###### Abstract

The collapse of massive stars is one of the most-studied paths to black hole
formation. In this chapter, we review black hole formation during the collapse
of massive stars in the broader context of single and binary stellar evolution
and the theory of supernova explosions. We provide a concise overview of the
evolutionary channels that may lead to black hole formation – the classical
route of iron core collapse, collapse due to pair instability in very massive
stars, and the hypothetical scenario of supermassive star collapse. We then
review the current understanding of the parameter space for black hole
formation and black hole birth properties that has emerged from theoretical
and computational modelling of supernova explosions and transient
observations. Finally, we discuss what the intricate interplay between stellar
evolution, stellar explosions, and binary interactions implies for the
formation of stellar-mass black holes.

##  Chapter 0 Black holes as the end state of stellar evolution:
Theory and simulations

###  1 Introduction

The concept of black holes has long been intimately tied to the theory of
stellar evolution. While black hole spacetimes had been discovered and
discussed as mathematical curiosities already shortly after the theory of
general relativity was formulated, they were first seriously considered as
astrophysical objects in the context of stellar collapse of sufficiently
massive stars, most notably in the seminal work of Oppenheimer and Snyder[1].
Stellar mass black holes formed as the end state of massive stars were also
the first ones to be discovered (Cygnus X-1[2]).

Since the early days of general relativity and even since the days of
Oppenheimer and Snyder, our understanding of stellar evolution and black hole
formation has, however, changed considerably. Today, the black holes are well
understood from the fundamental perspective of mathematical relativity, but
the astrophysics of stellar-mass black hole formation still poses many
deceptively simple questions despite considerable advances over the last few
decades: Which massive stars form black hole, which ones form neutron stars?
Do black holes form quietly, or are they sometimes born in supernova
explosions and perhaps receive a “kick” in the process? How fast do newly born
black holes spin? How do the fates of massive stars and the properties of
black holes depend on their environment and how do they shape their
environment through feedback processes?

In order to convey a comprehensive picture of our current theoretical
understanding of black holes as the end states of stellar evolution, it is
useful to approach the problem from three different angles. In this Chapter,
we first review the evolution of massive single stars up to the point of
collapse in Section 2. We next discuss the current state of supernova theory
with a focus on the outcomes of the collapse (neutron star formation, black
hole formation with and without a supernova explosion) in Section 3. The
closely related topic of black hole birth properties is discussed in Section
4. Section 5 reviews the interplay between black hole formation and binary
evolution. A brief summary can be found in Section 6.

###  2 Single star evolution up to the supernova

Figure 1: Cartoon sketch of remnant mass (red line) and stellar mass at “time
of remnant formation” (bluxe line) as functions of initial mass for non-
rotating single stars of about solar composition. For stars of initial mass up
to $\sim 8\,M_{\odot}$ we show, crudely, the mass before the onset of the
asymptotic giant branch (AGB) phase. They leave behind carbon-oxygen (CO) or
neon-magnesium-oxygen (NeO) white dwarfs. Dash-double-dotted lines indicate
the masses of the helium and CO cores at that evolution stage. At an initial
mass of about $30\,M_{\odot}$ the hydrogen envelope is lost due to stellar
winds and the star becomes a Wolf-Rayet star (e.g., early-type Wolf-Rayet
stars: WE, which could be nitrogen-rich sub-type WNE, carbon-rich sub-type WC,
or oxygen-rich sub-type WO) prior to explosion - making Type I b/c supernovae,
otherwise the star may explode as hydrogen-rich Type II supernova. At higher
masses the star may have strong mass loss already during the hydrogen-burning
phase undergoing evolution as late-type nitrogen-rich Wolf-Rayet (WNL) star or
a luminous blue variable (LBV). At high initial mass the mass loss rates are
highly uncerffig:tain and hence the final outcome is not reliably predicted,
hence we use dashed lines. Regimes of black hole formation may be interspersed
with islands of neutron star formation even at relatively high masses. We
indicate one such island as a representation in this cartoon figure.

The life and fate of stars is predominately determined by their mass at birth
(Figure 1). Forming from a cloud of usually mostly molecular gas, objects with
at least around $0.08$ times the mass of the sun (${\mathrm{M}_{\odot}}$) can
ignite hydrogen burning in their cores. Stars with initial masses of $\lesssim
1\,{\mathrm{M}_{\odot}}$ experience hydrogen burning powered by the proton-
proton (PP) chains[3], above that by the carbon-nitrogen-oxygen (CNO)
cycle[3].

The low-mass stars are faint red dwarf stars that can live a very long time:
below a mass of $0.8\,{\mathrm{M}_{\odot}}$ they live as long as the current
age of the universe; if not destroyed by some interaction, all single stars
below this mass limit ever formed are still around. Of these, stars with
initial mass of up to $\sim 0.6\,{\mathrm{M}_{\odot}}$ will end their lives as
helium white dwarfs; more massive stars can ignite helium burning, either in a
core helium flash (up to $\sim 2\,{\mathrm{M}_{\odot}}$ initial mass) or in
less violent manner. Stars with initial masses of up to $\sim
6\,{\mathrm{M}_{\odot}}$ develop a degenerate carbon-oxygen (CO) core and
leave behind a CO white dwarf (WD) of up to $1.1\,{\mathrm{M}_{\odot}}$[4].

More massive stars ignite carbon burning in their core. Stars with initial
masses of up to $\sim 8\,{\mathrm{M}_{\odot}}$ cannot ignite further burning
stages and leave behind oxygen-neon-magnesium (ONeMg) WDs of up to $\sim
1.4\,{\mathrm{M}_{\odot}}$, just below the Chandrasekhar mass (Eq. 1 with
$Y_{\mathrm{e}}\sim 0.5$ and $s\ll 1\,k_{\mathrm{B}}/\mathrm{nucleon}$). Just
above this upper mass limit for WD formation the stellar evolution can become
very complicated, e.g., leading to the formation of electron-capture
supernovae[5] or various off-centre advanced burning stages that may result in
violent burning flashes, but usually lead to the formation of an iron core
that undergoes core collapse[6] (§ 1). Above an initial mass of $\sim
10\,{\mathrm{M}_{\odot}}$ the core becomes massive enough for stellar
evolution to proceed in a more regular way (“textbook” case of Figure 2), also
making an iron core that undergoes core collapse, leaving behind a neutron
star or a black hole[7]. We refer to such stars that make an iron core and
collapse as _massive stars_[8].

At sufficiently low metallicity such that mass loss through winds can be
neglected, we may expect the following mass regimes. If the initial stellar
mass exceeds $\sim 90\,{\mathrm{M}_{\odot}}$, the stars encounter an
instability in the equation of state after core carbon burning due to the
production of electron-positron pairs, the _pair instability_[9]. Stars above
this mass limit we call _very massive stars_[8]. The pair instability cause
thermonuclear powered pulses that expel the outer layers of the star[10]. We
expect that these star usually should leave behind massive stellar black holes
of up to $\sim 45\,{\mathrm{M}_{\odot}}$. For stars with initial masses above
$\mathord{\sim}150\,{\mathrm{M}_{\odot}}$ the pulses can become so violent
that the entire star is disrupted, usually during the first pulse, resulting
in powerful supernova and no remnant[11]. Stars with initial masses above
$\sim 250\,{\mathrm{M}_{\odot}}-300\,{\mathrm{M}_{\odot}}$ encounter an
instability due to photo-disintegration of heavy nuclei and helium during the
pair-instability pulse, and collapse to a black hole rather than
exploding[11]. The resulting back hole masses are expected to be at least
$130\,{\mathrm{M}_{\odot}}$. Observations of high-red shift quasars have lead
to the speculation of the formation of primordial stars with the most extreme
masses[12]. For such stars of primordial (i.e., Big Bang) composition, there
is a hydrostatic upper mass limit of around
$150\mathord{,}000\,{\mathrm{M}_{\odot}}$ due to a general relativistic
instability[13, 14, 15]. These stars would collapse to black holes on a
thermal timescale, and stars beyond this mass limit we refer to as
_supermassive stars_.

####  1 Massive star evolution

Figure 2: Kippenhahn Diagram of the evolution of a non-rotating star of
initially $17\,{\mathrm{M}_{\odot}}$ and solar initial composition.[16] The
$x$-axis shows the logarithm (base 10) of the remaining time in years until
core collapse (core bounce) and the $y$-axis the enclosed mass (mass
coordinate) for a given spherical shell. The total mass of the star is
indicated by a solid black line in the upper part of the figure; it is reduced
by mass loss due to stellar winds. Note that the mass loss rate is actually
increasing toward the late evolution stages, however, the use of the
logarithmic time scale on the $x$-axis stretches out the curve. Green hatching
indicates convective regions - energy is transported by convection, and at the
same time keeps the connected regions close to chemically homogeneous. At the
end of the evolution the star becomes a red supergiant (RSG) and develops an
extended convective envelope. Blue shading indicates energy generation due to
nuclear burning. Purple shading indicates net energy loss - energy carried
away by neutrinos. For both, energy production and loss, each level of shading
corresponds to an increase of the specific energy generation or loss rate by
one order of magnitude. Black labels inside the figure indicate the respective
burning phase, starting with hydrogen burning (H), followed by helium burning
(He), then carbon (C), neon (Ne), oxygen (O) and finally (Si) burning. When
the fuel is depleted in the code, the burning can re-ignite in a shell. We
have marked the shells as such for hydrogen (H shell), helium (He shell), and
silicon (Si shell); for carbon, neon and oxygen we just use the chemical
symbol for both core and shell burning due to space constraints. Note that the
burning itself is usually highly concentrated toward the bottom of a
convective burning shell. After core silicon burning, this specific star
experienced a merger of the second oxygen shell with the neon and carbon
shells above (C+Ne+O shell merger). This has significant impact on the
structure of the star at the time of core collapse, with a fairly small core
of $1.6\,{\mathrm{M}_{\odot}}$ surround by a high-entropy layer of low
density, with the likely outcome being a neutron star not a black hole.
Earlier in the evolution, there is a sequence of thin convective regions -
each framed by a solid line - above the hydrogen- and helium-burning cores.
These are a result of semiconvection, which may cause only a modest amount of
mixing.

From its formation, the evolution of a massive star can be sketched as a
continuing path toward increasingly higher central temperature and density,
while central entropy decreases. This overall path, however, is interrupted by
stages in thermal equilibrium, powered by nuclear burning. This nuclear
burning, in turn, can trigger local instability and drive convection. In the
advanced burning stages, when the temperature exceeds some
$10^{9}\,\mathrm{K}$, energy loss due to thermal neutrinos can drive the
evolution timescale. Figure 2 shows the evolution of these quantities as a
function of time and of location inside the star. The different burning stages
in a massive star, in sequence, are:

  * •

Hydrogen burning. In massive stars this is powered by the CNO cycle,
12C(p,$\gamma$)13N($\beta^{+}$)13C(p,$\gamma$)14N(p,$\gamma$)15O($\beta^{+}$)15N(p,$\alpha$)12C
and converts hydrogen to helium, releasing about $6.5\,\mathrm{MeV}$ per
nucleon in nuclear binding energy.111We use a notation common in nuclear
astrophysics to show the reaction. Heavy ions are usually shown outside the
bracket. Within the bracket there is either one group or two groups separated
by a comma. If there is only one group, it either shows what is emitted due to
a spontaneous process or the process itself. For example, $\beta^{+}$ decay
emits a positron and an electron neutrino, which we also could have written as
$e^{+}\nu_{\mathrm{e}}$. If there are two groups separated by a comma, the
first group is the “ingoing” channel, i.e., the reactant(s), and the second
group is the “outgoing” channel, i.e., products. In case there is only an
ingoing channel, it is customary to write $\gamma$ for the outgoing channel.
About $7\,\%$ of that energy is radiated away in the form of neutrinos. This
energy generation is about one order of magnitude larger per unit mass than
any of the later nuclear burning stages, while the star typically also is less
luminous, making core hydrogen burning, often referred to as the main
sequence, the longest evolutionary phase of massive stars. When hydrogen is
exhausted in the core, it usually continues to burn in a shell outside the
helium core. In the advanced burning stages beyond helium burning (see below),
the hydrogen shell may become inactive, it may be “dredged up” (mixed with) a
convective envelope at the low-mass end for core collapse supernovae, of be
lost due to stellar winds or eruptions in the most massive stars.

  * •

Helium burning. This phase is started by the “triple alpha” reaction,
$3\,{}^{4}$He$\;\rightarrow{}^{12}$C. As 12C accumulates and 4He is depleted,
the 12C($\alpha$,$\gamma$)16O reaction starts to dominate. Since both
reactions have about the same temperature dependence at typical helium burning
conditions but different dependence on density, the resulting final carbon-to-
oxygen ratio varies. Low mass stars have higher densities and produce a larger
carbon mass fraction. The outcome also depends sensitively on the uncertainty
in these two reaction rates, which still have relevant experimental
uncertainties. At the very end of core helium burning, some trace of 20Ne or
heavier alpha nuclei (nuclei that are multiples of alpha particles) may be
made. After helium burning is depleted in the core, it can re-ignite in a
shell outside the CO core. For sufficiently large initial stellar masses, that
shell can become convective and even entrain some hydrogen from the envelope.
At the low-mass end for supernovae, instead, the shell may be dredged up by
the envelope (see above). As helium burning releases only about $10\,\%$ of
the energy per nucleon compared to hydrogen burning, the core helium burning
phase typically lasts about only $10\,\%$ of the time of core hydrogen
burning. There is some contribution of shell hydrogen burning during that
phase, but the stars also are typically more luminous.

  * •

Carbon burning. Carbon burning predominately starts by the
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ reactions. The resulting compound
nucleus of 24∗Mg excited state de-excites by particle emission of neutrons,
protons, and alpha particles.222The asterisk indicates an excited state of the
24Mg nucleus. These make secondary reactions. The typical outcome is
production of 20Ne and 16O. Most importantly, whether core carbon burning
proceeds as a convective phase, as in Figure 2, or radiative depends on the
carbon mass fraction left behind by core helium burning. When the burning is
convective, there is more time for both loss of entropy due to neutrinos as
well as for weak decays leading to a lower proton-to-neutron ratio (i.e.,
lower $Y_{\mathrm{e}}$, see below). This leads to a transition in the pre-
supernova structure, with typically stars on the high-mass side of this
transition being less likely to explode. At the transition itself, we find a
sequence of many small shells, leading to many discontinuous changes in the
stellar structure around the transition mass, which is about
$18\,{\mathrm{M}_{\odot}}$ for the solar composition models of Reference 16.
During core carbon burning, the star emits about $10\mathord{,}000$ times more
energy in neutrinos than in visible light. From this point on, the star has
effectively become a _neutrino star_. At the low-mass end of the core collapse
mass regime, carbon may ignite off-centre and burn inward in a convectively-
bounded flame. [6]

  * •

Neon burning. This phase is powered by a pair of reactions,
20Ne($\gamma$,$\alpha$)16O and 20Ne($\alpha$,$\gamma$)24Mg, effectively
burning 20Ne to 24Mg and 16O. The first of the two reactions is endothermic,
but the second reaction makes up for that by releasing about twice as much
energy as is needed to trigger the first reaction. It is usually a rather
brief and “flashy” phase, being induced by a photo-disintegration reaction
that causes strong positive self-feedback. It occurs briefly before each of
the oxygen burning phases, and often burns with the convective region less
extended in mass than the later oxygen burning. Similarly to carbon burning,
neon and subsequently oxygen may ignite off-centre and burn inward in a
convectively-bounded flame at the low-mass end of the core collapse mass
regime.[6]

  * •

Oxygen burning. The nuclear reactions powering oxygen burning proceed very
similar to carbon burning. The ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ reaction
produces a compound nucleus of 32∗S that predominately de-excites by particle
emission of neutrons, protons, and alpha particles that then induce secondary
reactions. The outcome is a mixture overwhelmingly consisting of 28Si and 32S.
The small mass fraction 24Mg, usually around $10\,\%$, left by neon burning is
consumed at the beginning of oxygen burning by photo-disintegration reactions.
With a typical oxygen mass fraction of around $80\,\%$, the phase is
relatively powerful and extended compared to neon burning. There are usually
one or two oxygen burning shells prior to core collapse, and during collapse
oxygen burning can become “explosive” with burning timescales of a fraction of
a second. It typically sets a specific entropy of $\gtrsim
4\,k_{\mathrm{B}}/\mathrm{nucleon}$, associated with a jump in density that
can have a critical role in inducing the neutrino-powered core collapse
supernova mechanism in some mass ranges.

  * •

Silicon burning. Silicon burning is dominated by a sequence of photo-
disintegrations and $\alpha$ captures. It usually lasts for just days. During
the burning, electron captures convert protons (inside nuclei) into neutrons,
decreasing the electron fraction,333This equals the fraction of protons
relative to all nucleons (neutrons and protons). $Y_{\mathrm{e}}$, below
$0.5$, and leaving behind a mixture of iron group isotopes. For not too
massive stars, the core silicon burning usually comprises a convective core of
about $1.05\,{\mathrm{M}_{\odot}}$, followed by at least one silicon burning
shell. At the low-mass end, below $\sim 12\,{\mathrm{M}_{\odot}}$ initial
mass, more complicated burning sequences may occur, e.g., silicon shell
burning igniting not at the bottom of silicon shell, leaving behind a layer of
unburnt silicon between two layers of iron.

  * •

Iron core collapse. The silicon continues to burn in shells until the critical
mass for collapse (Eq. 1) is exceeded. At this stage the iron core is very hot
and in nuclear statistical equilibrium (NSE), i.e., nuclear reactions are very
fast compared to the evolution time-scale of the star. Then further electron
captures combined with photo-disintegration soften the equation of state and
lead to the collapse of the iron core at about a quarter of the free-fall
acceleration. From the time that an infall velocity of $1\mathord{,}000\,$km/s
is reached, core bounce typically ensues within a fraction of a second.

During the collapse, remaining layers of silicon as well as the bottom of the
oxygen layer may undergo very fast, “explosive” burning on a timescale shorter
than the collapse timescale or any convective or sonic timescale. This may
seed large asymmetries in the infall flow.

Nuclear burning predominately proceeds in distinct convective layers, which
has a pronounced effect on distribution of outcomes as a function of initial
mass. For example, if at the end of silicon burning the iron core mass was
just below the critical mass for collapse, another shell of convective silicon
burning would be required, and, due to its finite mass, would lead to a much
larger iron core, well above the critical mass. A discontinuous jump occurs.
Similarly, earlier burning stages and their shells require a minimum mass for
ignition, leading to many jumps in properties of the pre-supernova structure
and the resulting outcomes (Figure 6). In particular, the transition of core
carbon burning from (i) being convective in large shells, to (ii) a sequence
of very small convective shells, to (iii) radiative burning at $\sim
18\,{\mathrm{M}_{\odot}}$ initial mass, and the non-linear impact of these
shells on the subsequent more advanced core and shell burning stages, leads to
many changes within a small range of initial mass, resembling almost a
“chaotic” behaviour444although not “chaotic” in the mathematical sense.
Stellar evolution solves a system of tightly-coupled non-linear partial
differential equations with non-trivial inhomogeneous functions, and hence a
first-principle prediction of outcomes such as “compactness” at the
presupernova stage is difficult. That said, the predictions between different
codes do usually agree well given the same input physics[17, 18].

Figure 3:  Similar to Figure 1 but for Population III stars - stars made of
pristine primordial material as left behind directly by the Big Bang. In
contrast to solar-composition stars, Population III stars may lose less mass
due to stellar winds. They reach large core masses at the end of their
evolution and can encounter the pulsational pair instability for initial
masses between $90\,M_{\odot}$ and $140\,M_{\odot}$ solar masses that sheds
the outer layers down to some threshold (Figure 8), and full pair instability
between $140\,M_{\odot}$ and $260\,M_{\odot}$ that may not leave behind any
remnant at all. For higher masses, the star collapses directly to a large
black hole. Above an initial mass of some $150\mathord{,}000\,M_{\odot}$ (not
shown) there does not exist a long-lived solution of a star in thermal
equilibrium. These “supermassive” stars would collapse into supermassive black
holes that could be the ideal seeds for quasars at high redshift.  Figure 4:
Schematic for regimes of compact remnants of single stars as a function of
initial mass ($x$-axis) and metallicity ($y$-axis). The metallicity scale is
approximate due largely to the uncertainty in stellar mass loss. We show white
dwarfs (white shading), neutron stars (light blue shading), black holes due to
fallback after an initial explosions (light red shading), direct collapse
(black shading), and a regime of no remnants (light gray shading) due to pair
instability supernovae. The green line shows where mass loss may remove the
hydrogen-rich envelope prior to explosion of the star; for low metallicity
between $100\,M_{\odot}$ and $140\,M_{\odot}$ initial mass it is due to
pulsational pair instability supernovae, otherwise due to stellar winds.

####  2 Stellar evolution physical parameters and uncertainties

In this section we briefly discuss some key dependencies on input physics for
single massive star evolution and their uncertainties. Each of these topics
could fill books by themselves. Uncertainties due to binary evolution are
discussed in Section 5.

##### Metallicity and Mass loss

Next to the initial mass of the star, metallicity is one of the key factors
impacting stellar evolution, stellar populations and outcomes. For example,
the first generation of stars forming from metal-free pristine gas as left
over by the Big Bang, so-called Population III stars, would not have strong
molecular cooling from complex molecules, resulting in higher Jeans mass and
hence typically making more massive stars than what we find in the more metal-
rich present-day universe[19, 20, 21]. Whereas the actual typical initial mass
function, which may depend on environment, is still subject to much debate,
some constraints have already become apparent: for example, we have not found
any Population III stars to date, whereas all single stars with initial masses
of $0.8\,M_{\odot}$ or below should still be around today as their lifetime
would exceed the current age of the universe. This could be taken as an
indication that the first generation of stars was typically more massive than
stars today, although this fact provides no actual constraints on the more
massive stars and black hole production.

Beyond the potentially different initial mass function of Population III
stars, their evolution is also different. Massive Population III stars, which
burn hydrogen via the CNO cycle, have to first produce carbon by the triple
alpha reaction in a primary fashion. They contract to high enough densities
and temperatures to start the helium fusion, until a carbon mass fraction of
$\lesssim 10^{-9}$ is reached. Then they produce enough energy by hydrogen
burning to balance energy losses from the surface and halt contraction. The
lack of initial metals, when preserved in the surface layers, likely leads to
much reduced mass loss due to stellar winds. The stars may retain most of
their initial mass and may follow different evolutionary paths to stars of
solar initial composition, for example, producing pair instability supernovae
(see below) or more massive black holes. Figure 3 depicts a schematic for the
potential evolution of non-rotating single Population III stars[22].

Other than some peculiarities in the burning of the metal-free and very low
metallicity stars, mass loss is one of the key drivers for different evolution
pathways and outcomes for massive stars: it reduces the mass of the star,
shrinking the resulting core sizes, or even ejects the entire hydrogen
envelope, changing the observational astronomical supernova type as well as
remnant type and mass (Figure 4)[23].

The mass loss itself, however, is still highly uncertain. Significant efforts
exist to constrain them from theory and observation. Asymmetries and clumping
in winds, dust in red supergiants[24, 25], line driving in hot stars and
optically thick winds[26, 27, 28, 29, 30, 31, 32, 33] with their metallicity
dependence[34, 35], episodic mass loss such as luminous blue variables (LBV)
or giant eruptions such as observed in Eta Carina (Figure 1)[36, 37, 38]
constitute sizeable challenges in these efforts.

##### Mixing, rotation, and magnetic fields

Mixing and transport processes play a key role in the evolution of stars.
Foremost among these is the transport of energy due to radiation, conduction,
or advection by fluid flows such as convection when other processes
(radiation, conduction, mechanical work, and neutrino losses) are
insufficient. Most critically, fluid flows also lead to transport and mixing
of different layers in the star, bringing fuel into burning regions and the
products of nuclear burning to the surface of the star. Whereas convection is
reasonably well described by the mixing length theory[39], many questions
remain with regards to double diffusive instabilities such as
semiconvection[40, 41, 42] and thermohaline convection[43, 44, 45, 46].
Semiconvection and thermohaline convection occur in dynamically stable
regions, with no Rayleigh-Taylor instability, but with buoyancy due to
composition gradients and thermal gradients pointing in opposite directions,
with the stabilizing gradient dominating. They are called “double diffusive”
instabilities because the diffusion coefficients for “heat” and “composition”
are vastly different in typical stellar conditions, with heat diffusing much
faster than chemical species (atomic nuclei). In particular, in
semiconvection, a destabilizing temperature gradient is stabilised against
dynamic instability by a larger stabilising composition gradient. Secular
instability leads to layer formation and eventually mixing driven by a slow
exchange though the layer boundaries and through the merging of the layers. In
thermohaline convection, a destabilising composition gradient (“heavy”
material above “lighter” material) is stabilised against dynamical instability
by a larger stabilising temperature gradient. “Fingers” of larger chemical
buoyancy may form and cool as they sink, though coherent structures, may be
destroyed by turbulence and (differential) rotation[46]. These mixing and
transport processes, however, are inherently three-dimensional with a vast
range of scales, requiring theoretical insight to model them in lower
dimensions over the entire evolution of a star.

A further quantity that is transported in stars is angular momentum. Rotation
can be an important aspect of stellar evolution[47]. In particular, massive
stars may be spinning rapidly throughout much of their lives. In close and
interacting binary stars, stellar and (vast) orbital angular momentum can be
exchanged, which can spin stars up or down. For single stars, mass loss due to
stellar winds can lead to significant braking as the surface layer of the star
has the highest specific moment of inertia. For magnetic stars – usually stars
with convective envelopes such as the Sun – magnetic fields force escaping
wind particles to remain in co-rotation with the surface out to large
distances from the star, which results in particularly efficient loss of
angular momentum. This process is known as magnetic braking[48]. Magnetic
fields in the stellar interior can also have significant impact on angular
momentum transport[49] and the resulting final spin of the stellar core at the
time of core collapse[50, 51].

Stellar rotation deforms the stars, leading to different temperature gradients
from the core to the surface at the pole than at the equator. This can drive
fluid flows, so-called meridional circulation due to its axisymmetry[52]. For
very rapid rotation, the mixing can be faster than the nuclear burning,
leading to chemically homogeneous evolution[53, 54]. When the star reaches the
end of core hydrogen burning, also outer layers are also depleted in hydrogen,
altering stellar structure, mass, and angular momentum loss.

##### Nuclear physics uncertainties

Stellar structure and evolution are driven by nuclear physics. Nuclear physics
defines the different evolution stages. Nuclear structure – and hence nuclear
reactions – are very complicated strongly-interacting quantum many body
systems and hence accurate first-principle calculations of stellar structure
is quite challenging. On the other hand, nuclear physics experiments _in the
relevant – usually low-energy – regime_ to directly measure reactions are also
very challenging. They require very low background experimental environments.
This becomes clear when you consider that stars may take millions of years for
some burning phases, hence little happens during a human lifetime. The
uncertainties can become an issue when one reaches branching points in the
nuclear reaction flows or has competing processes, and this can have
significant impact on stellar nucleosynthesis.

Nuclear reaction rates have very high temperature sensitivity, e.g., $\sim
T^{40}$ for helium burning at typical hydrostatic helium burning temperatures.
This means that if, e.g., a rate was changed by a factor two, changing
temperature by a factor $2^{-1/40}$ or $1.75\%$ would result in the same
burning rate. For helium burning in particular, however, there are two
competing reactions, triple alpha and 12C($\alpha$,$\gamma$)16O, that
determine the carbon mass fraction at the end of core helium burning, which,
in turn, impacts the carbon burning phases and ultimately the final stellar
fate. Since both reactions have about the same temperature dependence at the
relevant temperatures, it is the difference in their density dependence that
also plays a role. Realistically, we would like to know these two rates to
within some 5% accuracy[55, 56] but measurements are hard[57].

Whereas carbon production is the most prominent and likely most impactful[58,
59, 60], other reactions also have their key roles. This includes light
reactions in the CNO cycles, branching points in carbon burning[61, 62], and
weak reaction rates in silicon burning and in the iron core[63, 64].

###  3 Stellar collapse leading to the formation of black holes

####  1 The core collapse supernova mechanism

For massive stars with helium core masses below the somewhat uncertain
threshold value for pair instability supernovae (Section 4), compact object
formation proceeds through core collapse after hydrostatic burning stages up
to the formation of an iron core. In addition, there may be a narrow channel
of less massive supernova progenitors that proceed through carbon burning to
form a degenerate O-Ne-Mg core and already undergo dynamical collapse at this
stage due to electron captures on ${}^{20}\mathrm{Ne}$ and
${}^{24}\mathrm{Mg}$ [65, 66, 67, 68]. This progenitor channel invariably
produces neutron stars, however. For low-mass non-rotating progenitors,
baryonic remnant masses are expected to be close to $1.37M_{\odot}$ (resulting
in a gravitational mass of about $1.25M_{\odot}$). In the special case of
accretion-induced collapse of rotating white dwarfs, neutron stars masses may
be higher [69]. As far as black-hole formation is concerned, we need only
consider the standard scenario of iron core collapse.

In progenitors with an iron core, collapse occurs once the degenerate core
reaches its effective Chandrasekhar mass, $M_{\mathrm{Ch}}$, which is given
by[70]

$M_{\mathrm{Ch}}=1.45M_{\odot}\left(\frac{Y_{\mathrm{e}}}{0.5}\right)^{2}\left[1+\left(\frac{s}{\pi
Y_{\mathrm{e}}k_{\mathrm{B}}/\mathrm{nucleon}}\right)^{2}\right]$ (1)

including finite-temperature corrections. Here $Y_{\mathrm{e}}$ is the
electron fraction, and $s$ is the specific entropy $s$ of the core. While
$Y_{\mathrm{e}}\approx 0.44$ does not vary strongly across progenitors,
variations in core entropy between $0.5\,k_{\mathrm{B}}/\mathrm{nucleon}$ for
the lowest-mass stars and $1.5\,k_{\mathrm{B}}/\mathrm{nucleon}$ at high
masses (close to the pulsational pair instability regime) lead to substantial
variations in the final iron core mass. The contraction of the core
accelerates into a runaway collapse on a free-fall timescale because electron
captures on heavy nuclei and the small number of free protons further reduce
the degeneracy pressure; in case of higher core entropy, the reduction of
radiation pressure by photodisintegration of heavy nuclei is also relevant.

At core densities of about $10^{12}\,\mathrm{g}\,\mathrm{cm}^{-3}$, neutrinos
become trapped and electron captures can no longer reduce the lepton number of
the core. At this stage, the electron fraction of the core has decreased to
$Y_{\mathrm{e}}\approx 0.25$ [71]. Due to the loss of lepton number, the
effective Chandrasekhar mass of the core shrinks during collapse. Only the
inner core maintains sonic contact and remains in homologous collapse until it
reaches and overshoots nuclear saturation density [72]. Due to the stiffening
of the equation of state above nuclear density, the inner core rebounds
(“bounce”), and a shock wave is launched as the rebounding inner core crashes
into the supersonically collapsing shell of the outer core. At core bounce,
the newly formed compact remnant is still small with a mass of around
$0.45M_{\odot}$ (somewhat dependent on the nuclear equation of state). In
modern supernova simulations using up-to-date stellar progenitor models, the
iron core collapse of massive stars therefore never results in prompt black
hole formation; there is always at least a transient proto-neutron star phase.

Figure 5: Sketch of the supernova core prior to shock revival in a neutrino-
driven supernova. A fraction of the neutrinos emitted from the proto-neutron
star (grey, cyan) are reabsorbed in the “gain region” behind the stalled shock
further out. Neutrino heating drives convection in the gain region, and in
addition, the standing accretion shock instability (SASI) can lead to large-
scale shock oscillations. Neutrino heating in conjunction with the violent
non-spherical fluid motions eventually drives runaway shock expansion.
Neutrino cooling also drives convection inside the proto-neutron star. Figure
from Müller (2017)[73] reproduced with permission, ©Cambridge University
Press.

As the shock propagates through the outer core, its energy is quickly drained
by dissociation of heavy nuclei in the infalling shells and by neutrino losses
[74, 75, 76]. Within milliseconds after bounce, the shock turns into an
accretion shock (i.e., the post-shock velocity becomes negative) that still
reaches a radius of $100\texttt{-}200\,\mathrm{km}$ and then retracts again.
Unless the shock is “revived”, ongoing accretion will eventually lead to black
hole formation (although part of the star may still be ejected in this case;
see Section 1).

Various mechanisms for shock revival have been explored in the literature. In
the neutrino-driven paradigm (Figure 5), the partial absorption of neutrinos
from the proto-neutron star behind the shock increases the post-shock pressure
to allow the shock to expand. Once the volume and mass of the heating region
have been increased sufficiently, a runaway feedback cycle of stronger heating
and shock expansion can occur [77, 78, 79]. Since most of the neutrino
emission feeds on the accretion power of the infalling material, the neutrino-
driven mechanism is somewhat self-regulating and will roughly pump energy into
the post-shock matter until the incipient explosion is sufficiently energetic
to terminate further accretion [16]. Except for the least massive supernova
progenitors, hydrodynamic instabilities such as buoyancy-driven convection
[80, 81] and shock oscillations [82] (standing accretion shock instability,
SASI) play a crucial role in supporting neutrino heating by providing
additional turbulent pressure [81, 83] and transporting hot material from the
proto-neutron star to the shock [84].

In the case of rapid progenitor rotation, an explosion may instead be driven
by magnetic fields that tap the rotational energy of the proto-neutron
star[85, 86, 87, 88](magnetorotational mechanism). Similar mechanisms may also
operate after black hole formation in collapsar disks (Section 3), but it is
convenient to distinguish the collapsar scenario from magnetorotational
explosions proper. The interplay of rotation and magnetic fields in the
progenitor and after collapse are still topics of ongoing research, and hence
it is far from clear when the magnetorotational mechanism can operate.
Nevertheless, some robust features of the magnetorotational mechanism can be
identified. The energy reservoir for magnetorotational explosions is
determined by the free rotational energy in _differential_ rotation of the
proto-neutron star on short timescales [86] and by its entire rotational
energy on longer timescales [85]. Simulations of successful magnetorotational
explosions tend to exhibit bipolar jet-like outflows [86, 87, 88] that suggest
an association with broad-lined Ic supernovae whose polarisation indicates a
nearly axisymmetric bipolar structure of the bulk of the ejecta[89, 90].

In recent years, phase-transition driven explosions have been considered as
yet another alternative explosion mechanism. In this scenario, a first-order
phase transition to strange matter or quark matter triggers a second collapse
of the proto-neutron star. If this collapse is stopped before black hole
formation and a second bounce occurs, a second shock wave can be launched and
expel the outer shells in a potentially very powerful explosion [91]. This
scenario hinges on uncertain assumptions about the nuclear equation of state,
however, and its robustness and viability is still under debate [92].

####  2 Parameter space for black-hole formation – theory and observations

Naively, one may expect black holes to form if and only if there is no
successful explosion. In reality, this is only a somewhat useful
approximation; black holes may sometimes form in successful explosions as well
as we shall discuss in Sections 3 and 1). Nonetheless, it is useful to first
focus on the question of successful shock revival first in order to approach
the systematics of black hole formation in core-collapse supernovae.

Detailed multi-dimensional radiation (magneto)-hydrodynamics simulations have
now matured to the point that many of them show successful shock revival [93,
94, 95, 96, 97, 98, 99, 100, 101, 102] and are able to produce explosion and
remnant properties broadly in line with observational constraints [96, 97,
103, 104]. This first-principle approach is still of limited use for
understanding the systematics of the progenitor-remnant connection for two
reasons. The immense computational costs of self-consistent three-dimensional
simulations only allow for a limited exploration of the progenitor parameter
space (mass, metallicity, multiplicity, rotation). Only a few dozen such
simulations have been performed so far by different groups. Furthermore,
first-principle simulations are still beset with uncertainties and still
cannot perfectly reproduce observational constraints [105]. For this reason,
simpler phenomenological models with an appropriate calibration and
observations remain the most suitable means for determining the progenitor-
remnant connection and the parameter space for black hole formation in
particular.

Phenomenological models to determine the “explodability” of supernova
progenitors have so far exclusively considered the neutrino-driven scenario
for shock revival. A number of studies have used one-dimensional models with
various neutrino transport treatments and artificially enhanced neutrino
heating to study the parameter space for neutron star and black hole
formation, and in some cases the remnant mass distribution as well [106, 107,
108, 109, 110, 111, 112, 113]. The problem has also been studied using
different (semi-)analytic approaches [114, 16]. Large-scale parameter studies
are also possible in axisymmetry (2D) already [115], but these can only cover
the initial phase of the explosion and the assumption of axisymmetry severely
impacts the dynamics of shock revival and of the explosion for such models to
be considered substantially superior to the aforementioned approaches.
Similarly, attempts to incorporate multi-dimensional effects into one-
dimensional simulations [116] suffer from too many shortcomings to be
considered a major improvement [117] and are at odds with observational
constraints (see below).

It must be borne in mind that phenomenological models need to explicitly or
implicitly incorporate calibration points or constraints to predict the
landscape of neutron star and black hole formation. Common choices are to fix
the explosion parameters to those of SN 1987A [107] and possibly add extra
constraints for low explosion energies of the least massive supernova
progenitors [108, 110]. Other studies have used softer constraints such as
plausible limits on observed supernova explosion energies [16]. Further
observational constraints may inform the models, even if they are not
explicitly incorporated. Considerable care must therefore be taken to gauge
the predictive value of phenomenological supernova models. In some instances
they may furnish more of an interpretation or physically motivated
extrapolation from observations than firm theoretical predictions.

Despite the disparity of methods among phenomenological supernova models and
the calibration uncertainties, some robust features of the progenitor-remnant
connection have nonetheless emerged. The “explodability” of progenitors by the
neutrino-driven mechanism is strongly correlated with structural parameters of
the stellar core and its surrounding shells. A popular predictor for
explodability is the _compactness parameter_ $\xi$ [106], which is defined as

$\xi_{M}=\frac{M/M_{\odot}}{R(M)/1000\,\mathrm{km}}$ (2)

where $M$ is a fiducial mass coordinate (measured in solar masses) and $R(M)$
is the corresponding radius. Values of $\xi_{M}$ with $M$ in the range
$1.75\texttt{-}2.5\,M_{\odot}$ have been found to provide good proxies for
explodability [106, 107]. The threshold value of $\xi$ for black hole
formation is subject to empirical calibration; values of $0.2\texttt{-}0.45$
are commonly used [106, 16, 118].

While the compactness parameter has proved a popular measure for
explodability, it has no particular physical meaning to single it out as a
unique metric. Other structural parameters that are suitable predictors for
explodability also exist. The mass $M_{4}$ of the Fe-Si core is also a good
indicator for the outcome of core collapse, especially if combined with a
second parameter $\mu_{4}$ that essentially characterizes the density of the
oxygen shell [108], and can be linked to the mass accretion rate onto the
shock after the infall of the Si/O shell interface, which is often the point
at which and explosion develops in detailed simulations. A high binding energy
of the shells outside the silicon-oxygen shell boundary is also an indicator
for black hole formation because the binding energy tends to be strongly
correlated with both $M_{4}$ and $\mu_{4}$. All of these core parameters tend
to increase with initial stellar mass, the helium core mass after hydrogen
burning, and the CO core mass after helium burning, although the dependence is
not strictly monotonic (see Figure 7). Because of reduced mass loss, higher
values of $\xi_{M}$, $M_{4}$ and $\mu_{4}$ may be reached at collapse for a
given initial mass at lower metallicity, especially for initial masses
$\gtrsim 20M_{\odot}$ (cf. Figures 1 and 3).

Figure 6: Supernova explosion and remnant properties for massive single stars
of solar metallicity, based on an analytic explosion model including
fallback[16, 119]. The panels show a) the supernova explosion energy
$E_{\mathrm{expl}}$ (if an explosion occurs), b) the black hole mass, c) the
neutron star mass (depending on the type of remnant), and d) the ejected mass
of iron-group (IG) elements, comprising mostly radioactive
${}^{56}\mathrm{Ni}$. Models that explode, but form black holes by fallback
are indicated in red in panels a) and d).

Figure 6 illustrates the predicted outcomes of core collapse based on a study
that is fairly representative of the current phenomenological models. For
_single-star_ progenitors of solar metallicity, models usually predict robust
explosions as the final evolutionary stage of stars with birth masses up to
$18\texttt{-}20\,M_{\odot}$ [106, 107, 110, 16]. At higher masses, they
typically find black hole formation, often interspersed with “islands of
explodability” due to non-monotonicities in the progenitor core structure as a
function of stellar initial mass or, more generally, core mass (Section 4). In
particular, successful explosions of single stars with birth masses around
$25M_{\odot}$ have been found in several studies [107, 110, 16]. The
underlying structural reason for the collapsing star to explode is a local
minimum in the compactness parameter and Fe-Si core size in this mass range.
This is due to the complex interaction of core masses left behind by each core
burning phase and the sequence in which subsequent core and shell phases
ignite, e.g., whether a critical mass for ignition or core contraction and
collapse is reached. This is somewhat similar to the concept of the Schönberg-
Chandrasekhar Limit[120], which defines a threshold mass for the isothermal
core after hydrogen core burning on the main sequence, above which the core
must contract and subsequently ignite helium core burning. In particular, in
Reference 16 this is related to mixing of the oxygen burning shell, located
very close to the iron core, with the burning shells above (e.g., Ne and C
burning), and the more volatile fuel being transported closer to the core and
raising the entropy in these layers. Fine details of the landscape of neutron
star and black hole formation are less robust. Some studies indicate the
possibility of islands of black hole formation as low as $15M_{\odot}$ in
birth mass. At birth masses above $20M_{\odot}$, the islands of explodability
tend to appear in similar places regardless of the simulation methodology,
even though different models may disagree how big these islands are, or
whether a particular island of explodability is present at all. This behaviour
is easily understood by recognizing that the underlying explodability is set
by the stellar structure, but different phenomenological supernova models
differ in the effective threshold (or “water level”) for parameters like the
compactness $\xi$ or Fe-Si core mass $M_{4}$. When considering the final fate
of massive stars as a function of _initial_ mass, one should bear in mind that
the pattern of explodability also depends on mass loss by winds or eruptive
mass loss events, and on binary interactions (see below), which may still open
the possibility for successful explosions of stars with rather high birth
masses well above $20M_{\odot}$ in certain evolutionary scenarios.

Most phenomenological supernova models have focused on solar-metallicity
progenitors, but some have investigated sets of stellar models of different
metallicity [106, 112], typically including $Z=0$ and $Z=10^{-4}Z_{\odot}$.
Due to reduced progenitor mass loss, fewer (if any) islands of explodability
at high mass are predicted at these low metallicities compared to the solar
case.

Figure 7: Core compactness $\xi_{2.5}$ (left) and central specific entropy,
$s_{c}$ (right), as a function of CO core mass at helium exhaustion for single
stars and stars that experienced mass loss in binaries with different mass
transfer cases (late Case A near the end of main-sequence evolution, Case B on
or during the ascent to the red giant branch, Case C after core He burning).
The different C/O ratios after He core burning shift the compactness at
$7\texttt{-}8M_{\odot}$ and also considerably reduces compactness between
$11M_{\odot}$ and $15M_{\odot}$ in He core mass in case of Case A and Case B
mass transfer. Figure from Reference 121, reproduced with permission from
Astronomy & Astrophysics, ©ESO.

For stripped-envelope supernova progenitors that have undergone mass transfer
in _binary stars_ , the initial (pragmatic) assumption has been that such
progenitors will exhibit similar explodability as single-star progenitors with
the same helium core or C/O core mass [122, 123, 119]. This leads to the
expectation that stripped-envelope progenitors with a C/O core mass of
$\gtrsim 4.5M_{\odot}$ will mostly form black holes with the possibility of
some interspersed instances of neutron star formation at higher values. This
was initially confirmed in a study[109] that modelled stripped-envelope
progenitors as helium stars, although there were differences in detail between
helium stars and single stars, e.g., a higher threshold by about $1M_{\odot}$
in He core mass in the compactness peak that corresponds to the first island
of black hole formation. A more nuanced picture (that also accounts for the
shift in the compactness peak) emerges when considering the evolutionary phase
during which mass transfer occurs [121]. For Case A mass transfer (during core
H burning in the donor) and Case B mass transfer (after core H burning during
the red giant phase or the ascent to the red giant branch), the dependence of
compactness on C/O core mass is indeed systematically shifted (Figure 7) due
to composition differences that are imprinted onto the core during core He
burning. Since the He core can grow during He core burning due to hydrogen
shell burning in single stars but not in stripped stars, He burning in single
stars results in a higher O/C ratio[124] due to the extra supply of fresh
${}^{4}\mathrm{He}$, unless mass transfer occurs after He core burning has
already finished.

In addition to a shift of the first compactness peak up by
$\mathord{\sim}1M_{\odot}$ compared to the single-star case, there is also a
much wider low-compactness valley at $8\texttt{-}14M_{\odot}$ in C/O core
mass, giving rise to the possibility of successful explosions up to high C/O
core masses[125]. On the other hand, Case C mass transfer (after the end of
core He burning) results in a similar dependence of explodability on the C/O
core mass as in single stars. In terms of initial masses, the windows for
black hole formation for stars undergoing Case A or B mass transfer may be
considerably reduced; Schneider et al.[121] report only a small birth mass
range between about $31M_{\odot}$ and $35M_{\odot}$ at solar metallicity as
well as black hole formation at high initial masses $\gtrsim 70M_{\odot}$ for
these channels. As a result, the predicted merger rates for black-hole neutron
star and black hole-black hole binary systems may be reduced by taking into
account the structural effect of mass transfer on the pre-supernova
structure[121].

There are, however, some noteworthy counterpoints to the aforementioned
picture in supernova theory. Some 3D simulations [126, 127, 128, 99, 101, 129,
130] and tuned 1D simulations [116] found _shock revival_ at high compactness.
Shock revival does not guarantee a successful explosion, however. If shock
revival occurs in progenitors with massive cores, massive oxygen shells, and
very high compactness, the proto-neutron star will accrete considerable mass
after shock revival and likely form a black hole by fallback. Consistent
explosions in high-mass, high-compactness progenitors would also be at odds
with observational constraints (see below). Nonetheless, the possibility of
shock revival in high-compactness progenitors followed by black hole formation
due to fallback needs to be considered and will be highly relevant for the
distribution of black hole masses, kicks, and spins (Section 1).

The most direct _observational_ constraints on the parameter space for neutron
star and black hole formation come from the identification of supernova
progenitors. Progenitor masses have been inferred for a number of Type IIP
supernovae (hydrogen-rich supernovae with an extended luminosity plateau of
about $100\,\mathrm{d}$) from red supergiant progenitors by matching the
brightness and colour of pre-explosion images to stellar evolution tracks
[131]. The majority of these progenitors will have evolved as single stars
[132, 133]. The inferred birth masses of the progenitors are somewhat
dependent on the treatment of convection during hydrogen core burning in the
underlying stellar evolution models[134], but there is rather strong evidence
that most red supergiants above $15\texttt{-}18M_{\odot}$ do no explode [131,
135]. In view of possible alternative explanations for the lack of high-mass
red supergiant explosions and statistical uncertainties [136], surveys for
disappearing red supergiants have been suggested as a more direct means to
study the parameter space for black hole formation[137]. The observed
disappearance of a $25M_{\odot}$ star [138, 139] supports the hypothesis that
more massive progenitors mostly form black holes. Progenitor mass estimates
based on the nebular spectroscopy of Type IIP supernovae is also consistent
with a lack of explosions at high progenitor masses [140]. Unfortunately, the
most readily available data – supernova photometry – cannot provide strong
constraints on the progenitor mass due to parameter degeneracies [141].

The evidence from well-studied historic supernovae and young supernova
remnants requires more careful interpretation because of various types of
binary interaction. In principle, matching the type of the compact remnants
for historic supernovae to their progenitor or He core mass can help constrain
the progenitor-remnant connection. SN 1987A and the Cas A supernova have left
neutron star remnants [142, 143] and mass estimates place their progenitors at
initial masses of $16\texttt{-}22M_{\odot}$ for SN 1987A[144] and
$15\texttt{-}25M_{\odot}$ for Cas A[145]. However, these numbers are based
heavily on models, and in the case of SN 1987A, the attribution of a ZAMS mass
is questionable in the first place, as its progenitor was likely the product
of a late stellar merger. Mass loss definitely played a critical role in the
progenitor evolution of Cas A. By means of light echoes, Cas A was identified
as a Type IIb supernova[146] whose progenitor had undergone partial stripping
of the hydrogen envelope (possibly during the companion’s supernova, which
would make the progenitor estimates questionable[147]). Their inferred helium
core masses of $\sim 6M_{\odot}$ place them slightly below the first major
island of black hole formation as predicted by most phenomenological models.
The young remnant W49B originated in a stripped-envelope supernova (Type Ib/c)
from a progenitor with an inferred mass of $\mathord{\sim}25M_{\odot}$, and
persuasive arguments have been made that the explosion produced a black
hole[148], which might point towards a collapsar engine, or towards fallback
in an explosion driven by some other mechanism. Unfortunately, progenitor mass
estimates based on pre-explosion images for Type Ib/c supernovae are still
scarce and cannot strongly constrain the parameter space for black hole
formation among massive stars that have undergone mass loss in binaries. For
an identified progenitor system for the Ib supernova iPTF13bvn, a tentative
mass estimate has been formulated based on binary evolution modelling and
constraints from the photomoetry of the progenitor system and the supernova
itself, putting the pre-collapse mass (i.e., final helium star mass) at about
$3.5M_{\odot}$ [149, 150], but this estimate is not yet on par with those for
Type IIP supernovae.

Yet another, more indirect way to constrain dependence of explodability on
initial mass is to age-date the environments of supernova remnants. Such age-
dating of remnant environments in M31 and M13 qualitatively supports the
hypothesis of missing explosions at higher mass [151, 152].

####  3 Collapsars, Hypernovae, and Gamma-Rays Bursts

Black hole formation may be a crucial element in _hypernovae_ with unusually
high explosion energies up to $\sim 10^{52}\,\mathrm{erg}$ as opposed to the
typical core-collapse supernova explosion energy of $\sim
10^{51}\,\mathrm{erg}$. Such events make up about $1\%$ of the supernova
population in the local universe[153] and possibly up to $10\%$ in low-
metallicity environments [154]. Starting with SN 1998bw[155], it has been
recognized that long gamma-ray bursts (GRBs) are associated with such
hypernovae [89], although it is not clear whether all hypernovae produce long
GRBs.

The collapsar scenario[156], the characteristic features of hypernovae and
gamma-ray bursts are explained by the formation of a black hole and accretion
disk in the collapse of a rapidly rotating massive star. A non-relativistic
wind outflow from the disk provides the energy of the hypernova and abundant
radioactive ${}^{56}\mathrm{Ni}$ to power its light curve[156]. The formation
of the relativistic GRB jet likely involves the extraction of rotational
energy from the black hole or the disk by magnetohydrodynamics effects via the
Blandford-Znajek[157] or Blandford-Payne[158] mechanism. To date, we still
lack unambiguous observational evidence whether hypernovae and long GRB
involve black hole formation, or whether rapidly rotating neutron stars
(“millisecond magentars”) are behind the relativistic jet [159, 160] and the
hypernova explosion [85]. It is also possible that similar disk-power engines
akin to the collapsar scenario operate in some superluminous supernovae [161].
An extensive review of current research on hypernovae and long GRBs as
provided by recent reviews[89, 162] is beyond the scope of this chapter. It is
more pertinent to focus on how the collapsar scenario fits into the broader
picture of stellar evolutionary channels to black hole formation and black
hole birth properties.

After a black hole has formed in a rapidly rotating progenitor, feedback from
a collapsar-type engine will affect further accretion onto the black hole
roughly once the specific angular momentum $j$ of the infalling shells reaches
the critical specific angular momentum at the innermost stable circular orbit
($j_{\mathrm{c,Kerr}}\gtrsim 2/\sqrt{3}GM/c$ for a Kerr black hole,
$j_{\mathrm{c,NR}}\gtrsim 2\sqrt{3}GM/c$ for a non-rotating black hole).
Although the angular momentum of the black hole could be small in principle
when disk formation occurs, one usually expects the black to have a high spin
parameter $a=Jc/(GM^{2})\approx 1$ based on actual stellar evolution models
for hypernova progenitors [163]. Once an accretion-powered engine operates,
the outflows will extract energy and angular momentum from the disk and/or the
black hole, and the feedback from the engine may quench the accretion flow
[164, 165]. Qualitatively, one therefore expects a sub-population of black
holes with lower mass and high spin parameter from rapidly rotating progenitor
stars [164]. The quantitative evolution of the black hole mass and spin
parameter is somewhat more complicated and depends on how efficiently the
outflows extract energy and angular momentum from the system. Depending on
whether or not powerful magnetohydrodynamic jets form or not (which depends,
e.g., on the field geometry in the accretion disk), the black hole may lose or
gain energy (i.e., mass) and angular momentum [166].

Different scenarios have been proposed to account for progenitors with the
requisite rapid rotation at collapse to enable hypernova explosions. In a
scenario without binary interactions, sufficient angular momentum needs to be
retained in stars that are born with rapid rotation. Such rapid rotation could
also be the result of binary star mergers during the pre-main sequence or on
the main sequence. Whereas without angular momentum transport, stars may
easily reach critical rotation in their cores[53], the big challenge in models
is to retain sufficient angular momentum in their cores when angular momentum
transport is considered. The challenge includes the requirement that most pre-
supernova stellar cores need to rotate slowly enough to be compatible with
observed rotation rates of supernovae and supernova explosion energies555The
rotational energy of a nascent neutron star with a rotation period of a few
milliseconds would far exceed observed supernova explosion energies.. Next to
transport of angular momentum, a key factor is loss of angular momentum due to
stellar winds, in particular when the star has an extended red supergiant
envelope with its huge specific moment of inertia. One way around this is a
stellar evolution path where the star already rotates very rapidly on the main
sequence such that it remains fully mixed and undergoes what is called
chemically homogeneous evolution [53, 51, 167], avoiding the red supergiant
evolution phase. The scenario still requires low metallicity to avoid loss of
angular momentum due to extended Wolf-Rayet winds[51] and the contribution of
magnetic stresses due to dynamo action [49, 50] can be disfavourable to
angular momentum retention. This scenario, however, is not commonly realised
in nature and the predicted GRB/hypernova fraction of less than $0.1\%$ at low
metallicity[167] appears too low to account for all observed events. The
conditions for chemically homogeneous evolution may also be reached due to
spin-up by mass transfer in close-to-equal-mass binaries, with a likely break-
up of the system after the supernova of the mass donor [168]. Stellar mergers
of evolved stars present a possible channel to generate more rapidly spinning
helium stars than can be easily produced in a single-star scenario [169, 170,
89, 171]. This scenario is noteworthy because the ensuing collapsar would by
construction give birth to an isolated black hole unless the progenitor system
was a triple. Proposed merger pathways include, e.g., the merger of two helium
cores of evolved stars during common-envelope evolution [170], or the merger
of an evolved massive stars after He core burning with a low-mass star leading
to explosive ejection of the helium and hydrogen shell [171]. Tidal spin-up of
Wolf-Rayet stars in binaries, e.g., Reference 172, is discussed later in
Sections 3 and 5.

####  4 Pair-instability and pulsational pair instability supernovae.

Figure 8: Final stellar masses of pulsational pair instability supernovae as a
function of initial mass (red dots). The dash-dotted line indicates the helium
core mass. Above $\mathord{\sim}80\,M_{\odot}$ in initial mass the entire
hydrogen envelope is ejected, and above $\mathord{\sim}120\,M_{\odot}$ parts
of the carbon-oxygen core are ejected as well. The solid blue line shows the
mass of the star in the absence of mass loss. Pulsational pair instability
continues to operate as long as the star is above a critical mass limit, or
until a sufficiently large iron core is produced and the star collapses to a
black hole. The result is an effective upper mass limit for black holes that
can be made this way (orange line). Above $\mathord{\sim}140\,M_{\odot}$ the
first pair-instability pulse is already powerful enough to entirely disrupt
the star, and no compact remnant is left behind. The data for the plot is
taken from Reference 173; a somewhat different initial-final mass function for
rotating very massive stars is found in Reference 174.

When the mass of the helium core of a star at the end of core helium burning
exceeds about $\gtrsim 40M_{\odot}$, the star encounters the electron-positron
pair creation instability [175]. Shortly after core carbon burning, which is
radiative, the temperature exceeds $10^{9}\,$K and high-energy photons in the
tail of the Planck spectrum can create electron-positron pairs. The rest-mass
of these pairs is taken from the internal energy of the gas, leading to a
softening of the equation of state, i.e., lowering of the adiabatic index,
$\gamma_{\mathrm{ad}}$, beyond the limit of stability666The star is stable
when the condition is fulfilled.,

$\int_{0}^{M}\frac{P(m)}{\rho(m)}\,\left(\gamma_{\mathrm{ad}}(m)-\frac{4}{3}\right)\,\mathrm{d}m>0\;,$
(3)

where $M$ is the total mass of the star, $P$ is the pressure, $\rho$ is the
density,and $m$ is the mass coordinate. The star contracts rapidly on a
dynamical timescale and encounters “ _explosive_ ” (very rapid) nuclear
burning until the temperature has risen enough that new particles actually
contribute sufficiently positively to the gas pressure. The star re-expands
and a shock wave forms that ejects the outer layers of the star. After the
pulse, the star cools down, contracts, and may encounter further pair
instability pulses. Subsequent stages of neon, oxygen, and silicon burning may
be encountered in one or several pulses until a large enough iron core is
formed and the star collapses to a black hole. As the initial mass of the star
and its post-helium-burning core increase, and also as one gets to higher
pulse numbers in a sequence, the thermonuclear pulses get increasingly more
energetic. This leads to larger mass ejection, but also to larger post-pulse
entropy in the core. Larger entropies imply that the core takes longer to
cool, with a longer Kelvin-Helmholtz timescale before the next pulse can
occur. There is, indeed, a critical transition when the post-bounce
temperature drops much below $10^{9}\,$K: the gas becomes too cool to
efficiently emit neutrinos, and has to cool by photon emission from the
surface instead, resulting in inter-pulse cooling times as long as
$10\mathord{,}000\,$yr[173]. In contrast, the inter-pulse phase between low-
energy pulses may be as short as hours to days. When the inter-pulse phase is
of the order of years, for stars with initial mass of around $200\,M_{\odot}$,
the timescale may just be right to produce bright outbursts from collisions of
ejecta shells from subsequent pulses, as discussed below.

The production of neutron stars in the final collapse seems unlikely, and
should be very rare at best. The final collapse may yet trigger a final
hypernova explosion or gamma-ray burst [176] due to the collapsar
mechanism[177] (Section 3). The remarkable outcome of these pulses is that
they imply an upper mass limit for black holes that can be made though this
channel of around $\mathord{\sim}45\,M_{\odot}$[173] (Figure 8).

For initial stellar masses larger than $\mathord{\sim}140\,M_{\odot}$, i.e.,
helium core masses $\gtrsim 65\,M_{\odot}$, even the first pulse is already
energetic enough to entirely disrupt the star. _There is no compact remnant._
This is the domain of the full pair instability [178, 179, 22] regime. Kinetic
explosion energies can range from $4\times 10^{51}\,$erg to almost
$10^{53}\,$erg, and the nucleosynthesis can range from being basically free
from primary production of iron group elements to the production of more than
$60\,M_{\odot}$ of 56Ni,[22] the same isotope that is responsible for powering
the light curves of regular core collapse and Type Ia supernovae. The extreme
case is basically the equivalent of 100 Type Ia supernovae simultaneously in
one spot.

At the high-mass end for full pair instability the entropy in the core gets so
high that the 56Ni from silicon burning photo-disintegrates back into alpha
particles, however, when the star explodes these recombine to 56Ni for the
most part. At the upper mass limit for full pair instability, i.e., for
stellar masses of $\gtrsim 260\,M_{\odot}$, helium core masses $\gtrsim
130\,M_{\odot}$, the entropy in the core gets so high that even the alpha
particles photo-disintegrate into free nucleons. These photo-disintegrations
take out internal energy of the gas, reducing the pressure and thereby
softening the equation of state similar to the pair instability. In
particular, the instability caused by photo-disintegration of the alpha
particles is so strong that the collapse is not turned around by an outward
shock, and instead continues as a direct collapse to a black hole. Unless
there is sufficient angular momentum to form an accretion disk, the entire
star should collapse. The resulting black holes should have initial masses of
at least $130\,M_{\odot}$ for pure helium cores[22], larger for larger stars
or if the hydrogen envelope was not lost. If there is sufficient angular
momentum in the star at collapse, the collapsing core may become a strong
source of gravitational waves[180], or the angular momentum centrifugal
barrier may generate an accretion disk powering a long-duration gamma-ray
burst, leading to more mass ejection and reducing the black home mass (see
Section 3).

The consequence of the upper mass limit for black holes from pulsational pair
instability ($45\,M_{\odot}$) combined with the lower mass limit for black
holes beyond the full pair instability regime ($130\,M_{\odot}$) is a gap in
black hole birth mass function[181]. This straight-forward prediction has been
challenged by gravitational-wave observations that imply likely detections of
black holes with masses within this mass gap[182]. Suggested solutions for the
existence of mass-gap black holes include low-metallicity stars just below the
pair instability supernova limit that may be as massive as $70\,M_{\odot}$ at
the time of core collapse[183] or modifications to stellar physics such as key
nuclear reaction rates[59, 60], binary evolution, rotation, and accretion
after black hole formation[184, 185, 186, 60]. Whereas these may be able to
shift, and, in part, even narrow the mass gap, these works show that
eliminating the gap entirely remains a challenge. Another possibility is that
the merging black holes are products of earlier mergers, perhaps through
dynamical formation in globular or nuclear clusters[187, 188].

Transient observations have so far been unable to provide further insights on
the predicted pair instability mass gap. It is noteworthy, though, that there
is no unambiguous detection of a pair instability supernova yet. Due to the
large ejected mass of ${}^{56}\mathrm{Ni}$, pair-instability supernovae were
adduced as an explanation for superluminous supernovae early on [189], but the
observed events tend to differ markedly from model predictions in terms of
their light curves (especially in terms of the rise time before peak) and
spectra [190, 191, 192]. Different from pair-instability supernovae proper,
there is a considerable class of observed transients that fit the
characteristics of pulsational pair-instability supernovae. Whereas the
individual pulses may not be particularly powerful in terms of kinetic energy
compared to ordinary supernovae, collisions of shells from different pulses
can produce very bright transients [176] due to the high efficiency of
conversion of kinetic energy to observable photons. This suggests that some
_superluminous supernovae_ might be pulsational pair instability events.
Depending on the mass, metallicity, rotation rate, and prior mass loss
history, a wide variety of light curves can be produced, and both hydrogen-
rich Type IIn superluminous supernovae (with evidence for interaction in the
form of narrow emission) and Type I superluminous supernovae can be accounted
for [193, 194, 195, 173]. Observationally, the narrow emission lines in Type
IIn superluminous supernovae constitute strong evidence that these are
interaction-powered [196]. Bumps and undulations in the light curves of both
Type II and Type I superluminous supernovae[197, 198, 199, 200] can also be
interpreted as signs of interaction, although alternative interpretations are
often possible. In many cases, high ejecta masses have been inferred for such
interacting superluminous supernovae from light curve fitting [176, 194, 201,
202] and nebular spectroscopy[203], which is compatible with the pulsational
pair-instability scenario. The progenitor masses cannot be determined
sufficiently well to verify the nature of the progenitors, however, let alone
to constrain the mass range for the pulsational-pair instability channel.

###  4 Black holes at birth: masses, kicks and spins

####  1 Amount of mass loss during collapse

The dependence of “explodability” on stellar mass, rotation, and metallicity
is only one ingredient for understanding the observed population of stellar-
mass black holes. It has been recognized that in many instances of black hole
formation, partial mass ejection is likely to occur, which has important
implications for the birth distribution of black hole masses. The possibility
of partial mass ejection is most evident in the case of the collapsar scenario
(Section 3). However, the impact of partial mass ejection on black hole birth
parameters has been studied more extensively for other scenarios.

Already in the 1980s it was suggested[204] that massive stars may eject part
of their envelope after iron core collapse even if the shock is never revived.
The energy loss through neutrinos during the proto-neutron star phase reduces
the gravitational mass of the star, which disturbs the hydrostatic equilibrium
in the envelope. As a result, a sound pulse is launched, which may eject
tenuously bound envelope material. This idea, known as Nadyozhin-Lovegrove
mechanism, has been developed further in recent years using numerical
simulations [205, 206, 207] and analytic theory for the wave pulse launched by
the reduction of the gravitational mass [208, 209]. For plausible assumptions
about the energy loss through neutrinos and the black hole formation
timescale, the hydrogen envelope is likely to be ejected in the case of red
supergiant progenitors [206]. For blue supergiants with more compact
envelopes, only $\sim 0.1M_{\odot}$ will be lost, very little mass loss is
expected for Wolf-Rayet stars, and scenarios without mass ejection are also
conceivable [206]. Mass ejection due to this mechanism would give rise to a
long-lived red transient with a small energy of
$\mathord{\lesssim}10^{48}\,\mathrm{erg}$ [205, 210, 206], and may be followed
by faint emission due to fallback for up to several years [208]. In addition,
there will be a brighter and bluer luminosity peak from shock breakout ov
$3\texttt{-}70\,\mathrm{h}$ [210]. Observations have yet to positively
identify such a transient from the shedding of the envelope.

To approximately account for mass ejection by the Nadyozhin-Lovegrove
mechanism, recent phenomenological supernova models often assume that the
black hole mass will be given by the hydrogen-free mass of the progenitor,
although this may underestimate the black hole mass in some cases.

Genuine fallback supernovae present a more complicated case of partial mass
ejection. In fallback supernovae, the shock is successfully revived, but the
(proto-)neutron star eventually accretes enough mass later on to collapse to a
black hole. Several fallback scenarios can be distinguished. In the case of
early fallback, continuing accretion after early shock revival already drives
the neutron star to collapse during the first seconds to minutes of the
explosion. In the case of late fallback, accretion onto the neutron star is
initially quenched, but some of the ejecta fall back after they are
decelerated by one of the reverse shocks that form when the forward shock runs
across a shell interface [211, 212, 213]. In explosions of red supergiant
progenitors, a strong reverse shock forms when the forward shock crosses the
helium/hydrogen interface, transiently accelerates and then decelerates again
as it scoops up more material from the hydrogen envelope. Nonetheless,
fallback by deceleration in the reverse shock usually adds little mass onto
the remnant. Fallback masses were mostly limited to $\lesssim
10^{-2}M_{\odot}$ in phenomenological 1D supernova models of solar-metallicity
single-star progenitors [214].

Fallback can become much more dramatic, however, when the energy input by the
supernova engine exceeds the binding energy of the outer shells only by a
moderate margin [126, 127]. In this case, considerable fallback can occur
already during the early phase of the explosion because accretion downflows
are not quenched after shock revival [126, 128, 99]. At later stages, more
fallback can occur as the forward shock and the matter in its wake are slowed
down without the need to involve a reverse shock [126, 127]. The mechanisms
governing the final explosion energy and fallback mass in such marginal
explosions are only qualitatively understood at this point. After the
supernova engine has stopped, the initial energy of the blast wave will be
drained as the shock scoops up bound material. Once the shock becomes
sufficiently weak, it will turn into a sonic pulse that transports energy
through the star without transporting matter, and little further energy will
be lost from the pulse [126, 127, 215]. The final mass cut is set roughly by
the point where the shock leaves the weak shock regime again (i.e., post-shock
Mach numbers reach $\gtrsim 1$) as it proceeds to shells with smaller sound
speed [126]. For a marginal explosion to succeed, black hole formation must
not occur too early, however. If the black hole is formed when the shock has
not crossed the sonic point of the infall region yet, the incipient explosion
is likely completely stifled[130], and only part of the envelope may be
ejected by the Nadyozhin-Lovegrove mechanism.

Multi-dimensional effects are extremely important in marginal explosions with
early fallback. So far only a handful of multi-dimensional simulations of
fallback after black hole formation have been conducted[126, 127, 215]; these
have been helpful for identifying the aforementioned principles. They also
showed that fallback in marginal explosions can produce black holes over a
considerable mass range from close to the maximum neutron star mass to almost
complete collapse [127]. In particular, fallback can explain entities like the
$2.6M_{\odot}$ compact object in the merger event GW190814[216]. Estimating
the effect of fallback on black hole populations using phenomenological 1D
supernova simulations or analytic models is more difficult. Current models
show considerable variations in the fraction of stars affected by strong
fallback [214, 16, 119, 217]. There is, however, agreement that fallback will
produce a sizable number of low-mass black holes and populate the “mass gap”
that was formerly assumed to exist between at $2\texttt{-}5M_{\odot}$ between
the most massive neutron stars and the least massive black holes in known
X-ray binaries[218, 219]. The amount of ejected mass in the case of a
partially successful explosion of a rapidly rotating progenitor (which
includes the collapsar scenario) is less well understood.

Aside from GW190814, there is additional circumstantial evidence for partial
mass ejection after black hole formation. Dark objects with masses between
$2\texttt{-}5M_{\odot}$ have been observed in microlensing experiments[220,
221] and in some detached binaries[222]. Abundances in some ultra metal-poor
stars that were likely polluted by one or a few supernovae can best be
understood as resulting from the removal of the iron group elements and
sometimes some intermediate mass elements from the inner ejecta of a supernova
[223, 224]. The composition of the black-hole companion in Nova Scorpii also
suggests pollution by a fallback supernova that formed the black hole [225].
Evidence from transient observations is more dubious. Suggestions that faint
Type IIP supernovae with small ejected mass of ${}^{56}\mathrm{Ni}$ are black-
hole forming fallback events proved unlikely upon more recent analysis [226].
Some superluminous supernovae may be interpreted as being powered by fallback
[227, 161], but a smoking gun for this interpretation is lacking.

####  2 Asymmetry / kicks?

The multi-dimensional nature of the modern fallback scenario opens up the
possibility of strongly asymmetric mass ejection, which could result in
sizable black hole kicks due to momentum conservation. Initial analytic
estimates envisaged the possibility of similar kick velocities for black holes
and neutron stars [228]. Only two 3D simulations have yet addressed black-hole
kicks from fallback and present a more nuanced picture with a kick of
$\mathord{\sim}500\,\mathrm{km}\,\mathrm{s}^{-1}$ (i.e., slower than the
fastest neutron star kicks) for a case with moderate fallback, and a
relatively small kick of tens of $\mathrm{km}\,\mathrm{s}^{-1}$ for strong
fallback in a weak explosion. The physics evinced by these simulations
suggests that high-velocity black hole kicks are confined to black holes of
relatively low mass. Attempts to extrapolate these results to black hole
populations involve some calibration of the absolute scale of black hole
kicks. However, theory does point to a less populated high-kick tail and a
more pronounced low-velocity peak in the black-hole kick distribution compared
to the neutron star kick distribution with its peak at non-zero kick velocity.
Even in case of complete fallback, asymmetric neutrino emission prior to black
hole formation may still impart kicks of a few
$10\,\mathrm{km}\,\mathrm{s}^{-1}$ onto the black hole [229], although these
estimates are still based on 2D simulations that may somewhat overestimate the
emission anisotropy.

The observational evidence for black hole kicks is somewhat uncertain. It
appears that heavier black holes such as Cygnus X-1 were born with very low
kicks of $\lesssim 10$ km s-1 [230]. On the other hand, lighter black holes
have been inferred to have larger kicks of $\gtrsim 200$ km s-1 through
observations of the positions and velocities of BH low-mass X-ray binaries
(see section 5) [231, 232] (though see [233]), with further evidence
potentially provided by observed spin-orbit misalignments in BH binaries
[234].

####  3 Black hole spin

In the simplest picture of partial mass ejection, the black hole birth spins
will simply be determined by the amount of total angular momentum in the
progenitor star inside the mass cut (envisaged as a spherical demarcation line
between the black hole and the ejecta). In the case of rapidly rotating
progenitors, this is likely a good approximation.

It is not, however, trivial to form rapidly rotating black holes. Whereas
typical initial stellar rotation is easily sufficient to make Kerr black
holes, the amount of angular momentum actually present in the core largely
depends on how much of the initial angular momentum is retained to core
collapse. This, in turn, strongly depends on i) angular momentum loss from the
surface of the star and ii) the transport of angular momentum. Magnetic fields
may be quite efficient in angular momentum transport[50]; angular momentum
loss through winds is also efficient except for low metallicities[51]. Co-
rotation or spin-up by accretion in close binaries in early evolutionary
phases is still subject to angular momentum losses later in the evolution.
Some black hole progenitors, however, may be placed in very close binaries
after the envelope is removed, e.g., by a common-envelope event, so that the
naked helium core is efficiently tidally spun up and remains rapidly spinning
until core collapse, as discussed in Section 5. While this may limit angular
momentum loss from the surface, the spin-up of the star has to occur early
enough during the evolution to still impart enough angular momentum onto the
very core - as soon as the critical angular momentum for Kerr black holes
corresponds to surface rotation rates in excess of Keplerian rotation for a
rigidly rotating star, sufficient spin-up of the core is no longer possible.

For slowly rotating progenitors, the black hole can, however, also be spun up
by asymmetric fallback. Due to the large lever arm, relatively small amounts
of fallback and small non-radial velocities can impart significant angular
momentum onto the black hole. A recent 3D simulation showed that black hole
spin parameters of $\mathord{\sim}0.25$ are within reach for low-mass black
holes ($\mathord{\sim}3M_{\odot}$) [127]. Spin-up of black holes by asymmetric
fallback will undoubtedly exhibit large stochastic variations, and more
systematic theoretical and computational studies are required to predict its
effect on the distribution of black hole spins.

The spin of a black hole changes the radius of the innermost stable circular
orbit, which sets the inner radius of the accretion disk. Therefore, spins of
accreting stellar mass black holes can be inferred through the observations of
continuum X-ray flux or reflection lines from the accretion disk. Spin
measurements suggest a broad distribution of spins, from nearly non-spinning
to nearly maximally-spinning [235, 236], although all inferred spins are
model-dependent, so caution is warranted (see Chapter IV of this volume).
Gravitational-wave observations (see Section 5 and Chapter VIII of this
volume) also allow spins to be inferred, albeit with limited precision. There
is some debate in the literature regarding the spin distribution of merging BH
binaries [182, 237, 238, 239]; it is possible that some merging binary BHs
have negligible spins, while others have moderate combined spins
preferentially aligned with the direction of the orbital angular momentum.

###  5 Black holes in binaries

Most stellar-mass black holes are observed in binaries, through either mass
transfer from a non-degenerate companion onto the black hole (X-ray binaries),
detached binaries in which the presence of the black hole is inferred through
the orbital motion of the optical companion, or mergers observed via their
gravitational-wave signature. Moreover, massive stars that go on to form black
holes are typically born in binaries or systems with even more companions
[240]. Black hole progenitors in such systems frequently gain mass from their
companion, experience mass stripping by the companion, or are tidally spun up
by the companion [241]. These interactions play a crucial role in black-hole
formation and in determining the mass and spin of the black hole.
Consequently, even the single black holes, particularly those observed via
gravitational micro-lensing [220, 221] likely came from binaries and
experienced binary interactions [242]. With the exception of microlensing
observations, binaries are also responsible for all mass measurements of
stellar-mass black holes, as shown in Figure 9 and further discussed in
Chapter IV of this volume. In this section, we briefly summarise the
theoretical models of the impacts of binarity on the formation and properties
of black holes.

Figure 9: The masses of black holes measured through gravitational waves
(blue, in chronological order, with merger product masses shown) and
electromagnetic observations of X-ray binaries and detached binaries (red,
random order). Figure courtesy of Aaron Geller, Northwestern University and
LIGO–Virgo–KAGRA collaborations.

We can broadly divide binary interactions into two categories: those that
happen prior to BH formation and those that take place after the BH is formed.

Prior to BH formation, the key interaction mechanisms are mass transfer and
tides.

If either star in a binary is going to make a black hole, it is likely to be
the primary, the initially more massive star, though the secondary may also
follow. Mass ratio reversal through mass transfer can lead to the formation
of, say, a neutron star from the primary and a black hole from the initially
less massive secondary, but such outcomes are expected to be rare [243]. As
the primary expands at the later stages of main sequence evolution or,
particularly, after evolving off the main sequence, tidal gravity from the
secondary will increasingly distort it and may ultimately lead to mass
transfer (Roche lobe overflow). This will reduce the star’s mass and possibly
alter its structure and even affect future collapse outcomes [121]. Mass
transfer that removes the primary’s envelope, which contains the bulk of its
angular momentum, may leave behind a relatively slowly rotating core [244,
245, 246]. A stripped star may also experience particularly strong Wolf-Rayet
winds, driving further mass loss and spin-down. Of course, mass transfer can
also add mass to the companion and change the orbital separation, potentially
impacting future interactions.

In some cases, mass transfer may become dynamically unstable, leading to a
common-envelope phase [247], which may in turn result in a stellar merger. The
merger product may still form a black hole, but perhaps one with unusual
features, such as a non-standard ratio of the core and envelope masses; such
systems may, for example, explain (pulsational) pair instability supernova
candidates and unexpectedly large black hole masses in high-metallicity
environments [248, 249].

Tides primarily impact orbital evolution, synchronising and perhaps
circularising the binary. In the process, they can spin up the BH progenitor.
While that angular momentum may be removed with the envelope, in some cases,
tidal spin-up may accompany mass transfer in such a way that the envelope is
removed and the remaining core is spun up at the same time, possibly
explaining some rapidly spinning BHs such as Cygnus X-1 [250, 230].

Rapid rotation through tidal coupling may also cause efficient circulation
within the main-sequence BH progenitor [251, 252]. This could lead to mixing
throughout the star and, ultimately, chemically homogeneous evolution, in
which the entire stellar mass of hydrogen fuses into helium, not just the core
[253, 254]. This process may lead to the formation of close pairs of black
holes from over-contact binaries [255, 256].

Binarity may also be important during the formation of the BH itself if some
of the stellar material is torqued by the companion before falling back into
the BH, thus contributing to BH spin, though this process may rely on some
fine tuning [257, 258].

Following BH formation, mass transfer, winds, and possibly gravitational-wave
emission and/or dynamical interactions become key. Tides do not affect the BH
after its formation, though they can still impact subsequent binary orbital
evolution, and could be responsible for the tidal spin-up of the secondary
(which could go on to become another black hole [259, 260, 245, 261]).

Mass transfer onto a black hole may allow for electromagnetic observations of
BHs as the accreting material radiates in X-rays. BH X-ray binaries (XRBs) are
generally divided into low-mass and high-mass XRBs, with the former fed by
Roche lobe overflow from a low-mass companion and the latter by winds from a
high-mass companion.

Low-mass XRBs can be very long-lived (on timescales of Gyrs), though often
transient in nature as mass accretion stops and re-starts. The total mass
reservoir, however, is sufficiently small that the BH is unlikely to accrete a
lot of mass or spin, although some studies suggest that such low-mass XRBs may
be the evolutionary outcomes of intermediate-mass XRBs with significant mass
accretion and BH spin-up [262, 263].

Meanwhile, high-mass XRBs are necessarily short-lived, with lifetimes of order
a Myr or less due to the short lifetime of massive stars. This is much less
than the mass doubling time of a black hole accreting at the Eddington limit
($\gtrsim 30$ Myr), and hence the BH cannot accrete a significant fractional
amount of mass or get appreciably spun up [264] (but see Refs.[184, 265] for a
discussion of the impact of super-Eddington accretion).

In recent years, several black holes have also been observed in non mass-
transferring binaries [222, 266], with prospects for future detections through
Gaia data [267]. However, observations of such detached BH binaries are
notoriously challenging, with several recent candidates including LB-1 [268],
the ‘unicorn’ [269] and the ‘giraffe’ [270] ruled out by subsequent re-
analyses [271, 272, 273, 274].

Our discussion so far has focussed exclusively on isolated binaries. However,
additional channels for BH binary formation and evolution include dynamical
interactions in addition to stellar and binary evolution. Hierarchical triple
systems may experience Lidov-Kozai oscillations leading to enhanced inner
binary eccentricity [275, 276]. Interactions in dense stellar environments
such as globular and nuclear clusters can introduce BHs into binaries through
replacements and subsequently tighten those binaries [277, 278]. Similar
dynamical interactions of stellar-mass black holes in AGN discs additionally
include the possibility of significant accretion onto these BHs [279, 280].

If two black holes are sufficiently close, gravitational-wave emission can
drive them to merger. The timescale for this merger scales with the fourth
power of the separation and the inverse cube of the mass [281], so that the 30
$M_{\odot}$ BHs that merged in the first gravitational-wave detection GW150914
[282] would have needed to be separated by less than a quarter of an
astronomical unit in order to merger within the current age of the Universe.
Almost a hundred merging binary BHs have been detected as of 2022, providing
the largest catalog of known stellar-mass BHs [283, 182]. The implications of
these observations for the formation and evolution of stellar-mass black holes
are the topic of very active ongoing work, partly summarised in a set of
recent reviews [284, 285, 286]; see Chapter VIII of this volume for more
details.

###  6 Concluding Remarks

A complete end-to-end understanding of the physics of stars and eventually the
properties of black holes they make requires full three-dimensional
simulations covering the full range from the micro-physics to the integral
spatial and temporal scale of the stars, Such calculations do not seem
feasible for the foreseeable future based on current technology. Practical
approaches at our disposal include work toward understanding the physical
processes to the extent that they can be accurately modelled on long time-
scales, in lower dimensions, allowing to replace resolved microphysics by sub-
grid models to be included in simulations on the integral scale of the problem
– the stellar scale – or even on the system scale for multi-star studies.

The advances in computational modelling will thrive on insights obtained from
the growing stream of observations in multi-messenger astronomy that combines,
e.g., gravitational waves, cosmic rays and high energy neutrinos, and the
entire range of electromagnetic observations over a vast range of time-scales,
from sub-second transients to years-long light curves, to quasi-steady objects
preceding violent stellar deaths. New techniques for data analysis, such as
machine learning or, in the long run, quantum computing, may allow us to
better exploit the data to constrain theoretical models. Genuinely new
approaches are needed to better address key problems such as, e.g., black hole
formation in supernovae and mass transfer in binaries. We require a deeper
understanding of how the physics of single and binary star evolution and
supernova explosions connect to each other.

In the near future, a challenge lies in understanding the formation of black
holes from the first generation of stars, where direct observations of the
individual objects remain a challenge except for a few rare circumstances of
chance magnification due to strong gravitational lensing[287] and caustic
crossings with their huge magnifications on the order of $10\mathord{,}000$
[288].

### Acknowledgements

A.H. was supported by the Joint Institute for Nuclear Astrophysics through
Grant No. PHY-1430152 (JINA Center for the Evolution of the Elements) and by
the Australian Research Council (ARC) Centre of Excellence (CoE) for
Gravitational Wave Discovery (OzGrav) through project number CE170100004, and
by the ARC CoE for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) through
project number CE170100013. B. M. acknowledges support by ARC Future
Fellowship FT160100035. I.M. acknowledges support from the Australian Research
Council Centre of Excellence for Gravitational Wave Discovery (OzGrav),
through project number CE17010004. I.M. is a recipient of the Australian
Research Council Future Fellowship FT190100574. Part of this work was
performed at the Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-1607611. I.M.’s participation at the Aspen Center
for Physics was partially supported by the Simons Foundation.

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