Solar System Dynamics: Regular and Chaotic Motion 809
In this case, Roche’s limitrRocheis given by
rRoche= 31 /^3(
ρplanet
ρs) 1 / 3
Rplanet, (58)withρplanetandρsare the densities for the planet and satel-
lite, respectively, andRplanetis the planetary radius. When a
fluid moon is considered and flattening of the object due to
the tidal distortion is taken into account, the correct result
for a liquid moon (no internal strength) is
rRoche= 2. 456(
ρplanet
ρs) 1 / 3
Rplanet. (59)Most bodies have significant internal strength, which allows
bodies with sizes≤∼100 km to be stable somewhat inside
Roche’s limit. Mars’s satellite Phobos is well inside Roche’s
limit; it is subjected to a tidal force equivalent to that in
Saturn’s B ring.
4.Internal stresses caused by variations in tides on a
body in an eccentric orbit or not rotating synchronously
with its orbital period can result in significant tidal heat-
ing of some bodies, most notably in Jupiter’s moon Io. If
no other forces were present, this would lead to a decay of
Io’s orbital eccentricity. By analogy to the Earth–Moon sys-
tem, the tide raised on Jupiter by Io will cause Io to spiral
outward and its orbital eccentricity to decrease. However,
there exists a 2:1 mean-motion resonant lock between Io
and Europa. Io passes on some of the orbital energy and
angular momentum that it receives from Jupiter to Europa,
and Io’s eccentricity is increased as a result of this transfer.
This forced eccentricity maintains a high tidal dissipation
rate and large internal heating in Io, which displays itself in
the form of active volcanism. [See IO]
7.6 Tidal Evolution and Resonances
Objects in prograde orbits that lie outside thesynchronous
orbitcan evolve outward at different rates, so there may
have been occasions in the past when pairs of satellites
evolved toward anorbit–orbit resonance. The outcome
of such a resonant encounter depends on the direction from
which the resonance is approached. For example, capture
into resonance is possible only if the satellites are approach-
ing one another. If the satellites are receding, then capture
is not possible, but the resonance passage can lead to an
increase in the eccentricity and inclination. In certain cir-
cumstances it is possible to study the process using a sim-
ple mathematical model. However, this model breaks down
near the chaotic separatrices of resonances and in regions
of resonance overlap.
It is likely that the major satellites of Jupiter, Saturn,
and Uranus have undergone significant tidal evolution and
that the numerous resonances in the Jovian and Saturnian
systems are a result of resonant capture. The absence
of orbit–orbit resonances among the major moons in the
Uranian system is thought to be related to the fact that
the oblateness of Uranus is significantly less than that of
Jupiter or Saturn. In these circumstances, there can be large
chaotic regions associated with resonances and stable cap-
ture may be impossible. However, temporary capture into
some resonances can produce large changes in eccentricity
or inclination. For example, the Uranian satellite Miranda
has an anomalously large inclination of 4◦, which is thought
to be the result of a chaotic passage through the 3:1 res-
onance with Umbriel at some time in its orbital history.
Under tidal forces, a satellite’s eccentricity is reduced on
a shorter timescale than its inclination, and Miranda’s cur-
rent inclination agrees with estimates derived from a chaotic
evolution. [SeePlanetarySatellites.]8. Chaotic Rotation
8.1 Spin–Orbit Resonance
One of the dissipative effects of the tide raised on a natural
satellite by a planet is to cause the satellite to evolve toward
a state of synchronous rotation, where the rotational period
of the satellite is approximately equal to its orbital period.
Such a state is one example of a spin–orbit resonance, where
the ratio of the spin period to the orbital period is close to
a rational number. The time needed for a near-spherical
satellite to achieve this state depends on its mass and or-
bital distance from the planet. Small, distant satellites take
a longer time to evolve into the synchronous state than do
large satellites that orbit close to the planet. Observations
by spacecraft and ground-based instruments suggest that
most regular satellites are in the synchronous spin state, in
agreement with theoretical predictions.
The lowest energy state of a satellite in synchronous rota-
tion has the moon’s longest axis pointing in the approximate
direction of the planet–satellite line. Letθdenote the an-
gle between the long axis and the planet–satellite line in the
planar case of a rotating satellite (Fig. 21). The variation ofθ
with time can be described by equating the time variation of
the rotational angular momentum with the restoring torque.
The resulting differential equation isθ ̈+ω2
0
2 r^3sin 2(θ−f)= 0 , (60)whereω 0 is a function of the principal moments of iner-
tia of the satellite,ris the radial distance of the satellite
from the planet, andfis the true anomaly (or angular po-
sition) of the satellite in its orbit. The radius is an implicit
function of time and is related to the true anomaly by the