Annu. Rev. Astron. Astrophys. 1984. 22:
471-506
Copyright © 1984 by . All rights reserved |

Before focusing on specific properties of black holes, it is interesting to consider some general features of compact ultraluminous sources. Certain order-of-magnitude quantities are involved in any model.

A central mass *M* has a gravitational radius

(1) |

where *M*_{8} is the mass in units of
10^{8}
*M*_{}. The
characteristic minimum time scale for variability is

(2) |

A characteristic luminosity is the "Eddington limit," at which radiation pressure on free electrons balances gravity:

(3) |

Related to this is another time scale (112):

(4) |

This is the time it would take an object to radiate its entire
rest mass if its luminosity were *L*_{E}. The characteristic
blackbody
temperature if luminosity *L*_{E} is emitted from radius
*r*_{g} is

(5) |

We can further define a characteristic magnetic field, whose energy density is comparable with that of the radiation. Its value is

(6) |

The expected field strengths induced by accretion flows can be of this order. The corresponding cyclotron frequency is

(7) |

The Compton cooling time scale for a relativistic electron of
Lorentz factor
_{e} (equivalent to
the synchrotron lifetime in the field *B*_{E}) is

(8) |

The photon density
*n*_{} within
the source volume is
~ (*L*/*r*^{2}*c*) /
<*h*>. If a
luminosity
*f L*_{E} emerges in photons with
*h*
*m*_{e}
*c*^{2}, which can
interact (with a cross section
~ _{T}) to produce
electron-positron pairs
(40),
then these photons will interact before escaping if

(9) |

Several inferences now follow about the radiation processes,
given
only the assumption that a primary flux with
*L*
*L*_{E} is
generated within radii a few times *r*_{g}:

- Thermal radiation from
*optically thick*material would be in the far-ultraviolet or soft X-ray region; if, however, thermal gas in the region were hot enough to emit X-rays, reabsorption would be unimportant. - If the bulk of the luminosity
*L*were synchrotron radiation in a field*B**B*_{E}(Equation 6), then the self-absorption turnover would be (99)(10) (i.e. typically in the infrared). No significant

*radio*emission can come directly from*r**r*_{g}unless some coherent process operates at_{cE}. Synchrotron emission at ~_{sE}would require electrons with_{e}40*M*_{8}^{1/14}. - The synchrotron or inverse Compton lifetimes of relativistic
electrons is
<< (
*r*_{g}/*c*) under these conditions, so in any model involving such mechanisms, the radiating particles must be injected or repeatedly reaccelerated at many sites distributed through the source volume. - If a substantial fraction of the radiation were generated as gamma rays with energies 1 Mev, then electron-positron pairs would inevitably be produced.

This last point is less familiar than the previous three, and so it
may merit some elaboration. Photons with energies above 0.5 Mev will
experience an optical depth to pair production that exceeds unity
whenever (*Lf* /*r*) exceeds a value equivalent to
~ 5 × 10^{29}
erg s^{-1} cm^{-1}.
Moreover, the annihilation rate constant for these pairs is
~ _{T}*c* if
they are subrelativistic, and smaller by
~ _{e}^{2}
if they are
ultrarelativistic (104).
This has the important consequence that a
compact source that produces gamma rays (either thermally or
nonthermally) at a steady rate satisfying
(9) will shroud itself
within an optically thick "false photosphere" of electron-positron
pairs, which scatters and Comptonizes all lower-energy photons
(58).

**2.2. Processes in Ultrahot Thermal Plasma**

The only quantities entering into the above discussion have been
essentially those involving the electromagnetic energy densities. We
now consider the physical conditions in plasma near a collapsed
object. If thermal plasma can radiate efficiently enough, it can cool
(even at
*r*
*r*_{g}) to the relatively modest
temperature *T*_{e} (Equation 5). However, two-body
cooling processes can be inefficient at low
densities; for this reason, and also because the energy available in
the relativistically deep potential well may amount to 100 Mev
ion^{-1},
the plasmas in AGNs may get hotter than those familiarly encountered
elsewhere (even by astrophysicists).

At ion temperatures up to, say,
*kT*_{i} = 100 Mev the ions are of course
nonrelativistic, but the thermal electrons may be relativistic. The
main distinctive effects arise because the time scale for establishing
electron-ion equipartition via two-body processes, or even for setting
up a Maxwellian distribution among the electrons themselves, may
exceed the time scale for radiative cooling via the same two-body
effects. Moreover, other cooling processes may hold the electron
temperature to
1 Mev even if the
ions are much hotter. Detailed
discussions of these various time scales are given by Gould
(54 -
56)
and Stepney (121).

COMPTONIZATION If photons of energy hv are scattered by
electrons with temperature
*T*_{e} such that
*kT*_{e} >>
*h*, then there is a systematic
mean gain (67,
125)
in photon energy of
(
/
)
(*kT*_{e}/*m*_{e}*c*^{2}) until,
after many scatterings,
a Wien law is established. If soft photons are injected in an
optically thick
(_{T} > 1) source,
then the emergent spectrum depends
essentially on the parameter
*y* = _{T}^{2}(*kT*_{e} /
*m*_{e}*c*^{2}) : if *y* << 1,
nothing much
happens; if *y* >> 1, a Wien law is set up; but in the
intermediate case when
*y* 1, the emergent
spectrum has an approximate power-law
form. When
*kT*
*m*_{e} *c*^{2}, the energy change in each
scattering is too
large for a diffusion approximation to be valid, and Monte Carlo
methods are needed (57).

PAIR PRODUCTION EFFECTS When the electron energies on the
tail of the Maxwellian
distribution exceed a threshold of 0.5 Mev, collisional processes can
create not only gamma rays but also e^{+}e^{-}
pairs. These pairs then
themselves contribute to the cooling and opacity; the physical
conditions must therefore be computed self-consistently, with pairs
taken into account (22).
Discussions have been given by several
authors (22,
40,
69,
70,
132).

The main production/annihilation processes are summarized in
Table 1. Further high-energy processes can
operate above 50 Mev (46).
The fullest discussions of thermal balance in relativistic plasma that
take pairs into account are due to Lightman and collaborators
(7,
69,
70)
and Svensson (122 -
124).
There is a maximum possible equilibrium
temperature, of order 10 Mev; but if the heat input is raised beyond a
certain value, the increment in pair density is so great that the
temperature falls again toward 1 Mev. Note that to extend the usual
cooling function
(*T*_{e})
into the temperature range where pair
production is important, one must specify the column density
*n*_{i}*r* of
the source as a second parameter (*n*_{i} being the ion
density). When
*n*_{i}*r* << 1, the dominant pair production is via
e-p collisions; but for
sources of higher column density, relation (9) may be fulfilled, and
more pairs come from
+
encounters.

**2.3. Cyclotron/Synchrotron and Inverse Compton
Cooling**

Suppose that the magnetic energy is *q* times the rest mass density of
the plasma: we might expect
*q*
*kT*_{i} / *m*_{p} *c*^{2} for
accretion flows. The
ratio of the cyclotron cooling time (neglecting reabsorption) to the
bremsstrahlung time for a subrelativistic electron is
_{f}(*m*_{e} /
*m*_{p}) *q*^{-1}(*kT*_{e} /
*m*_{e} *c*^{2})^{-1/2},
which is << 1 for a plasma with
*kT*_{i}
1 Mev with an
equipartition field; for ultrarelativistic electrons the dominance of
synchrotron losses over bremsstrahlung is even greater. Analogously,
Compton losses can be very important: indeed, in any source where
Thomson scattering on electrons (or positrons) yields
_{T} > 1, the
requirement that the Compton
-parameter be
1 implies that the
electrons or positrons must be mostly subrelativistic.

The conventional distinction between thermal and nonthermal
particles becomes somewhat blurred in these contexts where two-body
coupling processes cannot necessarily maintain a Maxwellian
distribution. Various acceleration mechanisms (relativistic shocks,
reconnection, etc.) may, moreover, boost some small fraction of the
particles to high : such
mechanisms operate in many contexts in
high-energy astrophysics and should be even more efficient in an
environment where the bulk velocities and Alfvén speeds are both ~
*c*. These particles would then emit synchrotron or inverse Compton
radiation. Such acceleration would be "impulsive," in the sense that
its time scale is
<< *r*_{g} / *c*. The accelerating force would be
e*E*, where
*E*( *B*)
is the electric field "felt" by the charge. There is then a
characteristic peak energy attainable by such processes
(39), namely
that for which the radiative drag due to synchrotron and inverse
Compton emission equals e*B*. For
*B* = *B*_{E} (Equation 6), this yields
_{drag} = 4 ×
10^{5} *M*_{8}^{1/4}.
For acceleration along straight field lines,
synchrotron losses are evaded, and the terminal energy could be
~ *B*_{E}*r*_{g} (corresponding to
_{e} = 3 ×
10^{14}*M*_{8}^{1/2}) if
linear acceleration operated
over the whole scale of the source. Such limiting energies have
emerged from specific studies of accretion disk electrodynamics
(34,
72).
However, inverse Compton losses cannot be evaded in this way, and
they would set a limit not much greater than
_{drag}. (Individual * ions*,
not subject to radiative losses, could in principle get more energetic
than electrons.) The parameter
_{drag} scales as
*B*^{-1/2},
and electrons with this
energy emit synchrotron photons with
*h*
_{f}^{-1}
*m*_{e} *c*^{2} (i.e. 60 Mev)
(58,
99).
Inverse Compton radiation from the same electrons could of course
have photon energies right up to
_{drag}
*m*_{e} *c*^{2}. There is thus no
reason
why a (power-law?) spectrum should not extend up to the gamma-ray band.