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Abstract:
Orbital evolution of a two-body system (planet and satellite) tends
toward a synchronous orbit, where the angular velocity of each body equals the
orbital velocity of the satellite.
Orbital evolution occurs in this manner because gravitational forces
between the two bodies deform their spherical shapes, producing what is known
as a tidal bulge. This tidal bulge
disturbs the symmetry of the two-body system and allows angular momentum to be
redistributed between the bodies. Once
in synchronous orbit, the tidal bulge does not change within the planet and the
planet no longer experiences tidal heating, a major source of geological activity.
Orbital resonance in the Galilean
satellite system has prevented Io’s orbit from evolving in this typical
manner. Io, Europa, and Ganymede have
orbital periods that are proportional in whole number ratios. For every two times Io orbits, Europa orbits
once, and for every two times Europa orbits, Ganymede orbits once. Because of this, the planets are in
conjunction together at approximately the same longitude every cycle and
therefore the gravitational forces on each other also repeat at the same
longitude. These cyclical gravitational
tugs disturb the orbits of the bodies from their normal course of
evolution. Because of the orbital
resonance, Io has a far more eccentric orbit then it would by itself, which in
turn causes strong tides and tidal heating.
This tidal heating has made Io the solar system’s most volcanically
active planet.
Tidal Forces
A satellite experiences a
gravitational force from its accompanying planet and a centrifugal force from
its revolution around the planet. The
gravitational force depends on the mass of the moon, the mass of the planet,
and the distance between the two. This
is related by the following equation;
F=GMm/r2 (Eq 1)
The mass of the
satellite is m and the mass of the planet is M. The distance between the two is r and
G is the gravitational constant.
The gravitational force is directed from the center of the moon towards
the center of the planet. Because the
force changes by a factor of r-2, the force is very sensitive
to the distance between the two bodies.
The centrifugal force is in the
opposite direction of the gravitational force.
This centrifugal force is actually a pseudo force because it does not
originate from any field as normal forces do.
Rather, the centrifugal force is the result of a rotating frame of
reference. The gravitational force
pulls the satellite directly toward the planet, however, the satellite does not
fall toward the planet, rather it falls around the planet with centripetal
acceleration. The moon still feels the
pull toward the planet; in order to keep it from falling there must be some
induced force that cancels out the force of gravity. This induced force, or pseudo force, is the centrifugal force. The pull against the car door when making a
sharp turn is an example of a pseudo force similar to the centrifugal
force. Figure 1 shows the direction of
the centrifugal and gravitational forces.
Figure 1: Direction and magnitude of the
centrifugal and gravitational forces.
The satellite orbits around the planet with an angular velocity of n
degrees per day.
The centrifugal and gravitational
forces are equal in strength at the center of the satellite. Their opposite direction causes the forces
to cancel each other out, creating an equilibrium at the moon’s center. To either side of this center the
equilibrium is broken. The side of the
moon closest to the planet is overwhelmed by gravitational force. The gravitational force is strong enough to
warp the shape of the satellite and form a bulge directed at the center of the planet. On the opposite side of the moon, the centrifugal force does the
same and a bulge is formed that is directed away from the center of the planet. Figure 2 shows these bulges.
Figure 2: The centrifugal bulge is directed away
from the host planet while the gravitational bulge is directed toward the
planet’s center.
Rising and falling tides result when
the bulge changes location in respect to the moon’s longitude or the bulge
changes shape. The first type of tide
occurs because of the moon’s rotation and is referred to as a rotational
tide. The moon has a rotational angular
velocity of w, measured in degrees per day. If w differs from the moon’s revolutionary angular
momentum n, the side of the moon facing the planet will change. In figure 3, a man is located on the moon at
location 1. At this point, the tidal
bulge is strongest. As time goes by the
moon will rotate ant the man will be at position 2. At this point, the tidal bulge is weakest. The person has not moved in longitude, so it
is the bulge that has moved.
Figure 3: As the satellite rotates, the bulge
remains pointing at the planet, causing the bulge to rotate through the
satellite.
A tide caused by a change in the
shape of the bulge is called a radial tide.
Kepler's laws of planetary motion state that satellites with elliptical
orbits will revolve around a planet located at one of the foci of the ellipse. Satellites with highly elliptical orbits
will vary greatly in their distance from the planet. Kepler's second law states that a line from a satellite to a
planet sweeps over equal areas in equal amounts of time. This means the angular velocity around the
planet changes throughout the orbit as well.
The change in distance and the change in angular velocity correspond to
changes in gravitational and centrifugal forces respectively. The changes in these forces affect the
severity of the tidal bulge through out the orbit. In Figure 4, we see that the forces are weakest when the
satellite is at its apocenter, or furthest point from the planet. At this point, the tidal bulge is small or
nonexistent. At the pericenter of the
orbit, or point closest to the planet, the forces are strong and the tidal
bulge is severe. The radial tide,
therefore, changes over the course of the orbit rather than the rotational day
of the satellite.
Figure 4: A satellite in an elliptical orbit
experiences variations in gravitational and centrifugal forces. This leads to changes in the tidal bulge
throughout the orbit.
Tidal
Friction and Orbital Evolution
Satellites are rigid bodies. Even satellites with large amounts of fluid
have a great deal of rigidity. This
rigidity opposes the shape changing affects of tides, resulting in tidal
friction. Tidal friction is a major
source of geological heat and is responsible for the majority of a planets
geological activity.
Tidal friction also causes tidal
lag. We have described the tidal bulge
as being directly aligned with the line between the centers of the satellite
and the planet. This is only true in
the case of a frictionless satellite or if the angular rotation of the
satellite equals the angular revolution around the planet, which is a
frictionless condition in itself. In
the case when w does not equal n, the tidal bulge is moved away from the
line of centers by friction and the satellite rotation. Friction within the satellite keeps the
bulge from keeping up with the rotation.
The effect is that the bulge lags behind or ahead, depending on the
speed of rotation. In Figure 5 we see
the position of the tidal bulge for the three conditions where w=n,
w>n, and w<n.
Figure 5: The three possible relations between n
and w, and the resulting tidal lags.
Adapted from Burns [1977].
The tidal lag ultimately affects the
orbital elements of the satellite. In
the case of the rotational tidal lag, the lag creates an imbalance of mass
along the planet and satellite line of centers. The imbalance creates a torque that pulls the bulge toward its
ideal and frictionless position. For
real satellites this position is only attainable if w=n. Over a long period of time angular momentum
of the satellite is lost because of the tidal friction. In Figure 6 the satellite is depicted in an
early state; w¹n and the resulting torque vector is shown. The later state shows the synchronous rotation and the symmetry
of the mass. The characteristic feature
of synchronous rotation is that only one side of the moon faces the planet.
Figure 6: Torque resulting from tidal lag and the
eventual equilibrium between n and w.
A tidal lag also occurs with the radial
tide. At the point of pericenter in the
planets orbit, the tidal forces are the greatest. However, the tide does not reach its greatest magnitude until
after this point because frictional forces are delaying the change in
shape. This radial tidal lag also
affects the orbit; eventually tidal friction causes elliptical orbits to become
circular. This evolution is better
examined as a matter of work and energy rather than a change in angular
momentum.
A host planet creates a
gravitational well and the satellites position within this well depends on the
amount of energy the satellite has. If
a satellite gains energy, its orbit will become hyperbolic or parabolic and the
satellite will escape. However, if a
satellite loses energy its orbit becomes more circular. When the tidal bulge of the satellite
changes, energy is lost through friction.
As the orbit loses energy and becomes more circular, the difference in
distance throughout the orbit is reduced and therefore the difference in tidal
strain is reduced as well. As the orbit
becomes circular, the satellite develops one constant shape.
We have just seen how tidal friction
within a satellite can alter satellite’s orbits. The normal evolutionary trend is toward a
non-frictional/non-variable tide state.
Under normal conditions, a satellite will eventually have a circular
orbit with synchronous rotation. Our
own Moon is a perfect example of normal orbital evolution. The Moon’s rotation is the same as its
revolution around the Earth; this is why we only see one side of the Moon. The Moon does not experience variable tides
or frictional heating. Because of this,
the Moon is for the most part geologically inactive.
An important aspect of the
Earth/Moon system is that the Moon has already completed its evolution. In the five billion years the solar system
has been in existence, all satellites have had enough time to complete this
evolution. Because of this astronomers
have thought that all satellites should have circular/synchronous orbits and
should be for the most part geologically inactive. In most cases, this is true, but new theories and new
observations have presented some exceptions.
Io and Orbital Resonance
Jupiter’s moon Io, is a prime
example of one of these exceptions.
Early infrared observations showed heat variations on Io’s surface. Later detailed measurements of its orbit
showed the satellite to have an elliptical orbit. Under normal circumstances, a satellite of Io’s mass should have
developed a circular/synchronous orbit a long time ago. To solve this problem astronomers began
investigating the theory of orbital resonance.
Now it is commonly accepted that Io’s eccentricity is being fuelled by
its orbital resonance with two other Galilean moons, Europa and Ganymede.
Orbital
resonance occurs when satellites have orbital periods that are related by small
whole number ratios. For example, we
see in Figure 7 that satellite S2 makes two complete orbits for every one orbit
of S1. The periods are related by a 2:1
ratio. Because the ratio is on the
scale of a whole number, the satellites will be in conjunction at the same
point in their orbits every time.
Figure 7: Orbital periods have ratios of small
whole numbers. S2 goes around twice
for every orbit of S1.
When the satellites are in
conjunction, their gravitational fields will exert a strong force on each
other. To simplify our example, we will
consider the mass of S1 to be much greater than the mass of S2. This allows us to ignore any gravitational
affects on S1. Every time the planets
are in conjunction, the more massive S1 pulls on S2 with its gravitational
field. The effect is to pull S2 away
from the planet at that point in its orbit.
As more conjunctions occur, the distance between S2 and the planet
increases. Eventually S2 develops an
elliptical orbit, as illustrated in figure 8.
Figure 8: The mass of S1 is much greater than the
mass of S2. S1 eventually pulls S1
into an elliptical orbit.
The eccentricity is maintained by the orbit of S1. If S2 begins to become too eccentric, S2
will gain velocity and reach the conjunction point before S1. When this happens S1 will exert a restoring
force on S2 and pull it back. If S2
slows down too much, S1 then pulls S2 forward to the conjunction point. In this respect, the system acts much like a
spring, in the sense that there is a restoring force that pulls the system back
to equilibrium. Figure 9 shows the direction
of the restoring forces.
Figure 9: If S2 pulls ahead of the conjunction
point S1 exerts an adjusting force that brings S2 back. A similar effect occurs if S2 falls
behind.
The Galilean resonance system is
more complicated than the example we just discussed. The resonance system is made up of three satellites, Io, Europa,
and Ganymede. For every two orbits Io
makes, Europa makes one; for every two orbits Europa makes, Ganymede makes
one. Not all three satellites have one
conjunction point; rather, Io and Europa have a unique conjunction point and
Europa and Ganymede have a conjunction point 180 degrees from that. The Io and Europa conjunction is shown in
Figure 10.
Figure 10: The Galilean resonance system is shown
with ratios of the orbital periods.
Ganymede, being the outermost
satellite, forces Europa into an eccentric orbit. Europa then forces Io into an even more eccentric orbit. Table 1 compares the free eccentricities of
the satellites with the forced eccentricities [Pleale, 1986]. The free eccentricity is the eccentricity of
the satellite if there was no orbital resonance; the forced eccentricity is the
result of the resonance. Notice that
Ganymede’s forced eccentricity is actually lower than its free
eccentricity. Apparently, Ganymede
sacrifices its eccentricity to Europa, which trickles down to Io.
Table 1: Eccentricities of the Galilean resonance
system.
Io has a nearly synchronous
rotation, however its high eccentricity generates strong radial tides. This strong radial tide generates strong
tidal friction, which leads to strong tidal heating, thus explaining the intense
geological activity on Io. Geologists
estimate the amount of tidal heating through various models. One such model generates the following
equation for the energy dissipated by Io [Pleale 1979]; 
Eq. 2
In this equation p
refers to the density of Io, n is the revolutionary angular velocity, Rs
is Io’s radius, e is the eccentricity, m is the rigidity, and Q is the
specific dissipation function. Some of
these values, such as eccentricity and radius, can easily be determined. However, the rigidity, density, and dissipation
function can only be guessed at.
Because geologists cannot accurately determine these values they cannot
determine the amount of energy exchange in Io’s resonance with Europa. Without the energy function of the system,
an accurate picture of the history of the resonance system cannot be
determined. In fact, this is true of
all satellite systems. Because the
energy functions of these satellites cannot be accurately modeled, it is
impossible to know their true evolutionary history.
Orbital
modeling has made a great deal of headway in the past few decades, especially
with the use of computers. However, the
field is still young and the physics of orbital evolution is not fully
understood. Astronomers, physicists,
and geologists have made breakthroughs.
The theory of orbital resonance and its ability to explain Io’s
geological activity is a testament to that.
Unfortunately, questions such as the origin of our moon and its exact
history are still a long way from being answered.