In the analysis of BH formation, new important parameters appear. The first one is the threshold mass
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] –
; Woosley et al. [458] –
; Portegies Zwart, Verbunt, and Ergma [328] –
more than
. A simple physical argument usually put forward in the literature is that the mantle of
the main-sequence star with
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 [111
, 112].
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 the mass-range of
precursors of black holes is constrained to
, 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
are rather uncertain, especially for the most massive stars, mainly because of clumping in
the winds (see, e.g., [203, 70, 131]). 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 of the nascent BH. There are various studies as for what the
mass of the BH should be (see, e.g., [403
, 33
, 111, 114
]). 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
,
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
[297, 256, 346]. A continuous range of BH masses up to
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
by the fraction of the pre-supernova mass
that collapses into the BH:
. In fact, the pre-supernova mass
is directly related to
, 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
, where
itself depends on
. The parameter
can vary in a wide
range.
The third parameter, similar to the case of NS formation, is the possible kick velocity 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 [229
]) reads
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 ) 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
. 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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