Introduction
The synchronization of the orbital periods of the natural satellites of
planets resembles the synchronization of mechanical oscillators. When the
satellites are of comparable mass, each gravitationally influences the orbit
of the other satellites. Eccentricities of the orbits cause nonlinear
oscillations. The satellites can therefore modulate each other’s periods
slightly through altering each other’s eccentricities.
Variations in Io’s Perijove as a
Percent of Io’s Real Semimajor Axis
8
Example of Precession of
Large Libration Pattern:
This pattern makes a
complete rotation in about
40000 Europa orbits.
7
6
5
4
3
2
1
0
50
60
Resonances
In this study, Mathematica 7 was used to model a planet with only two
satellites, based on the orbital properties of Jupiter and two of its satellites,
Io and Europa. This was chosen because Io, Europa and Ganymede
actually are synchronized with orbital periods in ratios of 1:2:4. The orbits
are also phase-locked, with conjunctions between Io and Europa always
occurring at Io’s perijove, and Europa’s apojove (this equilibrium could
result from the unmodeled influence of Ganymede) . No tidal influences
were modeled, and eccentricities were exaggerated by a factor of 10.
The dependence of these libration patterns on initial conditions was also
investigated by changing the starting phase angle between the two orbits. The
libration pattern angular width and eccentricity were both found to increase as
the initial phase angle increased from 0o to 180o. They then decreased again as
the phase angle approached 360o.
Model Results
150o
Angular Width of Precession Loop
90
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
80
70
60
50
60
120
180
240
300
360
Diagram of 90o phase angle difference in orbits of
Io and Europa. (eccentricities are further
exaggerated for clarity).
The largest scale patterns in the libration were found to depend on the
mass of Jupiter. The period of these patterns increased with increasing
planet mass. This is analogous to observations of increased coupling of
synchronizing pendulums when the inertia of the coupling medium is
reduced.
Dependence of Precession Rate of
Europa's Apojove on Jupiter's Mass
20
10
0
60
6000
40000
5000
30000
Europa Orbits
Europa Orbits
35000
25000
20000
15000
4000
3000
2000
10000
1000
5000
0
0
0.2
0.4
0.6
0.8
1
Jupiter Mass
1.2
0
0
0.2
0.4
0.6
0.8
Jupiter Mass
120
180
240
300
360
Phase Angle (degrees)
In addition to large patterns in the libration, there are smaller scale structures as
well. There are small loops with periods on the order of 100 Europa orbits.
These form larger “omega” structures that can become small kinks or overlap into
larger loops. As initial-condition phase-angles increase to values closer to 90o,
these structures expand to become the large jagged patterns described
previously, with angular widths approaching 90o. In spite of all of this variation in
libration pattern structure, the precession of this structure proceeds smoothly with
a period of about 40000 Europa orbits, independent of the initial phase angle.
Example of Fine Structure in a Libration Pattern
Resonances
Large Precession Pattern Repeat Period
1
1:2
110
3:5
120
2:3
130
3:4 4:5
140
In the model, with exaggerated eccentricities, the orbit of Europa became
unstable when Io’s semimajor axis was increased beyond the 4:5
resonance. The 1:5 resonance was not visible. But as the graph shows,
significant perturbations occurred at the 1:4, 1:3, 2:5, 1:2, 3:5, 2:3, 3:4
and 4:5 resonances. The measured perturbations are changes in the
perijove distance of Io as a function of Io’s modeled semimajor axis
distance, with both measured as a percent of Io’s true semimajor axis.
The frequency and pattern of the perturbations also changes as the
distances approach a resonance, and these frequencies and patterns are
different for different resonances.
30
Phase Angle (degrees)
Segment of libration patterns showing alternating quick prograde libration of
one satellite while the other satellite retrogrades slowly. In this view, Europa
(green) retrogrades, then progrades, then retrogrades, while Io (blue) starts
In prograde libration, then retrogrades and then progrades again.
2:5
100
40
0
0
1:3
90
How much of the observed synchronization behaviors of satellites can be explained by
orbital considerations only, without tidal influences?
Does phase-lock cause or precede period synchronization (or vice versa)?
What is required to dampen the phase librations to achieve phase-lock?
Conjunction synchronization seems to move from the inner satellites outward. Does
orbital phase-lock do the same?
Does Ganymede alter the preferred equilibrium orbit positions?
Is outward migration of an inner satellite necessary to achieve synchronization?
Variations in Libration Pattern Due to Initial Phase Angle
Variation of Europa Eccentricity as a
Function of Initial Phase Angle
80
Future Investigations
330o
Angular Width (degrees)
We generated the libration by deliberately displacing Europa’s orbit with a
phase angle between Europa’s and Io’s perijoves of 90o. The libration is
not a simple oscillation; it evolves in a very complicated pattern. This
pattern is the combined effect of the libration itself, and of corresponding
changes in the eccentricities of the orbits. The biggest effects are on the
orbit of Europa.
30o
Eccentricity Variation
Although the orbits do not (in our timescale) achieve the total phase lock
observed in the real Galilean satellites, the orbits do not simply precess at
a constant rate. Instead, the orbits librate about a mutual axis that does
precess at a rate that appears fairly steady in the observed timescale.
Unlike the more complicated real situation, the libration is about the
equilibrium with both satellites at perijove.
1:4
70
Io’s Distance,
as a Percent of
Io’s Real
Semimajor Axis
Synchronization of orbital periods can only occur when the satellites are at a
handful of resonances. The usual 1:1 resonance of a pair of coupled
mechanical oscillators is not possible. Also, significant mutual perturbations of
the orbits of the satellites only occurs when the satellite orbits are very close to
one of a few other resonances, or when the orbits are so nearly similar that
they remain in close proximity.
Acknowledgements
OWU Physics & Astronomy Department
National Science Foundation
This study was supported by REU/RET NSF Grant #0648751
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