**8.2. Binary black hole formation**

Given the large frequency of galaxy encounters
and mergers, if there is a massive black hole in nearly
every galaxy, the formation of a binary
black hole should be a common phenomenon.
The successive physical processes able to brake
the two black holes in their relative orbit have been considered by
Begelman et al (1980).
Each black hole sinks first toward the merger remnant center
through dynamical friction onto stars. A binary is formed;
but the life-time of such a binary can be much larger
than a Hubble time, if there is not enough stars to
replenish the loss cone, where stars are able to interact
with the binary. Once a loss cone is created, it is replenished
only through the 2-body relaxation between stars,
and this can be very long (see section 2).
Modelising the merger remnant as an elliptical,
with a core of radius *r*_{c} and mass *M*_{c}
(and corresponding velocity *V*_{c}), the radius
where loss cone effects are significant is:
*r*_{lc} / *r*_{c} = (*M*_{bh} /
*M*_{c})^{3/4}.
The various time-scales involved, and
corresponding characteristic scales are defined by the following steps:

- the dynamical friction on stars, in less than a
galactic dynamical time,
- when the separation of the binary shrinks to
a value
*r*_{b}=*r*_{c}(*M*_{bh}/*M*_{c})^{1/3}, the two black holes become bound together - the binary hardens, with
*r*_{h}(*r*/*r*_{b})^{3/2} - but when
*r*<*r*_{lc}= (*M*_{bh}/*M*_{c})^{3/4}*r*_{c}, the stars available for the binary to interact with, are depleted through the loss cone effect, and replenished only by 2-body relaxation - gas infall can reduce the binary life-time
(whether the gas is flung out, or accreted, there is
a contraction of the binary) in
*t*_{gas} - gravitational radiation shrinks the orbit on
*t*_{GR}~ 0.3*Myr*(10^{8}/*M*_{bh})^{3}(*r*/ 0.003*pc*)^{4}if the two black holes have comparable masses

All these time-scales are represented on figure 25. If the binary life-time is too long, another merger with another galaxy will bring a third black-hole. Since a three-body system is unstable, one of the three black-holes will be ejected by the gravitational slingshot effect.

Since the life-time of the binary is not short,
there should be observable manifestations of massive black hole binaries.
One of the best tracer is to detect the periodicity
of the keplerian motion, with
the period P ~ 1.6yr r_{16}^{3/2}
M_{8}^{-1/2}. This is the case for the AGN
OJ 287 where eclipses have been monitored for a century
(Takalo 1994,
Lehto & Valtonen
1996,
Pietilä 1998).
Also, if the black holes are rotating, and their
spins have misaligned axes, they precess around the orbital one.
Plasma beams (aligned to the hole axis) precess,
and curved jets should be observed,
with periods between 10^{3} to 10^{7} yr.
This is frequently the case in radio structures observed with VLA and VLBI,
modified by Doppler boosting, and light travel time
(cf 3C 273, NGC 6251, 1928+738,
Kaastra & Roos
1992;
Roos et al 1993).
Finally, pairs of radio galaxies have been observed during
their merger with four radio jets (3C 75,
Owen et al 1985).

Numerical simulations have brought more precision in the determination of the life-time of the binary, although numerical artifacts have given rise to debates. Ebisuzaki et al (1991) claimed that the life-time of the binary should be much shorter if its orbit is excentric, since then the binary can interact with more stars and release the loss cone problem. The first numerical simulations tended to show that orbit excentricity should grow quickly through dynamical friction (Fukushige et al 1992). Mikkola & Valtonen (1992) and others found that the excentricity in fact grows only very slowly.

Numerical simulations suffer from a restricted number
of bodies N, and consequently of a large random velocity of the
binary (that shoud decrease in N^{-1/2}).
The binary then wanders in or even out of the loss cone,
and the effect of the loss cone depletion does not occur
(Makino et al 1993).
Also the 2-body relaxation time is shorter than
in the real system, contributing to replenish the cone.
Numerically, the life-time of the binary depends on the total
number of particles, i.e. the ratio between the
black hole to particle mass:

To summarize the conclusions of several numerical
computations, there is finally
little dependence on excentricity *e*, only in rare
cases, when *e* is large from the beginning
(Quinlan 1996).
Eventually, the wandering of the binary helps the merging
of the two black holes
(Quinlan &
Hernquist 1997).
The ejection out of the core of stars interacting with the binary
weakens the stellar cusp, while the
binary hardens. This may help to explain the
surprisingly weak stellar cusps in the center of giant ellipticals
observed recently with HST. Observations show that bright elliptical
galaxies have weak cusps, while faint
galaxies have strong cusps, with a power law slope
of density versus radius of up to 2.
A way to weaken the cusps is a sinking black hole
(Nakano & Makino
1999),
and this could be the case for giant galaxies that have experienced
many mergers in their life.