Solar System Formation & Dynamics

In Newtonian gravity, the two body problem is the hardest problem that can be solved exactly. Two Newtonian point masses in a bound orbit describe closed Keplerian ellipses, and are stable for all time.

For more complex N-body systems we would like to be able to assess their stability. There are various definitions of stability applicable to planetary systems, but a simple definition is that a planetary system is stable if there are never any close approaches between the planets (if there are close approaches, then usually collisions, ejections, or similarly violent rearrangements of the orbits follow soon after). As many famous mathematicians over the ages have found, rigorously proving that a complex planetary system like the Solar System is stable is extremely difficult, and it is also hard to even identify the precise dynamical mechanisms that lead to instability. Nonetheless, a few exact results are known for simple systems, and these allow us to develop heuristic understanding (backed up by numerical calculations) of the behavior of more complex systems.

The circular restricted three body problem is the simplest case of three body dynamics. We consider a star and a planet (of arbitrary mass) on circular obits, together with a “test particle” whose mass is so low as to be negligible for the dynamics (physically, something like an asteroid would qualify). In this setup, it is quite easy to derive the condition for the system to be Hill stable, which means that the test particle is guaranteed never to have a close approach to the planet for all time. If we consider, for example, test particles on initially circular exterior orbits, with,

,

then Hill stability is assured provided that the dimensionless measure of the separation,

.

A similar result holds for two planets, whose masses can be comparable to one another but which are both much smaller than that of the star. Stability requires,

.

Notice the dependence of the stability boundary on the planet masses to the one third power. This is the same dependence as the Hill radius, and it follows from essentially the same physics – gravitational perturbations between the planets are destabilizing and the strength of those perturbations is measured by the separation of the planets in units of their Hill radii.

These exact mathematical results apply only to systems of two planets. More complex planetary systems are invariably chaotic, and their stability or instability can only be assessed by means of numerical calculations. An example of the results of such calculations is show below. To make this plot, we placed three planets (of equal masses, but increasing from left to right through the panels) on initially circular obits, separated by a fixed number of Hill radii, and integrated their orbits numerically until the first close approaches occurred. We then plotted the median time scale to the first close approach as a function of the initial separation.

The details of the plot are fairly complex, but (especially for the low mass planets plotted in the left-most panel) the trends are simple. The time scale for instability to develop is a steep function of the initial separation of the planets, measured in Hill radii units. Roughly, the dependence is exponential in the initial separations. This has important implications. Even if we cannot prove that a multiple planet system is permanently stable, it may be very securely stable for practical purposes if the planets are well enough separated. This is the case for the Solar System, whose long term stability has been studied in great detail. Although the orbits of the planets in the inner Solar System are chaotic, and neighboring trajectories diverge on a surprisingly short time scale (just millions of years!), the probability of close approaches between planets is extremely small. Dynamically it appears possible that Mercury could be perturbed into a Venus-crossing orbit, but the chances of this happening during the remaining five billion years of the Sun’s lifetime is only around a percent.