Orbital Resonances
Caden Armstrong
Ka Chun Lau
Introduction
In ancient philosophy, there were many academics that were under the belief that the universe was perfect. This perfection came out in the form of perfect shapes (isometric), integer
numbers and many other ideals. The idea of perfection within the universe has a small connection to orbital resonances. Objects in orbital resonance will have integer ratios for orbital
period, for example 2 : 1, 3 : 2, etc. But beyond the philosophical implications to orbital
resonances, there are many other questions that arise. Orbital resonances are observed in
moons, planets, and asteroids, but significantly different implications from each. Observed
resonances in asteroid-planet interactions prove to be highly unstable, but exoplanet resonances show signs of stability with the high level of occurrence. For the sake of planetary
science, it is worth looking at the occurrences of resonant systems, how we observe them (
and how resonances will change observations), examining how resonant systems form, and
most importantly; what effects come from resonant systems?
Bode’s Law
In the late 1700’s, Johann Titius created a simple numerical sequence to represent mean
orbital distances of planets. This sequence was later popularized by Johann Bode, thus being
named
”Titius-Bode rule”. The rule was simple, planets should be found at Distancen (AU ) =
P
(4 + 3 · 2n ) where n = −∞, 0, 1, 2.... This search was solidified when the use of the law
found an asteroid (Ceres) at 2.8AU . While the planets up to Saturn, and even the predicition
of Ceres seem like solid evidence for Bode’s Law, it is generally considered a ”rule” because
of how poorly it predicts planets past Uranus. Considering it as a loose rule, there is still
little explanation as to why it works. A few theories suggest that it may stem from orbital
resonances, as a modified version of the law also works for the moons of Jupiter, Saturn and
Uranus. [Carroll and Ostlie, 2007, p. 717]
Even with the general acceptance that Bode’s rule is most likely a mathematical coincidence,
and that there is no physical reasoning for the rule (Carroll and Ostlie [2007]), astronomers
are still attempting to apply the rule to new discoveries, with little luck. A recent paper
Chang [2010], Bode’s law was fitted to exoplanetary systems in an attempt to find a relation.
It was concluded that Bode’s rule explaining the distance distribution of planetary systems
happens purely by chance.
Kirkwood Gaps
The Kirkwood gaps are areas in the asteroid belt (between Mars and Jupiter) that are almost
devoid of objects. These gaps appear as dips in the distribution of periods of objects in the
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asteroid belt. The interesting feature of these gaps is where they are located. The 4 main
gaps appear at a semi-major axis that is in 3:1, 5:2, 7:3 and 2:1 resonance with Jupiter’s
orbit. As seen in figure 1 [Beatty et al., 1999, p. 225]
Figure 1: Kirkwood gaps as seen in the distribution of asteroids by semi-major axis
Credit:NASA
Plutinos
In the far reaches of our solar system, there is a type of object who’s classification is based on
it’s orbital period. Object’s called ”Plutinos” are objects found in the Kuiper belt, found in a
3 : 2 mean motion resonance with Neptune. In 1951, Kuiper suggested that there would be a
number of minor bodies beyond the orbit of Neptuen, which are the remains of planetesimals
from the formation of our solar system. The objects were named after Pluto, which was the
first of the Plutinos to be discovered. To date, there are over 32 known plutinos Wan and
Huang [2001]. In figure 2, the largest of the Plutinos are shown.
Plutinos are a very interesting set of objects to study as they are an extreme example of the
existence of resonant bodies in planetary systems. Many numerical studies have been done
on these objects to test the stability of them. Simulations suggest that interactions with
Pluto can cause Plutinos to be forced out of resonance Wan and Huang [2001].
Resonance in Exoplanets
In the search for exoplanets, resonances can cause quite an issue. To look for an eclipsing
planet, one would look at periodic dips in the light curve of a star, measuring things like the
time between dips (among many other things). From these periodic dips, we can conclude
the existence of planets. In a similar fashion, planets can also be discovered through the
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Figure 2: Some of the largest and first discovered Plutinos compared in size and albedo
Credit:Wiki user Eurocommuter
Doppler shift of a star. As planets orbit a star, their gravitational influence causes the star
to also orbit (around a common centre of mass). This motion causes a red and blue shift,
from which we can interpolate the existence of planets.
Figure 3: Left: XKCD comic 1371, credit: Randall Munroe
Right: Sample light curve from Kepler mission showing a transit
But this leads to an issue with planets in resonance. Any resonances in period will translate
to resonances in observational data from the star. This can cause issues and possibly hide the
existence of planets. In Anglada-Escud´e et al. [2010], single planet systems were examined
and compared to systems containing 2 : 1 orbital resonances. In their analysis, they found
that nearly 35% of of eccentric single planet systems are indistinguishable from two planet
systems in 2 : 1 resonance. They also found that planets with a mass comparable to Earth
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could be hidden in the orbital solutions of eccentric Neptune mass planets. While we have
found lots of exoplanets, there may still be many hiding in the systems we have already
examined.
This is a bigger problem than it seems on the surface, as the number of multi-planet systems
found to be in resonance is staggering. Some studies have found that nearly a third of
multi-planet systems have resonance, with about half of them in a 2 : 1 resonance Petrovich
et al. [2013]. Taking these facts into consideration, the number of hiding planets could be a
significant number.
Entering and Leaving Orbital Resonances
In essence, orbital resonances of celestial objects (typically planets) have a close tie with their
orbital migrations during the early formation stage of a planetary system. A well accepted
theory of the formation of planetary systems state that planets are formed in a proto-stellar
disc, as a natural by-product of the formation of stars Rein [2010] Tidal interactions between
the planets and the proto-planetary disc allow the migration of the planets through the
exchange of angular momentum Rein [2010]. Depending on many parameters, such as the
masses of the planets and the density distribution of the disc, the dissipative forces acted on
the planets may cause different types of planet migrations, and this leads to gradual changes
in their semi-major axes, as well as their orbital periods, governed by the Kepler’s third law
Rein [2010]. A convergent migration between two planets may cause them to enter a Mean
Motion Resonance when the ratios of their orbital periods are close to a rational number
q/p, with q and p being small integers, for example 1/2, 1/3 or 2/3 Rein [2010]. They would
experience periodic gravitational forces from each other, and such mechanism is much like
a swing being pushed rhythmically at the exact right moment of each cycle, making it go
higher and higher [Beatty et al., 1999, p. 255].
Some resonances are stable configurations, for example, having a slow initial convergent
migration, the planets can eventually be locked into resonance, and migrate together in
the proto-planetary disc, keeping a constant ratio of their orbital periods Rein [2010]. On
the other hand, some resonances may be unstable and can end after a certain time, for
example, having the migration process only stalled partially by the resonance, one planet
still maintains a much faster migration rate, and the two planets start to diverge afterwards
Rein [2010]. In fact, the stability of a particular resonance has nothing to do with the
ratio of the orbital periods of the resonating planets, for example, a 1/2 resonance can be
either stable or unstable depending on many factors. So far the previous discussion has not
considered the effects of the turbulence in the proto-planetary disc. A more complete model
of the formation of a planetary system would have density fluctuations in the disc, driven by
Magneto-Rotational Instability. Planets (of low masses) embedded in the turbulent disc are
easily affected by stochastic migration forces, produced by such density fluctuations, whereas
massive planets, being able to sweep out a gap in the disc, are affected by stochastic forces
to a much smaller extent Rein [2010] Overall, stochastic migration is a diffusive process, and
it tends to destroy orbital resonances Rein [2010].
Usually satellites are formed in a very similar way, if not the same, like the planets did, but
on a much smaller scale. Therefore, it is reasonable to describe the start and the end of the
resonance patterns between satellites using the same process mentioned above.
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Effect of Orbital Resonance
Orbital resonance has pronounced effects in celestial mechanics, and can be seen in many
places in our solar system. One of such place would be the Asteroid Belt in our solar system.
As mentioned before, although the Titius-Bode law is generally considered as a rule, it still
manages to predict the locations of the planets in our solar system fairly well. The greatest
discrepancy of this rule, however, is the absence of a planet between Jupiter and Mars, and
orbital resonance may be the very reason responsible for this situation Beatty et al. [1999].
It is possible that during the early ages of our solar system, the orderly accretional assembly
of a planet in the Asteroid Belt was disrupted by the huge gravitational field of the protoJupiter[Beatty et al., 1999, p. 19]. Planetesimals whose orbital periods were simple integral
fractions of the orbital period of Jupiter, such as 1/2, or 2/5, would be in Mean Motion
Resonances with Jupiter. Those affected by unstable resonant perturbations, typically the
1/3 resonance, would have chaotic variations in their orbital eccentricity, and this ultimately
terminated their dynamic life [Beatty et al., 1999, p. 224]. They would either be diving too
deep into the solar system, striking a planet or the Sun, or be soaring out to a much higher
orbit, perhaps flying beyond the solar system [Beatty et al., 1999, p. 19]. In fact, Jupiter is
still removing objects from the Asteroid Belt in the same way today. And since the Asteroid
Belt is crowded with small planetesimals, in time these asteroids would have a change in
their orbital periods due to the mutual gravitational interactions that occur when they get
too close to each other [Beatty et al., 1999, p. 19]. This process provides means for asteroids
that are previously not near the unstable resonant regions, such as the Kikwood gaps, to be
eventually captured in resonance with Jupiter and leave the Asteroid Belt.
Another place in our solar system where one can see the effects of orbital resonances lies
beyond Neptune’s orbit, and is known as the Kuiper-belt. Some of the Kuiper-belt objects
experience a 2/3 Mean Motion Resonance with Neptune, and one of them is Pluto, which
is the most famous Kuiper-belt object, due to once being classified as a planet of our solar
system (demoted on 2006). Pluto, like some other Kuiper-belt objects have their orbits
overlapping that of Neptune, but scientists have shown, through computer simulations, that
they have never been close to Neptune, nor can they ever be [Beatty et al., 1999, p. 294].
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The dynamical lifetime of Pluto and these Kuiper-belt objects are prolonged by their Mean
Motion Resonance with Neptune [Beatty et al., 1999, p.66]
Orbital resonance is not an exclusive feature for only planets in our solar system. Resonance
patterns can also be found in the satellite systems, for example, the Galilean satellites Io,
Europe, and Ganymede have an orbital period ratio of 1 : 2 : 4, a typical example of the
Laplace resonance [Beatty et al., 1999, p.224] The geological and geophysical evolutions of
satellites are highly dependent on the availability of sources of energy. Such evolution can
be heavily influenced by orbital resonance, because resonating satellites are able to tap into
an additional energy source, tidal heating [Beatty et al., 1999, p.243] A periodic flexing in
the satellite’s structure, caused by the tidal forces from the planet, can produce friction
and heat inside the body of the satellite. However, such phenomenon does not happen
in a satellite that has a perfect circular orbit, because there would be essentially no work
done [Beatty et al., 1999, p.243]. In the case of Io, it has an almost perfect circular orbit,
however, due to being in Laplace resonance with the other two Galilean satellites, whenever
Io is in conjunction with Europa, their mutual gravitational interaction forces their orbital
eccentricity to go up. As a result, Io would be wobbling back and forth at an angle of about
1/2 degree from the axis facing Jupiter around its orbit, thus obtaining substantial tidal
heating that triggers the active volcanism on its surface, and the satellite may possibly have
a partially molten core [Beatty et al., 1999, p.245].
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As mentioned above, Europa is also affected by tidal flexing due to being a member of the
Laplace resonance between the Galilean moons. The cryovolcanism observed on the surface
of Europa is most likely a consequence of tidal heating. Scientists even suspect that the tidal
heating on Europa is sufficient to maintain a subsurface ocean within the satellite [Beatty
et al., 1999, p.275].
The last two members of the Galilean moons, Ganymede and Callisto share a lot of similarities, such as in origin, size, density and composition, yet they demonstrate huge difference in
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geological structure [Beatty et al., 1999, p.274] Ganymede may possess a globally differentiated interior, possibly a partially motlen core, while Callisto may have its interior remained
unchanged ever since its formation 4.5 billion years ago [Beatty et al., 1999, p.275]. Among
all the theories explaining the Ganymede-Callisto dichotomy, tidal heating is once again a
promising candidate. Although the current eccentricity of Ganymede’s orbit is too small
to cause enough tidal heating, Renu Malhotra’s calculations have shown that the satellite
could have experienced other resonance configurations in the past, thus might have attained
a much higher orbital eccentricity, which ultimately provided sufficient tidal heating to trigger global differentiation in the satellite’s interior (The New Solar System, 1999, P.275).
Callisto, on the other hand, took a completely different evolutionary path due to the lack of
tidal heating.
Conclusion
Orbital resonance is much more than a beautiful coincidence in nature. It has far reaching
influence in celestial mechanics. This can be well understood by studying the origin of
resonances, and its effects on different systems, from massive planets to tiny asteroids, as
shown in the previous sections. It is not a special feature exclusive only to our solar system.
In fact, the ever increasing discoveries of exo-planetary systems provide many more examples
of this interesting phenomenon. Meanwhile, by carefully analyzing the resonance patterns
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in our own solar system, valuable insights in planetary science can be gained, which helps
us further improve our techniques for the search of exo-planets. In fact, this paper does
not cover every single detail of this broad topic. For example, the ring systems of Saturn,
Jupiter, and Neptune exhibit resonance features, in a more complicated way.
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