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 L_{4} and L_{5}, 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.

###### Outline

The Trojan asteroids

These are dark bodies

Asymmetric populations

Numerical simulations

The Greek are more stable than the Trojans

Where are they now?

The study and its authors

## 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 L_{4} and L_{5} 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 L_{4} point (leading), and 2,439 at the L_{5} trailing point. These objects are named after characters of the Trojan War in the Iliad. L_{4} is populated by the Greeks, and L_{5} by the Trojans. There are actually two exceptions: (624) Hektor is in the Greek camp, and (617) Patroclus in the Trojan camp.

## 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 L_{4}. After that, it will reach the L_{5} 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 L_{4} point, and 2,439 of them at the L_{5} 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 L_{4} 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 L_{4} and 997 L_{5}, 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 (L_{4}), you are less likely to escape than if you are in the Trojan one (L_{5}). The rate of escape is 1.1 times greater at L_{5} than at L_{4}. But, remember the asymmetry in the populations: L_{4} is much more populated than L_{5}. 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 L_{4} 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:

- The webpage of Romina Paula di Sisto (in Spanish),
- the website of Ximena Soledad Ramos,
- and the webpage of Tabaré Gallardo.

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.