Encyclopedia of the Solar System 2nd ed

810 Encyclopedia of the Solar System

f

satellite

planet

periapse

orbit of satellite

r

FIGURE 21 The geometry used to define the orientation of a
satellite in orbit about a planet. The planet–satellite line makes
an anglef(the true anomaly) with a reference line, which is
taken to be the periapse direction of the satellite’s orbit. The
orientation angle,θ, of the satellite is the angle between its long
axis and the reference direction.

equation

r=

a(1−e^2 )

1 +ecosf

, (61)

whereaandeare the constant semimajor axis and the ec-
centricity of the satellite’s orbit, respectively, and the orbit
is taken to be fixed in space.
Equation (60) defines a deterministic system where the
initial values ofθandθ ̇determine the subsequent rotation
of the satellite. Sinceθandθ ̇define a unique spin position
of the satellite, a surface of section plot of (θ,θ ̇) once every
orbital period, say at everyperiapsepassage, produces a
picture of the phase space. Figure 22 shows the resulting
surface of section plots for a number of starting conditions
usinge=0.1 andω 0 =0.2. The chosen values ofω 0 and
eare larger than those that are typical for natural satel-
lites, but they serve to illustrate the structure of the surface
of section; large values ofeare unusual since tidal forces
also act to damp eccentricity. The surface of section shows
large, regular regions surrounding narrow islands associated
with the 1:2, 1:1, 3:2, 2:1, and 5:2 spin–orbit resonances at
θ ̇=0.5, 1, 1.5, 2, and 2.5, respectively. The largest island
is associated with the strong 1:1 resonance and, although
other spin states are possible, most regular satellites, in-
cluding Earth’s Moon, are observed to be in this state. Note
the presence of diffuse collections of points associated with
small chaotic regions at the separatrices of the resonances.
These are particularly obvious at the 1:1 spin–orbit state
atθ=π/2,θ ̇=1. Although this is a completely differ-
ent dynamical system compared to the circular restricted
three-body problem, there are distinct similarities in the
types of behavior visible in Fig. 22 and parts of Figs. 14
and 15.

FIGURE 22 Representative surface of section plots of the

orientation angle,θ, and its time derivative,θ ̇, obtained from the

numerical solution of Eq. (59) usinge=0.1 andω 0 =0.2. The

values ofθandθ ̇were obtained at every periapse passage of the

satellite. Four starting conditions were integrated for each of the

1:2, 1:1, 3:2, 2:1, and 5:2 spin–orbit resonances in order to

illustrate motion inside, at the separatrix, and on either side of

each resonance. The thickest “island” is associated with the

strong 1:1 spin–orbit stateθ=1, whereas the thinnest is

associated with the weak 5:2 resonance atθ=2.5.

In the case of near-spherical objects, it is possible to in-

vestigate the dynamics of spin–orbit coupling using analyti-

cal techniques. The sizes of the islands shown in Fig. 22 can

be estimated by expanding the second term in Eq. (60) and

isolating the terms that will dominate at each resonance.

Using such a method, each resonance can be treated in iso-

lation and the gravitational effects of nearby resonances can

be neglected. However, if a satellite is distinctly nonspher-

ical,ω 0 can be large and this approximation is no longer

valid. In such cases it is necessary to investigate the motion

of the satellite using numerical techniques.

8.2 Hyperion

Hyperion is a satellite of Saturn that has an unusual shape

(Fig. 23). It has a mean radius of 135 km, an orbital eccen-

tricity of 0.1, a semimajor axis of 24.55 Saturn radii, and a

corresponding orbital period of 21.3 days. Such a small ob-

ject at this distance from Saturn has a large tidal despinning

timescale, but the unusual shape implies an estimated value

ofω 0 =0.89.

The surface of section for asingletrajectory is shown in

Fig. 24 using the same scale as Fig. 22. It is clear that there is

a large chaotic zone that encompasses most of the spin–orbit

resonances. The islands associated with the synchronous

and other resonances survive but in a much reduced form.

Although this calculation assumes that Hyperion’s spin

axis remains perpendicular to its orbital plane, studies

have shown that the satellite should also be undergoing