© CAMBRIDGE UNIVERSITY PRESS 2000 |

**13.2. Epicyclic orbits**

In the limit of *quasi-circular orbits*, which can be quantified as

(13.17) |

we have
_{r}(*E*,
*J*)
(*r*_{0}),
_{}(*E*,
*J*)
(*r*_{0})
and the radial
periodic motion can be approximated by a harmonic oscillator with the *epicyclic frequency* defined by:

(13.18) |

(We also have *J*_{r} ~ [*E* -
*E*_{0}(*r*_{0})] /
(*r*_{0}),
which is analogous to
*v*_{}^{2} / *B* for a gyrating charged particle
in a magnetic field.) Thus if we separate out the motion of the guiding
center by writing *r*(*t*) = *r*_{0} +
*r*_{1}(*t*) and
(*t*) =
(*r*_{0})
+ _{1}(*t*), the
linearized equation for the conservation of angular momentum leads to:

(13.19) |

thus the epicycles are *ellipses* characterized by aspect ratio
2 /
(therefore they are
usually elongated in the direction of the motion), with
the star running in the opposite direction with respect to the guiding
center
(i.e., the motion in the epicycle is clockwise if the motion on the circular
orbit at *r*_{0} is counter-clockwise).

Note that from Eq. (18) the condition for the stability of circular
orbits (^{2} > 0)
formally coincides with the classical Rayleigh's
criterion for the stability of a rotating fluid
^{(4)}.
A few important special cases should be noted. A
pure harmonic potential (i.e., the mean field potential associated with a
homogeneous sphere) implies solid body rotation, in the sense that
^{2} =
4 *G*
/3 = constant;
in this case we have
=
2, and orbits are
closed in the form of ellipses centered at *r* = 0. A point mass
generates a Keplerian potential; from the third law of planetary motion
we see that in this case
=
, and thus orbits
are closed in the form of ellipses
with one focus at *r* = 0. For galaxy disks, since they are often
characterized by a flat rotation curve, the typical relation should be
2^{1/2}
, and orbits are
generally not closed. Some simple cases of
orbits with and
in rational ratio are
shown in Fig. 13.3.

In Chapter 14 we will show that the velocity distribution for a relatively
cool disk, because of the epicyclic constraints, has an anisotropic pressure
tensor for which the radial pressure exceeds the tangential pressure by the
ratio 4^{2} /
2.

For some purposes (e.g. for some detailed stellar dynamical studies of
density waves where an integration along the unperturbed orbits is
performed),
it is of interest to have a full description of the epicyclic expansion,
beyond the lowest order harmonic oscillator obtained by approximating the
potential
_{eff} with a
parabola in *r*_{0}. Such a systematic expansion
^{(5)}
is obtained by introducing an appropriate phase variable
. In order to do
this, it is first convenient to consider the transformation
(*E*, *J*)
(*a*,
*r*_{0}),
where the dimensionless epicyclic energy *a* is given by

(13.20) |

Thus the radial momentum can be expressed as a function

(13.21) |

Now the phase variable
is introduced by replacing the
radial velocity coordinate *p*_{r} with

(13.22) |

The complete epicyclic expansion is thus obtained by Taylor expansion
of Eq. (21) around *r* = *r*_{0}, which inserted in
Eq. (22) gives

(13.23) |

From here one gets the expression for
*dr* / *d*
as well. Then from

(13.24) |

(13.25) |

one gets the desired expressions for
=
(*t*) and
=
(*t*), which
completes the derivation. The first
terms of the full expansion can be summarized by noting

(13.26) |

(13.27) |

with *B*_{3} = - *A*_{3} + 2*A*_{2} -
1 and

(13.28) |

Here one can easily check that for the harmonic oscillator
*A*_{2} = 1/2 and
*A*_{3} = *B*_{3} = 0, while for the Keplerian
case
*A*_{2} = 0, *A*_{3} = 0, *B*_{3}
= -1. Note that for the *isochrone potential* (see Chapter 21)
*A*_{1} = 1 and all the other *A*_{n} vanish.

Many of these results find application in the study of the
dynamics of galaxies. In addition, they are also of interest in
some simple problems of celestial mechanics, where the potential
is
often close to being Keplerian. For example, the potential of the Earth, in
space (*r* > *r*_{T}), because of its flattening at
the poles, is approximately given by

(13.29) |

where we have retained only the quadrupole term in the general solution
to the Laplace equation (here at *r* = *r*_{T} the
quantity /2 -
represents the geographical latitude); for the Earth one has
*J*_{20}
10^{-3}. We recall the expression for the Legendre polynomial
*P*_{20}(*x*) = (3*x*2 - 1)/2.
The epicyclic theory easily allows us to study the precession of the
perigee of a satellite on the equatorial plane, where
=
/2. The precession
rate is proportional to the difference between
and
.

^{4} See, for example,
Chandrasekhar, S.
(1961), **Hydrodynamic and Hydromagnetic Stability**,
Oxford University Press,
Oxford; reprinted by Dover
Back.

^{5} Shu, F.H.
(1969), *Astrophys. J.*, **158**, 505;
Mark, J.W-K.
(1976), *Astrophys. J.*, **203**, 81.
The analysis of Mark removes
an undesired secular term present in the original derivation
Back.