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 pointmasses include some random element. To the contrary, theNewtonian equations describing their motion are deterministic (i.e., each orbit follows a well defined and unique trajectory)

How to quantify chaos?

0

)(ln1limd

tdtt

The Lyapunov exponent characterizes the rate of exponential divergence of nearby orbits

It is formally defined as:

Therefore, if d(t) = d0 exp n(t-t0), then λ = n, and λ = 0 otherwise

The rate of divergence may depend on the orientation of the d0 vector

trajectory

nearby trajectory

d0d(t)

Lyapunov ExponentThe 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 d0 vector, the computed λ corresponds to the direction of the largest stretching rateand is therefore called the Maximum Lyapunov Exponent or MLE.

The inverse of the MLE, Lyapunov time TLyap, 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 TLyap since 10-4e10 > 1. Even if your orbit were accurate at the (absurd) level of 10-46, you couldn’t predict where the body would be after 106 TLyap.

How can we study chaos?

There are several additional methods that can describe different aspectsof 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 tothe 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”

H (I ,θ;t)I 2

4π+ ε cosθ ⋅δ2π (t)

dθδt∂H∂I → In In−1 + ε sinθn−1

nnn IHdtdI

+ 1θθθ

http://www.scholarpedia.org/article/Chirikov_standard_map

↑ Kick parameter

Trojans of Jupiter (and Neptune)librate around L4 and L5.

Resonant angle = λ - λJ

ε = 0 (regular motion) ε = 9 (strong chaos)

Surfaces of Section for the Standard Map

No perturbation (ε = 0)

θn = θn−1 + In−1

In In−1

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 perturbationOne 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 AnalysisThe 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 complexdynamical 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 differentsolar system

Frequency Analysis of the Standard Map

ε= 1 1

1210log

fff

0 2 4 6

0

2

4

6

Red = fast diffusion

Frequency Analysis of the Solar System

Semi-major Axis (AU)

Ecc

entr

icity

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 ~10TLyap, 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 followedclones 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 = 1.324 AU, e = 0.280, i = 1.622o

Discovered 1998http://commons.wikimedia.org/wiki/File:Itokawa-orbit.svg

Example for asteroid ItokawaResidence time distributionsIndividual clones

This specific clone collides with the Earth at 3.5 My

Itokawa may impact the Earth in thedistant future, but we can’t say for sure.

Michel & Yoshikawa 2005

Fates of 39 Itokawa Clones

Fate Number of Clones

Sun 14

Mercury 1

Venus 14

Earth 4

Mars 0

Jupiter 1

Saturn and beyond 1

Survived 100 million years 4

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 TLyap, chaotic orbitstend 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 TLyap 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 slowlyleak out of the main belt via resonances such as υ6 or 3:1

Escape from the Main Belt via υ6 Resonance

Fate of Asteroids that Escape Main Belt

Residence Time Distribution

Results for υ6 resonance

Distribution shows where NEAs are statistically likely to spend their time

70% of NEAs from the υ6 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 functionof the impact energy

Impact energy Average time interval between impacts

Average projectile size

Observational completeness

1,000 MT 63,000 y 277 m 18%

10,000 MT 241,000 y 597 m 37%

100,000 MT 935,000 y 1,287 m 50%

1,000,000 MT 3,850,000 y 2,774 m 70%

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 2.5o

• Flash at 65 km altitude seen by satellite 20 hours later– Diameter 4 meters– Mass 83 tons = 8.3 × 104 kg– Hit Earth’s atmosphere at 12.4 km/s = 1.24 × 104 m/s– Kinetic energy = ½ mv2 = 6.4 × 1012 J = 1500 tons of TNT

P Jenniskens et al. Nature 458, 485-488 (2009) doi:10.1038/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 http:/www.skyandtelescope.com/skytel/beyondthepage/8594717.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