Tag Archives: resonances

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.

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.

The mass of Cressida from Uranus’ rings

Hi there! Today I will present you a new way to weigh an inner satellite of a giant planet. This is the opportunity for me to present you Weighing Uranus’ moon Cressida with the η Ring by Robert O. Chancia, Matthew M. Hedman & Richard G. French. This study has recently been accepted for publication in The Astronomical Journal.

The inner system of Uranus

Uranus is known to be the third planet of the Solar System by its radius, the 4th by its mass, and the 7th by its distance to the Sun. It is also known to be highly tilted, its polar axis almost being in its orbital plane. You may also know that it has 5 major satellites (Ariel, Umbriel, Titania, Oberon, and Miranda), and that it has been visited by the spacecraft Voyager 2 in 1986. But here, we are interested in its inner system. If we traveled from the center of Uranus to the orbit of the innermost of its major satellites, i.e. Miranda, we would encounter:

  • At 25,559 km: the location where the atmosphere reaches the pressure 1 bar. This is considered to be the radius of the planet.
  • Between 37,850 and 41,350 km: the ζ Ring,
  • At 41,837 km: the 6 Ring,
  • At 42,234 km: the 5 Ring,
  • At 42,570 km: the 4 Ring,
  • At 44,718 km: the α Ring,
  • At 45,661 km: the β Ring,
  • At 47,175 km: the η Ring,
  • At 47,627 km: the γ Ring,
  • At 48,300 km: the δ Ring,
  • At 49,770 km: the satellite Cordelia (radius: 20 km),
  • At 50,023 km: the λ Ring,
  • At 51,149 km: the ε Ring
  • At 53,790 km: the satellite Ophelia (radius: 22 km),
  • At 59,170 km: the satellite Bianca (radius: 26 km),
  • At 61,780 km: the satellite Cressida (radius: 40 km),
  • At 62,680 km: the satellite Desdemona (radius: 34 km),
  • At 64,350 km: the satellite Juliet (radius: 47 km),
  • At 66,090 km: the satellite Portia (radius: 68 km),
  • Between 66,100 and 69,900 km: the ν Ring,
  • At 69,940 km: the satellite Rosalind (radius: 36 km),
  • At 74,800 km: the satellite Cupid (radius: 9 km),
  • At 75,260 km: the satellite Belinda (radius: 45 km),
  • At 76,400 km: the satellite Perdita (radius: 15 km),
  • At 86,010 km: the satellite Puck (radius: 81 km),
  • Between 86,000 and 103,000 km: the μ Ring,
  • In the μ Ring, at 97,700 km: the satellite Mab (radius: 13 km)
  • At 129,390 km: the satellite Miranda (radius: 236 km).

The rings of Uranus are being discovered since 1977. Originally it was from star occultations observed from the Earth. Then Voyager 2 visited Uranus in 1986, which revealed other rings, and more recently the Hubble Space Telescope imaged some of them, permitting other discoveries.. Most of them have a width of ≈1 km.
All of the inner moons have been discovered on Voyager 2 images, except Cupid and Mab, which have been discovered in 2003, once more thanks to Hubble. On the contrary, the major moons have been discovered between 1787 and 1948.

Today we will focus only on

  • At 47,175 km: the η Ring,
  • At 61,780 km: the satellite Cressida (radius: 40 km).

The η Ring is very close to the 3:2 mean-motion resonance (MMR) with Cressida, which means that any particle of the η Ring makes 3 revolutions around Uranus while Cressida makes 2. As a consequence, Cressida has a strong gravitational action on the η Ring.

Gravitational interactions

How do we know the mass of planetary bodies? When we send a spacecraft close enough, the spacecraft is deviated, and from the deviation we have the gravity field, or at least the mass. If we cannot send a spacecraft, then we can invert, i.e. analyze, the interactions between different bodies. We know the mass of the Sun thanks to the orbits of the planets, we know the mass of Jupiter thanks to the orbits of its satellites, and the deviations of the spacecraft. We can also use MMR. For instance, in the system of Saturn, the mass ratios between Mimas and Tethys, between Enceladus and Dione, and between Janus and Epimetheus, were accurately known before the arrival of Cassini, thanks to resonant relations.

We can have resonant interactions between a satellite and a ring, as well. A ring is actually a cloud of small particles, and the way their motion is affected reveals the gravitational interaction with something. When you have a MMR, then the ring exhibits streamlines, which give a pattern with equally spaced corners. From the number of these corners you can determine the MMR involved, and from the size of the pattern you get the mass of the disturbing satellite. This is exactly what happens here, i.e. 3:2 MMR with Cressida affects the η Ring in such a way that you can read the mass of Cressida from the shape of this ring. But for that, you need to be accurate enough on the location of the ring.

The data

The authors used 49 observations, including 3 Voyager 2 ones, the other ones being star occultations by rings. Such an observation should be anticipated, i.e. the relative position of Uranus with respect to thousands of stars is calculated, then the star has to be observed where possible, i.e. in a place where it will be high enough in the sky, and of course at night. You measure the light flux coming from the star, which should be pretty constant… and is not because of the variability of the atmospheric thickness since the star is moving in the sky (remember: the Earth rotates in one day), so you have to compensate with other stars… and if you detect a flux drop, then this means that something is occulting the star. Possibly a ring.
Most of the observations were made in the K band, i.e. at an infrared wavelength of 2.2 μm, where Uranus is fainter than its rings. These observations have been made between 1977 and 1996. Since then, the opening of the rings is too small, i.e. we see Uranus by the edge, which reduces the chances to occult a star.


The authors made a least-square fit. This means that they fitted their corpus of observations with a shape of the ring as R-A cos (mθ), where R is a constant radius, A is an amplitude of distortion of the ring, θ is the angle (a longitude), and m is a factor giving the frequency of the distortion, which could be related to its cause, i.e. the orbital motion of the satellite affecting the ring. You fit R, A and m, i.e. you adjust them so as to reduce the difference (the error, which is mathematically seen as a distance) between your model and the observations. From R you have a ring (and you can check whether there should be a ring there), from A you have the mass of the satellite, and from m and have its frequency (and you can check whether a known satellite has this frequency).
The authors show that the highest effect of the inner satellites on the rings should be the effect of Cressida on the η Ring, thanks to the 3:2 MMR.


The authors find that Cressida should have a density of 0.86±0.16 g.cm-3, which is lighter than water. Usually these bodies are supposed to be kind of porous dirty ice, which would mean this kind of density. This is the first measurement of the density of an inner satellite of Uranus. A comparison with other systems shows that this is much denser than the inner satellites of Saturn. However, the inner satellite of Jupiter Amalthea has a pretty similar density.

Finally the authors say that they used this method on other rings, and that additional results should be expected, so we stay tuned. They also say that a spacecraft orbiting Uranus would help knowing these satellites. I cannot agree more. Some years ago, a space mission named Uranus Pathfinder has been proposed to ESA, and another one, named Uranus orbiter and probe, has been proposed to NASA.

The study and the authors

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.

A polar resonant asteroid

Hi there! Did you know that an asteroid could be resonant and in polar orbit? Yes? No? Anyway, one of them has been confirmed as such, i.e. this body was already discovered, known to be on a polar orbit, but it was not known to be in mean-motion resonance with Neptune until now. This is the opportunity for me to present you First transneptunian object in polar resonance with Neptune, by M.H.M. Morais and F. Namouni. This study has recently been accepted for publication in The Monthly Notices of the Royal Astronomical Society.

Polar asteroids

The planets of the Solar System orbit roughly in the same plane. In other words, they have small mutual inclinations. However, asteroids are much more scattered, and can have any inclination with respect to the ecliptic, i.e. the orbital plane of the Earth, even if low inclinations are favored.

Two angles are needed to orientate an orbit:

  • the ascending node, which varies between 0 and 360°, and which is the angle between a reference and the intersection between the ecliptic and the orbital plane,
  • the inclination, which is the angle between the ecliptic and the orbital plane. It varies between 0° and 180°.

So, an almost planar orbit means an inclination close to 0° or close to 180°. Orbits close to 0° are prograde, while orbits close to 180° are retrograde. However, when your inclination is close to 90°, then you have a polar orbit. There are prograde and retrograde polar orbits, whether the inclination is smaller (prograde) or larger (retrograde) than 90°.

There are 7 known Trans-Neptunian Objects with an eccentricity smaller than 0.86 and inclination between 65 and 115°, hence 7 known polar TNOs. You can find them below:

Semimajor axis Eccentricity Inclination Ascending node Period
(471325) 2011 KT19 (Niku) 35.58 AU 0.33 110.12° 243.76° 212.25 y
2008 KV42 (Drac) 41.44 AU 0.49 103.41° 260.89° 266.75 y
2014 TZ33 38.32 AU 0.75 86.00° 171.79° 237.20 y
2015 KZ120 46.07 AU 0.82 85.55° 249.98° 312.70 y
(127546)2002 XU93 67.47 AU 0.69 77.95° 90.39° 554.18 y
2010 WG9 52.90 AU 0.65 70.33° 92.07° 384.77 y
2017 CX33 73.97 AU 0.86 72.01° 315.88° 636.21 y

These bodies carry in their names their year of discovery. As you can see, the first of them has been discovered only 15 years ago. We should keep in mind that TNOs orbit very far from the Earth, this is why they are so difficult to discover, polar or not.

The last of them, 2017 CX33, is so recent that the authors did not study it. A recent discovery induces a pretty large uncertainty on the orbital elements, so waiting permits to stay on the safe side. Among the 6 remaining, 4 (Niku, Drac, 2002 XU93 and 2010 WG9) share (very) roughly the same orbit, 2 of them being prograde, while the others two are retrograde. This happened very unlikely by chance, but the reason for this rough alignment is still a mystery.

Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the x-y plane.
Orbits of the polar TNOs, in the y-z plane.
Orbits of the polar TNOs, in the y-z plane.

The study I present you today investigated the current dynamics of these bodies, and found a resonant behavior for one of them (Niku).

Behavior of the resonant asteroids

By resonant behavior, I mean that an asteroid is affected by a mean-motion resonance with a planet. This means that it makes a given (integer) number of revolutions around the Sun, while the planet makes another number of revolutions. Many outcomes are possible. It can slowly enhance the eccentricity and / or the inclination, which could eventually lead to a chaotic behavior, instability, collision… it could also protect the body from close encounters…

It usually translates into an integer combination of the fundamental frequencies of the system (orbital frequencies, frequencies of precession of the nodes and pericentres), which is null, and this results in an integer combination of angles positioning the asteroid of the planet, which oscillates around a given number instead of circulating. In other words, this angle is bounded.

Another point of interest is how the asteroid has been trapped into the resonance. A resonance is between two interacting bodies, but the mass ratio between an asteroid and a planet implies that the planet is insensitive to the gravitational action of the asteroid, and so the asteroid is trapped by the planet. The fundamental frequencies of the orbital motion are controlled by the semimajor axes of the two bodies, so a trapping into a resonance results from a variation of the semimajor axes. Models of formation of the Solar System suggest that the planets have migrated, this could be a cause. Another cause is close encounters between planets and asteroids, which result in abrupt changes in the trajectory of the asteroid. And this is probably the case here: Niku got trapped after a close encounter.

Numerical and analytical study

The authors used both numerical and analytical methods to get, understand, and secure their results.

Numerical study

The authors ran long-term numerical simulations of the orbital motion of the 6 relevant asteroids, perturbed by the planets. They ran 3 kinds of simulations: 2 with different integrators (algorithms) over 400 kyr and 100 Myr and 8 planets, and one over 400 Myr and the four giant planets. With less planets, you go faster. Moreover, since the inner planets have shorter orbital periods, removing them allows you to increase the time-step, and thus go further in time, inward and backward. In each of these simulations, the authors cloned the asteroids to take into consideration the uncertainty on the orbital elements. They used for that a well-known devoted code, MERCURY.

Analytical study

Numerical studies give you an idea of the possible dynamical states of a system, but you need to write down equations to fully understand it. Beside these numerical simulations, the authors wrote a dynamical theory of resonant polar orbits, in another paper (or here).

This consists in reducing the equations to the only terms, which are useful to reproduce the resonant dynamics. For that, you keep the secular variations, i.e. precessions of the nodes and pericentres, and the term involving the resonant argument. This is a kind of averaged dynamics, in which all of the small oscillations of the orbital elements have been dropped. To improve the relevance of the model, the authors used orbital elements which are based on the barycenter (center of mass) of the whole Solar System instead on the Sun only. This is a small correction, since the barycenter is at the edge of the Sun, but the authors mention that it improves their results.


Niku, i.e. (471325) 2011 KT19, is trapped into a 7:9 mean-motion resonance with Neptune. In other words, it makes 7 revolutions around the Sun (sorry: the barycenter of the Solar System) while Neptune makes 9. More precisely, its resonant argument is φ=9λ-7λN-4ϖ+2Ω, where λ and λN are the mean longitudes of the asteroid and of Neptune, respectively, ϖ is the longitude of its pericenter, and Ω is the one of its ascending node. Plotting this argument shows a libration around 180°. Niku has been trapped in this resonance after a close encounter with Neptune, and should leave this resonance in 16±11 Myr. This means that all of the numerical simulations involving Niku show a resonant object, however they disagree on the duration of the resonance.
Their might be another resonant object: a few simulations suggest that Drac, i.e. 2008 KV42 is in a 8:13 mean-motion resonance with Neptune.

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.