Monthly Archives: January 2017

Dwarf planets

Matter exchange in Pluto’s backyard

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Hi there! Today I will tell you about a recent study accepted for publication in The Monthly Notices of the Royal Astronomical Society. This study, by Rachel A. Smullen and Kaitlin M. Kratter, addresses The fate of debris in the Pluto-Charon system, and has been conducted at Steward Observatory, AZ (USA).

The Pluto-Charon system

Pluto has been discovered by Clyde Tombaugh in 1930 at Lowell Observatory in Flagstaff, Arizona. This was the first discovered object of the Kuiper Belt, and it has been considered as the 9th planet of the Solar System until 2006. Still in Flagstaff, its satellite Charon has been discovered in 1978. Later on, the next arrival of the New Horizons spacecraft motivated observing this system, and thanks to the Hubble Space Telescope, 4 other small moons were discovered: Nix and Hydra in 2005, and Styx and Kerberos in 2012. You can find below images of these 6 bodies taken by New Horizons, and some of their orbital and shape parameters.

Pluto (left) and Charon (right) seen by New Horizons. The white heart on Pluto’s surface is an ice-covered basin named Sputnik Planitia, while the dark spot on Charon’s north pole is named Mordor macula. Copyright: NASA
The small moons, seen by New Horizons. Copyright: NASA
Discovery Radius Distance
Pluto 1930 1187 km 0
Charon 1978 606 km 19571 km
Hydra 2005 23 km 64738 km
Nix 2005 18 km 48694 km
Kerberos 2012 5 km 57783 km
Styx 2012 5 km 42656 km

The pair Pluto-Charon is fascinating from a dynamical point of view, since they represent a case of double synchronous spin-orbit resonance. You know that the Moon is always showing the same face to the Earth, which is due to its synchronous rotation. This means that its orbital period around the Earth is exactly the same as its rotation period, this is a dynamical equilibrium which has been reached after tides had dissipated the rotational energy of the Moon. But the phenomenon goes further for Pluto-Charon, since not only Charon shows the same face to Pluto, but Pluto shows the same face to Charon! This is a consequence of the relative size of the two bodies, each of them being sufficiently large to affect the other one.
On the contrary, the small moons have a much more rapid rotation, which is less obvious to explain.

The system of Pluto has been visited in 2015 by the spacecraft New Horizons, which gave us invaluable data and the nice images I show you today.

The Plutinos

The orbit of this system around the Sun is interested as well. Not only it has a significant inclination (17.16° wrt ecliptic), but it is also in a 3:2 mean-motion resonance (MMR) with Neptune. This means that Pluto makes exactly two revolutions around the Sun while Neptune makes three. Moreover, this is a pretty stable dynamical zone. This is probably why Pluto and its satellites are not the only bodies in this zone. Beside the Pluto system, the first Plutino has been discovered in 1993 at the Mauna Kea Observatory, HI.
The following figure gives a repartition of the known Trans-Neptunian Objects with respect to their semimajor axis, the Plutinos represent a peak at 39 astronomical units.

Distribution of the Kuiper-Belt Objects, plotted from the data of the Minor Planet Center, consulted on January 28th 2017. We see the Plutinos as an accumulation of objects close to 39 AU, which corresponds to the 3:2 MMR with Neptune. The second peak, close to 44 AU, does not correspond to a known resonance. Copyright: The Planetary Mechanics Blog.

Formation of planetary debris disks

The last thing I would like to tell you before presenting the study itself is: how to make a debris disk around a pretty massive body? It is thought to come from an impact. An impactor impacts the target, is destroyed into very small parts, which coalesce into rings, before eventually reaccreting and / or being ejected. The most famous debris disk in the Solar System in the system of the Saturnian rings, but there are actually rings about the four giant planets of the Solar System, and the Centaurs (asteroids between the orbits of Jupiter and Neptune) Chariklo and possibly Chiron.
It is thought that the Moon is the consequence of such a process, i.e. there has been a debris disk around the Earth. And it is also thought that Charon has been created the same way.

This paper

This study aims at understanding the fate of the debris disk which has created Charon. Once enough debris accreted to create Charon, or a proto-Charon, debris remained, and have been ejected. There are at least two ways to model a disk: either you consider it as a gas, i.e. some fluid, or you see it as a cloud of many particles, which interact. These interactions are close encounters and collisions, with translate into viscosity if you model the disk as a gas.

A numerical study

In this study, the authors chose to model the debris disk as a cloud of particles, which is probably the only way to model the path of ejecta. They made several simulations involved 27060 test particles, over 27.3 kyr, i.e. 1.5 million orbits of Pluto and Charon about their common barycenter. Such a study requires high performance soft- and hardware. Their code was based on the integrator Mercury, which is a commonly used N-body code modeling the motion of N body which interact gravitationally and may collide. The test particles are massless, so they have no gravitational action, but they are under the action of Pluto and its 5 satellites. In some of the tests, a migration of Pluto, which is predicted by models of formation of the Solar System, has also been considered.
The hardware is the Super-computer El-Gato (Extremely LarGe Advanced TechnOlogy), based at the University of Arizona, and partly funded by the National Science Foundation.

Once the simulations have run, the authors got the results. And the results are… drum roll please…

Making craters on Charon

The New Horizons images show that Charon is craterized. In all of their simulations, the authors have collisions between Charon and the debris disk. They show that the impact rate is higher if Charon formed on a wide and eccentric orbit. Moreover, they have fewer impacts if secular migration of Pluto is considered.
An issue is: what could be the signature of such an impact now? We know from its synchronous rotation and from the ridges at its surface that Charon has been hot. Hot enough would mean that part of its surface could have been renewed, and then the older impacts would have no signature anymore. Moreover, it would be interesting, but I doubt the information is present in the New Horizons data, to map the impact on the whole surface of Charon. If Charon was synchronous during the most intense episode of impacts, then we would expect a hemispheric repartition of the craters.

Making Plutinos

The simulations show that the most probable destination of the ejected debris is the 3:2 MMR with Neptune. This means that the observed Plutinos could originate from the impact which created Charon. This would mean that the Plutinos are a collisional family, which could be test from their composition. It should be similar to Pluto’s.

And the small moons?

The simulations do not manage to form the small moons. So, the question of their origin is still open.

Some links…

And that’s it for today! New Horizons is en-route to the asteroid 2014 MU69, which would be the first object visited by a spacecraft which had been launched before its discovery. It should reach it either on December 31th, 2018, or January 1st 2019.
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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!.