Tag Archives: celestial mechanics

The fate of the Alkyonides

Hello everybody! Today, I will tell you on the dynamics of the Alkyonides. You know the Alkyonides? No? OK… There are very small satellites of Saturn, i.e. kilometer-sized, which orbit pretty close to the rings, but outside. These very small bodies are known to us thanks to the Cassini spacecraft, and a recent study, which I present you today, has investigated their long-term evolution, in particular their stability. Are they doomed or not? How long can they survive? You will know this and more after reading this presentation of Long-term evolution and stability of Saturnian small satellites: Aegaeon, Methone, Anthe, and Pallene, by Marco Muñoz-Gutiérrez and Silvia Giuliatti Winter. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

The Alkyonides

As usually in planetary sciences, bodies are named after the Greek mythology, which is the case of the four satellites discussed today. But I must admit that I cheat a little: I present them as Alkyonides, while Aegeon is actually a Hecatoncheires. The Alkyonides are the 7 daughters of Alcyoneus, among them are Anthe, Pallene, and Methone.

Here are some of there characteristics:

Methone Pallene Anthe Aegaeon
Semimajor axis 194,402 km 212,282 km 196,888 km 167,425 km
Eccentricity 0 0.004 0.0011 0.0002
Inclination 0.013° 0.001° 0.015° 0.001°
Diameter 2.9 km 4.4 km 2 km 0.66 km
Orbital period 24h14m 27h42m 24h52m 19h24m
Discovery 2004 2004 2007 2009

For comparison, Mimas orbits Saturn at 185,000 km, and the outer edge of the A Ring, i.e. of the main rings of Saturn, is at 137,000 km. So, we are in the close system of Saturn, but exterior to the rings.

Discovery of Anthe, aka S/2007 S4. Copyright: NASA.
Discovery of Anthe, aka S/2007 S4. Copyright: NASA.

These bodies are in mean-motion resonances with main satellites of Saturn, more specifically:

  • Methone orbits near the 15:14 MMR with Mimas,
  • Pallene is close to the 19:16 MMR with Enceladus,
  • Anthe orbits near the 11:10 MMR with Mimas,
  • Aegaeon is in the 7:6 MMR with Mimas.

As we will see, these resonances have a critical influence on the long-term stability.

Rings and arcs

Beside the main and well-known rings of Saturn, rings and arcs of dusty material orbit at other locations, but mostly in the inner system (with the exception of the Phoebe ring). In particular, the G Ring is a 9,000 km wide faint ring, which inner edge is at 166,000 km… Yep, you got it: Aegaeon is inside. Some even consider it is a G Ring object.

Methone and Anthe have dusty arcs associated with them. The difference between an arc and a ring is that an arc is longitudinally bounded, i.e. it is not extended enough to constitute a ring. The Methone arc extends over some 10°, against 20° for the Anthe arc. The material composing them is assumed to be ejecta from Methone and Anthe, respectively.

However, Pallene has a whole ring, constituted from ejecta as well.

Why sometimes a ring, and sometimes an arc? Well, it tell us something on the orbital stability of small particles in these areas. Imagine you are a particle: you are kicked from home, i.e. your satellite, but you remain close to it… for some time. Actually you drift slowly. While you drift, you are somehow shaken by the gravitational action of the other satellites, which disturb your Keplerian orbit around the planet. If you are shaken enough, then you may leave the system of Saturn. If you are not, then you can finally be anywhere on the orbit of your satellite, and since you are not the only one to have been ejected (you feel better, don’t you?), then you and your colleagues will constitute a whole ring. If you are lucky enough, you can end up on the satellite.

The longer the arc (a ring is a 360° arc), the more stable the region.

Frequency diffusion

The authors studied

  1. the stability of the dusty particles over 18 years
  2. the stability of the satellites in the system of Saturn over several hundreds of kilo-years (kyr).

For the stability of the particles, they computed the frequency diffusion index. It consists in:

  1. Simulating the motion of the particles over 18 years,
  2. Determining the main frequency of the dynamics over the first 9 years, and over the last 9 ones,
  3. Comparing these two numbers. The smaller the difference, the more stable you are.

The numerical simulations is something I have addressed in previous posts: you use a numerical integrator to simulate the motion of the particle, in considering an oblate Saturn, the oblateness being mostly due to the rings, and several satellites. Our four guys, and Janus, Epimetheus, Mimas, Enceladus, and Tethys.

How resonances destabilize an orbit

When a planetary body is trapped in a mean-motion resonance, there is an angle, which is an integer combination of angles present in its dynamics and in the dynamics of the other body, which librates. An example is the MMR Aegaeon-Mimas, which causes the angle 7λMimas-6λAegaeonMimas to librate. λ is the mean longitude, and ϖ is the longitude of the pericentre. Such a resonance is supposed to affect the dynamics of the two satellites but, given their huge mass ratio (Mimas is between 300 and 500 millions times heavier than Aegaeon), only Aegaeon is affected. The resonance is at a given location, and Aegaeon stays there.
But a given resonance has some width, and several resonant angles (we say arguments) are associated with a resonance ratio. As a consequence, several resonances may overlap, and in that case … my my my…
The small body is shaken between different locations, its eccentricity and / or inclination can be raised, until being dynamically unstable…
And in this particular region of the system of Saturn, there are many resonances, which means that the stability of the discovered body is not obvious. This is why the authors studied it.

Results

Stability of the dusty particles

The authors find that Pallene cannot clear its ring efficiently, despite its size. Actually, this zone is the most stable, wrt the dynamical environments of Anthe, Methone and Aegaeon. However, 25% of the particles constituting the G Ring should collide with Aegaeon in 18 years. This probably means that there is a mechanism, which refills the G Ring.

Stability of the satellites

From long-term numerical simulations over 400 kyr, i.e. more than one hundred millions of orbits, these 4 satellites are stable. For Pallene, the authors guarantee its stability over 64 Myr. Among the 4, this is the furthest satellite from Saturn, which makes it less affected by the resonances.

A perspective

The authors mention as a possible perspective the action of the non-gravitational forces, such as the solar radiation pressure and the plasma drag, which could affect the dynamics of such small bodies. I would like to add another one: the secular tides with Saturn, and the pull of the rings. They would induce drifts of the satellites, and of the resonances associated. The expected order of magnitude of these drifts would be an expansion of the orbits of a few km / tens of km per Myr. This seems pretty small, but not that small if we keep in mind that two resonances affecting Methone are separated by 4 km only.

This means that further results are to be expected in the upcoming years. The Cassini mission is close to its end, scheduled for 15 Sep 2017, but we are not done with exploiting its results!

To know more…

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Measuring the tides of Mercury

Hi there! I have already told you about the tides. If you follow me, you know that the tides are the deformations of a planet from the gravitational action of its parent star (the Sun for Mercury), and that a good way to detect them is to measure the variations of the gravity field of a planet from the deviations of a spacecraft orbiting it. From periodic variations we should infer a coefficient k2, known as the potential Love number, which represents the response of the planet to the tides…

That’s all for today! Please feel free to comment… blablabla…

Just kidding!

Today, I will tell you about another way to measure the tides, from the rotation of Mercury. For this, I will present you a study entitled Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque, which was recently published in The Monthly Notices of the Royal Astronomical Society. This is a collaboration between English and Italian mathematicians, i.e. Michele Bartuccelli, Jonathan Deane, and Guido Gentile. In planetary sciences mathematics can lead to new discoveries. In this case, the idea is: tides slow down the rotation of a planetary body, which eventually reaches an equilibrium rotation (or spin). For the Moon, the equilibrium is the synchronous rotation, while for Mercury it is the 3:2 spin-orbit resonance. Very well. A very good way to describe this final state is to describe the equilibrium rotation, i.e. in considering that the tides do not affect the spin anymore. But this is just an approximation. The tides are actually still active, and they affect the final state. In considering it, the authors show that the variations of the spin rate of Mercury should be composed of at least two sinusoids, i.e. two periodic effects, the superimposition of these two periods being quasi-periodic… you now understand the title.

The rotation of Mercury

I have already presented you Mercury here. Mercury is the innermost planet of the Solar System, with a semimajor axis which is about one third of the one of the Earth, i.e. some 58 million km, and a surprisingly large orbital eccentricity, which is 0.206. These two elements favor a spin-orbit resonance, i.e. the rotation rate of Mercury is commensurate with its orbital rate. Their ratio is 3/2, Mercury performing a revolution about the Sun in 88 days, while a rotation period is 58 days. You can notice a 3/2 ratio between these two numbers.

The 3:2 spin-orbit resonance of Mercury
The 3:2 spin-orbit resonance of Mercury

Why is this configuration possible as an equilibrium state? If you neglect the dissipation (the authors do not) and the obliquity (the authors do, and they are probably right to do it), you can write down a second-degree ODE (ordinary differential equation), which rules the spin. In this equation, the triaxiality of Mercury plays a major role, i.e. Mercury spins the way it spins because it is triaxial. Another reason is its orbital eccentricity. This ODE has equilibriums, i.e. stable spin rates, among them is the 3:2 spin-orbit resonance.

And what about the obliquity? It is actually an equilibrium as well, known as Cassini State 1, in which the angular momentum of Mercury is tilted from the normal to its orbit by 2 arcminutes. This tilt is a response to the slow precessing motion (period: 300,000 years) of the orbit of Mercury around the Sun.

Let us forget the obliquity. There are several possible spin-orbit ratios for Mercury.

Possible rotation states

If you went back to the ODE which rules the spin-rate of Mercury, you would see that there are actually several equilibrium spin rates, which correspond to p/2 spin-orbit resonances, p being an integer. Among them are the famous synchronous resonance 1:1 (p=2), the present resonance of Mercury (p=3), and other ones, which have never been observed yet.

If we imagine that Mercury initially rotated pretty fast, then it slowed down, and crossed several resonances, e.g. the 4:1, the 7:2, 3:1, 5:2, 2:1… and was trapped in none of them, before eventually being trapped in the present 3:2 one. Or we can imagine that Mercury has been trapped for instance in the 2:1 resonance, and that something (an impact?) destabilized the resonance…
And what if Mercury had been initially retrograde? Why not? Venus is retrograde… In that case, the tides would have accelerated Mercury, which would have been trapped in the synchronous resonance, which is the strongest one. This would mean that this synchronous resonance would have been destabilized, to allow trapping into the 3:2 resonance. Any worthwhile scenario of the spin evolution of Mercury must end up in the 3:2 resonance, since it is the current state. The scenario of an initially retrograde Mercury has been proposed to explain the hemispheric repartition of the observed impacts, which could be a signature of a past synchronous rotation. Could be, but is not necessarily. Another explanation is that the geophysical activity of Mercury would have renewed the surface of only one hemisphere, making the craters visible only on the other part.

Anyway, whatever the past of Mercury, it needed a dissipative process to end up in an equilibrium state. This dissipative process is the tides, assisted or not by core-mantle friction.

The tides

Because of the differential attraction of the Sun on Mercury, you have internal friction, i.e. stress and strains, which dissipate energy, and slow down the rotation. This dissipation is enforced by the orbital eccentricity (0.206), which induces periodic variations of the Sun-Mercury distance.
An interesting question is: how does the material constituting Mercury react to the tides? A critical parameter is the tidal frequency, i.e. the way you dissipates depends on the frequency you shake. A derivation of the tidal torque raised by the Sun proves to be a sum of periodic excitations, one of them being dominant in the vicinity of a resonance. This results in an enforcement of all the spin-orbit resonances, which means that a proper tidal model is critical for accurate simulations of the spin evolution.
A pretty common way to model the tides is the Maxwell model: you define a Maxwell time, which is to be compared with the period of the tidal excitation (the shaking). If your excitation is slow enough, then you will have an elastic deformation, i.e. Mercury will have the ability to recover its shape without loss of energy. However, a more rapid excitation will be dissipative. Then this model can be improved, or refined, in considering more dissipation at high frequencies (Andrade model), or grain-boundary slip (Burgers model)… There are several models in the literature, which are supported by theoretical considerations and lab experiments. Choosing the appropriate one depends on the material you consider, under which conditions, i.e. pressure and temperature, and the excitation frequencies. But in any case, these physically realistic tidal models will enforce the spin-orbit resonances.

Considering only the tides assumes that your body is (almost) homogeneous. Mercury has actually an at least partially molten outer core, i.e. a global fluid layer somewhere in its interior. This induces fluid-solid boundaries, the outer one being called CMB, for core-mantle boundary, and you can have friction there. The authors assumed that the CMB was formed after the trapping of Mercury into its present 3:2 spin-orbit resonance, which is supported by some studies. This is why they neglected the core-mantle friction.

This paper

This paper is part of a long-term study on the process of spin-orbit resonance. The authors studied the probabilities of capture (when you slow down until reaching a spin-orbit resonance, will you stay inside or leave it, still slowing down?), proposed numerical integrators adapted to this problem…
In this specific paper, they write down the ODE ruling the dynamics in considering the frequency-dependent tides (which they call realistic), and solve it analytically with a perturbation method, i.e. first in neglecting a perturbation, that they add incrementally, to eventually converge to the real solution. They checked their results with numerical integrations, and they also studied the stability of the solutions (the stable solutions being attractors), and the probabilities of capture.

In my opinion, the main result is: the stable attractor is not periodic but quasi-periodic. Fine, but what does that mean?

If we neglect the influence of the other planets, then the variations of the spin rate of Mercury is expected to be a periodic signal, with a period of 88 days. This is due to the periodic variations of the Sun-Mercury distance, because of the eccentricity. This results in longitudinal librations, which are analogous to the librations of the Moon (we do not see 50% of the surface of the Moon, but 59%, thanks to these librations). The authors say that this solution is not stable. However, a stable solution is the superimposition of these librations with a sinusoid, which period is close to 15 years, and an amplitude of a few arcminutes (to be compared to 15 arcminutes, which is the expected amplitude of the 88-d signal). So, it is not negligible, and this 15-y period is the one of the free (or proper) oscillations of Mercury. A pendulum has a natural frequency of oscillations, here this is exactly the same. But contrarily to a pendulum, the amplitude of these oscillations does not tend to 0. So, we could hope to detect it, which would be a direct observation of the tidal dissipation.

Measuring the rotation

What can we observe? We should first keep in mind that the authors addressed the early Mercury, when being trapped into the 3:2 spin-orbit resonance, which was pretty homogeneous. The current Mercury has a global fluid layer, which means a larger (about twice) amplitude of the 88-d signal, and a different dissipative process, the tides being assisted by core-mantle friction. As a consequence, there is no guarantee that the 15-y oscillation (actually a little shorter, some 12 years, because of the fluid core) would still exist, and that would require a dedicated study. But measuring it would be an information anyway.

How to measure it? The first observations of the rotation of Mercury in 1965 and of the librations in 2007 were Earth-based, radar observations, which are sensitive to the velocity. This means that they are more likely to detect a rapid oscillation (88 d, e.g.) than a slow one (12 years…). Observations of the surface of Mercury by the spacecraft MESSENGER confirmed those measurements. In 2018 the ESA/JAXA (Europe / Japan) joint mission Bepi-Colombo will be sent to Mercury, for orbital insertion in 2025 and hopefully a 2-y mission, with a better accuracy than MESSENGER. So, we could hope a refinement of the measurements of the longitudinal motion.

Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.
Purple: The 88-d oscillation. Green: Superimposed with the 15-y one. Keep in mind that Bepi-Colombo will orbit Mercury during some 2 years.

To know more

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

On the stability of Chariklo

Hi there! Do you remember Chariklo? You know, this asteroid with rings (see this post on their formation). Today, we will not speak on the formation of the rings, but of the asteroid itself. I present you the paper entitled The dynamical history of Chariklo and its rings, by J. Wood, J. Horner, T. Hinse and S. Marsden, which has recently been published in The Astronomical Journal. It deals with the dynamical stability of the asteroid Chariklo as a Centaur, i.e. when Chariklo became a Centaur, and for how long.

(10199)Chariklo

Chariklo is a large asteroid orbiting between the orbits of Saturn and Uranus, i.e. it is a Centaur. It is the largest known of them, with a diameter of ~250 km. It orbits the Sun on an elliptic orbit, with an eccentricity of 0.18, inducing variations of its distance to the Sun between 13.08 (perihelion) and 18.06 au (aphelion), au being the astronomical unit, close to 150 millions km.
But the main reason why people are interested in Chariklo is the confirmed presence of rings around it, while the scientific community expected rings only around large planets. These rings were discovered during a stellar occultation, i.e. Chariklo occulting a distant star. From the multiple observations of this occultation in different locations of the Earth’s surface, 2 rings were detected, and announced in 2014. Since then, rings have been hinted around Chiron, which is the second largest one Centaur, but this detection is still doubtful.
Anyway, Chariklo contributes to the popularity of the Centaurs, and this study is focused on it.

Small bodies populations in the Solar System

The best known location of asteroids in the Solar System is the Main Belt, which is located between the orbits of Mars and Jupiter. Actually, there are small bodies almost everywhere in the Solar System, some of them almost intersecting the orbit of the Earth. Among the other populations are:

  • the Trojan asteroids, which share the orbit of Jupiter,
  • the Centaurs, which orbit between Saturn and Uranus,
  • the Trans-Neptunian Objects (TNOs), which orbit beyond the orbit of Neptune. They can be split into the Kuiper Belt Objects (KBOs), which have pretty regular orbits, some of them being stabilized by a resonant interaction with Neptune, and the Scattered Disc Objects (SDOs), which have larger semimajor axes and high eccentricities
  • the Oort cloud, which was theoretically predicted as a cloud of objects orbiting near the cosmological boundary of our Solar System. It may be a reservoir of comets, these small bodies with an eccentricity close to 1, which can sometimes visit our Earth.

The Centaurs are interesting from a dynamical point of view, since their orbits are not that stable, i.e. it is estimated that they remain in the Centaur zone in about 10 Myr. Since this is very small compared to the age of our Solar System (some 4.5 Gyr), the fact that Centaurs are present mean that the remaining objects are not primordial, and that there is at least one mechanism feeding this Centaur zone. In other words, the Centaurs we observe were somewhere else before, and they will one day leave this zone, but some other guys will replace them.

There are tools, indicators, helpful for studying and quantifying this (in)stability.

Stability, Lyapunov time, and MEGNO

Usually, an orbiting object is considered as “stable” (actually, we should say that its orbit is stable) if it orbits around its parent body for ever. Reasons for instability could be close encounters with other orbiting objects, these close encounters being likely to be favored by a high eccentricity, which could itself result from gravitational interactions with perturbing objects.
To study the stability, it is common to study chaos instead. And to study chaos, it is common to actually study the dependency on initial conditions, i.e. the hyperbolicity. If you hold a broom vertically on your finger, it lies in a hyperbolic equilibrium, i.e. a small deviation will dramatically change the way it will fall… but trust me, it will fall anyway.
And a good indicator of the hyperbolicity is the Lyapunov time, which is a timescale beyond which the trajectory is so much sensitive on the initial conditions that you cannot accurately predict it anymore. It will not necessarily become unstable: in some cases, known as stable chaos, you will have your orbit confined in a given zone, you do not know where it is in this zone. The Centaur zone has some kind of stable chaos (over a given timescale), which partly explains why some bodies are present there anyway.
To estimate the Lyapunov time, you have to integrate the differential equations ruling the motion of the body, and the ones ruling its tangent vector, i.e. tangent to its trajectory, which will give you the sensitivity to the initial conditions. If you are hyperbolic, then the norm of this tangent vector will grow exponentially, and from its growth rate you will have the Lyapunov time. Easy, isn’t it? Not that much. Actually this exponential growth is an asymptotic behavior, i.e. when time goes to infinity… i.e. when it is large enough. And you have to integrate over a verrrrry loooooooong time…
Fortunately, the MEGNO (Mean Exponential Growth of Nearby Orbits) indicator was invented, which converges much faster, and from which you can determine the Lyapunov time. If you are hyperbolic, the Lyapunov time is contained in the growth rate of the MEGNO, and if not, the MEGNO tends to 2, except for pretty simple systems (like the rotation of synchronous bodies), where it tends to zero.

We have now indicators, which permit to quantify the instability of the orbits. As I said, these instabilities are usually physically due to close encounters with large bodies, especially Uranus for Centaurs. This requires to define the Hill and the Roche limits.

Hill and Roche limits

First the Roche limit: where an extended body orbits too close to a massive object, the difference of attraction it feels between its different parts is stronger than its cohesion forces, and it explodes. As a consequence, satellites of giant planets survive only as rings below the Roche limit. And the outer boundary of Saturn’s rings is inner and very close to the Roche limit.

Now the Hill limit: it is the limit beyond which you feel more the attraction of the body you meet than the parent star you both orbit. This may result in being trapped around the large object (a giant planet), or more probably a strong deviation of your orbit. You could then become hyperbolic, and be ejected from the Solar System.

This paper

This study consists in backward numerical integrations of clones of Chariklo, i.e. you start with many fictitious particles (the authors had 35,937 of them) which do not interact with each others, but interact with the giant planets, and which are currently very close to the real Chariklo. Numerical integration over such a long timespan requires accurate numerical integrators, the authors used a symplectic one, i.e. which presents mathematical properties limiting the risk of divergence over long times. Why 1 Gyr? The mean timescale of survival (called here half-life, i.e. during which you lose half of your population) is estimated to be 10 Myr, so 1 Gyr is 100 half-lives. They simulated the orbits and also drew MEGNO maps, i.e. estimated the Lyapunov time with respect to the initial orbital elements of the particle. Not surprisingly, the lower the eccentricity, the more stable the orbit.

And the result is: Chariklo is in a zone of pretty stable chaos. Moreover, it is probably a Centaur since less than 20 Myr, and was a Trans-Neptunian Object before. This means that it was exterior to Neptune, while it is now interior. In a few simulations, Chariklo finds its origin in the inner Solar System, i.e. the Main Belt, which could have favored a cometary activity (when you are closer to the Sun, you are warmer, and your ice may sublimate), which could explain the origin of the rings. But the authors do not seem to privilege this scenario, as it supported by only few simulations.

What about the rings?

The authors wondered if the rings would have survived a planetary encounter, which could be a way to date them in case of no. But actually it is a yes: they found that the distance of close encounter was large enough with respect to the Hill and Roche limits to not affect the rings. So, this does not preclude an ancient origin for the rings… But a specific study of the dynamics of the rings would be required to address this issue, i.e. how stable are they around Chariklo?

To know more

And that’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.

Chaotic dynamics of asteroids

Hi there! Today’s post deals with the fate of an asteroid family. You remember Datura? Now you have Hungaria! Datura is a very young family (< 500 kyr), now you have a very old one, i.e. probably more than 1 Gyr, and you will see that such a long time leaves room for many uncertainties... The paper I present is entitled Planetary chaos and the (In)stability of Hungaria asteroids, by Matija Ćuk and David Nesvorný, it has recently been accepted for publication in Icarus.

The Hungaria asteroids

Usually an asteroid family is a cluster of asteroids in the space of the orbital elements (semimajor axis, eccentricity, inclination), which share, or a supposed to share, a common origin. This suggests that they would originate from the same large body, which would have been destroyed by a collision, its fragments then constituting an asteroid family. Identifying an asteroid family is not an easy task, because once you have identified a cluster, then you must make sure that the asteroids share common physical properties, i.e. composition. You can get this information from spectroscopy, i.e. in comparing their magnitudes in different wavelengths.

The following plot gives the semimajor axis / eccentricity repartition of the asteroids in the inner Solar System, with a magnitude smaller than 15.5. We can clearly see gaps and clusters. Remember that the Earth is at 1 UA, Mars at 1.5 UA, and Jupiter at 5.2. The group of asteroids sharing the orbit of Jupiter constitute the Trojan population. Hungaria is the one on the left, between 1.8 and 2 AU, named after the asteroid 434 Hungaria. The gap at its right corresponds to the 4:1 mean-motion resonance with Jupiter.

Distribution of the asteroids in the inner Solar System, with absolute magnitude < 15.5. Reproduced from the data of The Asteroidal Elements Database. Copyright: planetary-mechanics.com

If we look closer at the orbital elements of this Hungaria population, we also see a clustering on the eccentricity / inclination plot (just below).

Eccentricity / Inclination of the asteroids present in the Hungaria zone. Copyright: planetary-mechanics.com

This prompted Anne Lemaître (University of Namur, Belgium) to suggest in 1994 that Hungaria constituted an asteroid family. At that time, only 26 of these bodies were identified. We now know more than 4,000 of them.

The origin of this family can be questioned. The point is that these asteroids have different compositions, which would mean that they do not all come from the same body. In other words, only some of them constitute a family. Several dynamics studies, including the one I present today, have been conducted, which suggest that these bodies are very old (> 1 Gyr), and that their orbits might be pretty unstable over Gyrs… which suggests that it is currently emptying.

This raises two questions:

  1. What is the origin of the original Hungaria population?
  2. What is the fate of these bodies?

Beside the possible collisional origin, which is not satisfying for all of these bodies since they do not share the same composition, it has been proposed that they are the remnants of the E-Belt, which in some models of formation of the Solar System was a large population of asteroid, which have essentially been destabilized. Another possibility could be that asteroids might pass by and eventually be trapped in this zone, feeding the population.

Regarding the fate, the leaving asteroids could hit other bodies, or become Trojan of Jupiter, or… who knows? Many options seem possible.

The difficulty of giving a simple answer to these questions comes partly from the fact that these bodies have a chaotic dynamics… but what does that mean?

Chaos, predictability, hyperbolicity, frequency diffusion, stability,… in celestial dynamics

Chaos is a pretty complicated mathematical and physical notion, which has several definitions. A popular one is made by the American mathematician Robert L. Devaney, who said that a system is chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set will eventually hit the other set), and its periodic orbits form a dense set.

Let us make things a little simpler: in celestial mechanics, you assume to have chaos when you are sensitive to the initial conditions, i.e. if you try to simulate the motion of an object with a given uncertainty on its initial conditions, the uncertainties on its future will grow exponentially, making predictions impossible beyond a certain time, which is related to the Lyapunov time. But to be rigorous, this is the definition of hyperbolicity, not of chaos… but never mind.

A chaotic orbit is often thought to be unstable. This is sometimes true, especially if the eccentricity of your object becomes large… but this is not always the same. Contrarily, you can have stable chaos, in which you know that your object is not lost, it is in a given bounded zone… but you cannot be more accurate than that.

Chaos can also be related to the KAM theory (for Kolmogorov-Arnold-Moser), which says that when you are chaotic, you have no tores in the dynamics, i.e. periodic orbits. When your orbit is periodic, its orbital frequency is constant. If this frequency varies, then you can suspect chaos… but this is actually frequency diffusion.

And now, since I have confused you enough with the theory, comes another question: what is responsible for chaos? The gravitational action of the other bodies, of course! But this is not a satisfying answer, since a gravitational system is not always chaotic. There are actually many configurations in which a gravitational system could be chaotic. An obvious one is when you have a close encounter with a massive object. An other one is when your object is under the influence of several overlapping mean-motion resonances (Chirikov criterion).

This study is related to the chaos induced by the gravitational action of Mars.

The orbit of Mars

Mars orbits the Sun in 687 days (1.88 year), with an inclination of 1.85° with respect to the ecliptic (the orbit of the Earth), and an eccentricity of 0.0934. This is a pretty large number, which means that the distance Mars – Sun experiences some high amplitude variations. All this is valid for now.

But since the Hungaria asteroids are thought to be present for more than 1 Gyr, a study of their dynamics should consider the variations of the orbit of Mars over such a very long time-span. And this is actually a problem, since the chaos in the inner Solar System prevents you from being accurate enough over such a duration. Recent backward numerical simulations of the orbits of the planets of the Solar System by J. Laskar (Paris Observatory), in which many close initial conditions were considered, led to a statistical description of the past eccentricity of Mars. Some 500 Myr ago, the eccentricity of Mars was most probably close to the current one, but it could also have been close to 0, or close to 0.15… actually it could have taken any number between 0 and 0.15.

The uncertainty on the past eccentricity of Mars leads uncertainty on the past orbital behavior of Solar System objects, including the stability of asteroids. At least two destabilizing processes should be considered: possible close encounters with Mars, and resonances.

Among the resonances likely to destabilize the asteroids over the long term are the gi (i between 1 and 10) and the fj modes. These are secular resonances, i.e. involving the pericentres (g-modes) and the nodes (f-modes) of the planets, the g-modes being doped by the eccentricities, and the f-modes by the inclinations. These modes were originally derived by Brouwer and van Woerkom in 1950, from a secular theory of the eight planets of the Solar System, Pluto having been neglected at that time.

The eccentricity of Mars particularly affects the g4 mode.

This paper

This paper consists of numerical integrations of clones of known asteroids in the Hungaria region. By clones I mean that the motion of each asteroid is simulated several times (21 in this study), with slightly different initial conditions, over 1 Gyr. The authors wanted in particular to test the effect of the uncertainty on the past eccentricity of Mars. For that, they considered two cases: HIGH and LOW.

And the conclusion is this: in the HIGH case, i.e. past high eccentricity of Mars (up to 0.142), less asteroids survive, but only if they experienced close encounters with Mars. In other words, no effect of the secular resonance was detected. This somehow contradicts previous studies, which concluded that the Hungaria population is currently decaying. An explanation for that is that in such phenomena, you often have a remaining tail of stable objects. And it seems make sense to suppose that the currently present objects are this tail, so they are the most stable objects of the original population.

Anyway, this study adds conclusions to previous ones, without unveiling the origin of the Hungaria population. It is pretty frustrating to have no definitive conclusion, but we must keep in mind that we cannot be accurate over 1 Gyr, and that there are several competing models of the evolution of the primordial Solar System, which do not affect the asteroid population in the same way. So, we must admit that we will not know everything.

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Mathematics of the spin-orbit resonance

Hi there! Today things are a little bit different. The paper I present you is not published in a journal of astronomy, nor of planetary sciences, but of mathematics. It is entitled Hamiltonian formulation of the spin-orbit model with time-varying non-conservative forces, by Ioannis Gkolias, Christos Efthymiopoulos, Giuseppe Pucacco and Alessandra Celletti, and it has been recently published in Communications in Nonlinear Science and Numerical Simulation. It deals with a mathematical way to express and solve the spin-orbit problem. This mathematical way is the Hamiltonian formulation.

The spin-orbit problem

It is something I already discussed on this blog, but never mind. Imagine you have a triaxial body orbiting a largest one… e.g. the Moon orbiting the Earth… or a satellite orbiting a giant planet. Usually the satellite always show the same face to the planet, which is a consequence of a synchronous rotation, which you can call 1:1 spin-orbit resonance. It can be shown that this synchronous resonance is a dynamical equilibrium, i.e. the fact that the angular momentum of the satellite is almost orthogonal to its orbit, and the long axis always points to the parent planet, is a stable position. This is makes the synchronous rotation ubiquitous in the Solar System. Initially the satellite had some rotation, which could have had any spin and orientation. And then, the dissipations of energy, mostly tides raised by the planet, have damped the rotation until reaching the synchronous rotation. At this point, the energy given by the gravitational torque of the planet is large enough to compensate the tides. Since it is a stable equilibrium, then the system stays there, i.e. the rotation remains synchronous.

Hamiltonian formulation

Let us start from conservative mechanics, i.e. in the absence of dissipation. Neglecting the dissipation might be a priori surprising, but this approximation is used since centuries. In planetary systems, dissipation can be easily seen from geysers, volcanoes…, but its effects on the orbital and rotational dynamics are very small, and hence difficult to measure. Lunar Laser Ranging have shown us that the Earth-Moon distance is increasing by some 3.9 cm / yr, as a consequence of the dissipation. We have measurements of such an effect in the system of Jupiter since 2009, and in the system of Saturn since 2011. Moreover, if we assume that the equilibrium has been reached, then we can consider that the loss of energy is compensated by the energy exchanges between the parent planet and the satellite. This is why neglecting the dissipation is sometimes allowed… even if the paper I present you does not neglect it.

So, in conservative mechanics, the total energy of the system is conserved. The total energy of the system is the sum of the kinetic and potential energies of all of the bodies involved. This total energy depends on the variables of the system, i.e. the orbital and rotational variables. It can be shown that convenient sets of variables exist, i.e. canonical variables, which time derivatives are the partial derivatives of the total energy, written with this set of variables, which respect to their conjugate variables. In that case, the formulation of the total energy is called Hamiltonian of the system, and the ensuing equations are the Hamilton equations.

The Hamiltonian formulation is very convenient from a mathematical point of view. Its properties make the dynamics of the system easier to interpret. For instance, in manipulating the Hamiltonian, you can determine its equilibrium, their stability, and the small oscillations (librations) around it. This mathematical structure can also be used to construct dedicated numerical integrators, called symplectic integrators, which solve the equations numerically. Symplectic integrators are reputed for their numerical stability.

Viscoelasticity and tides

Let us talk now on the dissipation. The main source of dissipation is the tides raised by the parent planet. Since its gravitational torque felt by the satellite is not homogeneous over its volume, as distance-dependent, then the satellite experiences stress and strains which alter its shape and induces energy loss. So, the tides have two consequences: loss of energy and variation of the shape. The paper proposes a way to consider these effects in a Hamiltonian formalism.

This paper

As the authors honestly admit, it is somehow inaccurate to speak of Hamiltonian formulation when you have dissipation. Their paper deals with the dissipative spin-orbit problem, so their “Hamiltonian” function is not an Hamiltonian strictly speaking, but the ensuing equations have a symplectic structure.

They assume that the dissipation is contained in a function F, which depends on the time t, and discuss the resolution of the problem with respect to the form of F: either a constant dissipation, or a quasi-periodic one, or the sum of a constant and a quasi-periodic one.

Of course, this paper is very technical, and I do not want to go too deep into the details. I would like to mention their treatment of the quasi-periodic case. Quasi-periodic means that the function F, i.e. the dissipation, can be written under a sum of sines and cosines, i.e. oscillations, of different frequencies. This is physically realistic, in the sense that the material constituting the satellite has a different response with respect to the excitation frequency, and the time evolution of the distance planet-satellite and a pretty wide spectrum itself.
In that case, the dissipation function F depends on the time, which is a problem. But it is classically by-passed in assuming the time to be a new variable of the problem, and in adding to the Hamiltonian a dummy conjugate variable. This is a way to transform a non-autonomous (time-dependent) Hamiltonian into an autonomous one, with an additional degree of freedom.
Once this is done, the resolution of the problem is made with a perturbative approach. It is assumed, which is physically realistic, that the amplitudes of the oscillations which constitute the F function are of different orders of magnitudes. This allows to classify them from the most important to the less important ones, with the help of a virtual book-keeping parameter λ. This is a small parameter, and the amplitude of the oscillations will be normalized by λq, q being an integer power. The largest is q, the smallest is the amplitude of the oscillations. The resolution process is iterative, and each iteration multiplies the accuracy by λ.

It is to be noted that such algorithms are usually written as formal processes, but their convergence is not guaranteed, because of potential resonances between the different involved frequencies. When two frequencies become too close to each other, the process might be destabilized. But usually, this does not happen before a reasonable order, i.e. before a reasonable number of iterations, and this is why such methods can be used. The authors provide numerical tests, which prove the robustness of their algorithm.

Potential applications

Such a study is timely, since dissipation can now be observed. For instance, the variations of the shapes of planetary bodies have been observed by measurements of variations of their gravity fields, which give the tidal Love number k2. k2 has been measured for Mercury, Venus, the Earth, the Moon, Mars, Saturn, and Titan, thanks to space missions. Moreover, its dissipative counterpart, i.e. k2/Q, has been measured for the Earth, Mars, Jupiter and Saturn. This means that conservations models for the spin-orbit problem are not sufficient anymore.

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