacity of the infalling material, which is likely to be ionized hydrogen (so = 0.4 cm2 g1) and is the efficiency for converting mass into (photon) energy via the accretion process. In general, if the accretion rate grows, the luminosity increases and self-regulates the accretion rate to values below the Eddington limit. However, if the accretion rate grows to values which exceed this limit by a couple of orders of magnitude, then photons become trapped. Chevalier (1981, 1989, 1990) and later Brown & Weingartner (1994) found that when the diffusion time-scale of the photons generated by the accretion process is longer than the dynamical timescale of the accreting material, the photons become trapped and a shock forms (at rsh) inside the photon-trapping radius (rph). The shock diminishes the kinetic energy of the accreting material by a factor of 50 and converts it into thermal energy. When the temperature reaches T 1 MeV, e e+ pairs are created. These pairs annihilate into neutrinoantineutrino pairs which can easily escape and allow the transport of energy out of the accreting black hole (BH). This prevents the accretion luminosity from exceeding the photon LEdd as long as rsh < rph. However, M Edd is exceeded by a factor of 104 (Brown 1995). This is known in the literature as hypercritical mass accretion. Mass accretion into a BH instead of on to a neutron star (NS), such as in the cases calculated by Chevalier (1981, 1989, 1990) and Brown & Weingartner (1994), might make an even stronger case for hypercritical accretion. This is because for a BH there is no surface on to which the infalling material can collide and radiate, but rather an event horizon through which matter passes uninhibitedly. Therefore the Eddington luminosity depends strictly on the efficiency to produce photons by the accreted material as it falls towards the event horizon. On top of this, most scenarios, at least in binaries, do not involve spherical mass accretion but rather accretion through a disc, so the Eddington limit might not even be the most accurate prescription. On the other hand, the material being accreted has angular momentum which must be lost before it can reach the event horizon of the BH. Angular momentum loss from the accreting material is an important problem that will need to be addressed. This is beyond the scope of this paper. Here we will only point out the need of hypercritical mass accretion in models which evolve massive binaries into the observed BH binaries, such as those of Lee, Brown & Wijers (2002), or De Mink et al. (2008, 2009a,b), Moreno Mendez et al. (2011). Lee et al. (2002) and later Moreno Mendez et al. (2011) have modelled the evolution of 15 Galactic BH binaries. They start from wide binaries (i.e. the initial orbital separation is ai 1500 R ) allowing the primary star in the binary to evolve as if it were a single star until it starts helium-shell burning. At this point, the primary fills its Roche lobe (RL) and starts to transfer mass to the secondary. Mass transfer during He-shell burning is known in the literature as Case C mass transfer (e.g. Van den Heuvel 1994). The mass transfer to the less massive secondary star shrinks the orbit until a common envelope sets in. The secondary star spirals in while expelling the hydrogen envelope of the primary until the Roche lobe overflow (RLOF) is stopped and the two stars orbit each other in a much tighter orbit (a few R ). In such an orbit, and within a time-scale of 102104 yr, the helium star becomes tidally synchronized before it collapses into a BH. Therefore the spin periods Pspin of the binary components coincide with the orbital period Porb. Assuming angular momentum is conserved during the collapse, knowing the orbital period allows a good estimate of the natal Kerr parameter of the BH, a = Jc/GM2 (see e.g. fig. 1 of Brown, Lee & Moreno 2007), where J is the angular momentum and M is the mass of the collapsed object. Here we will assume that the star is rotating as a solid body at the moment of collapse.1 This provides an upper limit to the available angular momentum, which translates into an upper limit to the natal Kerr parameter. Lee et al. (2002) and Moreno Mendez et al. (2011) estimated the Kerr parameters of 15 Galactic BH binaries. This was done by modelling the evolution of the orbital periods from their current to their pre-explosion conditions. Measurements of the Kerr parameters on several Galactic BHs as well as LMC X-1 and M33 X-7 have been performed by fitting the X-ray continuum. This was done with a fully relativistic model of a thin disc around a Kerr BH whose plane is that of the binary and which assumes that the inner edge of the disc is at the innermost stable circular orbit. These measurements rely squarely on the model of the disc and still need confirmation via other methods. However, at present, they are the only available data and we will assume they have been determined meaningfully. Among the measurements of the Kerr parameters of Galactic BHs there are those of GRO J165540 (XN Sco 94), 4U 154347 (Il Lupi) (Shafee et al. 2006) and more recently, Steiner et al. (2010) have measured the Kerr parameter of XTE J1550564, utilizing the aforementioned method as well as by modelling the Fe K line shape. The match between the measurements of these three systems and the predictions suggests that the model of Lee et al. (2002) and Moreno Mendez et al. (2011)2 represents a viable approach to study the formation and evolution of the Kerr parameter in BH binaries. Extending the model from the Galactic binaries to the massive binary in M33 X-7, Moreno Mendez et al. (2008) conc (...truncated)