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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.

Figure 1

Figure 1. Flow chart illustrating the main factors determining the dynamical state of black hole accretion flows. The specific angular momentum relative to GM / c determines whether the flow is centrifugally choked or not, and the same quantity, relative to the Keplerian angular momentum, determines whether the flow resembles a disk or a quasi-spherical ("starlike") envelope. Radiative inefficiency is a necessary but not sufficient condition for strongly sub-Keplerian rotation. Additional factors, such as net magnetic flux and the spin of the central black hole, mainly affect the rate of angular momentum transport and the production of jets and winds.

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.

3.1. Disklike Accretion

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.

3.2. Starlike Accretion

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 R1/2. 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: LE = 4π GMc / κ. The accretion rate capable of producing such a luminosity is dot{M}E = LE/ є c2, 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 dot{M} > dot{M}E are literally radiatively inefficient because radiation is trapped and advected inward by the large optical depth at radii R < (dot{M} / dot{M}E) GM / c2 (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 ρ2) 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 TiTe. [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 dot{M} < α2 dot{M}E (very roughly), where α ≪ 1 is a widely-used parameter that describes the rate of angular momentum transport. [18]

At high (> dot{M}E) and intermediate (α2 dot{M}E < dot{M} < dot{M}E) accretion rates, the thermal state of accretion is uniquely determined by dot{M}. 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.

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