Chaos II

Last week I described the background to my recent paper, which investigates how planets destabilise bodies on nearby orbits, and the implications for attempting to characterise planets by studying how they interact with debris discs. Here I’ll go into a little more detail on what we actually did.

We were attempting to find out under what conditions orbits of small bodies (henceforth “particles”–they can be comets or asteroids) near planets are unstable. Since instability can often take a long time to become manifest (as an extreme example, there is a change that the planets of the inner Solar System will become unstable in several billion years), we chose as a proxy whether or not the orbits were chaotic, which can be measured on shorter timescales; in our case, we used around 10,000 orbits. It is fairly easy to tell, by plotting the evolution of orbit elements as a function of time, whether or not an orbit is chaotic:

Regular and Chaotic orbits

The evolution of orbital eccentricity as a function of time (measured by the number of conjunctions with the planet) for particles on a regular orbit (top) and a chaotic orbit (bottom).

The argument then is that the chaotic orbits are unstable because they are free to wander through a large set of values, rather than being restricted in the way that non-chaotic, or regular, orbits are. They may then come close enough to the planet to collide with it, or experience a very strong perturbation that flings them out of the system or into the star. This is not strictly true, but is a good approximation.

There are basically three relevant parameters in this problem: the ratio of the masses of the planet and the star, the difference between the orbital semi-major axes of the planet and the particle, and the eccentricity of the particle’s orbit. The work of Wisdom (1980) showed that, as the planet mass is increased, orbits at greater semi-major axes become unstable. We then investigated the role of eccentricity. We followed the orbits of many thousands of particles, and produced plots such as these:

The chaotic zone

The chaotic zone, as a function of planet:star mass ratio mu, eccentricity e, and semi-major axis ratio epsilon=(a-a_pl)/a_pl.* In each plot, the planet lies to the left. White regions are populated with chaotic orbits; black with regular.

Here we show, for a fixed planet mass, where orbits with different semi-major axes (along the x-axis) and eccentricities (along the y-axis) are regular or chaotic. At each grid cell we followed the orbits of 100 particles; in white cells, all 100 were chaotic, while in black cells, none were. The planet lies to the left of the plot, so we see that orbits closer to the planet are chaotic whereas orbits further away are not. The vertical line on the plot shows the result for the extent of the chaotic zone derived by Wisdom in 1980. At low eccentricities, it underestimates the extent slightly, but on the whole does a good job. However, we see that for particles at higher eccentricities the chaotic zone extends considerably beyond this.

It is still, however, the same basic mechanism at work. Recall that the chaos here is driven by the overlap of mean motion resonances, and that these resonances have a width that grows with the eccentricity of the particle. We derived an improved condition for the overlap of these resonances, accounting for their increasing width, and the results are plotted as red lines. These match the edge of the chaotic zone at higher eccentricities extremely well, over 5 orders of magnitude of planet mass! The width of the chaotic zone changes from 1.3\mu^{2/7} to 1.8e^{1/5}\mu^{1/5}. It now includes the eccentricity dependence, and the mass dependence changes slightly. Our new result works for higher eccentricities while Wisdom’s is valid for lower.

So we see there are two regimes: for low eccentricities the chaotic zone width is given by the classical Wisdom result. However, for larger eccentricities, the chaotic zone can be significantly larger. These eccentricities need not be very large: the boundary separating the regimes is around 0.01 for a planet of Jupiter’s mass (\mu=10^{-3}). Since objects such as Pluto in the Kuiper Belt can have eccentricities significantly in excess of this, this is potentially important for understanding the interactions of such bodies with planets.

Disc edge profiles of HR8799

Calculated profiles of the inner edge of the debris disc of HR 8799. The density of particles is plotted as a function of semi-major axis. Solid black lines show the results using our new criterion for the edge of the chaotic zone. Red lines show the results from the Wisdom criterion, ignoring (vertical) or including (sloping) the smearing-out effects of eccentricity. Lines are shown for planet masses of 2, 4, 6, 8 and 10 times Jupiter's, increasing from left to right.

To show the importance of this effect, we took a real system. The young star HR 8799 is orbited by at least four planets and two planetesimal discs. The outermost planet is around 68AU from the star, while the disc’s inner edge is not exactly known but estimated at 90AU. We compared the expected shape of the edge of the debris disc, if it is made up of eccentric particles, using our new result (black lines in the above plot) to those using the Wisdom result (red lines; the sloping ones include the smearing-out effect of the particles’ eccentricities while the vertical do not). Since the planet mass is not known with certainty, we computed the profiles for a range of planet masses, from twice to ten times Jupiter’s. Notice how the new result requires a smaller planet mass to attain the same degree of clearing as the Wisdom result.

The masses of the planets are very uncertain, because they must be derived from theoretical models of their interior structure and cooling history. Independent limits on the masses, as are provided by dynamical studies such as this, are very valuable. The hope is that, if the disc edge and planetary separation are known, the mass of the outermost planet can be estimated from the size of the hole it has cleared, in the same way as in the Fomalhaut system. At the moment, this is not possible, because of the uncertainties both on the planet’s orbit and the location of the planetesimals. However, we revealed an additional complication: the size of the clearing for a given mass, or the planet’s mass for a given clearing, depends on the eccentricities of the particles in the disc. Only very detailed images of the disc’s edge can fully determine the planet’s mass. It is to be hoped that future observations will provide this.

*Does anyone know how to get LaTeX into figure captions? If I put the BBcode in it gets really messed up…

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