A test particle moving at velocity **v** along a trajectory that
passes a stationary field star of mass *m* with impact parameter
*b*
is deflected by the attraction of the field star. For a distant
passage, it acquires a transverse velocity component
|**v**_{}|
2*Gm* /
(*bv*) to first order (BT08 eq. 1.30). Encounters at
impact parameters small enough to produce deflections where this
approximation fails badly are negligibly rare and relaxation is driven
by the cumulative effect of many small deflections.

If the density of field stars is *n* per unit volume, the test
particle will encounter
*n* =
2
*b* *b*
*nv* stars per unit
time with impact parameters between *b* and *b* +
*b*. Assuming
stars to have equal masses, each encounter at this impact parameter
produces a randomly directed
**v**_{} that
will cause a mean square net deflection per unit time of

(A1) |

The total rate of deflection from all encounters is the integral over impact parameters, yielding

(A2) |

where ln
ln(*b*_{max} / *b*_{min}) is the Coulomb
logarithm. Typically one chooses the lower limit to be the impact
parameter of a close encounter, *b*_{min}
2*Gm* /
*v*^{2}, for
which |**v**_{}|
is overestimated by the linear formula, while the
upper limit is, say, the half-mass (or effective) radius, *R*, of the
stellar distribution beyond which the density decreases rapidly. The
vagueness of these definitions is not of great significance to an
estimate of the overall rate because we need only the logarithm of
their ratio. The Coulomb logarithm implies equal contributions to the
integrated deflection rate from every decade in impact parameter
simply because the diminishing gravitational influence of more distant
stars is exactly balanced by their increasing numbers.

Note that the first order deflections that give rise to this steadily increasing random energy come at the expense of second order reductions in the forward motion of the same particles that we have neglected (Hénon 1973). Thus the system does indeed conserve energy, as it must.

We define the **relaxation time** to be the time needed for
*v*_{}^{2}
*v*^{2},
where *v* is the typical velocity of a star. Thus

(A3) |

To order of magnitude, a typical velocity *v*^{2}
*GNm* /
*R*, where *N* is the number of stars each of mean mass
*m*, yielding
*N*. Defining the
dynamical time to be
_{dyn} = *R* /
*v* and setting *N* ~ *R*^{3} *n*, we have

(A4) |

which shows that the collisionless approximation is well satisfied in
galaxies, which have 10^{8}
*N*
10^{11}
stars. Including the effect of a smooth dark matter component in this
estimate would increase the typical velocity, *v*, thereby further
lengthening the relaxation time.

This standard argument, however, assumed a pressure-supported
quasi-spherical system in several places.
Rybicki (1972)
pointed out
that the flattened geometry and organized streaming motion within
disks affects the relaxation time in two important ways. First, the
assumption that the typical encounter velocity is comparable to the
orbital speed *v* = (*GNm* / *R*)^{1/2} is
clearly wrong; stars move past each
other at the typical random speeds in the disk, say
*v* with
~ 0.1,
causing larger deflections and decreasing the
relaxation time by a factor
^{3}.

Second, the distribution of scatterers is not uniform in 3D, as was
implicitly assumed in eq. (A1). Assuming a razor thin
disk, changes the volume element from
2 *v*
*b* *b* for 3D to
2*v* *b* in
2D, which changes the integrand in eq.& (A2)
to *b*^{-2} and replaces the Coulomb logarithm by the
factor (*b*_{min}^{-1} -
*b*_{max}^{-1}). In 2D therefore, relaxation is
dominated by close encounters. Real galaxy disks are neither razor
thin, nor spherical, so the spherical dependence applies at ranges up
to the typical disk thickness, *z*_{0}, beyond which the
density of stars drops too quickly to make a significant further
contribution to the relaxation rate. Thus we should use
*z*_{0} /
*b*_{min} for disks. More significantly, the local mass
density is also higher, so that *N* ~ *R*^{2}
*z*_{0}*n*. These considerations shorten the
relaxation time by the factor (*z*_{0} / *R*)
ln(*R* / *z*_{0}). An additional
effect of flattened distribution of scatterers is to determine the
shape of the equilibrium velocity ellipsoid, as discussed in
Section 10.3.

A third consideration for disks is that the mass distribution is much less smooth than is the case in the bulk of pressure supported galaxies. A galaxy disk generally contains massive star clusters and giant molecular clouds whose influence on the relaxation rate turns out to be non-negligible (see Section 10).