Transcript Frequency Analysis

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:
1 d (t )
  lim ln
t  t
d0
d0
d(t)
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
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 d0 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 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 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”
2
I
H (I,;t) 
  cos   2 (t)
4 ↑ Kick parameter
d H


 I n  I n 1   sin  n 1
dt
I
dI
H


 n   n 1  I n
dt

http://www.scholarpedia.org/article/Chirikov_standard_map
Trojans of Jupiter (and Neptune)
librate around L4 and L5.
Resonant angle = λ - λJ
Surfaces of Section for the Standard Map
= 0 (regular motion)
= 9 (strong chaos)
No perturbation (ε = 0)
I n  I n1
 n   n 1  I n 1
Action stays constant.
Angle changes at a constant rate.
Like the two-body problem.
Planet goes around Sun forever in the same orbit.
 =0
 =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
Red = fast diffusion
2
4
6
=1
0

log 10
f 2  f1
f1
0
2
4
6
Eccentricity
Frequency Analysis of the Solar System
Semi-major Axis (AU)
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
Lyapunov times are ~
the interval between
“close” approaches to
planets.
Stadium billiards
Tancredi (1999)
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
Itokawa’s long axis is 535 meters long
Hayabusa back to Earth (with sample?)
a = 1.324 AU, e = 0.280, i = 1.622o
Discovered 1998
http://commons.wikimedia.org/wiki/File:Itokawa-orbit.svg
Example for asteroid Itokawa
Individual clones
Residence time distributions
Itokawa may impact the Earth in the
distant future, but we can’t say for sure.
This specific clone collides
Michel & Yoshikawa 2005
with the Earth at 3.5 My
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 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 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 slowly
leak 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
Implications for Earth Impact Hazard
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 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%
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/b
eyondthepage/8594717.html
Laskar (1994)
Distribution of Planetary Eccentricities
Laskar (2008)
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