Next Previous Previous

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 Phi 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

Equation 8 (13.8)

so that the energy integral can be written as

Equation 9 (13.9)

Figure 13.2

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 Phieff exhibits one minimum at r = r0 (see Fig. 13.2), which identifies the radius of circular orbits with angular momentum J. If we take J > 0, and define

Equation 10 (13.10)

the guiding center radius is related to the specific angular momentum by

Equation 11 (13.11)

which is generally one-to-one; in order for Phieff to exhibit a minimum at r0, the function J = J(r0) defined by Eq. (11) must be monotonically increasing. Typically, for a given value of J bound orbits are associated with energies in the range E0 < E < 0, with

Equation 12 (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 rin(E, J) < r0 < rout(E, J). A radial action variable can thus be set

Equation 13 (13.13)

with the property

Equation 14 (13.14)

Here the radial frequency is defined as Omegar = 2pi / taur, with the bounce time given by

Equation 15 (13.15)

In turn, the angular frequency is defined by

Equation 16 (13.16)

Orbits are closed (in the inertial frame of reference) if the ratio between the two frequencies is rational.

Next Previous Previous