© CAMBRIDGE UNIVERSITY PRESS 2000 |
13.1. Action and angle variables
For the dynamics of disk galaxies the basic situation to be considered is that of a mean field potential that is stationary and axisymmetric around the z-axis. Note that such potential is very different from the Keplerian potential generated by a point mass located at r = 0. In the equatorial plane defined by z = 0, the calculation of orbits is reduced to a one-dimensional problem by introducing an effective potential
(13.8) |
so that the energy integral can be written as
(13.9) |
Figure 13.2. Sketch of the effective potential for equatorial orbits in an axisymmetric field. |
Thus the radial momentum (in our case this is identified with the radial velocity) can be expressed as a function of r and of the integrals of the motion E and J, with J the specific angular momentum. For a large class of potentials, the function _{eff} exhibits one minimum at r = r_{0} (see Fig. 13.2), which identifies the radius of circular orbits with angular momentum J. If we take J > 0, and define
(13.10) |
the guiding center radius is related to the specific angular momentum by
(13.11) |
which is generally one-to-one; in order for _{eff} to exhibit a minimum at r_{0}, the function J = J(r_{0}) defined by Eq. (11) must be monotonically increasing. Typically, for a given value of J bound orbits are associated with energies in the range E_{0} < E < 0, with
(13.12) |
the minimum energy which corresponds to the circular orbit. In the radial coordinate the motion is periodic and takes place between two turning points r_{in}(E, J) < r_{0} < r_{out}(E, J). A radial action variable can thus be set
(13.13) |
with the property
(13.14) |
Here the radial frequency is defined as _{r} = 2 / _{r}, with the bounce time given by
(13.15) |
In turn, the angular frequency is defined by
(13.16) |
Orbits are closed (in the inertial frame of reference) if the ratio between the two frequencies is rational.