Category Archives: Mean-motion resonances

Asteroids: Trans-Neptunian Objects, Mean-motion resonances

Does Neptune have binary Trojans?

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Hi there! Jupiter, Uranus and Neptune are known to share their orbits with small bodies, called Trojans. This is made possible by a law of celestial mechanics, which specifies that the points located 60° ahead and behind a planet on its orbit are stable. Moreover, there are many binary objects in the Solar System, but no binary asteroid have been discovered as Trojans of Neptune. This motivates the following study, Dynamical evolution of a fictitious population of binary Neptune Trojans, by Adrián Brunini, which has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society. In this study, the author wonders under which conditions a binary Trojan of Neptune could survive, which almost means could be observed now.

Outline

The coorbital resonance
The Trojans of Neptune
Binary asteroids
Numerical simulations
Survival of the binaries
Challenging this model
The study and its author

The coorbital resonance

The coorbital resonance is a 1:1 mean-motion resonance. This means that the two involved bodies have on average the same orbital frequency around their parent one. In the specific case of the Trojan of a planet, these two objects orbit the Sun with the same period, and the mass ratio between them makes that the small body is strongly affected by the planet, however the planet is not perturbed by the asteroid. But we can have this synchronous resonance even if the mass ratio is not huge. For instance, we have two coorbital satellites of Saturn, Janus and Epimetheus, which have a mass ratio of only 3.6. Both orbit Saturn in ~16 hours, but in experiencing strong mutual perturbations. They are stable anyway.

In the specific problem of the restricted (the mass of the asteroid is negligible), planar (let us assume that the planet and the asteroid orbit in the same plane), circular (here, we neglect the eccentricity of the two orbits) 3-body (the Sun, the planet and the asteroid) problem, it can be shown that if the planet and the asteroid orbit at the same rate, then there are 5 equilibriums, for which the gravitational actions of the planet and the Sun cancel out. 3 of them, named L1, L2 and L3, are unstable, and lie on the Sun-planet axis. The 2 remaining ones, i.e. L4 and L5, lag 60° ahead and behind the planet, and are stable. As a consequence, the orbits with small oscillations around L4 and L5 are usually stable, even if the real configuration has some limited eccentricity and mutual inclination. Other stable trajectories exist theoretically, e.g. horseshoe orbits around the point L4, L3 and L5. The denomination L is a reference to the Italian-born French mathematician Joseph-Louis Lagrange (1736-1813), who studied this problem.

The Lagrange points, in a reference frame rotating with Neptune.
The Lagrange points, in a reference frame rotating with Neptune.

At this time, 6,701 Trojans are known for Jupiter (4269 at L4 and 2432 at L5), 1 for Uranus, 1 for the Earth, 9 for Mars, and 17 for Neptune, 13 of them orbiting close to L4.

The Trojans of Neptune

You can find an updated list of them here, and let me gather their main orbital characteristics:

Location Eccentricity Inclination Magnitude
2004 UP10 L4 0.023 1.4° 8.8
2005 TO74 L4 0.052 5.3° 8.3
2001 QR322 L4 0.028 1.3° 7.9
2005 TN53 L4 0.064 25.0° 9.3
2006 RJ103 L4 0.031 8.2° 7.5
2007 VL305 L4 0.060 28.2° 7.9
2010 TS191 L4 0.043 6.6° 8.0
2010 TT191 L4 0.073 4.3° 7.8
2011 SO277 L4 0.015 9.6° 7.6
2011 WG157 L4 0.031 22.3° 7.1
2012 UV177 L4 0.071 20.9° 9.2
2014 QO441 L4 0.109 18.8° 8.3
2014 QP441 L4 0.063 19.4° 9.3
2004 KV18 L5 0.187 13.6° 8.9
2008 LC18 L5 0.079 27.5° 8.2
2011 HM102 L5 0.084 29.3° 8.1
2013 KY18 L5 0.121 6.6° 6.6

As you can see, these are faint bodies, which have been discovered between 2001 and 2014. I have given here their provisional designations, which have the advantage to contain the date of the discovery. Actually, 2004 UP10 is also known as (385571) Otrera, a mythological Queen of the Amazons, and 2005 TO74 has received the number (385695).

Their dynamics is plotted below:

Dynamics of the Trojans of Neptune, at the Lagrangian points L4 and L5 (squares).
Dynamics of the Trojans of Neptune, at the Lagrangian points L4 and L5 (squares).

Surprisingly, the 4 Trojans around L5 are outliers: they are the most two eccentric, the remaining two being among the three more inclined Trojans. Even if the number of known bodies may not be statistically relevant, this suggests an asymmetry between the two equilibriums L4 and L5. The literature has not made this point clear yet. In 2007, a study suggested an asymmetry of the location of the stable regions (here), but the same authors said one year later that this was indeed an artifact introduced by the initial conditions (here). In 2012, another study detected that the L4 zone is more stable than the L5 one. Still an open question… In the study I present today, the author simulated only orbits in the L4 region.

Binary asteroids

A binary object is actually two objects, which are gravitationally bound. When their masses ratio is of the order of 1, we should not picture it as a major body and a satellite, but as two bodies orbiting a common barycenter. At this time, 306 binary asteroids have been detected in the Solar System. Moreover, we also know 14 triple systems, and 1 sextuple one, which is the binary Pluto-Charon and its 4 minor satellites.

The formation of a binary can result from the disruption of an asteroid, for instance after an impact, or after fission triggered by a spin acceleration (relevant for Near-Earth Asteroids, which are accelerated by the YORP effect), or from the close encounter of two objects. The outcome is two objects, which orbit together in a few hours, and this system evolves… and then several things might happen. Basically, it either evolves to a synchronous spin-spin-orbit resonance, i.e. the two bodies having a synchronous rotation, which is also synchronous with their mutual orbit (examples: Pluto-Charon, the double asteroid (90) Antiope), or the two components finally split… There are also systems in which only one of the components rotates synchronously. Another possible end-state is a contact binary, i.e. the two components eventually touch together.

At this time, 4 binary asteroids are known among the Trojans asteroids of Jupiter. None is known for Neptune.

Numerical simulations

The author considered fictitious binary asteroids close to the L4 of Neptune, and propagated the motion of the two components, in considering the planetary perturbations of the planets, over 4.5 Byr, i.e. the age of the Solar System. A difficulty for such long-term numerical studies is the handling of numerical uncertainties. Your numerical scheme includes a time-step, which is the time interval between the simulated positions of the system, i.e. the locations and velocities of the two components of the binary. If your time-step is too large, you will have a mathematical uncertainty in your evaluation. However, if you shorten it, you will have too many iterations, which means a too long calculation time, and the accumulations of round-off errors due to the machine epsilon, i.e. rounding in floating point arithmetic.
A good time step should be a fraction of the shortest period perturbing the system. Neptune orbits the Sun in 165 years, which permits a time step of some years, BUT the period of a binary is typically a few hours… which is too short for simulations over the age of the Solar System. This problem is by-passed in averaging the dynamics of the binary. This means that only long-term effects are kept. In this case, the author focused on the Kozai-Lidov effect, which is a secular (i.e. very long-term) raise of the inclination and the eccentricity. Averaging a problem of gravitational dynamics is always a challenge, because you have to make sure you do not forget a significant contribution.
The author also included the tidal interaction between the two components, i.e. the mutual interaction triggering stress and strain, and which result in dissipation of energy, secular variation of the mutual orbits, and damping of the rotation.
He considered three sets of binaries: two with components of about the same size, these two samples differing by the intensity of tides, and in the third one the binary are systems with a high mass ratio, i.e. consisting of a central body and a satellite.

Survival of the binaries

The authors find that for systems with strong tides, about two thirds of the binaries should survive. The tides have unsurprisingly a critical role, since they tend to make the binary evolve to a stable end-state, i.e. doubly synchronous with an almost circular mutual orbit. However, few systems with main body + satellite survive.

Challenging this model

At this time, no binary has been found among the Trojans of Neptune, but this does not mean that there is none. The next years shall tell us more about these bodies, and once they will be statistically significant, we would be able to compare the observations with the theory. An absence of binaries could mean that they were initially almost absent, i.e. lack of binaries in that region (then we should explain why there are binaries in the Trans-Neptunian population), or that the relevant tides are weak. We could also expect further theoretical studies, i.e. with a more complete tidal dynamics, and frequency-dependent tides. Here, the author assumed a constant tidal function Q, while it actually depends on the rotation rate of the two bodies, which themselves decrease all along the evolution.

So, this is a model assisting our comprehension of the dynamics of binary objects in that region. As such, it should be seen as a step forward. Many other steps are to be expected in the future, observationally and theoretically (by the way, could a Trojan have rings?).

The study and its author

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Mean-motion resonances

The dynamics of the Quasi-Satellites

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Hi there! After reading this post, you will know all you need to know on the dynamics of quasi-satellites. This is the opportunity to present you On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited, by Alexandre Pousse, Philippe Robutel and Alain Vienne. This study has recently been published in Celestial Mechanics and Dynamical Astronomy.

Outline

The 1:1 mean-motion resonance at small eccentricity
At high eccentricities: Quasi-satellites
Known quasi-satellites
Understanding the dynamics
Perspectives
To know more…

The 1:1 mean-motion resonance at small eccentricity

(see also here)

Imagine a pretty simple case: the Sun, a planet with a keplerian motion around (remember: its orbit is a static ellipse), and a very small third body. So small that you can neglect its mass, i.e. it does not affect the motion of the Sun and the planet. You know that the planet has no orbital eccentricity, i.e. the static ellipse serving as an orbit is actually a circle, and that the third body (let us call it the particle) has none either. Moreover, we want the particle to orbit in the same plane than the planet, and to have the same revolution period around the Sun. These are many conditions.
Under these circumstances, mathematics (you can call that celestial mechanics) show us that, in the reference frame which is rotating with the planet, there are two stable equilibriums 60° ahead and astern the planet. These two points are called L4 and L5 respectively. But that does not mean that the particle is necessary there. It can have small oscillations, called librations around these points, the resulting orbits being called tadpole orbits. It is even possible to have orbits enshrouding L4 and L5, this results in large librations orbits, called horseshoe orbits.

All of these configurations are stable. But remember: the planet is much less massive than the Sun, the particle is massless, the orbits are planar and circular… Things become tougher when we relax one of these assumptions. And the authors assumed that the particle had a significant eccentricity.

At high eccentricities: Quasi-satellites

Usually, increasing the eccentricity destabilizes you. This is still true here, i.e. co-orbital orbits are less stable when eccentric. But increasing the eccentricity also affects the dynamical structure of your problem in such a way that other dynamical configurations may appear. And this is the case here: you have an equilibrium where your planet lies.

Ugh, what does that mean? If you are circular, then your particle is at the center of your planet… Nope, impossible. But wait a minute: if you oscillate around this position without being there… yes, that looks like a satellite of the planet. But a satellite is under the influence of the planet, not of the star… To be dominated by the star, you should be far enough from the planet.

I feel the picture is coming… yes, you have a particle on an eccentric orbit around the star, the planet being in the orbit. And from the star, this looks like a satellite. Funny, isn’t it? And such bodies exist in the Solar System.

Orbit of a quasi-satellite. It follows the planet, but orbits the star.
Orbit of a quasi-satellite. It follows the planet, but orbits the star.

Known quasi-satellites

Venus has one known quasi-satellite, 2002 VE68. This is a 0.4-km body, which has been discovered in 2002. Like Venus, it orbits the Sun in 225 days, but has an orbital eccentricity of 0.41, while the one of Venus is 0.007. It is thought to be a quasi-satellite of Venus since 7,000 years, and should leave this configuration in some 500 years.

The Earth currently has several known quasi-satellites, see the following table:

(277810) 2006 FV350.387.1°10,000 y2013 LX280.4550°40,000 y2014 OL3390.4610.2°1,000 y(469219) 2016 HO30.107.8°400 y

Known quasi-satellites of the Earth
Name Eccentricity Inclination Stability
(164207) 2004 GU9 0.14 13.6° 1,000 y

These bodies are all smaller than 500 meters. Because of their significant eccentricities, they might encounter a planet, which would then affect their orbits in such a way that the co-orbital resonance would be destabilized. However, significant inclinations limit the risk of encounters. Some bodies switch between quasi-satellite and horseshoe configurations.

Here are the known quasi-satellites of Jupiter:

Known quasi-satellites of Jupiter
Name Eccentricity Inclination Stability
2001 QQ199 0.43 42.5° > 12,000 y
2004 AE9 0.65 1.6° > 12,000 y
329P/LINEAR-Catalina 0.68 21.5° > 500 y
295P/LINEAR 0.61 21.1° > 2,000 y

329P/LINEAR-Catalina and 295P/LINEAR being comets.

Moreover, Saturn and Neptune both have a confirmed quasi-satellite. For Saturn, 2001 BL41 should leave this orbit in about 130 years. It has an eccentricity of 0.29 and an inclination of 12.5°. For Neptune, (309239) 2007 RW10 is in this state since about 12,500 years, and should stay in it for the same duration. It has an orbital eccentricity of 0.3, an inclination of 36°, and a diameter of 250 km.

Understanding the dynamics

Unveiling the dynamical/mathematical structure which makes the presence of quasi-satellites possible is the challenge accepted by the authors. And they succeeded. This is based on mathematical calculation, in which you write down the equations of the problem, you expand them to retain only what is relevant, in making sure that you do not skip something significant, and you manipulate what you have kept…

The averaging process

The first step is to write the Hamiltonian of the restricted planar 3-body problem, i.e. the total energy of a system constituted by the Sun, the planet, and the massless particle. The dynamics is described by so-called Hamiltonian variables, which allow interesting mathematical properties…
Then you expand and keep what you need. One of the pillars of this process is the averaging process. When things go easy, i.e. when your system is not chaotic, you can describe the dynamics of the system as a sum of sinusoidal contributions. This is straightforward to figure out if you remember that the motions of the planets are somehow periodic. Somehow means that these motions are not exactly sinusoidal, but close to it. So, you expand it in series, in which other sinusoids (harmonics) appear. And you are particularly interested in the one involving λ-λ’, i.e. the difference between the mean longitude of the planet and the particle. This makes sense since they are in the co-orbital configuration, that particular angle should librate with pretty small oscillations around a given value, which is 60° for tadpole orbits, 180° for horseshoes, and 0° for quasi-satellites. Beside this, you have many small oscillations, in which you are not interested. Usually you can drop them in truncating your series, but actually you just average them, since they average to 0. This is why you can drop them.
To expand in series, you should do it among a small parameter, which is usually the eccentricity. This means that your orbit looks pretty like a circle, and the other terms of the series represent the difference with the circle. But here there is a problem: to get quasi-satellite orbits, your eccentricity should be large enough, which makes the analytical calculation tougher. In particular, it is difficult to guarantee their convergence. The authors by-passed this problem in making numerical averaging, i.e. they computed numerically the integrals of the variables of the motion over an orbital period.

Once they have done this, they get a simplified system, based on one degree-of-freedom only. This is a pair of action-angle variables, which will characterize your quasi-satellite orbit. This study also requires to identify the equilibriums of the system, i.e. to identify the existing stable orbits.

Perspectives

So, this study is full of mathematical calculations, aiming at revisiting this problem. The authors mention as possible perspective the study of resonances between the planets, which disturb the system, and the proper frequency of the quasi-satellite orbit. This is the oscillating frequency of the angle characterizing the orbit, and if it is equal to a frequency already present in the system, it could have an even more interesting dynamics, e.g. transit between different states (quasi-satellite / horsehoe,…).

To know more…

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Mean-motion resonances

Resonances around the giant planets

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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=pλA-(p+q)λA+q1ϖA+q2ϖB+q3ΩA+q4ΩB, 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 eAq1eBq2iAq3iBq4. 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:

  • λA-2λB+ϖA (order 1), which would force eA,
  • λA-2λB+ϖB (order 1), which would force eB,
  • 2λA-4λB+ϖA+ϖB (order 2), which would force eA and eB,
  • 2λA-4λB+2ΩA (order 2), which would force iA,
  • 2λA-4λB+2ΩB (order 2), which would force iB,
  • 2λA-4λB+2ΩA+ΩB (order 2), which would force iA and iB.

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, kcrit, 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!.