The dynamical properties of black-hole accretion flows
are affected most strongly by their angular momentum distributions and
secondarily by the efficiency of radiative losses. In almost all
astrophysically interesting circumstances the flow is expected to be
*centrifugally choked*, meaning that the specific angular momentum
of the gas supply exceeds a small multiple of *GM* /
*c*. Various possible modes of accretion, discussed on more detail
below, are summarized in Fig. 1.

Hypothetical flows which are not centrifugally choked should resemble spherically symmetric Bondi accretion [17], which may or may not have a high radiative efficiency, depending on the presence or absence of a dynamically significant magnetic field.

Without significant pressure support, gas will follow approximately Keplerian orbits. Since energy is generally thought to be dissipated more quickly than angular momentum is transported away, a radiatively efficient flow will circularize to form a thin disk [18]. Gas then slowly spirals towards the black hole as it gives up angular momentum, until it reaches the ISCO and plunges into the hole with relatively little additional dissipation. The energy dissipated at every radius is radiated away locally, and is a well-defined function of mass accretion rate Ṁ, radius and black hole spin that can be converted into a run of effective temperatures and (assuming a large disk optical depth, which usually applies) a continuum spectrum. A key strength of accretion disk theory is that these results do not depend on the angular momentum transport mechanism. Fitting the continuum spectrum close to the inner edge of the disk is the basis of the continuum fitting method for determining black hole spins [19]. However, portions of the spectrum originating farther out often seem hotter than predicted, probably due to irradiation of the disk, poorly understood optical depth effects, or nonlocal dissipation.

At lower radiative efficiency, the disk will thicken as more internal
energy is retained, leading to a *slim disk*
[20],
in which radial pressure forces depress the angular momentum at every
radius below the Keplerian value and radial advection of energy becomes
appreciable.

It is important to note that the energy dissipated locally in an
accretion disk is *not* the same as the gravitational binding
energy liberated locally. Far from the black hole where the angular
momentum is close to the Keplerian value, the dissipation rate is three
times the local rate of energy liberation because two-thirds of the
dissipated energy is transported from closer in by the same torques that
transport angular momentum outward. Overall energy conservation is
maintained because the dissipation rate close to the ISCO is lower than
the local rate at which energy is liberated. This means that the outer
parts of an accretion disk will accumulate internal energy and becomes
unbound unless at least two-thirds of the dissipated energy is radiated
away.

This problem was noted by Narayan & Yi
[21,
22]
in their models for *advection-dominated accretion flows* (ADAFs),
which assume steady accretion with low radiative
efficiency. Self-similar ADAF models exhibit positive Bernoulli function
*B* (where *B* is the sum of gravitational potential energy,
kinetic energy and specific gas enthalpy), implying that elements of gas
in the flow are able to unbind neighboring elements by doing work on
them. Eventually the entire flow should disperse. Possible resolutions
that preserve disklike structure include *adiabatic inflow-outflow
solutions* (ADIOS)
[6],
*convection-dominated accretion flows* (CDAFs)
[23,
24],
and models with large-scale matter circulation
[7].
These models operate by suppressing the accretion rate relative to the
mass supply and/or providing an escape route for energy and angular
momentum that avoids the accreting gas. All of them require some
unspecified mechanism for separating accreting matter from outflowing
mass, energy and angular momentum. Despite these uncertainties, various
numerical simulations
[25,
26,
27]
suggest that such a separation can take place naturally.

The dynamical significance of the Bernoulli parameter can be
demonstrated by considering a sequence of two-dimensional, axisymmetric,
self-similar models of quasi-Keplerian rotating flows with
pressure. Quasi-Keplerian here means that the specific angular momentum
scales with radius as *R*^{1/2}. If *B* is either
constant or has a radial scaling ∝ 1 / *R*, then as *B*
approaches zero from below the surface of the disklike flow closes up to
the rotation axis, and the flow becomes *starlike*.
[21,
28]
If *B* becomes positive, the flow becomes unbounded. This suggests
that disklike ADAF models without some kind of escape valve (such as
outflow or circulation) are untenable. Could such flows alternatively
resolve their energy crisis by expanding and becoming nearly spherical?

It turns out that the ratio of the specific angular momentum to the
Keplerian angular momentum is the parameter that decides whether an
accretion flow is disklike or starlike. Any of the disklike, radiatively
inefficient flows have relatively "flat" density distributions as a
function of radius, ρ ∝ *R*^{-n} with
1/2 < *n* < 3/2. But in order for such flows to remain in
dynamical equilibrium with *B* < 0, the specific angular
momentum has to maintain a fairly large fraction of the Keplerian
value. For the interesting case of a radiatively inefficient accretion
flow (RIAF) dominated by radiation pressure, meaning that the adiabatic
index γ is 4/3, the minimum specific angular momentum compatible
with disklike flow ranges between 74% and 88% of Keplerian, as *n*
ranges from 3/2 to 1/2. Any lower value of angular momentum leads to
starlike flow. (This result is sensitive to γ, with the constraint
relaxed for larger values of this parameter. For the limiting case of a
gas pressure-dominated RIAF with γ = 5/3 and *n* = 3/2,
disklike flow is formally possible for any specific angular momentum,
but even in this limit an angular momentum 30% of Keplerian forces the
disk surface to close up to within 5^{∘} of the rotation
axis.)

The outer boundary conditions for the mass supply probably play the dominant role in determining whether the angular momentum is large enough to keep the flow disklike. Gas that is supplied from the outside, in the form of a thin disk, probably holds on to enough angular momentum to remain disklike all the way to the black hole. Accretion flows in X-ray binaries fall into this category. On the other hand, flows that start out with a finite supply of angular momentum, such as debris from tidal disruption events or the interiors of stellar envelopes, might become starved of angular momentum (particularly if the envelope expands as it absorbs energy released by accretion) and forced into a starlike state.

The presumption for disklike flows is that the accretion rate adjusts itself so that the circulation, outflow, convection — or whatever it is that relieves the energy crisis — is able to carry away any accretion energy that isn't lost to radiation. This is possible because the radial density and pressure gradients are rather flat, giving the outer flow a high "carrying capacity" for excess energy, compared to the inner flow where most of this energy is liberated. But this presumption fails in the case of starlike accretion flows, where the density and pressure profiles are forced to steepen in order to keep the gas bound while satisfying dynamical constraints. This means that matter is more centrally concentrated around the black hole in a starlike flow, and if this matter is swallowed with an energy efficiency typical for the ISCO it will liberate much more energy than can be carried away by the outer parts of the flow.

One can envisage several outcomes of this situation: 1) there is no equilibrium configuration, and the flow either blows itself up or becomes violently unsteady, with successive episodes of energy accumulation and release; 2) the gas close to the black hole is pushed inward by the pressure until it approaches the marginally bound orbit, in which case the large accretion rate releases very little energy; and 3) the inner flow finds a way to release most of the accretion energy locally, without forcing it to propagate through the outer flow. This third possibility, for example, could involve the production of powerful jets that escape through the rotational funnel. It is hard to decide by pure thought which, if any, of these possibilities is most likely, and simulations have not yet addressed this problem. Nevertheless, there are observational indications that some tidal disruption events, at least, choose the third option (Sec. 4.3).

**3.3. Causes of Radiative Inefficiency**

The dynamical properties of black hole accretion flows depend on whether
the gas radiates efficiently or not, but do not depend on the mechanisms
that determine radiative efficiency. From an observational point of
view, however, these details are crucial because radiatively inefficient
flows can be either very faint or, paradoxically, very luminous. The
fiducial mass flux that governs radiative efficiency is related to the
Eddington limit, which is the luminosity (assumed to be isotropic) at
which radiation pressure force balances gravity for a gas with opacity
κ: *L*_{E} = 4π *GMc* / κ. The
accretion rate capable of producing such a luminosity is
_{E} =
*L*_{E}/ є
*c*^{2}, where є is the radiative efficiency of
accretion. Radiation escaping from such a flow will exert outward
pressure forces competitive with centrifugal force, thus creating the
dynamical conditions prevalent in radiatively inefficient flow. In fact,
flows with >
_{E} are
literally radiatively inefficient because radiation is trapped and
advected inward by the large optical depth at radii *R* <
( /
_{E})
*GM* / *c*^{2} (to within a factor ∼ є).
[29]

Thus, accretion flows with high accretion rates are radiatively inefficient because they liberate more energy than they can radiate, but they are also very luminous because they radiate at close to the Eddington limit. Such flows are strongly dominated by radiation pressure and should be modeled using an adiabatic index of 4/3.

Accretion flows with low accretion rates can also be radiatively
inefficient, because their densities are so low that radiative processes
(which typically scale as ρ^{2}) cannot keep up with
dissipation (which scales as ρ). Because electrons cool much more
rapidly than ions, such flows are expected to develop a
"two-temperature" thermal structure, with *T*_{i}
≫ *T*_{e}.
[30] They would be optically thin, dominated by
thermal gas pressure, and characterized by an adiabatic index 5/3. If
thermal coupling between ions and electrons is provided by Coulomb
interactions, such flows can exist for accretion rates
<
α^{2}
_{E} (very
roughly), where α ≪ 1 is a widely-used parameter that
describes the rate of angular momentum transport.
[18]

At high (>
_{E}) and
intermediate (α^{2}
_{E} <
<
_{E})
accretion rates, the thermal state of accretion is uniquely determined
by . But flows with
low accretion rates may exist in either a radiatively efficient (thin
disk) or inefficient (two-temperature) state. It is not understand what
would trigger a thin disk in this regime to transition to the
radiatively inefficient state, or vice-versa, but there is evidence that
such transitions do occur in X-ray binaries.