4.2 Black hole formation parameters

So far, we have considered the formation of NSs and binaries with NSs. It is believed that very massive stars end up their evolution with the formation of stellar mass black holes. We will discuss now their formation.

In the analysis of BH formation, new important parameters appear. The first one is the threshold mass Mcr beginning from which a main-sequence star, after the completion of its nuclear evolution, can collapse into a BH. This mass is not well known; different authors suggest different values: van den Heuvel and Habets [428]40 M ⊙; Woosley et al. [458]60M ⊙; Portegies Zwart, Verbunt, and Ergma [328] – more than 20 M ⊙. A simple physical argument usually put forward in the literature is that the mantle of the main-sequence star with M > Mcr ≈ 30M ⊙ before the collapse has a binding energy well above 1051 erg (the typical supernova energy observed), so that the supernova shock is not strong enough to expel the mantle [111Jump To The Next Citation Point112].

The upper mass limit for BH formation (with the caveat that the role of magnetic-field effects is not considered) is, predominantly, a function of stellar-wind mass loss in the core-hydrogen, hydrogen-shell, and core-helium burning stages. For a specific combination of winds in different evolutionary stages and assumptions on metallicity it is possible to find the types of stellar remnants as a function of initial mass (see, for instance [140]). Since stellar winds are mass (or luminosity) and metallicity-dependent, a peculiar consequence of mass-loss implementation in the latter study is that for Z ≃ Z ⊙ the mass-range of precursors of black holes is constrained to M ≈ (25 –60) M ⊙, while more massive stars form NSs because of heavy mass loss. The recent discovery of the possible magnetar in the young stellar cluster Westerlund 1 [265] hints to the reality of such a scenario. Note, however, that the estimates of ˙ M are rather uncertain, especially for the most massive stars, mainly because of clumping in the winds (see, e.g., [20370131]). Current reassessment of the role of clumping generally results in the reduction of previous mass-loss estimates. Other factors that have to be taken into account in the estimates of the masses of progenitors of BHs are rotation and magnetic fields.

The second parameter is the mass MBH of the nascent BH. There are various studies as for what the mass of the BH should be (see, e.g., [403Jump To The Next Citation Point33Jump To The Next Citation Point111114Jump To The Next Citation Point]). In some papers a typical BH mass was found to be not much higher than the upper limit for the NS mass (Oppenheimer–Volkoff limit ∼ (1.6– 2.5)M ⊙, depending on the unknown equation of state for NS matter) even if the fall-back accretion onto the supernova remnant is allowed [403]. Modern measurements of black hole masses in binaries suggest a broad range of BH masses of the order of 4– 17M ⊙ [297256346]. A continuous range of BH masses up to 10 –15 M ⊙ was derived in calculations [114]. Since present day calculations are still unable to reproduce self-consistently even the supernova explosion, in the further discussion we have parameterized the BH mass MBH by the fraction of the pre-supernova mass M ∗ that collapses into the BH: kBH = MBH ∕M ∗. In fact, the pre-supernova mass M ∗ is directly related to Mcr, but the form of this relationship is somewhat different in different scenarios for massive star evolution, mainly because of different mass-loss prescriptions. According to our parameterization, the minimal BH mass can be min M BH = kBHM ∗, where M ∗ itself depends on Mcr. The parameter kBH can vary in a wide range.

The third parameter, similar to the case of NS formation, is the possible kick velocity wBH imparted to the newly formed BH (see the end of Section 3.4). In general, one expects that the BH should acquire a smaller kick velocity than a NS, as black holes are more massive than neutron stars. A possible relation (as adopted, e.g., in calculations [229Jump To The Next Citation Point]) reads

wBH M ∗ − MBH 1 − kBH -----= -----------= -------------, (55 ) wNS M ∗ − MOV 1 − MOV ∕M ∗
where MOV = 2.5 M ⊙ is the maximum NS mass. When MBH is close to MOV, the ratio wBH ∕wNS approaches 1, and the low-mass black holes acquire kick velocities similar to those of neutron stars. When MBH is significantly larger than MOV, the parameter kBH = 1, and the BH kick velocity becomes vanishingly small. The allowance for a quite moderate wBH can increases the coalescence rate of binary BH [229].

The possible kick velocity imparted to newly born black holes makes the orbits of survived systems highly eccentric. It is important to stress that some fraction of such binary BH can retain their large eccentricities up to the late stages of their coalescence. This signature should be reflected in their emitted waveforms and should be modeled in templates.

Asymmetric explosions accompanied by a kick change the space orientation of the orbital angular momentum. On the other hand, the star’s spin axis remains fixed (unless the kick was off-center). As a result, some distribution of the angles between the BH spins and the orbital angular momentum (denoted by J) will be established [331]. It is interesting that even for small kicks of a few tens of km/s an appreciable fraction (30 – 50%) of the merging binary BH can have cosJ < 0. This means that in these binaries the orbital angular momentum vector is oriented almost oppositely to the black hole spins. This is one more signature of imparted kicks that can be tested observationally. These effects are also discussed in [181].

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