Tag Archives: celestial mechanics

An interstellar asteroid

Hi there! You may have heard this week of our Solar System visited by an asteroid probably formed in another planetary system. This is why I have decided to speak about it, so this article will not be based on a peer-reviewed scientific publication, but on good science anyway. The name of this visitor is for now A/2017 U1.

History of the discovery

Discovering a new object usually consists in

  1. Taking a picture of a part of a sky. Usually these are parts of the order of the degree, maybe much less… so, small parts. And this also requires to treat the image, to correct for atmospheric (brightness of the sky, wind,…) and instrumental (dead pixels…) effects,
  2. Comparing in with the objects, which are known to be in that field.

If there is an unexpected object, then it could be a discovery. Here is the history of the discovery of A/2017 U1:

  • Oct. 19, 2017: Robert Weryk, a researcher of the University of Hawaii, discovers a new object while searching for Near-Earth Asteroids with the Pan-STARRS 1 telescope. An examination of images archives revealed that the object had already been photographed the day before.
  • Oct. 25, 2017: The Minor Planet Center (Circular MPEC 2017-U181) gives orbital elements for this new object, from 34 observations over 6 days, from Oct. 18 to 24. Surprisingly, an eccentricity bigger than 1 (1.1897018) is announced, which means that the trajectory follows a hyperbola. This means that if this object would be affected only by the Sun, then it would come from an infinite distance, and would leave us for infinity. In other words, this object would not be fated to remain in our Solar System. That day, the object was thought to be a comet, and named C/2017 U1. 10 observation sites were involved (once an object has been detected and located, it is easier to re-observe it, even with a smaller telescope).
  • Oct. 26, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U185), using 47 observations from Oct. 14 on. The object is renamed A/2017 U1, i.e. from comet “C” to asteroid “A”, since no cometary activity has been detected. Same day: the press release announcing the first confirmed discovery of an interstellar object. New estimation of the eccentricity: e = 1.1937160.
  • Oct. 27, 2017: Update by the Minor Planet Center (Circular MPEC 2017-U234), using 68 observations. New estimation of the eccentricity: e = 1.1978499.

And this is our object! It has an absolute magnitude of 22.2 and a diameter probably smaller than 400 meters. These days, spectroscopic observations have revealed a red object, alike the KBOs (Kuiper Belt Objects). It approached our Earth as close as 15 millions km (0.1 astronomical unit), i.e. one tenth of the Sun-Earth distance.

The trajectory of A/2017 U1.
The trajectory of A/2017 U1.

What are these objects?

The existence of such objects is predicted since more than 40 years, in particular by Fred Whipple and Viktor Safronov. This is how they come to us:

  1. A protoplanetary disk creates a star, planets, and small objects,
  2. The small objects are very sensitive to the gravitational perturbations of the planets. As a consequence, they may be ejected from their planetary system, and become interstellar objects,
  3. They visit us.

Calculations indicate that A/2017 U1 comes roughly from the constellation Lyra, in which the star Vega is (only…) at 25 lightyears from our Sun. It is tempting to assume that A/2017 U1 was formed around Vega, but that would be only speculation, since many perturbations could have altered its trajectory. Several studies will undoubtedly address this problem within next year.

Maybe not the first one

Here we have an eccentricity, which is significantly larger (some 20%) than 1. Moreover, our object has a very inclined orbit, which means that we can neglect the perturbations of its orbit by the giant planets. In other words, it entered the Solar System on the trajectory we see now. But a Solar System object can get a hyperbolic orbit, and eventually be ejected. This means that when we detect an object with a very high eccentricity, like a long-period comet, it does not necessary mean that it is an interstellar object. In the past, some known objects have been proposed to be possible interstellar ones. This is for example the case for the comet C/2007 W1 (Boattini), which eccentricity is estimated to be 1.000191841611794±0.000041198 at the date May 26, 2008. It could be an IC (Interstellar Comet), but could also be an Oort cloud object, put on a hyperbolic orbit by the giant planets.

Detecting interstellar objects

A/2017 U1 object has been detected by the Pan-STARRS (for Panoramic Survey Telescope and Rapid Response System) 1 telescope, which is located at Haleakala Observatory, Hawaii. Pan-STARRS is constituted of two 1.8 m Ritchey–Chrétien telescopes, with a field-of-view of 3°. This is very large compared with classical instruments, and it is suitable for detection of bodies. It operates since 2010.

Detections could be expected from the future Large Synoptic Survey Telescope (LSST), which should operate from 2022 on. This facility will be a 8.4-meter telescope based in Chile, and will conduct surveys with a field-of-view of 3.5°. A recent study by Nathaniel Cook et al. suggests that LSST could detect between 0.001 and 10 interstellar comets during its nominal 10 year lifetime. Of course, 0.001 detection should be understood as the result of a formula. The authors give a range of 4 orders of magnitude in their estimation, which reflects how barely constrained the theoretical models are. This also means that we could be just lucky to have detected one.

What Pan-STARRS can do, LSST should be able to do. In a few years, i.e. in the late 2020s, the number or absence of new discoveries will tell us something on the efficiency of creation of interstellar objects in the nearby stars. Meanwhile, let us enjoy this exciting discovery!

The press release and its 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.

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.

Methodology

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.

Results

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.

How the Planet Nine would affect the furthest asteroids

Hi there! You have heard of the hypothetical Planet Nine, which could be the explanation for an observed clustering of the pericentres of the furthest asteroids, known as eTNOS for extreme Trans-Neptunian Objects. I present you today a theoretical study investigating in-depth this mechanism, in being focused on the influence of the inclination of this Planet Nine. I present you Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth by Melaine Saillenfest, Marc Fouchard, Giacomo Tommei and Giovanni B. Valsecchi. This study has recently been accepted for publication in Celestial Mechanics and Dynamical Astronomy.

Is there a Planet Nine?

An still undiscovered Solar System planet has always been dreamed, and sometimes even hinted. We called it Tyche, Thelisto, Planet X (“X” for mystery, unknown, but also for 10, Pluto having been the ninth planet until 2006). Since 2015, this quest has been renewed after the observation of clustering in the pericentres of extreme TNOS. Further investigations concluded that at least 5 observed dynamical features of the Solar System could be explained by an additional planet, now called Planet Nine:

  1. the clustering of the pericentres of the eTNOs,
  2. the significant presence of retrograde orbits among the TNOs,
  3. the 6° obliquity of the Sun,
  4. the presence of highly inclined Centaurs,
  5. the dynamical detachment of the pericentres of TNOs from Neptune.

The combination of all of these elements tends to rule out a random process. It appears that this Planet Nine would be pretty like Neptune, i.e. 10 times heavier than our Earth, that its pericentre would be at 200 AU (while Neptune is at 30 AU only!), and its apocentre between 500 AU and 1200 AU. This would indeed be a very distant object, which would orbit the Sun in several thousands of years!

Astronomers (Konstantin Batygin and Michael Brown) are currently trying to detect this Planet Nine, unsuccessfully up to now. You can follow their blog here, from which I took some inspiration. The study I present today investigates the secular dynamics that this Planet Nine would induce.

The secular dynamics of an asteroid

The secular dynamics is the one involving the pericentre and the ascending node of an object, without involving its longitude. To make things clear, you know that a planetary object orbiting the Sun wanders on an eccentric, inclined orbit, which is an ellipse. When you are interested in the secular dynamics, you care of the orientation of this ellipse, but not of where the object is on this ellipse. The clustering of pericentres of eTNOs is a feature of the secular dynamics.

This is a different aspect from the dynamics due to mean-motion resonances, in which you are interested in objects, which orbital periods around the Sun are commensurate with the one of the Planet Nine. Some studies address this issue, since many small objects are in mean-motion resonance with a planet. Not this study.

The Kozai-Lidov mechanism

A notable secular effect is the Kozai-Lidov resonance. Discovered in 1961 by Michael Lidov in USSR and Yoshihide Kozai in Japan, this mechanism says that there exists a dynamical equilibrium at high inclination (63°) for eccentric orbits, in the presence of a perturber. So, you have the central body (the Sun), a perturber (the planet), and your asteroid, which could have its inclination pushed by this effect. This induces a libration of the orientation of its orbit, i.e. the difference between its pericentre and its ascending node would librate around 90° or 270°.

This process is even more interesting when the perturber has a significant eccentricity, since the so-called eccentric Kozai-Lidov mechanism generates retrograde orbits, i.e. orbits with an inclination larger than 90°. At 117°, you have another equilibrium.

Now, when you observe a small body which dynamics suggests to be affected by Kozai-Lidov, this means you should have a perturber… you see what I mean?

Of course, this perturber can be Neptune, but only sometimes. Other times, the dynamics would rather be explained by an outer perturber… which could be the Planet Nine, or a passing star (who knows?)

Methodology

Before mentioning the results of this study I must briefly mention the methodology. The authors made what I would call a semi-analytical study, i.e. they manipulated equations, but with the assistance of a computer. They wrote down the Hamiltonian of the restricted 3-body problem, i.e. the expression of the whole energy of the problem with respect to the orbital elements of the perturber and the TNO. This energy should be constant, since no dissipation is involved, and the way this Hamiltonian is written has convenient mathematical properties, which allow to derive the whole dynamics. Then this Hamiltonian is averaged over the mean longitudes, since we are not interested in them, we want only the secular dynamics.

A common way to do this is to expand the Hamiltonian following small parameters, i.e. the eccentricity, the inclination… But not here! You cannot do this since the eccentricity of the Planet Nine (0.6) and its inclination are not supposed to be small. So, the authors average the Hamiltonian numerically. This permits them to keep the whole secular dynamics due to the eccentricity and the inclination.

Once they did this, they looked for equilibriums, which would be preferential dynamical states for the TNOs. They also detected chaotic zones in the phase space, i.e. ranges of orbital elements, for which the trajectory of the TNOs would be difficult to predict, and thus potentially unstable. They detected these zones in plotting so-called Poincaré sections, which give a picture of the trajectories in a two-dimensional plane that reduces the number of degrees-of-freedom.

Results

And the authors find that the two Kozai-Lidov mechanisms, i.e. the one due to Neptune, and the one due to the Planet Nine, conflict for a semimajor axis larger than 150 AU, where orbital flips become possible. The equilibriums due to Neptune would disappear beyond 200 AU, being submerged by chaos. However, other equilibriums appear.

For the future, I see two ways to better constrain the Planet Nine:

  1. observe it,
  2. discover more eTNOs, which would provide more accurate constraints.

Will Gaia be useful for that? Anyway, this is a very exciting quest. My advice: stay tuned!

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.

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.

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.

The dynamics of the Quasi-Satellites

Hi there! After reading this post, you will know all you need to know on the dynamics of quasi-satellites. This is the opportunity to present you On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited, by Alexandre Pousse, Philippe Robutel and Alain Vienne. This study has recently been published in Celestial Mechanics and Dynamical Astronomy.

The 1:1 mean-motion resonance at small eccentricity

(see also here)

Imagine a pretty simple case: the Sun, a planet with a keplerian motion around (remember: its orbit is a static ellipse), and a very small third body. So small that you can neglect its mass, i.e. it does not affect the motion of the Sun and the planet. You know that the planet has no orbital eccentricity, i.e. the static ellipse serving as an orbit is actually a circle, and that the third body (let us call it the particle) has none either. Moreover, we want the particle to orbit in the same plane than the planet, and to have the same revolution period around the Sun. These are many conditions.
Under these circumstances, mathematics (you can call that celestial mechanics) show us that, in the reference frame which is rotating with the planet, there are two stable equilibriums 60° ahead and astern the planet. These two points are called L4 and L5 respectively. But that does not mean that the particle is necessary there. It can have small oscillations, called librations around these points, the resulting orbits being called tadpole orbits. It is even possible to have orbits enshrouding L4 and L5, this results in large librations orbits, called horseshoe orbits.

All of these configurations are stable. But remember: the planet is much less massive than the Sun, the particle is massless, the orbits are planar and circular… Things become tougher when we relax one of these assumptions. And the authors assumed that the particle had a significant eccentricity.

At high eccentricities: Quasi-satellites

Usually, increasing the eccentricity destabilizes you. This is still true here, i.e. co-orbital orbits are less stable when eccentric. But increasing the eccentricity also affects the dynamical structure of your problem in such a way that other dynamical configurations may appear. And this is the case here: you have an equilibrium where your planet lies.

Ugh, what does that mean? If you are circular, then your particle is at the center of your planet… Nope, impossible. But wait a minute: if you oscillate around this position without being there… yes, that looks like a satellite of the planet. But a satellite is under the influence of the planet, not of the star… To be dominated by the star, you should be far enough from the planet.

I feel the picture is coming… yes, you have a particle on an eccentric orbit around the star, the planet being in the orbit. And from the star, this looks like a satellite. Funny, isn’t it? And such bodies exist in the Solar System.

Orbit of a quasi-satellite. It follows the planet, but orbits the star.
Orbit of a quasi-satellite. It follows the planet, but orbits the star.

Known quasi-satellites

Venus has one known quasi-satellite, 2002 VE68. This is a 0.4-km body, which has been discovered in 2002. Like Venus, it orbits the Sun in 225 days, but has an orbital eccentricity of 0.41, while the one of Venus is 0.007. It is thought to be a quasi-satellite of Venus since 7,000 years, and should leave this configuration in some 500 years.

The Earth currently has several known quasi-satellites, see the following table:

(277810) 2006 FV350.387.1°10,000 y2013 LX280.4550°40,000 y2014 OL3390.4610.2°1,000 y(469219) 2016 HO30.107.8°400 y

Known quasi-satellites of the Earth
Name Eccentricity Inclination Stability
(164207) 2004 GU9 0.14 13.6° 1,000 y

These bodies are all smaller than 500 meters. Because of their significant eccentricities, they might encounter a planet, which would then affect their orbits in such a way that the co-orbital resonance would be destabilized. However, significant inclinations limit the risk of encounters. Some bodies switch between quasi-satellite and horseshoe configurations.

Here are the known quasi-satellites of Jupiter:

Known quasi-satellites of Jupiter
Name Eccentricity Inclination Stability
2001 QQ199 0.43 42.5° > 12,000 y
2004 AE9 0.65 1.6° > 12,000 y
329P/LINEAR-Catalina 0.68 21.5° > 500 y
295P/LINEAR 0.61 21.1° > 2,000 y

329P/LINEAR-Catalina and 295P/LINEAR being comets.

Moreover, Saturn and Neptune both have a confirmed quasi-satellite. For Saturn, 2001 BL41 should leave this orbit in about 130 years. It has an eccentricity of 0.29 and an inclination of 12.5°. For Neptune, (309239) 2007 RW10 is in this state since about 12,500 years, and should stay in it for the same duration. It has an orbital eccentricity of 0.3, an inclination of 36°, and a diameter of 250 km.

Understanding the dynamics

Unveiling the dynamical/mathematical structure which makes the presence of quasi-satellites possible is the challenge accepted by the authors. And they succeeded. This is based on mathematical calculation, in which you write down the equations of the problem, you expand them to retain only what is relevant, in making sure that you do not skip something significant, and you manipulate what you have kept…

The averaging process

The first step is to write the Hamiltonian of the restricted planar 3-body problem, i.e. the total energy of a system constituted by the Sun, the planet, and the massless particle. The dynamics is described by so-called Hamiltonian variables, which allow interesting mathematical properties…
Then you expand and keep what you need. One of the pillars of this process is the averaging process. When things go easy, i.e. when your system is not chaotic, you can describe the dynamics of the system as a sum of sinusoidal contributions. This is straightforward to figure out if you remember that the motions of the planets are somehow periodic. Somehow means that these motions are not exactly sinusoidal, but close to it. So, you expand it in series, in which other sinusoids (harmonics) appear. And you are particularly interested in the one involving λ-λ’, i.e. the difference between the mean longitude of the planet and the particle. This makes sense since they are in the co-orbital configuration, that particular angle should librate with pretty small oscillations around a given value, which is 60° for tadpole orbits, 180° for horseshoes, and 0° for quasi-satellites. Beside this, you have many small oscillations, in which you are not interested. Usually you can drop them in truncating your series, but actually you just average them, since they average to 0. This is why you can drop them.
To expand in series, you should do it among a small parameter, which is usually the eccentricity. This means that your orbit looks pretty like a circle, and the other terms of the series represent the difference with the circle. But here there is a problem: to get quasi-satellite orbits, your eccentricity should be large enough, which makes the analytical calculation tougher. In particular, it is difficult to guarantee their convergence. The authors by-passed this problem in making numerical averaging, i.e. they computed numerically the integrals of the variables of the motion over an orbital period.

Once they have done this, they get a simplified system, based on one degree-of-freedom only. This is a pair of action-angle variables, which will characterize your quasi-satellite orbit. This study also requires to identify the equilibriums of the system, i.e. to identify the existing stable orbits.

Perspectives

So, this study is full of mathematical calculations, aiming at revisiting this problem. The authors mention as possible perspective the study of resonances between the planets, which disturb the system, and the proper frequency of the quasi-satellite orbit. This is the oscillating frequency of the angle characterizing the orbit, and if it is equal to a frequency already present in the system, it could have an even more interesting dynamics, e.g. transit between different states (quasi-satellite / horsehoe,…).

To know more…

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