Tag Archives: stability

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.

A constantly renewed ring of Saturn

Hi there! The outstanding Cassini mission ended last September with its Grand Finale, and it gave us invaluable data, which will still be studied for many years. Today I present you a study which has recently been published in The Astrophysical Journal: Particles co-orbital to Janus and Epimetheus: A firefly planetary ring, by a Brazilian team composed of Othon C. Winter, Alexandre P.S. Souza, Rafael Sfair, Silvia M. Giuliatti Winter, Daniela C. Mourão, and Dietmar W. Foryta. This study tells us how the authors characterized a dusty ring in the system of Saturn, studied its stability, and investigated its origin.

The rings of Saturn

As you may know, Saturn is the ringed planet, its rings being visible from Earth-based amateur telescopes. Actually, the 4 major planets of our Solar System have rings, and some dwarf planets as well, i.e. Chariklo, Haumea, and possibly Chiron. But Saturn is the only one with so dense rings. I summarize below the main relevant structures and distances, from the center of Saturn:

Distance Structure
60,268 km The atmospheric pressure of Saturn reaches 1 bar.
This is considered as the equatorial radius of Saturn.
66,900 – 74,510 km D Ring
74,658 – 92,000 km C Ring
92,000 – 117,580 km B Ring
117,580 – 122,170 km Cassini Division
122,170 – 136,775 km A Ring
133,589 km Encke Gap
140,180 km F Ring
151,500 km Orbits of Janus and Epimetheus
189,000 km Orbit of Mimas
1,222,000 km Orbit of Titan

The A and B Rings are the densest ones. They are separated by the Cassini Division, which appears as a lack of material. It actually contains some, arranged as ringlets, but they are very faint. The Encke Gap is a depletion of material as well, in which the small satellite Pan confines the boundaries. Here we are interested in a dusty ring enshrouding the orbits of Janus and Epimetheus, i.e. outside the dense rings. The discovery of this ring had been announced in 2006, this study reveals its characteristics.

The rings of Saturn seen by Cassini. From right to left: the A Ring with the Encke Gap, the Cassini Division, the B Ring, the C Ring, and the D Ring. © NASA
The rings of Saturn seen by Cassini. From right to left: the A Ring with the Encke Gap, the Cassini Division, the B Ring, the C Ring, and the D Ring. © NASA

Janus and Epimetheus

The two coorbital satellites Janus and Epimetheus are a unique case in the Solar System, since these are two bodies with roughly the same size (diameters: ~180 and ~120 km, respectively), which share the same orbit around Saturn. More precisely, they both orbit Saturn in 16 hours, i.e. at the same mean orbital frequency. This is a case of 1:1 mean-motion resonance, involving peculiar mutual gravitational interactions, which prevent them from colliding. They swap their orbits every four years, i.e. the innermost of the two satellites becoming the outermost. The amplitudes of these swaps (26 km for Janus and 95 for Epimetheus) have permitted to know accurately the mass ratio between them, which is 3.56, Janus being the heaviest one.

Interestingly, Epimetheus is the first among the satellites of Saturn for which longitudinal librations have been detected. As many natural satellites, Janus and Epimetheus have a synchronous rotation, showing the same face to a fictitious observer at the surface of Saturn. For Epimetheus, large librations have been detected around this direction, which are a consequence of its elongated shape, and could reveal some mass inhomogeneities, maybe due to variations of porosity, and/or to its pretty irregular shape.

Janus and Epimetheus seen by Cassini (mosaic of 2 images). © NASA
Janus and Epimetheus seen by Cassini (mosaic of 2 images). © NASA

Images of a new ring

So, Cassini images have revealed a dusty ring in that zone. To characterize it, the authors have first extracted images likely to contain it. Such images are made publicly available on NASA’s Planetary Data System. Since that ring had been announced to have been observed on Sept 15th 2006 (see the original press release), the authors restricted to 2 days before and after that date. The data they used were acquired by the ISS (Imaging Science Subsystem) instrument of Cassini, more precisely the NAC and WAC (Narrow- and Wide-Angle-Camera). They finally found 17 images showing the ring.

The images are given as raw data. The authors needed to calibrate their luminosity with a tool (a software) provided by the Cassini team, and sometimes to smooth them, to remove cosmic rays. Moreover, they needed to consider the position of the spacecraft, to be able to precisely locate the structures they would see.

One of the Cassini images used by the authors. I have added red stars at the location of the ring. © NASA / Ciclops
One of the Cassini images used by the authors. I have added red stars at the location of the ring. © NASA / Ciclops

It appears that the ring presents no longitudinal brightness variation. In other words, not only this is a whole ring and not just an arc, but no density variation is obvious. However, it presents radial brightness variations, over a width of 7,500 km, which is wider than the 5,000 km announced in the 2006 press release.

The next step is to understand the dynamics of this ring, i.e. its stability, its origin, the properties of the particles constituting it… Let us start with the stability.

The ring is removed in a few decades

The authors ran N-body simulations, i.e. numerical integrations of the equations ruling the motion of a ring particle, which would be gravitationally perturbed by the surrounding bodies, i.e. Saturn, and the Janus, Epimetheus, Mimas, Enceladus, Tethys, Dione, and Titan. Moreover, for a reason that I will tell you at the end of this article, the authors knew that the particles were smaller than 13 μm. The motions of such small particles are affected by the radiation pressure of the Sun, in other words the Solar light pushes the particles outward.

The authors simulated 14 times the motion of 18,000 particles equally distributed in the rings. Why 14 times? To consider different particle sizes, i.e. one set with 100 μm-sized particles, and the other sets with sizes varying from 1μm to 13μm.
And it appears that these particles collide with something in a few decades, mostly Janus or Epimetheus. This leaves two possibilities: either we were very lucky to be able to take images of the ring while it existed, or a process constantly feeds the ring. The latter option is the most probable one. Let us now discuss this feeding process.

Renewing the ring

The likeliest sources of material for the rings are ejecta from Janus and Epimetheus. The question is: how were these ejecta produced? By impacts, probably. This study show that Janus and Epimetheus are impacted by the particles constituting the rings, but the impact velocities would not permit to produce ejecta. This is why the authors propose a model, in which interplanetary particles collide with the satellites, generating ejecta.

A firefly behavior

And let us finish with something funny: the ring seems to behave like a firefly, i.e. sometimes bright, and sometimes dark, which means undetectable while present.
To understand what happens, figure out how the light would cross a cloud of particles. If the cloud is dense enough, then it would reflect the light, and not be crossed. But for dust, the light would be refracted, i.e. change its direction. This depends on the incidence angle of the Solar light, i.e. on the geometrical configuration of the Sun-Saturn-ring system. The Solar incidence angle is also called phase. And this phase changes with the orbit of Saturn, which results in huge brightness variations of the ring. Sometimes it can be detected, but most of the time it cannot. This can be explained and numerically estimated by the Mie theory, which gives the diffusion of light by small particles. This theory also explains the creation of rainbows, the Solar light being diffracted by droplets of water.

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 and Facebook.

Does Neptune have binary Trojans?

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

The coorbital resonance

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

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

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

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

The Trojans of Neptune

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

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

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

Their dynamics is plotted below:

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

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

Binary asteroids

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

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

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

Numerical simulations

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

Survival of the binaries

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

Challenging this model

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

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

The study and its author

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.

Triton: a cuckoo around Neptune

Hi there! Did you know that Neptune had a prominent satellite, i.e. Triton, on a retrograde orbit? This is so unusual that it is thought to have been trapped, i.e. was originally an asteroid, and has not been formed in the protoneptunian nebula. The study I present you today, Triton’s evolution with a primordial Neptunian satellite system, by Raluca Rufu and Robin M. Canup, explains how Triton ejected the primordial satellites of Neptune. This study has recently been published in The Astronomical Journal.

The satellites of Uranus and Neptune

We are tempted to see the two planets Uranus and Neptune as kinds of twins. They are pretty similar in size, are the two outermost known planets in the Solar System, and are gaseous. A favorable orbital configuration made their visitation possible by the spacecraft Voyager 2 in 1986 and 1989, respectively.

Among their differences are the high obliquity of Uranus, the presence of rings around Uranus while Neptune displays arcs, and different configurations in their system of satellites. See for Uranus:

Semimajor axis Eccentricity Inclination Radius
Miranda 5.12 Ru 0.001 4.338° 235.8 km
Ariel 7.53 Ru 0.001 0.041° 578.9 km
Umbriel 10.49 Ru 0.004 0.128° 584.7 km
Titania 17.20 Ru 0.001 0.079° 788.9 km
Oberon 23.01 Ru 0.001 0.068° 761.4 km
Puck 3.39 Ru 0 0.319° 81 km
Sycorax 480.22 Ru 0.522 159.420° 75 km

I show on this table the main satellites of Uranus, and we can see that the major ones are at a reasonable distance (in Uranian radius Ru) of the planet, and orbit almost in the same plane. The orbit of Miranda is tilted thanks to a past mean-motion resonance with Umbriel, which means that it was originally in the same plane. So, we can infer that these satellites were formed classically, i.e. the same way as the satellites of Jupiter, from a protoplanetary nebula, in which the planet and the satellites accreted. An exception is Sycorax, which is very far, highly inclined, and highly eccentric. As an irregular satellite, it has probably been formed somewhere else, as an asteroid, and been trapped by the gravitational attraction of Uranus.

Now let us have a look at the system of Neptune:

Semimajor axis Eccentricity Inclination Radius
Triton 14.41 Rn 0 156.865° 1353.4 km
Nereid 223.94 Rn 0.751 7.090° 170 km
Proteus 4.78 Rn 0 0.075° 210 km
Larissa 2.99 Rn 0.001 0.205° 97 km

Yes, the main satellite seems to be an irregular one! It does not orbit that far, its orbit is (almost) circular, but its inclination is definitely inconsistent with an in situ formation, i.e. it has been trapped, which has been confirmed by several studies. Nereid is much further, but with a so eccentric orbit that it regularly enters the zone, which is dynamically perturbed by Triton. You can also notice the absence of known satellites between 4.78 and 14.41 Neptunian radii. This suggests that this zone may have been cleared by the gravitational perturbation of a massive body… which is Triton. The study I present you simulates what could have happened.

A focus on Triton

Before that, let us look at Triton. The system of Neptune has been visited by the spacecraft Voyager 2 in August 1989, which mapped 40% of the surface of Triton. Surprisingly, it showed a limited number of impact craters, which means that the surface has been renewed, maybe some 1 hundred of millions of years ago. Renewing the surface requires an activity, probably cryovolcanism as on the satellite of Saturn Enceladus, which should has been sustained by heating. Triton was on the action of the tides raised by Neptune, but probably not only, since tides are not considered as efficient enough to have circularized the orbit. The tides have probably been supplemented by gravitational interactions with the primordial system of Neptune, i.e. satellites and / or disk debris. If there had been collisions, then they would have themselves triggered heating. As a consequence of this heating, we can expect a differentiated structure.

Moreover, Triton orbits around Neptune in 5.877 days, on a retrograde orbit, while the rotation of Neptune is prograde. This configuration, associated with the tidal interaction between Triton and Neptune, makes Triton spiral very slowly inward. In other words, it will one day be so close to Neptune that it will be destroyed, and probably create a ring. But we will not witness it.

A numerical study with SyMBA

This study is essentially numerical. It aimed at modeling the orbital evolution of Triton, in the presence of Nereid and the putative primordial satellites of Neptune. The authors assumed that there were 4 primordial satellites, with different initial conditions, and considered 3 total masses for them: 0.3, 1, and 3 total masses of the satellites of Uranus. For each of these 3 masses, they ran 200 numerical simulations.

The simulations were conducted with the integrator (numerical code) SyMBA, i.e. Symplectic Massive Body Algorithm. The word symplectic refers to a mathematical property of the equations as they are written, which guarantee a robustness of the results over very long timescales, i.e. there may be an error, but which does not diverge. It may be not convenient if you make short-term accurate simulations, for instance if you want to design the trajectory of a spacecraft, but it is the right tool for simulating a system over hundreds of Myrs (millions of years). This code also handles close encounters, but not the consequences of impacts. The authors bypassed this problem in treating the impacts separately, determining if there were disrupting, and in that case estimated the timescales of reaccretion.


The authors found, consistently with previous studies, that the interaction between Triton and the primordial system could explain its presently circular orbit, i.e. it damped the eccentricity more efficiently than the tides. Moreover, the interaction with Triton caused collisions between the primordial moons, but usually without disruption (hit-and-run impacts). In case of disruption, the authors argue that the reaccretion would be fast with respect of the time evolution of the orbit of Triton, which means that we could lay aside the existence of a debris disk.

Moreover, they found that the total mass of the primordial system had a critical role: for the heaviest one, i.e. 3 masses of the Uranian system, Triton had only small chances to survive, while it had reasonable chances in the other two cases.

Something frustrating when you try to simulate something that happened a few hundreds of Myrs ago is that you can at the best be probabilistic. The study shows that a light primordial system is likelier to have existed than a heavy one, but there are simulations with a heavy system, in which Triton survives. So, a heavy system is not prohibited.

The study and its authors

  • The study, which is available as free article. The authors probably paid extra fees for that, many thanks to them! You can also look at it on arXiv.
  • A conference paper on the same study,
  • The ResearchGate profile of Raluca Rufu,
  • The Homepage of Robin M. Canup.

Before closing this post, I need to mention that the title has been borrowed from Matija Ćuk (SETI, Mountain View, CA), who works on this problem as well (see these two conference abstracts here and here).

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.