Hi there! Today the release of the paper Classification of satellite resonances in the Solar System, by Jing Luan and Peter Goldreich, is the opportunity for me to present you the mean-motion resonances in the system of satellites of the giant planets. That paper has recently been published in The Astronomical Journal, but the topic it deals with is present in the literature since more than fifty years. This is why I need to detail some of the existing works.

The mean-motion resonances (MMR)

Imagine that you have a planet orbited by two satellites. In a convenient case, their orbits will be roughly elliptical. The ellipse results from the motion of a small body around a large spherical one; deviations from the exact elliptical orbit come from the oblateness of the central body and the gravitational perturbation of the other satellite. If the orbital frequencies of the two satellites are commensurate, i.e. if Satellite A accomplishes N revolutions around the planet, while Satellite B accomplishes (almost exactly) M revolutions, i.e. M orbits, N and M being integers, then the 2 satellites will be in a configuration of mean-motion resonance. It can be shown that the perturbation of A on B (respectively of B on A) will not average to 0 but have a cumulative effect, due to the repetition, at the same place, of the smallest distance between the two bodies, the smallest distance meaning the highest gravitational torque. A consequence of a MMR is the increase of the eccentricity of one of the satellites, or of both of them, and / or their inclinations… or only the inclination of one of them. In the worst case, this could result in the ejection of one of the satellites, but it can also have less catastrophic but not less interesting consequences, like the heating of a body, and the evolution of its internal structure… We will discuss that a little later.

A mean-motion resonance can be mathematically explained using the orbital elements, which describe the orbit of a satellite. These elements are

• The semimajor axis a,
• the eccentricity e. e=0 means that the orbit is circular, while e<1 means that the orbit is elliptical. For planetary satellites, we usually have e<0.05. With these two elements, we know the shape of the orbit. We now need to know its orientation, which is given by 3 angles:
• the inclination i, with respect to a given reference plane. Usually it is the equatorial plane of the parent planet at a given date, and the inclination are often small,
• the longitude of the ascending node Ω, which orientates the intersection of the orbital plane with the reference plane,
• the longitude of the pericentre ϖ, which gives you the pericentre, i.e. the point at which the distance planet-satellite is the smallest. With these 5 elements, you know the orbit. To know where on its orbit the satellite is, you also need
• the mean longitude λ.

Saying that the Satellites A and B are in a MMR means that there is an integer combination of orbital elements, such as φ=pλ_A-(p+q)λ_A+q₁ϖ_A+q₂ϖ_B+q₃Ω_A+q₄Ω_B, which is bounded. Usually an angle is expected to be able to take any real value between 0 and 2π radians, i.e. between 0 and 360°, but not our φ. The order of the resonance q is equal to q₁+q₂+q₃+q₄, and q₃+q₄ must be even. Moreover, it stems from the d’Alembert rule, which I will not detail here, that a strength can be associated with this resonance, which is proportional to e_A^q₁e_B^q₂i_A^q₃i_B^q₄. This quantity also gives us the orbital elements which would be raised by the resonance.

In other words, if the orbital frequency of A is twice the one of B, then we could have the following resonances:

• λ_A-2λ_B+ϖ_A (order 1), which would force e_A,
• λ_A-2λ_B+ϖ_B (order 1), which would force e_B,
• 2λ_A-4λ_B+ϖ_A+ϖ_B (order 2), which would force e_A and e_B,
• 2λ_A-4λ_B+2Ω_A (order 2), which would force i_A,
• 2λ_A-4λ_B+2Ω_B (order 2), which would force i_B,
• 2λ_A-4λ_B+2Ω_A+Ω_B (order 2), which would force i_A and i_B.

Higher-order resonances could be imagined, but let us forget them for today.

The next two figures give a good illustration of the way the resonances can raise the orbital elements. All of the curves represent possible trajectories, assuming that the energy of the system is constant. The orbital element which is affected by the resonance, can be measured from the distance from the origin. And we can see that the trajectories tend to focus around points which are not at the origin. These points are the centers of libration of the resonances. This means that when the system is at the exact resonance, the orbital element relevant to it will have the value suggested by the center of libration. These plots are derived from the Second Fundamental Model of the Resonance, elaborated at the University of Namur (Belgium) in the eighties.

The Second Fundamental Model of the Resonance for order 1 resonances, for different parameters. On the right, we can see banana-shaped trajectories, for which the system is resonant. The outer zone is the external circulation zone, and the inner one is the internal circulation zone. Inspired from Henrard J. & Lemaître A., 1983, A second fundamental model for resonance, Celestial Mechanics, 30, 197-218.

The Second Fundamental of the Resonance for order 2 resonances, for different parameters. We can see two resonant zones. On the right, an internal circulation zone is present. Inspired from Lemaître A., 1984, High-order resonances in the restricted three-body problem, Celestial Mechanics, 32, 109-126.

Here, I have only mentioned resonances involving two bodies. We can find in the Solar System resonances involving three bodies… see below.

It appears, from the observations of the satellites of the giant planets, that MMR are ubiquitous in our Solar System. This means that a mechanism drives the satellite from their initial position to the MMRs.

Driving the satellites into resonances

When the satellites are not in MMR, the argument φ circulates, i.e. it can take any value between 0 and 2π. Moreover, its evolution is monotonous, i.e. either constantly increasing, or constantly decreasing. However, when the system is resonant, then φ is bounded. It appears that the resonance zones are levels of minimal energy. This means that, for the system to evolve from a circulation to a libration (or resonant zone), it should loose some energy.

The main source of energy dissipation in a system of natural satellites is the tides. The planet and the satellites are not exactly rigid bodies, but can experience some viscoelastic deformation from the gravitational perturbation of the other body. This results in a tidal bulge, which is not exactly directed to the perturber, since there is a time lag between the action of the perturber and the response of the body. This time lag translates into a dissipation of energy, due to tides. A consequence is a secular variation of the semi-major axes of the satellites (contraction or dilatation of the orbits), which can then cross resonances, and eventually get trapped. Another consequence is the heating of a satellite, which can yield the creation of a subsurface ocean, volcanism…

Capture into a resonance is actually a probabilistic process. If you cross a resonance without being trapped, then your trajectories jump from a circulation zone to another one. However, if you are trapped, you arrive in a libration zone, and the energy dissipation can make you spiral to the libration center, forcing the eccentricity and / or inclination. It can also be shown that a resonance trapping can occur only if the orbits of the two satellites converge.

The system of Jupiter

Jupiter has 4 large satellites orbiting around: J1 Io, J2 Europa, J3 Ganymede, and J4 Callisto. There are denoted Galilean satellites, since they were discovered by Galileo Galilei in 1610. The observations of their motion has shown that

• Io and Europa are close to the 2:1 MMR,
• Europa and Ganymede are close to the 2:1 MMR as well,
• Ganymede and Callisto are close to the 7:3 MMR (De Haerdtl inequality)
• Io, Europa and Ganymede are locked into the Laplace resonance. This is a 3-body MMR, which resonant argument is φ=λ₁-3λ₂+2λ₃. It librates around π with an amplitude of 0.5°.

This Laplace resonance is a unique case in the Solar System, to the best of our current knowledge. It is favored by the masses of the satellites, which have pretty the same order of magnitude. Moreover, Io shows signs of intense dissipation, i.e. volcanism, which were predicted by Stanton Peale in 1979, before the arrival of Voyager I in the vicinity of Jupiter, from the calculation of the tidal effects.

The system of Saturn

Besides the well-known rings and a collection of small moons, Saturn has 8 major satellites, i.e.

• S1 Mimas,
• S2 Enceladus,
• S3 Tethys,
• S4 Dione,
• S5 Rhea,
• S6 Titan,
• S7 Hyperion,
• S8 Iapetus,

and resonant relations, see the following table.

Satellite 1	Satellite 2	MMR	Argument φ	Libration center	Libration amplitude	Affected quantities
S1 Mimas	S3 Tethys	4:2	2λ1-4λ3+Ω1+Ω3	0	95°	i1,i3
S2 Enceladus	S4 Dione	2:1	λ2-2λ4+ϖ2	0	0.25°	e2
S6 Titan	S7 Hyperion	4:3	3λ6-4λ7+ϖ7	π	36°	e7

The amplitude of the libration tells us something about the age of the resonance. Dissipation is expected to drive the system to the center of libration, where the libration amplitude is 0. However, when the system is trapped, the transition from circulation to libration of the resonant argument φ induces that the libration amplitude is close to π, i.e. 180°. So, the dissipation damps this amplitude, and the measured amplitude tells us where we are in this damping process.

This study

This study aims at reinvestigating the mean-motion resonances in the systems of Jupiter and Saturn in the light of a quantity, k_crit, which has been introduced in the context of exoplanetary systems by Goldreich & Schlichting (2014). This quantity is to be compared with a constant of the system, in the absence of dissipation, and the comparison will tell us whether an inner circulation zone appears or not. In that sense, this study gives an alternative formulation of the results given by the Second Fundamental Model of the Resonance. The conclusion is that the resonances should be classified into two groups. The first group contains Mimas-Tethys and Titan-Hyperion, which have large libration amplitudes, and for which the inner circulation zone exists (here presented as overstability). The other group contains the resonances with a small amplitude of libration, i.e. not only Enceladus-Dione, but also Io-Europa and Europa-Ganymede, seen as independent resonances.

A possible perspective

Io-Europa and Europa-Ganymede are not MMR, and they are not independent pairs. They actually constitute the Io-Europa-Ganymede resonance, which is much less documented than a 2-body resonance. An extensive study of such a resonance would undoubtedly be helpful.

Some links

• The paper, i.e. Luan J. & Goldreich P., 2017, Classification of satellite resonances in the Solar System, The Astronomical Journal, 153:17.
• The web page of Jing Luan at Berkeley.
• The web page of Peter Goldreich at Princeton.
• The Second Fundamental Model of the Resonance, for order 1 resonances and for higher orders.
• A study made in Brazil by Nelson Callegary and Tadashi Yokoyama, on the same topic: Paper 1 Paper 2, also made available by the authors here and here, thanks to them for sharing!.