Hyperion seen by Cassini. © NASA

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.

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.

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