© CAMBRIDGE UNIVERSITY PRESS 2000 |

**13.4. Trapping at the Lagrangian points**

We now consider a weakly non-axisymmetric potential of a form that is
seen as
stationary in a suitable rotating frame (rotating at angular velocity
_{p})

(13.31) |

For simplicity (and in view of a number of simple applications) we take

(13.32) |

with *A*, *m* > 0. The number *m* is integer. If
the amplitude *A*(*r*) varies slowly and the phase
(*r*) is instead
rapidly varying, the potential
_{s} may
describe an *m*-armed
spiral pattern with local pitch angle *i* determined by

(13.33) |

where we have set
*k* = *d* /
*dr*. With respect to the azimuthal
coordinate, at a given radial location the perturbing potential presents
*m* maxima and *m* minima.

The perturbing potential
_{s} is taken to
be small, i.e. we consider the case when the
non-axisymmetric forces are small:

(13.34) |

Here _{0} is
defined in terms of Eq. (10) based on
_{0}. We choose
our frame of reference so that
(*r*_{co}) = 0,
with *r*_{co} the corotation radius.

Given the stationarity of
_{s}, the Jacobi
integral is conserved, while
in the inertial frame the energy *E* and the angular momentum
*J* vary along the orbits, with
=
_{p}
.

In general the orbits are very complicated. However, in the rotating frame
we may easily identify 2*m* points that are *stationary*,
i.e. points where a particle released at rest can in principle stay in
equilibrium. These are a
generalization of the *Lagrangian points* that are known in the
context of celestial mechanics. The equations of the motion show that
these points occur at the stationary points of
_{s} with
respect to the azimuthal coordinate
(*m* -
) =
*b*, with
*b* = 0, 1, 2...(2*m* - 1), so that the
tangential force vanishes. The vanishing of the radial force requires
that the following relation be satisfied:

(13.35) |

where *b* is even or odd depending on whether
_{s} is a
maximum or a minimum (with respect to the azimuthal coordinate). Thus we see
that the Lagrangian points occur in the vicinity of the corotation circle.

A curious behavior is found by checking the stability of these equilibrium
points. For simplicity, one may focus on a case where *A* =
constant, so that the stationary points are at
*r* = *r*_{co} and at
=
*b*/*m*. Then,
if we work out the linearized stability analysis for the motion in the
neighborhood of *r* = *r*_{co},
= 0 (which under the above
assumptions corresponds to a
*maximum* of the perturbing potential along the corotation circle),
we find the following relation for the eigen-frequencies
of the normal
modes of oscillations:

(13.36) |

where =
/
_{p}, and
all the *r*-dependent quantities are evaluated at the corotation
circle. Thus we see that under the
normal condition of monotonically decreasing
(*r*) the point of
*maximum* of the perturbing potential is *stable*. For small
values of
_{A}, one of
the two frequencies derived from Eq. (36)
corresponds to a modification of the (fast) radial epicyclic
oscillation, while
the new solution, which corresponds to the symmetry breaking and
modifies the neutral displacements along the corotation circle at
= 0, is a *slow
libration* frequency, given by

(13.37) |

Note the scaling
_{L} ~
_{A}^{1/2}. In contrast,
a similar analysis carried out at the perturbing potential *minima* at
corotation would show that they are generally *unstable*.

The above study of the stationary points for a non-axisymmetric rigidly
rotating
potential is analogous to the classical *restricted 3-body problem* in
celestial mechanics
^{(6)}. A two-star or a
star-planet (in circular orbit) configuration gives
rise to five Lagrangian points. Two points are at 60 degrees with
respect to
the two masses on opposite sides on the plane of the orbit. The other three
points are on the axis passing through the two masses. If the mass ratio
is small (such as
for the Sun-Jupiter case, for which the mass ratio is
10^{-3}),
then all the five points are close to the "corotation
circle", i.e. the circle of the orbit of the lighter mass. Then two of the
three axial Lagrangian points are at a distance
*O*(^{1/3})*r*_{co} from
the lighter mass, while the third is on the opposite side with respect
to the heavier mass. If
is sufficiently
small (below a threshold value which is
0.04), then the two
Lagrangian points off axis are
stable, as is well known from the "trapping" of two separate families
of asteroids in the solar system. The *libration* period scales as
^{-1/2} also
in this case.

Another application is found in the study of *geostationary
satellites*, many of which are used for telecommunication
purposes. In this case, the modification to the Earth's basically Keplerian
potential occurs via a quadrupole term which fits in the description of
Eq. (32) with = 0 and
*A*(*r*) ~ 1/*r*3 (as can also be checked
from the potential theory of the classical ellipsoids; see Chapter 10). This
is because the mass distribution of the Earth is not perfectly
axisymmetric. The rotating frame for which such
non-axisymmetric perturbation is stationary is obviously the one that makes
one turn in 24 hours. The weakness of the non-axisymmetric field is
quantfied by the fact that the libration period about the two stable
potential maxima is of
200 days.

The trapping of orbits close to a resonant circle is a rather general
process which would be best described by use of action and angle
variables ^{(7)}.
For example, at a Lindblad resonance
for an *m* = 2
perturbation a trapping phenomenon occurs which refers to the major axis
of the elliptical orbits that are closed in the axisymmetric case, but
that are not closed because of the presence of
_{s}. Thus for
an *m* = 2 stationary
perturbation the major axis slowly librates and is trapped around an
appropriate direction, while it precesses away from a direction at 90
degrees from that.

**13.4.1. Trapping of a charged particle on the top of a
hill**

The above discussion should be a reminder that the presence of rotation may
bring in some counter-intuitive features in the properties of orbits. A
simple problem in mechanics, where the presence of a magnetic field
replaces the role of rotation, presents some analogies with the above
case. It should also
be stressed that the concept of stability investigated here refers to cases
where no dissipation is present. From these examples it could also be
appreciated why, in some circumstances, *dissipation may have a
destabilizing role*
^{(8)} (see also Chapter 10).

Consider a charged particle (charge *q*, and mass *m*)
constrained to move on
a "hill" described by an axisymmetric surface *z* =
*z*(*r*) in the presence of a
constant vertical magnetic field
**B** = *B*_{0}**e**_{z} and constant
gravity **g** = - *g***e**_{z}. Then in polar cylindrical
coordinates the energy is

(13.38) |

which, because of the conservation of the canonical momentum associated with the axial symmetry:

(13.39) |

can be written as

(13.40) |

with

(13.41) |

Suppose

(13.42) |

with 1 < < 2 and
*C* > 0. Then the orbit of the particle stays
close to the top of the hill, which behaves as a stable equilibrium
point even
though it corresponds to the *maximum* of the potential energy.

^{6} Contopoulos, G.
(1973), *Astrophys. J.*, **181**, 657
Back.

^{7} Lynden-Bell, D. (1973), in
*Dynamical Structure and Evolution of Stellar
Systems*, edited by L. Martinet and M. Mayor, Geneva Observatory
Back.

^{8} M.D. Kruskal, as quoted in Coppi,
B. (1966), in
*Non-Equilibrium Thermodynamics, Variational Techniques, and
Stability*, ed. by R.J. Donnelly, R. Herman, and I. Prigogine,
Chicago University Press, Chicago, p. 259
Back.