Tag Archives: numerical methods

Pluto-Charon is dynamically packed

Hi there! Today, we leave the comets for the system of Pluto-Charon. Of course, you know Pluto. Formerly the 9th planet of our Solar System, until 2006, it remains an object of interest. So interesting that it has been visited by the spacecraft New Horizons in July 2015. You know, the same spacecraft which gave us these amazing images of Ultima Thule (also known as 2014 MU69).
Anyway, we are not done with Pluto. It has a large satellite, Charon, which makes Pluto-Charon a binary object, i.e. Pluto and Charon orbit about a common barycenter, which is significantly outside of Pluto. And around this binary, you have (at least) 4 small satellites, which are Styx, Nix, Kerberos and Hydra. I say at least, because the authors of the study I present today address the following question: could there be more? I mean, if you add a satellite somewhere, will it survive? If no, then you can say that the system is dynamically packed. This the opportunity for me to present A Pluto-Charon sonata: The dynamical architecture of the circumbinary satellite system, by Scott J. Kenyon and Benjamin C. Bromley. This study has recently been published in The Astronomical Journal.

The binary Pluto-Charon

I guess you have already heard of the discovery of Pluto by Clyde Tombaugh in 1933 (see here). It appeared that Pluto had been observed at least 16 times before, the first of these precoveries dating back to 1909.
The launch of the spacecraft New Horizons in 2006 motivated the observations of the binary Pluto-Charon by the most efficient observing facilities, in particular the Hubble Space Telescope. This telescope permitted the discoveries of 4 moons of the binary: Nix and Hydra in 2005, Kerberos in 2011, and Styx in 2012. You can find some of their properties below.

Discovery Diameter Semimajor axis Orbital period Spin period
Pluto 1933 2376.6 km 39.48 AU 248 years 6.39 days
Charon 1978 1212 km 19591 km 6.39 days 6.39 days
Styx 2012 16x9x8 km 42656 km 20.16 days 3.24 days
Nix 2005 53x41x36 km 48694 km 24.85 days 1.83 d
Kerberos 2011 19x10x9 km 57783 km 32.17 days 5.31 days
Hydra 2005 65x45x25 km 64738 km 38.20 days 10.3 hours

As you can see, the binary Pluto-Charon is doubly synchronous, i.e. Pluto and Charon have the same spin (rotation) period, and Charon has that same orbital period around Pluto. It would be accurate to say that Pluto and Charon have both this orbital period around their common barycenter. It can be shown that this state corresponds to a dynamical equilibrium, which itself results from the dissipation of rotational and orbital energy by the tidal interaction between Pluto and Charon.

However, the four other moons are much smaller, and much further from Charon. They spin much faster than they orbit, which means that the tides were not efficient enough to despin them until synchronization. Hydra spins in hours, while the others ones, which are closer to the binary, spin in days. So, they may have despun a little after all, but not enough.

Hydra as seen from NASA’s New Horizons spacecraft. © NASA/JHUAPL/SwRI
Hydra as seen from NASA’s New Horizons spacecraft. © NASA/JHUAPL/SwRI

No additional moon has been discovered since, even by New Horizons. The authors wonder whether that would be possible or not. For that, they ran intensive numerical simulations.

Simulations with Orchestra

They disposed of the numerical code Orchestra, which they developed themselves. This code is composed of several modules, permitting

  • N-body simulations,
  • to simulating planetary formation, especially the growth of the accreting bodies.

For this specific study, the authors considered only the N-body simulations. For that, they added massless particles in the binary, i.e. these particles were perturbed by the gravitational action of Pluto, Charon, and their four small moons. The simulations were ran over several hundreds of Myr.

I would like the reader to be aware that the stability, i.e. survival, of such particles is not trivial at all. You can imagine that if you come too close to a satellite, then you might be ejected. But this is not the only possible cause for ejection.

In such a system, you have many mean motion resonances. Imagine, for instance, that you are a massless particle (happy to be massless, aren’t you? trust me, it is not that fun), and that you orbit around Pluto-Charon exactly twice faster than Hydra (this is just an example). Every two orbits, your closest distance with Hydra will be at the same place. This will result in cumulative effects of Hydra on you, and since you are massless, you are very sensitive to these effects (which are actually a gravitational perturbation). And the outcome is: you might be ejected. Let us see now the results of the simulations.

Probably nothing inside the orbit of Hydra

Yes, because of these resonances, most of the massless particles orbiting inner to Hydra are unstable. In fact, some of them may survive, but in specific locations: either inner to the orbit of Styx, which is the innermost of the small moons, or outside the orbit of Hydra, i.e. outside of the known boundaries of the binary. In-between, you may have some particles, which would be coorbital to the small moons. This phenomenon of 1:1 mean-motion resonances appears in several locations of the Solar System. For instance, Jupiter has its Trojan asteroids, with which it shares its orbit. This also happens among the satellites of Saturn. Why not around Pluto-Charon? Well, you have to see them to be convinced they exist. These simulations just give you a theoretical possibility, i.e. this is not impossible.
Anyway, the preferred locations for yet-undiscovered moons is outside the orbit of Hydra. The challenge would be to discover such objects. Inside, the system appears to be dynamically packed.

Could there be something outside?

The authors present a discussion on the future possibility to detect them. First, they mention the stellar occultations.
Imagine the system of Pluto-Charon gets aligned between a terrestrial observer and a distant star. Then you can hope that, if there is something which is still unknown in that system, then it may occultate the light of the star, at least to some terrestrial observers. Of course, this may vary on from where on Earth you observe. For such a discovery to happen, you must be very lucky. But remember that the rings of Chariklo and Haumea were discovered that way.

Another hope for discovery is in the future instruments. The authors mention the JWST (Jawes Webb Space Telescope), which should be launched in March 2021. A kind of upgrade to HST (Hubble), its primary having a diameter of 6.5 meters, instead of 2.4 for Hubble. Moreover, it will be more efficient in the infrared, but unable to observe in the ultraviolet.

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.

The Oort remembers

Hi there! You may know that in the Solar System, we have long period comets. These are comets which visit us, i.e. visit the vicinity of the Sun and the Earth, but on orbits which suggest that they will not come back before some centuries, maybe more. The Dutch astronomer Jan Oort hypothesized in 1950 that these comets originate from a hypothetical, I mean unobserved, cloud, which is now known as the Oort cloud. It is supposed to lie between 2,000 and 200,000 astronomical units (AU).
In the study we discuss today, The “memory” of the Oort cloud, by Marc Fouchard, Arika Higuchi, Takashi Ito and Lucie Maquet, the authors wonder how the original Oort cloud was like. For that, they investigate whether the present observations of the comets originating from it contain any information on its original shape. This study has recently been published in Astronomy and Astrophysics.

The Oort cloud

As I said, the Oort cloud as a reservoir for long-period comets had been suggested by Jan Oort in 1950. Actually, its existence had been hinted 18 years before, in 1932 by the Estonian astronomer Ernst Öpik, but he did not think that the small bodies constituting this cloud could eventually become observable comets, in the sense that they would have anyway orbited too far from the Sun, even at perihelion.

We now think that the Oort cloud consists of two parts: an inner and an outer cloud. The inner cloud would have the shape of a torus, limiting the inclination of its constituents. It would lie between 2,000 and 20,000 AU (remember: Neptune orbits at only 30 AU). However, the outer cloud, or isotropic cloud, would have a spherical distribution. It would lie between 20,000 and 50,000 AU, and be much less dense than the inner one.

The observable comets

The information we dispose of come from the orbits of observable comets. A comet is a small icy body, which presents a cometary activity, i.e. outgassing. This comes from the sublimation of the ice.
This activity is favored by the temperature, which is directly linked to the distance to the Sun. This is particularly striking for comets, which have significantly elongated (eccentric) orbits around the Sun. When an orbit is eccentric, you have significant variations of the distance between the Sun and the body, in other words, significant variations of the temperature, and consequently of the cometary activity.
Dynamically, a comet can be characterized by its orbital elements. The most interesting one is, in my opinion, the semimajor axis, which gives you the period (the time interval between two approaches of the comet to the Sun).
Some comets have periods smaller than 20 years, and are called Jupiter-family comets. From 20 to 200 years, you have the Halley-type comets (after the well-known comet 1P/Halley), and beyond that limit you have the long-period comets. These are the comets, which are of interest for us, i.e. they are supposed to originate from the Oort cloud.
In fact, there are comets which orbits are even longer than that… in the sense that these comets may never return. These are comets with very high orbital eccentricities (>0.99), they are almost parabolic… and some of them are even hyperbolic, i.e. they are not dynamically bound to the Sun. Those ones may come from an extrasolar system, but this is another story…

Anyway, we speak about the long-period comets. And the question is: what information do their orbits contain on the primordial Oort cloud?

Numerical simulations

To understand how this information is preserved, the authors ran simulations of the orbits of more than 200 million comets. These are fictitious comets, evolving under the influence of

  • planetary perturbations,
  • stellar passages,
  • the Galactic tide.

Planetary perturbations

Planetary perturbations are the gravitational actions of the four giant planets (Jupiter, Saturn, Uranus, and Neptune). They may have dramatic consequences in case of close encounter: the comet is such a small body with respect to a giant planet that it could have almost every orbit after the encounter. Some comets might even be destroyed (remember Shoemaker-Levy 9).

Stellar passages

These comets, initially in the proto-Oort cloud, orbit very far from the Sun. This means that they are only weakly dynamically bound to it, and potentially sensitive to perturbations from other stellar systems. In particular if one of them passes by. The authors considered this effect in adding random passing stars. The velocities of the stars measured by the astrometric satellite Gaia permit to constrain the most recent stellar passages, but far from all of them.

The Galactic tide

The Galactic tide is the deformation of our Milky Way under the gravitational influence of the other galaxies. Previous studies have shown that it has a significant influence on the Oort cloud. The gravitational force exerted by the Sun is there weak enough for the Galactic tide to be significant.

Galactic tide can actually be seen on images of galaxies, which are close enough. This results in galaxies with irregular shape.

Tidal interaction between two galaxies, seen by the Hubble Space Telescope.
Tidal interaction between two galaxies, seen by the Hubble Space Telescope.

Four classes of observable comets

Before presenting the way the authors addressed that question, I would like to mention that they considered 4 different sub-classes of these long-period, observable comets.

First, let us define an observable comet: an observable comet has a perihelion at less than 5 AU of the Sun. The perihelion is the point of the orbit, which is the closest to the Sun, and 5 AU roughly corresponds to the orbit of Jupiter. Among these observable comets, the authors called

  • jumpers the comets which perihelion was larger than 10 AU during the previous passage,
  • and creepers the other ones.

And among these jumpers and creepers, the authors identified the comets, prefixed KQ, which required the assistance of a close encounter with a giant planet (a planetary kick) to push them outward, making them then sensitive enough to the stellar passages and the galactic tide to be injected into the observable zone.
The letters K and Q come from the two guys who identified this phenomenon, i.e. Nathan Kaib and Thomas Quinn, in 2009.

So, the four classes of observable long-period comets that the authors distinguished are

  • the jumpers,
  • the KQ-jumpers,
  • the creepers,
  • the KQ-creepers.

The reason why they distinguished these four classes is that they have different behaviors. So, different outcomes regarding the dynamics may be expected.

Two models of cloud

So, the question is: when you start from a given proto-Oort cloud, how will the observable comets look like? I mean, how many of them will be observable? How will their perihelions be distributed? How inclined will they be?

And this depends (I should rather say: is assumed to depend) on the structure of your initial proto-Oort cloud. For that, the authors considered two models:

  • A disk-like distribution, in which the inclinations are limited to 20°,
  • an isotropic cloud, in which the comets may have any inclination. As such, it looks like the shell of an empty sphere.

And among these two models, the authors used several sets of initial conditions or their comets, in changing the distribution of orbital energy from one set to another.

Now, let us discuss the results.

The disk remembers

Unsurprisingly, the disc model results in 4 to 8 times more observable comets than the isotropic one. This should have been expected, since the giant planets have limited inclinations. So, you should have a limited inclination yourself to receive the assistance of a planet to become observable. Since it is not a sine qua non condition, you can have observable comets with high inclinations anyway, thanks to the Galactic tide and stellar passages.

Another outcome of the paper is that the KQ objects are preferably retrograde. This maximizes their odds to survive, i.e. not to be ejected from the Solar System, in being less sensitive to planetary perturbations. This is not an original result, since Kaib and Quinn already met this conclusion, but it always gives confidence to find a result, which was already known. It suggests that your study is right.

The new result is in the memory. The present study shows that, if you started from an isotropic disk, then stellar passages have wiped out its structure. However, the observable comets would keep from an initial disk (and here I quote the paper):

  • a concentration of comets along the ecliptic plane for semimajor axes smaller than 7,000 AU,
  • the typical wave structure of the Galactic tide.

Now, we should determine whether the initial Oort cloud was more like a disk, a more like a shell. This actually depends on the whole process of formation of the Solar System. Several scenarios compete, which means that we currently do not know. Anyway, this study suggests that counting the observable comets could give a clue on the nature of the original distribution (disk-like or shell-like), and if it is a disk, then we could be able to guess part of its structure.

The future can only bring us more information, thanks to the observational data of comets to be discovered.

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.

The fate of Jupiter’s Trojans

Hi there! Today we discuss about the Trojans of Jupiter. These are bodies which orbit on pretty the same orbit as Jupiter, i.e. at the same distance of the Sun, but 60° before or behind. These asteroids are located at the so-called Lagrange points L4 and L5, where the gravitational actions of the Sun and of Jupiter balance. As a consequence, these locations are pretty stable. I say “pretty” because, on the long term, i.e. millions of years, the bodies eventually leave this place. The study I present today, The dynamical evolution of escaped Jupiter Trojan asteroids, link to other minor body populations, by Romina P. Di Sisto, Ximena S. Ramos and Tabaré Gallardo, addresses the fate of these bodies once they have left the Lagrange points. This study made in Argentina and Uruguay has recently been published in Icarus.

The Trojan asteroids

Jupiter orbits the Sun at a distance of 5.2 AU (astronomical units), in 11.86 years. As the largest (and heaviest) planet in the Solar System, it is usually the main perturber. I mean, planetary objects orbit the Sun, they may be disturbed by other objects, and Jupiter is usually the first candidate for that.

As a result, it creates favored zones for the location of small bodies, in the sense that they are pretty stable. The Lagrange points L4 and L5 are among these zones, and they are indeed reservoirs of populations. At this time, the Minor Planet Center lists 7,039 Trojan asteroids, 4,600 of them at the L4 point (leading), and 2,439 at the L5 trailing point. These objects are named after characters of the Trojan War in the Iliad. L4 is populated by the Greeks, and L5 by the Trojans. There are actually two exceptions: (624) Hektor is in the Greek camp, and (617) Patroclus in the Trojan camp.

Location of the Lagrange points.
Location of the Lagrange points.

These are dark bodies

The best way to know the composition of a planetary body is to get there… which is very expensive and inconvenient for a wide survey. Actually a NASA space mission, Lucy, is scheduled to be launched in 2021 and will fly by the Greek asteroids (3548) Eurybates, (15094) Polymele, (11351) Leucus, and (21900) Orus in 2027 and 2028. So, at the leading Lagrange point L4. After that, it will reach the L5 point to explore the binary (617) Patroclus-Menetius in 2033. Very interesting, but not the most efficient strategy to have a global picture of the Trojan asteroids.

Fortunately, we can analyze the light reflected by these bodies. It consists in observing them from the Earth, and decompose the light following its different wavelengths. And it appears that they are pretty dark bodies, probably carbon-rich. Such compositions suggest that they have been formed in the outer Solar System.

Asymmetric populations

We currently know 4,600 Trojan asteroids at the L4 point, and 2,439 of them at the L5 one. This suggests a significant asymmetry between these two reservoirs. We must anyway be careful, since it could be an observational bias: if it is easier to observe something at the L4 point, then you discover more objects.

The current ratio between these two populations is 4,600/2,439 = 1.89, but correction from observational bias suggests a ratio of 1.4. Still an asymmetry.

Numerical simulations with EVORB

The authors investigated the fate of 2,972 of these Trojan asteroids, 1,975 L4 and 997 L5, in simulating their trajectories over 4.5 Gyr. I already told you about numerical integrations. They consist in constructing the trajectory of a planetary body from its initial conditions, i.e. where it is now, and the equations ruling its motion (here, the gravitational action of the surrounding body). The trajectory is then given at different times, which are separated by a time-step. If you want to know the location at a given time which is not one considered by the numerical integration, then you have to interpolate the trajectory, in using the closest times where your numerical scheme has computed it.

When you make such ambitious numerical integrations, you have to be very careful of the accuracy of your numerical scheme. Otherwise, you propagate and accumulate errors, which result in wrong predictions. For that, they used a dedicated integrator, named EVORB (I guess for something like ORBital EVolution), which switches between two schemes whether you have a close encounter or not.

As I say in previous articles like this one, a close encounter with a planet may dramatically alter the trajectory of a small body. And this is why it should be handled with care. Out of any close encounter, EVORB integrates the trajectory with a second-order leapfrog scheme. This is a symplectic one, i.e. optimized for preserving the whole energy of the system. This is critical in such a case, where no dissipative effect is considered. However, when a planet is encountered, the scheme uses a Bulirsch-Stoer one, which is much more accurate… but slower. Because you also have to combine efficiency with accuracy.

In all of these simulations, the authors considered the gravitational actions of the Sun and the planets from Venus to Neptune. Venus being the body with the smallest orbital period in this system, it rules the integration step. They authors fixed it to 7.3 days, which is 1/30 of the orbital period of Venus.

And these numerical simulations tell you the dynamical fate of these Trojans. Let us see the results!

The Greek are more stable than the Trojans

It appears that, when you are in the Greek camp (L4), you are less likely to escape than if you are in the Trojan one (L5). The rate of escape is 1.1 times greater at L5 than at L4. But, remember the asymmetry in the populations: L4 is much more populated than L5. The rates of escape combined with the overall populations make than there are more escapes from the Greek camp (18 per Myr) than from the Trojan one (14 per Myr).

Where are they now?

What do they become when they escape? They usually (90% of them) go in the outer Solar System, first they become Centaurs (asteroids inner to Neptune), and only fugitives from L4 may become Trans-Neptunian Objects. And then they become a small part of these populations, i.e. you cannot consider the Lagrange points of Jupiter to be reservoirs for the Centaurs and the TNOs. However, there are a little more important among the Jupiter-Family Comets and the Encke-type comets (in the inner Solar System). But once more, they cannot be considered as reservoirs for these populations. They just join them. And as pointed out a recent study, small bodies usually jumped from a dynamical family to another.

The study and its authors

You can find the study here. The authors made it freely available on arXiv, many thanks to them for sharing!

And now, the 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.

The Solar System is a mess: Using Big Data to clear it up

Hi there! When you look at our Solar System, you wonder how it came to be that way. I mean, it formed from a nebula, in which grain accreted to create the Sun, the planets,… and all these small bodies. Most of them have disappeared since the origin, they were either ejected, or accreted on a planet, or on the Sun… anyway still many of them remain. You have the Asteroid Main Belt, the Centaurs, the Kuiper Belt, the Oort cloud, the comets… Several studies have tried to determine connections between them, i.e. where does this comet come from? Was it originally a Centaur, a Kuiper Belt Object, something else? And how did it change its orbit? A close approach with a giant planet, maybe?
And to address this question, you simulate the trajectories… which is not straightforward to do. It is pretty classical to simulate a trajectory from given initial conditions, but to answer such a question, you need more.
You need more because you do not know how reliable are your initial conditions. Your comet was there that day… very well. How sure are you of that? You observe a position and a velocity, fine, but you have uncertainties on your measurements, don’t you?
Yes, you have. So, you simulate the trajectories of many comets, which initial conditions are consistent with your observations. That’s better. And you hope that the outcome of the trajectories (trajectories simulated backward, if you want to know the origin) will be pretty much the same, since the initial conditions are very close to each other…
But they are not! This is what we call sensitivity to the initial conditions. This often means chaos, but I do not want to detail this specific notion. But basically, when a comet swings by a giant planet, its trajectory is dramatically deviated. And the deviation is highly sensitive to the location of the comet. So sensitive that at some point, you lose the information given by your initial conditions. C’est la vie.
As a result, there are in the literature many studies presenting their simulations, and which conclusions are sometimes inconsistent with each other.
The study we discuss today, It’s Complicated: A Big Data Approach to Exploring Planetesimal Evolution in the Presence of Jovian Planets, by Kevin R. Grazier, Julie C. Castillo-Rogez, and Jonathan Horner, suggests another approach to clear up this mess. It considers that all of these possible trajectories constitute a reservoir of Big Data. This study has recently been published in The Astronomical Journal.

Architecture of the Solar System

You know the 8 planets of our Solar System, from the closest to the outermost one:

  • Mercury,
  • Venus,
  • Earth,
  • Mars,
  • Jupiter,
  • Saturn,
  • Uranus,
  • Neptune.

And these planets are accompanied by many small bodies, which constitute

  • the Near-Earth Asteroids, which orbit among the 4 terrestrial planets (from Mercury to Mars),
  • the Main Belt Asteroids, which orbit between Mars and Jupiter,
  • the Centaurs, which orbit between Jupiter and Neptune,
  • the Kuiper Belt, which extends between 30 and 50 AU (astronomical units) from the Sun. So, its inner limit is the orbit of Neptune,
  • the scattered disc, which extends to 150 AU from the Sun. These objects are highly inclined. Eris is the largest known of them.
  • the detached objects, like Sedna. This population is very poorly known, and we do not even know if it is truly a population, or just some objects,
  • the hypothetical Oort cloud, which could be as far as one light-year, or 50,000 AU.

Of course, this list is not exhaustive. For instance, I did not mention the comets, which could originate from any of those populations of small objects.

In this study, the authors limit themselves to the orbit of Neptune. They consider 3 populations of objects between the orbits of Jupiter and Saturn, between Saturn and Uranus, and between Uranus and Neptune. And the question is: how do these populations evolve, to the current state? For that, planetary encounters appear to be of crucial importance.

Planetary encounters

Imagine a small body flying by Jupiter. It approaches Jupiter so closely that it enters its sphere of influence, in which the gravity of Jupter dominates the one of the Sun. Virtually, the object orbits Jupiter, but usually this orbit cannot be stable, since the approach is too fast. Locally, its orbit around Jupiter is hyperbolic, and the object does not stay there. Jupiter ejects it, and you do not know where, because the direction of the ejection is highly sensitive to the velocity of the object during its approach. It also depends on the mass of Jupiter, but this mass is very well known. Sometimes, the action of Jupiter is so strong that it fragments the object, as it did for the comet Shoemaker-Levy 9, in July 1994. And you can have this kind of phenomenon for any of the giant planets of the Solar System.

This is how planetary encounters could move, disperse, eject,… entire populations in the Solar System.

The Big Data approach

With so many objects (the authors considered 3 ensembles of 10,000 test particles, the ensembles being the 3 zones between two consecutive giant planets) and so many potential planetary encounters (the trajectories were simulated over 100 Myr), you generate a database of planetary encounters… how to deal with that? This is where the Big Data approach enters the game.

The authors performed it into two stages. The first one consisted to determine close encounter statistics and correlations, for instance with changes of semimajor axis, i.e. how a planetary encounter displaces an object in the Solar System. And the second stage aimed at reconstituting the path of the particles.

And now, the results.

Random walk from one belt to another

It appears that the particles could easily move from one belt to another. Eventually, they can be ejected. As the authors say, the classification of a particle into a population or another is ephemeral. It depends on when you observe it. In other words, a small object you observe in the Solar System could have been formed almost anywhere else. Even in situ. Now let us talk about specific examples.

The origin of Ceres

For instance, Ceres. You know, this is the largest of the Main Belt Asteroids, and the first to have been discovered, in 1801. It has recently been the target of the mission Dawn, which completed in October 2018.

Ceres seen by Dawn. © NASA
Ceres seen by Dawn. © NASA

Ceres is rich in volatiles like ammonia and carbon dioxides, as are other asteroids like Hygeia. Hygeia is itself a large Main Belt Asteroid. Knowing the origin of Ceres could give you the origin of these volatiles… but they could have been partly accreted after the migration… You see, it is difficult to be 100% sure.

Ceres could have formed in situ, i.e. between Mars and Jupiter, but this study shows that it could have originated from much further in the Solar System, and migrated inward.

The origin of trapped satellites

Most of the main satellites of the giant planets are thought to have been formed with the planet, in the protoplanetary nebula.
But in some cases, you have satellites, which orbit far from the parent planet, on an irregular orbit, i.e. a significantly inclined and eccentric one. In such a case, the body has probably not been formed in situ, but has been trapped by the planet. Among them are Saturn’s Phoebe and Neptune’s Triton, which are large satellite. I have discussed the case of Triton here. The trapping of Triton probably ejected mid-sized satellites of Neptune, which are now lost.

Phoebe seen by <i>Cassini</i> in August 2017 © NASA/ESA/JPL/SSI
Phoebe seen by Cassini in August 2017 © NASA/ESA/JPL/SSI
Mosaic of Triton taken by Voyager 2 in 1989. © NASA
Mosaic of Triton taken by Voyager 2 in 1989. © NASA

Phoebe and Triton entered the sphere of influence of their parent planet, but did not leave it. And where did they come from?

It seems probable that Triton was a Trans-Neptunian Object (TNO) before. In that part of the Solar System, the velocities are pretty low, which facilitate the captures. However, several scenarios are possible for Phoebe. The study show that it could have originated from an inner or from an outer orbit, and have jumped to Saturn from close encounters with Jupiter / Uranus / Neptune.

Something frustrating with such a study, which goes back to the origins, is that you lose some information. As a consequence, you can only conclude by “it is possible that”, but you cannot be certain. You have to admit it.

A way to secure some probabilities is to cross the dynamics with the physical properties, i.e. if you see that element on that body, and if that element is thought to have formed there, then you can infer something on the body, and the authors discuss these possibilities as well. But once more, you cannot be 100% sure. How do you know that this element has been formed there? Well, from the dynamics… which is chaotic… And when you see an element at the surface of a planetary body, does it mean that it is rich in it, or just coated by it, which means it could have accreted after the migration?

You see, you cannot be certain…

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.

Weighing the Kuiper Belt

Hi there! Today, we are back to the Solar System, and more especially the Kuiper Belt. You know, all these small bodies, which orbit beyond Neptune. Instead of speaking of specific cases, like Pluto, Haumea or Sedna, we will see the Kuiper Belt as a whole.
The study I present, Mass of the Kuiper Belt, by E.V. Pitjeva and N.P. Pitjev, constrains its total mass with planetary ephemerides. This study has been recently published in Celestial Mechanics and Dynamical Astronomy.

The Kuiper Belt

I have presented the Kuiper Belt many times. These are objects, orbiting beyond the orbit of Neptune. This zone is named after the Dutch-American astronomer Gerard Kuiper, who hypothesized that it could have been a reservoir of comets, even if he thought that it would be almost clear. At that time, the only known Kepler Belt Object was Pluto. Now, more than 2,000 of them are known, and many more are probably to be discovered.
Most of these objects orbit between 30 and 50 AU (astronomical units) from the Sun.

This study wants to constrain the total mass of the Kuiper Belt, from the motion of the planets. For that, the authors built and used planetary ephemerides.

Planetary ephemerides

Planetary ephemerides give the location of the Solar System objects, especially the planets, at given dates. They have been of strategical importance during centuries for celestial navigation. Now, we still need them, for instance to identify potentially hazardous objects, to guide spacecraft, to detect new objects,…

I can cite 3 institutions, which provide ephemerides:

  • NASA’s JPL,
  • IMCCE, Paris Observatory, France,
  • Institute of Applied Astronomy, Russian Academy of Sciences.

JPL stands for Jet Propulsion Laboratory. It is located near Pasadena, CA, and is associated with the Californian Institute of Technology (CalTech). As part of NASA, it is associated with the American spacecraft.
The IMCCE, for Institute of Celestial Mechanics, is responsible for the French ephemerides. It has been founded in 1795 as the Bureau des Longitudes, in a context of rivalry between France and England. Its goal was then to assist France, to regain control of the seas.
And the Institute of Applied Astronomy, in Russia, is the place where this study has been conducted.

These 3 institutions provide their own ephemerides, i.e. solutions for the orbital motion of the planets, their satellites, the asteroids,… Now, let us see how to include the Kuiper Belt.

The Kuiper Belt as a ring

The orbital motion of planetary bodies come from the numerical integration of the gravitational equations, in which each body is perturbed by all the other ones… this makes many of them. So many that a common computer cannot handle that, when it comes to 2,000 of them. Moreover, there are probably many more Kuiper Belt Objects, which are not discovered yet, but which perturb the motion of the planets…

The authors by-passed this problem in modeling the Kuiper Belt as a ring. Not the whole Kuiper Belt actually. The 31 most massive of these objects are modeled as point masses, ans the remaining ones are embedded into a fictitious rotating ring, which mass perturbs the planets.

If you know the perturbation, you know the mass… Easy, isn’t it? Well, not that easy, actually…

As many data as possible

The authors maintain their ephemerides since many years, and each new version is enriched with new data. The current version, EPM2017, uses about 800,000 positional observations of planets and spacecraft, ranging from 1913 to 2015. Many of the observations of planets are Earth-based astrometric observations, while spacecraft observations include MESSENGER (mission to Mercury), Venus Express (to Venus), Cassini (to Saturn), and the Martian missions Viking-1 & 2, Pathfinder, Mars Global Surveyor, Odyssey, Mars Reconnaissance Orbiter, and Mars Express.

Very small objects like spacecraft are very sensitive to planetary perturbations, this is why their navigation data may be invaluable.

Observed and fitted parameters

Making ephemerides consists in fitting a dynamical model to data, i.e. observed positions. The dynamical model is mainly composed of the gravitational interactions between the planetary bodies, with some relativistic corrections (Einstein-Infeld-Hoffmann equations). These interactions use the masses of the objects as parameters.

When you want to fit the model to the data, you fit the initial conditions, i.e. the location of the objects at the beginning of the simulation, and some of the parameters. Why only some of them? It depends on how well you know them.

For instance, in this case, the mass of (1)Ceres is assumed to be accurately known, thanks to the Dawn mission (just finished, by the way). This means that fitting this mass would be counterproductive.

So, the authors have to make critical choices between what they fit and what they don’t, and also how they ponder the observations between each other.

A light Kuiper Belt

From formation models of the Solar System, the initial Kuiper Belt should be as massive as ten times our Earth. However,
fitting the ephemerides gives much smaller numbers. You can find below the outcomes of the previous studies and this last one, by the same team.

Year Kuiper Belt mass (in Earth mass)
2010 0.0258
2013 0.0263
2014 0.0197
2017 0.0228 ± 0.0046
2018 0.0197 ± 0.0035

As you can see

  • the current Kuiper Belt is by far much lighter than the original one. This means that this region of the Solar System has probably been depleted by the gravitational action of the main planets, only few objects remaining,
  • the numbers do not converge very fast, but they converge. In particular, each new measurement is consistent with the previous one, and the uncertainty tends to shrink. Slowly, but it shrinks.

This number of 0.02 Earth mass makes the Kuiper Belt about 2 orders of magnitudes (i.e. between 10 and 1,000) heavier than the Asteroid Main Belt, but some 3 orders of magnitude lighter than the proposed Planet Nine.

The Planet Nine would have a limited influence

You remember the Planet Nine? It is a yet undiscovered body, which is supposed to exist anyway. It should orbit far behind the orbit of Neptune, should be as massive as 10 Earth masses, and would be responsible for the clustering of the pericentres of the Trans-Neptunian Objects (the Kuiper Belt), and for the obliquity of the Sun.

In this study, the authors benefited from the very accurate navigation data of the space mission Cassini, which orbited Saturn until September 2017. And for Cassini, the Kuiper Belt has a much stronger influence than the hypothetical Planet Nine. This makes me think that the author believe that using such ephemerides is not a good strategy for constraining the Planet Nine.

Actually, the planetologists looking for the Planet Nine focus on the individual trajectories of the Kuiper Belt Objects, because these are the most sensitive to it.

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.