794 Encyclopedia of the Solar System
plane:
μ^2 = 2 n^2 −κ^2. (29)
For a perfectly spherically symmetric planet,μ=κ=n.
Since Saturn and the other ringed planets are oblate,μ
is slightly larger andκis slightly smaller than the orbital
frequencyn.
Using Eqs. (24)–(29), one can show that the orbital and
epicyclic frequencies can be written as
n^2 =
GM
r^3
[
1 +
3
2
J 2
(
R
r
) 2
−
15
8
J 4
(
R
r
) 4
+
35
16
J 6
(
R
r
) 6
+···
]
, (30)
κ^2 =
GM
r^3
[
1 −
3
2
J 2
(
R
r
) 2
+
45
8
J 4
(
R
r
) 4
−
175
16
J 6
(
R
r
) 6
+···
]
, (31)
μ^2 =
GM
r^3
[
1 +
9
2
J 2
(
R
r
) 2
−
75
8
J 4
(
R
r
) 4
+
245
16
J 6
(
R
r
) 6
+···
]
. (32)
Thus, the oblateness of a planet causes apsides of particle
orbits in and near the equatorial plane to precess in the
direction of the orbit and lines of nodes of nearly equatorial
orbits to regress.
Resonances occur where the radial (or vertical) fre-
quency of the ring particles is equal to the frequency of
a component of a satellite’s horizontal (or vertical) forcing,
as sensed in the rotating frame of the particle. In this case
the resonating particle is always near the same phase in
its radial (vertical) oscillation when it experiences a partic-
ular phase of the satellite’s forcing. This situation enables
continued coherent “kicks” from the satellite to build up
the particle’s radial (vertical) motion, and significant forced
oscillations may thus result. The location and strengths of
resonances with any given moon can be determined by de-
composing the gravitational potential of the moon into its
Fourier components. The disturbance frequency,ω, can
be written as the sum of integer multiples of the satellite’s
angular, vertical, and radial frequencies:
ω=jns+kμs+
κs, (33)
where the azimuthal symmetry number,j, is a nonnegative
integer, andkand are integers, withkbeing even for hor-
izontal forcing and odd for vertical forcing. The subscript s
refers to the satellite. A particle placed at distancer=rL
will undergo horizontal (Lindblad) resonance ifrLsatisfies
ω−jn(rL)=±κ(rL). (34)
It will undergo vertical resonance if its radial positionrv,
satisfies
ω−jn(rL)=±μ(rv). (35)
When Eq. (34) is valid for the lower (upper) sign,rLis
referred to as the inner (outer) Lindblad or horizontal reso-
nance. The distancervis called an inner (outer) vertical res-
onance if Eq. (35) is valid for the lower (upper) sign. Since
all of Saturn’s large satellites orbit the planet well outside the
main ring system, the satellite’s angular frequencynsis less
than the angular frequency of the particle, and inner reso-
nances are more important than outer ones. Whenm =1,
the approximationμ≈n≈κmay be used to obtain the ratio
n(rL,v)
ns
=
j+k+
j− 1
. (36)
The notation (j+k+ )/(j−1) or (j+k+ ):(j−1) is
commonly used to identify a given resonance.
The strength of the forcing by the satellite depends,
to lowest order, on the satellite’s eccentricity,e, and in-
clination,i,ase|^ |[sini]|k|. The strongest horizontal res-
onances havek= =0, and are of the formj:(j−1). The
strongest vertical resonances havek=1, =0, and are of
the form (j+1):(j−1). The location and strengths of such
orbital resonances can be calculated from known satellite
masses and orbital parameters and Saturn’s gravity field.
Most strong resonances in the Saturnian system lie in the
outer A ring near the orbits of the moons responsible for
them. Ifn=μ=κ, the locations of the horizontal and ver-
tical resonances would consider:rL=rv. Since, owing to
Saturn’s oblateness,μ>n>κ, the positionsrLandrvdo
not coincide:rv<rL. A detailed discussion of spiral den-
sity waves, spiral bending waves, and gaps at resonances
produced by moons is presented elsewhere in this encyclo-
pedia. [SeePLANETARY RINGS.]
4. Chaotic Motion
4.1 Concepts of Chaos
In the nineteenth century, Henri Poincar ́e studied the
mathematics of the circular restrictedthree-body prob-
lem. In this problem, one mass (the secondary) moves in a
fixed, circular orbit about a central mass (the primary), while