Hi! Today I will tell you about a study recently accepted for publication in Astronomy & Astrophysics, by Rodney Gomes and David Nesvorný, on the survival of the asteroids which precede and follow Neptune on its orbit.

The coorbital resonance

In the Solar System, the mean motion resonances are ubiquitous. When the orbital frequencies of two bodies are commensurate, interesting phenomena might happen: they could have a more stable orbit, or they could experience a permanent forcing which raise their eccentricity and / or inclination, and in some cases could result in an ejection. A resonance has particularly strong effects on a small body which orbit resonates with the one of a large planet. This is for instance how the giant planets shaped the asteroid belt.

Here, we deal with the coorbital resonance, which is a very specific and interesting case. This happens between two bodies which have on average the same orbital frequencies, and the perturbations associated result in some zones of stability. In particular, there are five equilibrium positions for the coorbital restricted 3-body problem, i.e. if we consider the Sun, a planet, here Neptune, and a small body. These equilibriums are known as Lagrangian points, and the most remarkable of them are denoted L₄ and L₅. They precede and follow the planet at an angular distance of 60°, and are stable equilibriums. As a consequence, they are likely to accumulate several small bodies, and this is verified by the observations, which have detected asteroids which coorbit with Jupiter, Uranus and Neptune.

At this time, 6,288 of these objects have been detected for Jupiter, 1 for Uranus, and 17 for Neptune.

The planetary migration

Since 2004 and the first version of the Nice model, the giant planets are assumed to have formed closer to the Sun than they are now, and have migrated to their current orbit. The reason for this migration is that they were form in a large proto-planetary disk, full of planetesimals which drove migration. The asteroids are some of these planetesimals. This raises the following question: could the coorbital (or Lagrangian) asteroids survive this migration?

Long-term numerical integrations

Addressing this problem requires long-term and intensive numerical simulations. The issue is this: you need to simulate the evolution of the Solar System over 4.5 Gyr. For that, you write down the gravity equations ruling the motion of the planets and the planetesimals (these are many objects… the authors considered 60,000 of them), and you propagate them numerically.

To propagate them, you start from a given position and velocity of each of your bodies (initial conditions), and the equations give you the time-derivative at this point. You then use it to extrapolate the trajectory in the time, and you reiterate…

Of course, this algorithm does not give exact results. To lower the error, you should take a small time-step, but a too small time-step requires more iterations, and at each iteration you add an error due to the internal accuracy of the computer. To make your life easier, numerical integrators have been developed to improve the accuracy for a given time-step. In this study, the authors use two very well-known tools, SWIFT and MERCURY, dedicated to the integration of the motion of the planets and asteroids.

In this paper

The authors show it is difficult to get Trojans of Neptune that survive over the lifetime of the Solar System. In a first numerical integration, they do get captures, but none of them survive. Then they consider planetesimals which are very close to the observed Trojan, and they get some captures.

Something interesting is that they show that the orbital inclination of these Trojans can be excited during the migration process. For that, the migration should be slow enough, i.e. over 150 Myr, while previous studies, which assumed a migration ten times faster, did not excite the inclinations up to observed values.

Some perspectives

Even if it is now accepted that the planets have migrated, several competing scenarios exist (Nice, Nice 2, Grand Tack,…) and some are probably to come, just because there are many ranges of initial conditions which are possible, many possible assumptions on the initial state of the proto-planetary nebula… and these scenarios should of course impact the capture of Trojans of the giant planets.

Links

• The study, Gomes R. & Nesvorný D., 2016, Neptune trojan formation during planetary instability and migration, Astronomy and Astrophysics, in press
• The web site of David Nesvorný
• The page of Rodney Gomes on ResearchGate
• The integrator SWIFT
• The Minor Planet Center, which provides an up-to-date catalog of the known asteroids