Chaotic Dynamics of Near Earth Asteroids

Chaos Sensitivity of orbital evolution to a tiny change of the initial orbit is the defining property of chaos – your integrations show that the dynamics of Near Earth Asteroids (NEAs) is chaotic! This property is fundamental: the initial orbit is never known exactly (observation errors, limitations of the orbit determination algorithm, what else?) Note that chaos does not appear because the dynamics of point masses include some random element. To the contrary, the Newtonian equations describing their motion are deterministic (i.e., each orbit follows a well defined and unique trajectory)

How to quantify chaos? The Lyapunov exponent characterizes the rate of exponential divergence of nearby orbits It is formally defined as: Therefore, if d(t) = d 0 exp n(t-t 0 ), then λ = n, and λ = 0 otherwise The rate of divergence may depend on the orientation of the d 0 vector trajectory nearby trajectory d0d0 d(t)

Lyapunov Exponent The Lyapunov exponent can be computed numerically by following nearby trajectories with SwIFT, but problems may occur since d(t) must be infinitesimally small. (These problems can be avoided by using renormalization or variational equations.) Also, for a random choice of the initial orientation of the d 0 vector, the computed λ corresponds to the direction of the largest stretching rate, and is therefore called the Maximum Lyapunov Exponent or MLE. The inverse of the MLE, Lyapunov time T Lyap, tells us the time scale on which the orbits become unpredictable. For example, if the initial precision of orbits is 10 -4, macroscopic changes of order of unity will occur due to chaos on ~10 T Lyap since e 10 > 1. Even if your orbit were accurate at the (absurd) level of , you couldn’t predict where the body would be after 106 T Lyap.

How can we study chaos? There are several additional methods that can describe different aspects of chaos, such as the Surface of Section, Frequency Analysis, etc. To illustrate these methods, we use the fact that complex dynamical systems (such as, e.g., NEA dynamics) can often be reduced to an algebraic mapping that captures the behavior of the main variables Specifically, a 1-dimensional mapping can be constructed for any 2-dimensional Hamiltonian system by following intersections of a trajectory with some surface (called a Surface of Section or SOS) For example, near-resonant asteroidal dynamics can be reduced to the so-called Standard Map that describes behavior of action I (semi-major axis or eccentricity) and angle θ (proxy for resonant angle)

Standard Map – fast and easy to program! “Kicked Rotor” ↑ Kick parameter

Trojans of Jupiter (and Neptune) librate around L4 and L5. Resonant angle = λ - λ J

ε = 0 (regular motion) ε = 9 (strong chaos) Surfaces of Section for the Standard Map

No perturbation (ε = 0) Angle changes at a constant rate. Action stays constant. Like the two-body problem. Planet goes around Sun forever in the same orbit.

ε = 0 ε = 9

Weak perturbation One or two dominant frequencies of variation

Θ I

ε = 1.4

Regular orbit, ε= 1.4

ε = 1.4

Chaotic orbit, ε= 1.4

Frequency Analysis The Surface of Section (SOS) is useful because it can easily reveal the global behavior of a dynamical system and its dependence on parameters The disadvantage is that it can be difficult to reduce a complex dynamical system (e.g., NEA dynamics) to 2 degrees of freedom Frequency Analysis is also a global method but, unlike SOS, allows us to study chaos in complex systems directly by using N-body simulations For convenience, we will illustrate Frequency Analysis on the Standard Map. Examples of its application to asteroid (& Kuiper Belt) dynamics can be found in Robutel & Laskar (2001, Icarus 152, 4-28)

Real solar system Slightly different solar system

Frequency Analysis of the Standard Map ε = Red = fast diffusion

Frequency Analysis of the Solar System Semi-major Axis (AU) Eccentricity

Dynamics of Near Earth Asteroids NEA orbits are typically strongly chaotic, with the Lyapunov time, the inverse of the maximum Lyapunov exponent, being only ~ 100 years! Chaos appears due to a combination of frequent encounters of NEAs with the terrestrial planets and various orbital resonances As chaos typically causes gross changes in orbits in ~10T Lyap, the evolution of individual NEAs is unpredictable on time scales >1000 years (annoying but still better than a weather forecast) So what’s the point of following trajectories for > 1000 years???

Lyapunov Times of Near-Earth Asteroids Tancredi (1999) Lyapunov times are ~ the interval between “close” approaches to planets. Stadium billiards

Dynamics of Near Earth Asteroids What’s the point of following trajectories over > 1000 years??? The point is that the future evolution can be predicted statistically, i.e., by identifying all possible trajectories and finding their probabilities of occurrence To obtain such a statistical description, you used an N-body integrator and, in addition to the nominal orbit, you also followed clones that started from slightly different orbits All these trajectories are correct and equally probable. You may not have had enough time to obtain full statistical description. Projects may take many months of CPU time to complete.

Example for asteroid Itokawa Hayabusa spacecraft at Itokawa Hayabusa back to Earth (with sample?) Itokawa’s long axis is 535 meters long

a = AU, e = 0.280, i = o Discovered

Example for asteroid Itokawa Residence time distributionsIndividual clones This specific clone collides with the Earth at 3.5 My Itokawa may impact the Earth in the distant future, but we can’t say for sure. Michel & Yoshikawa 2005

Fates of 39 Itokawa Clones FateNumber of Clones Sun14 Mercury1 Venus14 Earth4 Mars0 Jupiter1 Saturn and beyond1 Survived 100 million years4

Dynamics of Near Earth Asteroids Orbits of NEAs are short-lived. Mean dynamical lifetime is ~4 My, i.e., ~1000x shorter than the age of the solar system! Why they are not long gone??? Observed NEAs must have been inserted into their present orbits only a few My ago. How? From where? What would happen if, instead of following the orbits into the future, we would follow them into the past??? Could such integrations be used to determine where NEAs come from? Unfortunately not! On timescales exceeding T Lyap, chaotic orbits tend to explore all available space in much the same way molecules would expand into empty space from a leaking oxygen tank

Dynamics of Near Earth Asteroids Backward integration over times significantly exceeding T Lyap is in fact equivalent to the forward integration over the same time interval So, how can we learn anything about the origin of NEAs??? First we need to identify potential sources… can you guess?

Main Asteroid Belt: ~1 million bodies with diameters >1 km between orbits of Mars and Jupiter

Main Asteroid Belt as a source of NEAs: objects slowly leak out of the main belt via resonances such as υ 6 or 3:1

Escape from the Main Belt via υ 6 Resonance

Fate of Asteroids that Escape Main Belt

Residence Time Distribution  Results for υ 6 resonance  Distribution shows where NEAs are statistically likely to spend their time  70% of NEAs from the υ 6 resonance reach orbits with a < 2 AU

Debiased Orbital and Size Distribution  There are ~ 970 NEAs with diameter > 1 km [H < 18, a < 7.4 AU]  44% of them have been found so far  60% come from the inner main belt (a < 2.5 AU). 10 km → 1 km

The model can be extended to obtain a statistical description of the NEA size distribution and calculate Earth impact rate as a function of the impact energy Impact energy Average time interval between impacts Average projectile size Observational completeness 1,000 MT63,000 y277 m18% 10,000 MT241,000 y597 m37% 100,000 MT935,000 y1,287 m50% 1,000,000 MT3,850,000 y2,774 m70% Implications for Earth Impact Hazard

2008 TC3 Discovered night of 2008 October 5/6 – Semi-major axis 1.31 AU – Perihelion distance 0.90 AU – Inclination 2.5 o Flash at 65 km altitude seen by satellite 20 hours later – Diameter 4 meters – Mass 83 tons = 8.3 × 10 4 kg – Hit Earth’s atmosphere at 12.4 km/s = 1.24 × 10 4 m/s – Kinetic energy = ½ mv 2 = 6.4 × J = 1500 tons of TNT

P Jenniskens et al. Nature 458, (2009) doi: /nature07920

Other Examples of Chaos in the Solar System Orbits of the inner planets – weak chaos Rotation of Saturn’s moon Hyperion – strong chaos Orbits of giant planets in the early Solar System – strong chaos Nice model animation at eyondthepage/ html

Laskar (1994)

Laskar (2008) Distribution of Planetary Eccentricities

Summary of Dynamics of NEAs  The orbital dynamics of NEAs is chaotic and in individual cases unpredictable on time scales >1000 years.  There are a number of useful tools that help us to deal with chaos such as Lyapunov exponents, surfaces of section, and frequency analysis.  Future evolution of a NEA can be described statistically by following a number of clones of the initial orbit. This may help us to determine things such as its dynamical lifetime, ultimate fate (impact with Sun or a planet, ejection from the Solar System), and Earth impact probability  The origin of NEAs can be studied by forward integrations of orbits from various sources (main belt, cometary reservoirs)  By calibrating the model to observations, we learn things about the relative contribution of sources, observational incompleteness, and the overall Earth impact hazard from NEAs