© CAMBRIDGE UNIVERSITY PRESS 2000 |

The theory of orbits is an important ingredient in the study of the
dynamics of
galaxies. Because the relaxation times for the relevant processes are very
long (see Chapter 7), much of the interest lies in the study of orbits
in a *mean field*. What is learned from these studies applies beyond
the case of
single particles, for the evolution operator in a continuum description
(either
in a fluid or in a stellar dynamical description, see Chapter 8) can
be essentially identified with the Hamiltonian that governs single particle
orbits.

Galaxy disks are relatively "cool" systems, that is most star (or gas
cloud) orbits are very close to being circular, so that a typical star
(or gas cloud) velocity at a given location is very close to the average
velocity of rotation of the disk. A proper description of a collection
of orbiting particles should
be given in terms of a distribution function (see Chapter 14). On the
other hand, the physically intuitive properties of such cool disks can
be easily traced back
to the characteristics of quasi-circular orbits of individual particles,
which
will be outlined below in this Chapter. The deviations from circular orbits
are called *epicycles*. To lowest order they correspond to a harmonic
oscillator in the radial direction characterized by the epicyclic frequency
. In cool disks the
typical radial velocity of a star *c* is much
smaller than the local average velocity *u*. Thus the epicycle is
small, because the typical radial excursion *c* /
is much smaller than
*r*. Note that the typical restoring force associated with the radial
oscillations
(which is basically determined by the conservation of angular momentum)
is *c*, which turns
out to be of the order of the vertical force due to the
self-gravity of the disk
2 *G*
. This point will be further
discussed in relation to the problem of density waves in Chapter 15.

The epicyclic motion is analogous to the Larmor oscillation of a charged
particle in a magnetic field (see Fig. 13.1).
The analogy is the basis
for the development of common tools of investigation, as will be
described at the end of
this Chapter. These studies separate the orbit in two parts, the orbit
of a *guiding center*, which reacts to perturbing forces
perpendicular to the
gyration axis in terms of *drifts* rather than accelerations, and
the rapid oscillations (Larmor or epicyclic) around it. One useful
concept of classical
dynamics that has found wide applications in the description of the
motion of charged particles in plasma physics is that of
*adiabatic invariant*
^{(1)}.
A well known example is that of the ratio of the energy
associated with the Larmor oscillation and the cyclotron frequency. Under
appropriate conditions, this quantity is approximately conserved when the
particle moves in an inhomogeneous and/or time-dependent field, which is
the key
to a simple interpretation of a number of interesting phenomena, such as the
trapping of particles in a magnetic mirror. This has a simple
counterpart in the study of star orbits in a cool disk.

From the empirical point of view, the best chance we have for a direct
measurement of the properties of individual stellar orbits in a galaxy
disk is that provided by the solar neighborhood. On the other hand, the
systematic motions occurring in the disk of our Galaxy are not easily
derived
from the observations, just because the Sun is part of such general flow. A
classical study by Oort
^{(2)}
managed to provide the empirical
demonstration that our Galaxy is in a state of differential rotation, as
proposed earlier by B. Lindblad. The basis for that study was the idea of
expanding the average velocity field (flow field) in Taylor series
around the location of the Sun, from which one could easily express the
properties of the
average motion relative to the Sun expected in the various directions in the
sky, along the galactic plane, and at various distances from the Sun. In
formulae

(13.1) |

(13.2) |

where *R* and *l* denote distance from the Sun location and
galactic longitude. For a generic flow field **u**, the four
constants that result from the Taylor expansion can be written as:

(13.3) |

(13.4) |

(13.5) |

(13.6) |

Since the disk is cool, by observing the motion of groups of stars in
various directions one can easily check that the flow is
basically a pure differental rotation, i.e. one finds
*K* 0 and
*C*
0. Furthermore, the parameters *A* and *B*, called the Oort
constants, obtained by fitting the data give the local properties of the
galactic rotation
curve. Note that for a pure differential rotation *u*_{r} =
0 and *u*_{} ~ *r*
(*r*), so that

(13.7) |

These and other kinematical studies provide a rather detailed picture of the
orbits of stars in the solar neighborhood
^{(3)}, where it is found
<
<
_{z}
(
26 km / (sec kpc) so
that the rotation period is
240 Myr; based on the
Oort limit for the
local mass density, described in Chapter 14, the vertical period of
oscillation is
70 Myr). The vertical
frequency of oscillation about the galactic plane is thus the fastest.

^{1} See
Kulsrud, R.M. (1957), *Phys. Rev.*, **106**, 205;
Kruskal, M. (1962), *J. Math. Phys.*, **3**, 806;
Northrop, T.G. (1963), **The Adiabatic Motion of Charged Particles**,
Interscience, New York
Back.

^{2} Oort, J.H.
(1927), *Bull. Astron. Inst. Netherlands*, **3 (no.120)**, 275
Back.

^{3} The precise numbers are
subject to several sources of uncertainties; see
Kerr, F.J., Lynden-Bell, D.
(1986), *Mon. Not. Roy. Astr. Soc.*, **221**,
1023
Back.