New chaos indicators

Hi there! Today it is a little bit different. I will not tell you about something that has been observed but rather of a more general concept, which is the chaos in the Solar System. This is the opportunity to present you Second-order chaos indicators MEGNO2 and OMEGNO2: Theory, by Vladimir A. Shefer. This study has been originally published in Russian, but you can find an English translation in the Russian Physics Journal.

To present you this theoretical study, I need to define some useful notions related to chaos. First is the sensitivity to the initial conditions.

Sensitivity to the initial conditions

Imagine you are a planetary body. I put you somewhere in the Solar System. This somewhere is your initial condition, actually composed of 6 elements: 3 for the position, and 3 for the velocity. So, I put you there, and you evolve, under the gravitational interaction of the other guys, basically the Sun and the planets of the Solar System. You then have a trajectory, which should be an orbit around the Sun, with some disturbances of the planets. What would have happened if your initial condition would have been slightly different? Well, you expect your trajectory to have been slightly different, i.e. pretty close.

Does it always happen this way? Actually, not always. Sometimes yes, but sometimes… imagine you have a close encounter with a planet (hopefully not the Earth). During the encounter, you are very sensitive to the gravitational perturbation of that planet. And if you arrive a little closer, or a little further, then that may change your trajectory a lot, since the perturbation depends on the distance to the planet. In such a case, you are very sensitive to the initial conditions.

What does that mean? It actually means that if you are not accurate enough on the initial condition, then your predicted trajectory will lack of accuracy. And beyond a certain point, predicting will just be pointless. This point can be somehow quantified with the Lyapunov time, see a little later.

An example of body likely to have close encounters with the Earth is the asteroid (99942) Apophis, which was discovered in 2004, and has sometimes close encounters with the Earth. There was one in 2013, there will be another one in 2029, and then in 2036. But risks of impact are ruled out, don’t worry. 🙂

Let us talk now about the problem of stability.


A stable orbit is an orbit which stays around the central body. A famous and recent example of unstable orbit is 1I/’Oumuamua, you know, our interstellar visitor. It comes from another planetary system, and passes by, on a hyperbolic orbit. No chaos in that case.

But sometimes, an initially stable orbit may become unstable because of an accumulation of gravitational interactions, which raise its eccentricity, which then exceeds 1. And this is where you may connect instability with sensitivity to initial conditions, and chaos. But this is not the same. And you can even be stable while chaotic.

Now, let us define a related (but different) notion, which is the diffusion of the fundamental frequencies.

Diffusion of the Fundamental Frequencies

Imagine you are on a stable, classical orbit, i.e. an ellipse. The Sun lies at one of its foci, and you have an orbital frequency, a precessional frequency of your pericenter, and a frequency related to the motion of your ascending node. All of these points have a motion around the Sun, with constant velocities. So, the orbit can be described with 3 fundamental frequencies. If your orbit is perturbed by other bodies, which have their own fundamental frequencies, then you will find them as additional frequencies in your trajectory. Very well. If the trajectories remain constant, then it can be topologically said that your trajectories lies on tori.

Things become more complicated when you have a drift of these fundamental frequencies. It is very often related to chaos, and sometimes considered as an indicator of it. In such a case, the tori are said to be destroyed. And we have theorems, which address the survival of these tori.

The KAM and the Nekhoroshev theorems

The most two famous of them are the KAM and the Nekhoroshev theorems.

KAM stands for Kolmogorov-Arnold-Moser, which were 3 famous mathematicians, specialists of dynamical systems. These problems are indeed not specific to astronomy or planetology, but to any physical system, in which we neglect the dissipation.

The KAM theorem says that, for a slightly perturbed integrable system (allow me not to develop this point… just keep in mind that the 2-body problem is integrable), some tori survive, which means that you can have regular (non chaotic) orbits anyway. But some of them may be not. This theorem needs several assumptions, which may be difficult to fulfill when you have too many bodies.

The Nekhoroshev theory addresses the effective stability of destroyed tori. If the perturbation is small enough, then the trajectories, even not exactly on tori, will remain close enough to them over an exponentially long time, i.e. longer than the age of the Solar System. So, you may be chaotic, unstable… but remain anyway where you are.

Chaos is related to all of these notions, actually there are several definitions of chaos in the literature. Consider it as a mixture of all the elements I gave you. In particular the sensitivity to the initial conditions.

Chaos in the Solar System

Chaos has been observed in the Solar System. The first observation is the tumbling rotation of the satellite of Saturn Hyperion (see featured image). So, not an orbital case. Chaos has also been characterized in the motion of asteroids, for instance the Main-Belt asteroid (522) Helga has been proven to be in stable chaos in 1992 (see here). It is in fact swinging between two mean-motion resonances with Jupiter (Chirikov criterion), which confine its motion, but make it difficult to predict anyway. The associated Lyapunov time is 6.9 kyr.

There are also chaotic features in the rings of Saturn, which are due to the accumulation of resonances with satellites so close to the planet. These effects are even raised by the non-linear self-dynamics of the rings, in which the particles interact and collide. And the inner planets of the Solar System are chaotic over some 10s of Myr, this has been proven by long-term numerical integrations of their orbits.

To quantify this chaos, you need the Lyapunov time.

The maximal Lyapunov exponent

The Lyapunov time is the invert of the Lyapunov exponent. To estimate the Lyapunov exponent, you numerically integrate the trajectory, and its tangent vector. When the orbit is chaotic, the norm of this vector will grow exponentially, and the Lyapunov exponent is the asymptotic limit of the divergence rate of this exponential growth. It is strictly positive in case of chaos. Easy, isn’t it?

Not that easy, actually. The exponential growth makes that this norm might be too large and generate numerical errors, but this can be fixed in regularly, i.e. at equally spaced time intervals, renormalizing the tangent vector. Another problem is in the asymptotic limit: you may have to integrate over a verrrrrry long time to reach it. To bypass this problem of convergence, other indicators have been invented.

To go faster: FLI and MEGNO

FLI stands for Fast Lyapunov Indicators. There are several variants, the most basic one consists in stopping the integration at a given time. So, you give up the asymptotic limit, and you give up the Lyapunov time, but you can efficiently distinguish the regular orbits from the chaotic ones. This is a good point.

Another chaos detector is the MEGNO, for Mean Exponential Growth of Nearby Orbits. This consists to integrate the norm of the time derivative of the tangent vector divided by the norm of the tangent vector. The result tends to a straight line, which slope is half the maximal Lyapunov exponent. And this tool converges very fast. The author of the study I present you wishes to improve that tool.

This study presents MEGNO2

And for that, he presents us MEGNO2. This works like MEGNO, but with an osculating vector instead of a tangent one. Tangent means that this vector fits to a line tangent to the trajectory, while osculating means that it fits to its curvature as well, i.e. second order derivative. In other words, it is more accurate.

From this, the author shows that, like MEGNO, MEGNO2 tends to a straight line, but with a larger slope. As a consequence, he argues that it permits a more efficient detection of the chaotic orbits with respect to the regular ones. However, he does not address the link between this new slope and the Lyapunov time.

Something that my writing does not render, is that this paper is full of equations. Fair enough, for what I could call mathematical planetology.

The study and its author

As it often happens for purely theoretical studies, this one has only one author.

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, Facebook, Instagram, and Pinterest.

2010 JO179: a new, resonant dwarf planet

Hi there! Today I present you the discovery of a Trans-Neptunian Object, you know, these objects which orbit beyond the orbit of Neptune. And I particularly like that one, since its orbit resonates with the one of Neptune. Don’t worry, I will explain you all this, keep in mind for now that this object is probably one of the most stable. Anyway, this is the opportunity to present you A dwarf planet class object in the 21: 5 resonance with Neptune by M.J. Holman and collaborators. This study has recently been accepted for publication in The Astrophysical Journal Letters.

The Trans-Neptunian Objects

The Trans-Neptunians Objects are small bodies, which orbit beyond the orbit of Neptune, i.e. with a semimajor axis larger than 30 AU. The first discovered one is the well-known Pluto, in 1930. It was then, and until 2006, considered as the ninth planet of the Solar System. It was the only known TNO until 1992. While I am writing this, 2482 are listed on the JPL small-body database search engine.

The TNOs are often classified as the Kuiper-Belt objects, the scattered disc objects, and the Oort cloud. I do not feel these are official classifications, and there are sometimes inconsistencies between the different sources. Basically, the Kuiper-Belt objects are the ones, which orbits are not too much eccentric, not too inclined, and not too far (even if these objects orbit very far from us). The scattered disc objects have more eccentric and inclined orbits, and these dynamics could be due to chaotic / resonant excitation by the gravitational action of the planets. And the Oort cloud could be seen as the frontier of our Solar System. It is a theoretical cloud located between 50,000 and 200,000 Astronomical Units. Comets may originate from there. Its location makes it sensitive to the action of other stars, and to the Galactic tide, i.e. the deformation of our Galaxy.

The object I present you today, 2010 JO179, could be a scattered disc object. It has been discovered in 2010, thanks to the Pan-STARRS survey.

The Pan-STARRS survey

Pan-STARRS, for Panoramic Survey Telescope and Rapid Response System, is a systematic survey of the sky. Its facilities are located at Haleakala Observatory, Hawaii, and currently consist of two 1.8m-Ritchey–Chrétien telescopes. It operates since 2010, and discovered small Solar System objects, the interstellar visitor 1I/’Oumuamua… It observes in 5 wavelengths from infrared to visible.

The Pan-STARRS1 telescope. © Pan-STARRS
The Pan-STARRS1 telescope. © Pan-STARRS

The data consist of high-accuracy images of the sky, containing a huge amount of data. Beyond discoveries, these data permit an accurate astrometry of the object present on the images, which is useful for understanding their motion and determining their orbits. They also allow a determination of the activity of variable objects, i.e. variable stars, a study of their surface from their spectrum in the five wavelengths, and (for instance) the measurement of their rotation. A very nice tool anyway!

Pan-STARRS delivered its first data release in December 2016, while the DR2 (Data Release 2) is scheduled for mid-2018… pretty soon actually.

Among the discovered objects are the one we are interested in today, i.e. 2010 JO179.

Identifying the new object

The first observation of 2010 JO179 dates back from May 2010, and it has been detected 24 times during 12 nights, until July 2016. The detections are made in comparing the Pan-STARRS data from the known objects. Once something unknown appears in the data, leaving what the authors call a tracklet, its motion is extrapolated to predict its position at different dates, to investigate whether it is present on other images, another time. From 3 detections, the algorithm makes a more systematic search of additional tracklets, and in case of positive additional detection, then an orbit is fitted. The orbital characteristics (and other properties) are listed below.

Semimajor axis 78.307±0.009 AU
Eccentricity 0.49781±0.00005
Inclination 32.04342±0.00001 °
Orbital period 6663.757±0.002 yr
Diameter 600-900 km
Absolute magnitude 3.4±0.1

You can notice the high accuracy of the orbital parameters, which almost looks like a miracle for such a distant object. This is due to the number of detections, and the accuracy of the astrometry with Pan-STARRS. Once an object is discovered, you know where it is, or at least where it is supposed to be. Thanks to this knowledge, it was possible to detect 2010 JO179 on data from the Sloan Digital Sky Survey, taken in New Mexico, and on data from the DECalS survey, taken in Chile. Moreover, 2010 JO179 was intentionally observed with the New Technology Telescope (NTT) in La Silla, Chile.

The spectroscopy (analysis of the reflected light at different wavelengths) of 2010 JO179 revealed a moderately red object, which is common for TNOs.

Measuring its rotation

This is something I have already evoked in previous articles. When you record the light flux reflected by the surface of a planetary body, you should observe some periodic variability, which is linked to its rotation. From the observations, you should extract (or try to) a period, which may not be an easy task regarding the sparsity and the accuracy of the observations.

In using the so-called Lomb-Scargle algorithm, the authors detected two possible periods, which are 30.6324 hours, and 61.2649 hours… i.e. twice the former number. These are best-fits, i.e. you try to fit a sinusoid to the recorded light, and these are the periods you get. The associated amplitudes are variations of magnitude of 0.46 and 0.5, respectively. In other words, the authors have two solutions, they favor the first one since it would imply a too elongated asteroid. Anyway, you can say that twice 30.6324 hours is a period as well, but what we call the spin period is the smallest non-null duration, which leaves the light flux (pretty) invariant. So, the measured spin period of 2010 JO179 is 30.6324 hours, which makes it a slow rotator.

Mean-motion resonances

Let us make a break on the specific case of 2010 JO179 (shall we give it a nickname anyway?), since I would like to recall you something on the mean-motion resonances before.

When two planetary bodies orbit the Sun, they perturb each other. It can be shown that when the ratio of their orbital periods (similarly the ratio of their orbital frequencies) is rational, i.e. is one integer divided by another one, then you are in a dynamical situation of commensurability, or quasi-resonance. A well known case is the 5:2 configuration between Jupiter and Saturn, i.e. Jupiter makes 5 orbits around the Sun while Saturn makes 2. In such a case, the orbital perturbations are enhanced, and you can either be very stable, or have a chaotic orbit, in which the eccentricities and inclinations could raise, the orbit become unpredictable beyond a certain time horizon (Lyapunov time), and even a body be ejected.

Mathematically, an expansion of the so-called perturbing function, or the perturbing mutual gravitational potential, would display a sum of sinusoidal term containing resonant arguments, which would have long-term effects. These arguments would read as pλ1-(p+q)λ2+q1ϖ1+q2ϖ2+q3Ω1+q4Ω2, with q=q1+q2+q3+q4. The subscripts 1 and 2 are for the two bodies (in our case, 1 will stand for Neptune, and 2 for JO 2010179), λ are their mean longitudes, ϖ their longitudes of pericentres, and Ω the longitudes of their ascending nodes.

In a perturbed case, which may happen for high eccentricities and inclinations, resonances involving several arguments may overlap, and induce a chaotic dynamics that could be stable… or not. You need to simulate the long-term dynamics to know more about that.

A resonant long-term dynamics

It appears that Neptune and 2010 JO179 are very close to the 21:5 mean-motion resonance (p=5, q=16). To inquire this, the authors ran 100 numerical simulations of the orbital motion of 2010 JO179, with slightly different initial conditions which are consistent with the uncertainty of the observations, over 700 Myr. And they saw that 2010 JO179 could be trapped in a resonance, with argument 5λ1-21λ2+16ϖ2. In about 25% of the simulations, JO179 remains trapped, which implies that the resonant argument is librating, i.e. bounded, all over the simulation. As a consequence, this suggests that its orbit is very stable, which is remarkable given its very high eccentricity (almost 0.5).

The study and its authors

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, Facebook, Instagram, and Pinterest.

Fracturing the crust of an icy satellite

Hi there! You may know that the space missions to the systems of giant planets have revealed that the surface of several of theirs satellites are fractured. We dispose of images of such structures on Jupiter’s Europa and Ganymede, Saturn’s Enceladus (the famous tiger stripes at its South Pole), and even on Uranus’ Miranda, which has been visited by Voyager II. These satellites are thought to be icy bodies, with an icy crust enshrouding a subsurface, global ocean (maybe not for Miranda, but certainly true for the other guys).

The study I present you today, Experimental constraints on the fatigue of icy satellite lithospheres by tidal forces, by Noah P. Hammond, Amy C. Barr, Reid F. Cooper, Tess E. Caswell, and Greg Hirth, has recently been accepted for publication in Journal of Geophysical Research: Planets. The authors particularly tried to produce in labs the process of fatigue, which would weaken a material after a certain number of solicitations, i.e. it would become easier to break.

Cycloids on Europa

The Galilean satellite of Jupiter Europa may be the most interesting satellite to focus on, since it is the most fractured, at least to the best of our knowledge. The observation of the surface of Europa, first by Voyager I and II in 1979, and after by Galileo between 1995 and 2003, revealed many structures, like lineae, i.e. cracks, due to the geophysical activity of the satellite. This body is so active that only few craters are visible, the surface having been intensively renewed since the impacts. Something particularly appealing on Europa is that some of these lineae present a cycloidal pattern, which would reveal a very small drift of the orientation of the surface. Some interpret it has an evidence of super-synchronous rotation of Europa, i.e. its rotation would not be exactly synchronous with its orbital motion around Jupiter.

Cycloids on Europa, seen by the spacecraft Galileo. © NASA
Cycloids on Europa, seen by the spacecraft Galileo. © NASA

Beside Europa, fractures have also been observed on Ganymede, but with less frequency. For having such fractures, you need the surface to be brittle enough, so that stress will fracture it. This is a way to indirectly detect a subsurface ocean. But you also need the stress. And this is where tides intervene.

Fractures on Ganymede. © Paul M. Schenk
Fractures on Ganymede. © Paul M. Schenk

Tides can stress the surface

You can imagine that Jupiter exerts a huge gravitational action on Europa. But Europa is not that small, and its finite size results in a difference of Jovian attraction between the point which is the closest to Jupiter, and the furthest one. The result of this differential attraction is stress and strain in the satellite. The response of the satellite will depend on its structure.

A problem is that calculations suggest that the tidal stress may be too weak to generate alone the observed fractures. This is why the authors suggest the assistance of another phenomenon: fatigue crack growth.

The phenomenon of fatigue crack growth

The picture is pretty intuitive: if you want to break something… let’s say a spoon. You twist it, bend it, wring it… once, twice, thrice, more… Pretty uneasy, but you do not give up, because you see that the material is weakening. And finally it breaks. Yes you did it! But what happened? You slowly created microcracks in the spoon, which weakened it, the cracks grew… until the spoon broke.

For geophysical materials, it works pretty much the same: we should imagine that the tides, which vary over an orbit since the eccentricity of the orbit induces variations of the Jupiter-Europa distance, slowly create microcracks, which then grow, until the cracks are visible. To test this scenario, the authors ran lab experiments.

Lab experiments

The lab experiments consisted of Brazil Tests, i.e. compression of circular disks of ice along their diameter between curved steel plates. The resulting stress was computed everywhere in the disk thanks to a finite-element software named Abaqus, and the result was analyzed with acoustic emissions, which reflections would reveal the presence of absence of microcracks in the disk. The authors ran two types of tests: both with cyclic loading, i.e. oscillating loading, but one with constant amplitude, and the other one with increasing amplitude, i.e. a maximum loading becoming stronger and stronger.

But wait: how to reproduce the conditions of the real ice of these satellites? Well, there are things you cannot do in the lab. Among the problems are: the exact composition of the ice, the temperature, and the excitation frequency.

The authors conducted the experiments in assuming pure water ice. The temperature could be below 150 K (-123°C, or -189°F), which is very challenging in a lab, and the main period of excitation is the orbital one, i.e. 3.5 day… If you want to reproduce 100,000 loading cycles, you should wait some 1,000 years… unfeasible…

The authors bypassed these two problems in constraining the product frequency times viscosity to be valid, the viscosity itself depending on the temperature. This resulted in an excitation period of 1 second, and temperatures between 198 and 233 K (-75 to -40°C, or -103 to -40°F). The temperature was maintained thanks to a liquid nitrogen-cooled, ethanol bath cryostat.

And now the results!

No fatigue observed

Indeed, the authors observed no fatigue, i.e. no significant microcracks were detected, which would have altered the material enough, to weaken it. This prompted the authors to discuss the application of their experiments for understanding the crust of the real satellites, and they argue that fatigue could be possible anyway.

Why fatigue may still be possible

As the authors recall, these experiments are not the first ones. Other authors have had a negative result with pure water ice. However, fatigue has been detected on sea ice, which could mean that the presence of salt favors fatigue. And the water ice of icy satellite may not be pure. Salt and other chemical elements may be present. So, even if these experiments did not reveal fatigue, there may be some anyway.

But the motivation for investigating fatigue is that a process was needed to assist the tides to crack the surface. Why necessarily fatigue? Actually, other processes may weaken the material.

How to fracture without fatigue

The explanation is like the most (just a matter of taste) is impacts: when you impact the surface, you break it, which necessarily weakens it. And we know that impacts are ubiquitous in the Solar System. In case of an impact, a megaregolith is created, which is more likely to get fractured. The authors also suggest that the tides may be assisted, at least for Europa, by the super-synchronous rotation possibly suggested by the geometry of the lineae (remember, the cycloids). Another possibility is the large scale inhomogeneities in the surface, which could weaken it at some points.

Anyway, it is a fact that these surfaces are fractured, and the exact explanation for that is still in debate!

The study and its authors

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, Facebook, Instagram, and Pinterest.