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…

Chaos I

Today we return to celestial mechanics, and I’d like to discuss the background to a paper I’ve written with my Ph.D. supervisor Mark Wyatt that has just been accepted to the Monthly Notices of the Royal Astronomical Society. In it we investigated one way in which a planet can destabilise nearby bodies in the same planetary system, and the implications of this for estimating the masses of planets in extra-Solar systems. Before describing our work I need to say a little about both the astronomical and mathematical background.

Astronomy

It is clear in our Solar System that planets clear asteroids and comets from orbits that are too close to them. For example, there are no stable populations of bodies (with the exception of the Trojans) between the orbits of Jupiter and Neptune, while inside Jupiter’s orbit and beyond Neptune’s there are stable populations — the Asteroid and Kuiper Belts. While the proximate cause of bodies being removed from the unstable regions is by coming very close to a planet and being scattered onto a very different orbit, the region of space over which this is effective is smaller than the cleared region. Some other dynamical mechanism is at work to move some particles onto orbits where they will encounter a planet, but without moving others.

The Asteroid Belt, its outer edge truncated by Jupiter. Image credit: Wikipedia

Similar clearings are seen in the debris discs around other stars. These debris discs are made of the dust formed in collisions between asteroids or comets, and there are many that have been imaged at a variety of wavelengths (see here for a gallery). Many are seen to have holes in the centre; a good example is that around Fomalhaut, which has a very sharp inner edge at around 130AU. Fomalhaut also hosts a planet which has been detected with the Hubble Space Telescope, as seen in the Figure below. The planet’s existence and general properties had been predicted by Alice Quillen (2007), who calculated what the nature of the planet must be in order to account for the shape of the disc edge. The planet was discovered in 2008 by Paul Kalas et al., at a location in very good agreement with the theoretical predictions. (This method of predicting an unknown planet based on the orbits of known bodies has a venerable tradition going back to the discovery of Neptune in the 19th Century.) Indeed, the mass of the planet as estimated by the amount of light received from it is very uncertain, and currently the best estimate of the planet’s mass is from the dynamical models (Chiang et al., 2009).

The planet and disc of Fomalhaut, as seen by HST. The star's light was blocked with a coronograph, allowing the much fainter planet and disc to be seen. The planet's orbit lies just interior to the disc, and orbital motion over a two-year period can be seen. Image credit: Kalas et al., Science, 322, 1345.

When material is destabilised and scattered by the planet, it may suffer one of three ultimate fates: collision with the star, collision with the planet, or ejection from the system. This leads to another scenario in which whether material can be destabilised is important: destabilised material colliding with the star has been invoked as an explanation for the unusual compositions of some White Dwarf atmospheres. White Dwarfs are the remnants of low-mass stars at the end of their lives, and their high density (around the mass of the Sun in a volume the size of the Earth) gives them a strong gravitational field at their surface. This gravitational field should act to separate the elements in the atmosphere, so instead of different species being mixed together, as is the case in Earth’s atmosphere, the heavier should sink to the bottom, as sand does in water. Hence, only the lightest elements, hydrogen or helium, should be seen. However, some White Dwarfs show spectroscopic evidence of “pollution” by heavier elements (called metals by astronomers), for which there must be an ongoing source to replace those that are sinking. A leading hypothesis is that they originate from destabilised asteroids or comets that have collided with the star.

We might expect orbits to be less stable around White Dwarfs than around their progenitors. This is becasue, during the giant phases of a star’s evolution that precede the white dwarf phase, the star can lose a significant quantity of its mass—around half, for a star such as the Sun. Therefore, the planets around White Dwarfs are more massive, relative to the star, than when they were around the progenitor. Since most dynamical effects depend on the ratio of planetary to stellar mass, these effects will become stronger as the star loses mass. Hence, unstable regions around planets would be expected to grow. This was something we explored in a paper earlier this year, finding that the amount of material scattered from typical planet–disc systems could broadly acccount for the amount of metal pollution in White Dwarf atmospheres of various ages. If this is the correct explanation, then it is telling us about the long-term fate of planets and asteroids as their stars age and die — including our own.

Mathematics

NB: See this post for definitions of orbital elements.

Although there are several ways by which one or more planets might destabilise other bodies, the one with the widest applicability is the overlap of mean motion resonances. These resonances occur when two bodies’ orbital periods are in a simple integer ratio, such as 2:1. From Kepler’s Law, P^3\propto a^2, this occurs at a ratio of semi-major axes of around 1.6. So if one planet is located at 1AU, a planet at 1.6AU would be at the 2:1 resonance. Resonances with Neptune in the Kuiper Belt are shown schematically here:

Resonances in the Kuiper Belt. Note that the resonances become more closely spaced the closer they are to Neptune. Image Credit: Wikipedia

Now, although the actual period ratio only occurs at one specific semi-major axis ratio, the resonance also affects nearby orbits. The elements of these orbits oscillate, with a maximum amplitude of oscillation that depends on the mass of the planet and the particular resonance being considered. In particular, the more massive the planet, the more powerful the resonance, and the greater the range of semi-major axes affected by it. If a resonance acts alone on an orbit, the result is a regular oscillation of the orbital elements.

However, since the resonances have a width, and bunch up more closely the closer you are to the planet, there comes a point where the resonances overlap. When this happens, the evolution of an orbit becomes chaotic and unpredictable—crudely, instead of being confined to one resonance, you can imagine the particle being passed amongst many since their regions of oscillation overlap. This overlap of resonances driving chaotic behaviour was described by Chirikov (1979) and applied to the stability of orbits in the Solar System’s Asteroid Belt by Wisdom (1980).

It is useful to have a simple formula to give the boundary of the chaotic region for any planet mass. This Wisdom derived assuming low particle eccentricities, finding \delta a = 1.3 a \mu^{2/7}, where \delta a is the width of the cleared chaotic zone and \mu is the ratio of planet mass to stellar mass. I.e., more massive planets clear out wider regions around their orbit, which is what one intuitively expects. This result is now used to estimate the mass of planets truncating debris discs, since if the planet and disc location are known, one can solve for \mu.

In our previous paper, we had used the Wisdom result to estimate the amount of material that would destabilised as the star loses mass (this changes \mu and so changes the width of the chaotic zone), and compared it to more accurate numerical results using both integrations of the full equations of motion, and a simpler numerical investigation using a dynamical map (described in Duncan, Quinn & Tremaine 1989). We considered a disc in which particles had moderate eccentricities, up to 0.1, since in our Solar System the Kuiper Belt bodies have eccentricities in this range. We found that the chaotic zone width increases with increasing eccentricity—this is also seen in Wisdom’s original paper, although Wisdom’s analytical result was only valid for low eccentricities. In our latest paper, we followed up these investigations, deriving a simple formula for the chaotic zone width at non-zero eccentricity.

In the next post, I’ll describe what exactly we did in our new paper, and the implications for the interactions of planets with planetesimal discs. Concisely, we found that if particles are on eccentric orbits, the width of the chaotic zone is no longer given by the Wisdom (1980) formula, and thus any attempt to estimate a planet mass in the way described above will give an erroneous result.