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3.2. Dissipative quasi-spherical accretion

When material with low specific angular momentum falls into a black hole, the resulting radiative output may be negligible (i.e., epsinon << 1) if the inflow is laminar. However, if the inflow is irregular, dissipative processes (turbulence, shock waves, etc.) could in principle liberate an amount of energy ~ c2 (r/rs)-1 per unit mass at radii gtapprox r. Provided that the cooling timescale is short enough to ensure that, at all r, the gas can radiate this energy on the inflow timescale, the efficiency can be just as high for this mode of inflow (which I shall term "quasi-spherical dissipative accretion") [33, 34] as for the well-known accretion disk.

Defining Lambda(T), the usual "cooling function", such that the power emitted per unit volume is (ne2) Lambda(T), the gas temperature T(r) at which cooling balances dissipative heating is given by

Equation 5 (5)

The infall timescale is tinfall (which may be longer than tfree fall if pressure gradients or rotation are important), and during that time each electron must radiate an energy ~ kTvirial appeq (GMhmp / r). If the dissipation results from each element of gas being shock-heated one or more times during the infall, then the time-averaged value of T appears on the LHS of eq. (5).

If cooling is very efficient, most of the gas will stay at ~ 104 K, the radiation emerging as lines and continuum in the optical or near ultraviolet (though harder radiation may emerge from elements of gas just behind shock fronts). But there is an obvious thermodynamic requirement that T > Tmin, where Tmin is the temperature of a black body which radiates L(< r) / 4pir2 per unit area. L(< r) is defined as the luminosity emerging from radii < r; and so

Equation 6 (6)

When accretion onto a black hole of < 108 Msun (i.e., rs 3 x 1013 cm) generates a quasar-level luminosity, then, even if cooling is very efficient, the temperature cannot be as low as ~ 104 K in the central regions (r ltapprox 100rs) where most of the energy is generated. Moreover, the radiation emerging from these regions may not escape directly, because a corollary of the assumed efficient cooling is that the opacity would be high as well. The energy would then all emerge from an effective photosphere at radius such as Tmin appeq 104 K.

It is clear from (5) that the cooling will be maximally efficient when Mdot is high and/or the infall is "lumpy" (so that < ne2 > >> < Ne >2 (If, as in disk accretion, the infall timescale is much longer than the free fall timescale, the cooling is of course efficient enough to guarantee T(r) << Tvirial for a wider range of parameters.)

But in the other extreme, when bremmstrahlung and line cooling are inefficient, the gas will heat up almost adiabatically (i.e., T appeq Tvirial) until at r ltapprox 103 rs the electrons get relativistic. At this stage a variety of more efficient cooling processes - e.g., pair production, synchrotron, Compton, etc. - can come into play. Some of these cooling processes depend on the radiation density, magnetic field, etc, and they do not necessarily depend on the square of ne; but in general the effective value of Lambda(T) swoops sharply upwards when T gtapprox 1010 K, guaranteeing that the overall efficiency of dissipative accretion (i.e., the amount radiated before gas falls into the hole) is always high, even for low Mdot. We would then expect the radiation to be emitted from regions near the hole by non-thermal processes (i.e., in the form of photons with hnu << kT. The absence of cool surrounding gas means that this radiation could escape without being absorbed.

The sharp bifurcation between the form of T(r) for the cases of "efficient" and "inefficient" cooling is a consequence of the form of the cooling curve: the value of Lambda(T) is high at 104-105 K, and does not rise to a higher value until T gtapprox 109 K, when relativistic processes come into play. If cooling cannot balance the dissipative heating at 104-105, a rise to relativistic temperatures is inevitable. Furthermore it is "touch and go" whether or not efficient non-relativistic cooling can occur. The relevant parameter is Q = (tcool(T) / tinfall) x (Tvirial / T), which must be ltapprox 1 for some T ltapprox Tvirial if cooling is efficient. When kT ltapprox mec2, and the dominant cooling is bremsstrahlung,

Equation 7 (7)

Note that M does not appear explicitly. Thus, even when Mdot appeq Mdotcrit, efficient cooling demands epsilon << 1, substantial "clumping" of the gas, or an inflow velocity slower than free fall. Note also that the electron-scattering optical depth taues (> r) is another quantity which depends on Mdot / Mdotcrit. Its value is

Equation 8 (8)

A further quantity of interest in accretion theory is the "trapping radius" rtrap, within which radiation is transported inward with the flow faster than it can diffuse out relative to the matter. This is the radius such that tau(> rtrap) appeq (Vinfall (rtrap) / c)-1, and its value is given by

Equation 9 (9)

Equation (7) shows that, for any dissipative accretion flow when L appeq LEdd and tinfall appeq tfree, the gas may be only marginally able to radiate its binding energy on the infall timescale. If cooling is adequate, then the gas remains at ~ 104 until the black body limit (6) forces a higher temperature; and the radiation emerges from a photosphere at ~ 104. But if cooling is inadequate at ~ 104, the gas must heat up until it becomes relativistic. The radiation will then emerge via synchrotron-type processes and we shall observe directly non-thermal radiation emitted near the hole, there being no surrounding gas to absorb it.

One can then get variability on timescales down to ~ rs / c(~ 1 hr for Mh appeq 108 Msun), the continuum being due to the Blandford-Znajek mechanism, magnetic flares, etc.; and highly polarized if the field is ordered. Hard X-rays produced by ordinary thermal bremsstrahlung could also originate from the region r ltapprox 103 rs.

One might conjecture that the so-called "Lacertids" are objects where cooling is inefficient (Q gtapprox 1); so that no emission lines are seen and any rapidly varying and polarized optical non-thermal continuum emitted near the hole can escape without reabsorption.

Equation (6) yields a lower limit to the size of a region from which emission lines can originate. (This will be the radius of the photosphere in the case of quasi-spherical dissipative accretion, but is a general limit insensitive to any details of the model.) The characteristic virial velocity or escape velocity, ~ (rmin / rs)-1/2 c, is thus ~ 104 km s-1. A large value could result only if the central mass exceeded 108 Msun. If the width of the emission lines is indeed comparable with the orbital or escape velocity from the photospheres, then we conclude that they are compatible with the model, but that any significantly broader lines would post problems. Conversely, one could perhaps adduce the observed line widths as evidence for a high central mass so compact that it could not be anything other than a black hole. This conclusion would be strengthened if one could decisively exclude the (currently conceivable) possibility that the high gas velocities represent outflow with speeds much exceeding the escape velocity. The "photospheric" region from which broad emission lines may originate could be comparable to the photosphere of a hot star in terms of surface brightness, but it would be much less smooth and quiescent. One would expect large-amplitude inhomogeneities, and macroscopic motions up to ~ 104 km s-1 (i.e., highly supersonic).

The specific angular momentum of the infalling material, and the extent to which the resulting spin axis possesses long-term stability, depends on the origin of the accreted gas. However, it seems in any case unlikely that the inflow will be smooth or laminar. If Q << 1 (eq. (7)), one would expect the gas whether in a disk or not - to develop a "two phase" structure: cool clouds at ~ 104 K in pressure balance with gas at ~ Tvirial (the fraction in the latter phase adjusting itself so that, for it, Q appeq 1). This possibility would seem to merit further investigation.

There is obviously a tendency for accretion-driven sources to stabilize at a value comparable with the "Eddington limit" LEdd appeq 1.3 x 1046 (Mh / 108 Msun) erg s-1. Indeed, this consideration led Zeldovich and Novikov [8] to propose as early as 1964 that quasars were powered by accretion onto black holes of ~ 108 Msun. In principle LEdd can be exceeded in a variety of ways in any non-spherical models. The inflow may resemble a very convective "settling solution" with "buoyant bubbles": a jet with high (P/ rho) may escape along the rotation axis, driven by buoyancy (i.e., higher P/ rho); or one could have "two phase" quasi-spherical accretion with one phase blowing out and the other falling in (provided, of course, that the net flux of matter is inward). If Mdot >> Mdotcrit, then energy liberated near the hole (r < rtrap, eq. (9)) cannot escape unless it is trapped in gas which is moving bodily outward.

Supercritical accretion can also readily occur when stellar destruction provides the gas supply. Radiation pressure has a negligible effect on the bulk motion of a star (though it may, for instance, cause evaporation from its surface - a further interesting process that could augment the gas supply). If the stellar density were high enough, tidal disruption could supply, in effect, an arbitrarily powerful "shell source" at r appeq rT. If this exceeded LEdd then some of the material would be blown off as a wind by the pressure of a supercritical radiation flux, the gravitational energy released by inflow of the remainder providing the power. The photon luminosity can exceed LEdd when there is a radiation-driven wind; and the kinetic energy of the wind can be converted into radiation when deceleration occurs at larger distances from the hole, thereby yielding additional luminosity.

Gas at ~ 104 K can of course be ejected by radiation pressure even if L < LEdd, because of the opacity due to lines and the photoionization continuum. This process is discussed in detail in the contributions by Goldreich and Kippenhahn. Such effects would be particularly effective in the high-density photospheric region envisaged here. When "two phase" accretion occurs, one could have a flow pattern where the hot material is falling in, but the cool clouds (on which radiation pressure acts more effectively) are accelerated outwards. Note however that even electron-scattering opacity can lead to very high terminal velocities if the acceleration starts at r appeq rs (where, for a settling solution, the effective sound speed is itself ~ c).

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