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KIRKWOOD GAPS AND INSTABILITY FOR THREE-BODY
PROBLEMS
NOTES OF A TALK GIVEN BY PROF. VADIM KALOSHIN
BY HONGYUAN ZHAN AND ZHIYING XU
Abstract. This is a summary of the presentation given by Prof. Vadim
Kaloshin at the MASS colloquium on October 17th, 2013. We would like to
thank Professor Kaloshin for providing the presentation slides on which this
summary is based.
1. Introduction
The talk introduced the study of the dynamics in the Newtonian Sun-JupiterAsteroid problem, which could be modeled as a three-body problem. A problem
related to the distribution of the asteroids was presented, which stated that there
are several gaps in the distribution of asteroids, the so called Kirkwood gaps. Relation between the instability of the asteroid motion mechanism and the presence
of Kirkwood gaps was explained. Further conjectures about the stochastic aspects
of the dynamics in near mean motion resonances were also discussed.
2. motivation
2.1. Laskar Simulation.
The key problem addressed in this talk concerned the following: Is the Solar
System stable? Numerical simulations by Sussman, Wisdom and Laskar [1] [2]
showed that collisions and ejections of inner planets are probable over the life span
of the Sun, and even over just a few million years. The Solar System, as well as
the extra-solar system, are believed to be unstable.
2.2. The Asteroid Belt.
The asteroid belt is located between the orbits of Mars and Jupiter, it consists of
millions of objects from asteroids of 950 kilometers in diameter to dust particles.
The mass of Jupiter is approximately 2960 masses of Mars, therefore, one could
neglect the influence of Mars on the asteroids. The influence of Saturn is also
omitted in the discussion. Hence, we study the N-body problem of the Sun, Jupiter
and asteroids.
Let m0 be the mass of the Sun, m1 be the mass of Jupiter, mi , 2 ≤ i ≤ N − 1 be the
masses of the asteroids, m0 >> m1 >> m2 , m3 , .....mN −1 , and let q0 , q1 , ..., qN −1 ∈
R2 be the positions of these objects. Then, the positions of the asteroids satisfy
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NOTES OF A TALK GIVEN BY PROF. VADIM KALOSHIN BY HONGYUAN ZHAN AND ZHIYING XU
the following differential equations:
X
q0 − qi
q1 − qi
qj − qi
q¨i = m0
+ m1
+
mj
.
3
3
||q0 − qi ||
||q1 − qi ||
||qj − qi ||3
j6=i,j>1
Letting mj , j > 1, tend to zero and N=3, we recover the Sun-Jupiter-asteroid three
body problem, where m0 and q0 represent the mass and the position of the sun,
m1 and q1 represent the mass and the position of Jupiter, and m2 and q2 for (one)
asteroid correspondingly.
2.3. The Two Body Problem and Kepler’s Law.
Figure 1. The orbit of an asteroid in a two body problem
In the remaining sections, we are going to use these notations: Let g denote the
arguement of perihelion in the orbit of the asteroid (the point in the orbit where it
is nearest to the sun), l denote the mean anomaly (a parameter relating position
and time for a body moving in a Kepler orbit), a denote the semi-major axis of the
orbit of the asteroid, and e denote its eccentricity. Let us take a look at the two
body problem for the moment, which Prof. Kaloshin talked about before he moved
on to the Sun-Jupiter-Asteroid three body problem. When we are neglecting the
influence of Jupiter, the Sun-asteroid two body problem satisfies: m0 q0 + mq = 0,
where q and m are the position and the mass of the asteroid. Further more, in
this two body problem, the Kepler’s Laws give the following: 1. Orbits are conic
sections. Bounded orbits are circular or elliptic. 2. Period of the elliptic asteroid
orbit is 2πa3/2 and independent of eccentricity. 3. Angular momentum of the
asteroid is constant.
Lemma 1. (Kepler) Semi-major axis a and eccentricity e are constants of motion,
ġ and l˙ = a−3/2 .
2.4. The Three Body Problem.
In this section, the influence of Jupiter is taken into account, which leads to the
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Sun-Jupiter-asteroid three body problem. We define µ = m1m+m
as the mass ratio
0
for Jupiter and the Sun. We consider the case 0 < µ << 1 (approximately 10−3
in reality). After normalizing the mass of the Sun and Jupiter as 1 − µ and µ, the
orbit of asteroid q(t) obeys:
q̈(t) = (1 − µ)
q0 (t) − q(t)
q1 (t) − q(t)
+µ
.
||q0 (t) − q(t)||3
||q1 (t) − q(t)||3
KIRKWOOD GAPS AND INSTABILITY FOR THREE-BODY PROBLEMS
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Lemma 2. (Lagrange-Laplace) Provided that the orbits are well sepererated and µ
is small, the semi-major axis of the asteroid satisfies
1
|a(t) − a(0)| . µ, ∀|t| . .
µ
Also, a theorem from KAM theory tells us that most orbits of nearly integrable
systems are quarsi-periodic, and combined with the following theorem by Arnold,
Herman and Fejoz it gives some information about the stability of asteroid orbit.
Theorem 1. (Arnold, Herman, Fejoz) Provided that unperturbed orbits of Jupiter
and asteroid are well seperated and have properly independent periods, for small µ
many orbits are quasiperiodic.
Figure 2. Asteroid Main-Belt Distribution
2.5. Kirkwood Gaps.
Figure 2 displays the distribution of asteroids in terms of their semi-major axes.
For most of semi-major axes in the asteroid belt, majority of the motions are stable
(yellow regions between the gaps). Yet, it could be observed from the picture that
there are several gaps in the distributions, which are called Kirkwood gaps. Such
gaps are believed to be caused by instability of the mechanism; the major work
associated with this talk by Prof. Kaloshin was to provide a rigorous mathematical
framework to examine the instability.
3. Explaination
In this section we summarize the heuristic explanation of the Kirkwood gaps given
by Prof. Kaloshin during the talk. Mean motion resonances occur when the ratio
between the period of Jupiter and the period of the asteroid is rational. In particular, Kirkwood gaps occur at the ratios of 3:1, 5:2 and 7:3. It is conjectured
and confirmed by numerical data in Wisdom’s work [1] that the eccentricity of the
asteroid placed in the Kirkwood gaps change by a magnitude of order one. The
semi-major axis of the asteroid a(t) is nearly constant while the eccentricity of the
asteroid e(t) grows. For example, consider the mean motion resonance 3:1 where
one of the Kirkwood gaps occurs. Then a3/2 ∼ 31 and so a ∼ 0.48. According to
the Lagrange-Laplace lemma, the semi-major axis will stay nearly constant for a
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NOTES OF A TALK GIVEN BY PROF. VADIM KALOSHIN BY HONGYUAN ZHAN AND ZHIYING XU
long time. The perihelion, a(t)(1 − e(t)) ∼ 0.48(1 − e(t)), gets closer and closer to
the origin. When e(t) reaches approximately 3/8, a close encounter between the
asteroid and Mars happens and eventually the asteroid is ejected from the asteroid
belt.
4. Main Result
The main results Prof. Kaloshin presented stated that for certain mean motion
resonances, there are unstable motions which lead to significant changes in the
eccentricities [3]. The results rely on numerical computations and concern with two
particular resonances, 1:7 and 3:1.
Main Result 1. (resonance 1:7) Consider the elliptic three-body problem with
mass ratio µ = 10−3 , and eccentricity of Jupiter e1 > 0. Assume it is in general
position. Then for e1 small enough, there exists a time T > 0 and a trajectory of
the asteroid whose eccentricity e(t) satisfies
e(0) < 0.48 and e(T ) > 0.67
while
|a(t) − 72/3 | ≤ 0.027 and t ∈ [0, T ].
Main Result 2. (resonance 3:1) Consider the elliptic three-body problem with
mass ratio µ = 10−3 , and eccentricity of Jupiter e1 > 0. Assume it is in general
position. Then for e1 small enough, there exists a time T > 0 and a trajectory of
the asteroid whose eccentricity e(t) satisfies
e(0) < 0.59 and e(T ) > 0.91
while
|a(t) − 3−2/3 | ≤ 0.149 and t ∈ [0, T ].
Figure 3. Transition from the instant ellipse of eccentricity e =
0.48 to instant ellipse of eccentricity e = 0.67. When the perihelion gets closer to the origin, the asteroid is kicked away by Mars.
However, this is just a heuristic representation of the orbit. Actual
diffusing orbit is much more complicated.
KIRKWOOD GAPS AND INSTABILITY FOR THREE-BODY PROBLEMS
5
Hence, from the two results above, there exist orbits of asteroids whose change in
eccentricity is above 0.3.
5. Conjecture on Stochasticity
Towards the end of the presentation, a conjecture on the stochasticity was posed.
Before stating the conjecture, let us look at a theorem from probability theory.
Theorem 2. (Central Limit Theorem) Suppose {X1 , X2 , ......} is a sequence of
independent and indentically distributed P
(i.i.d.) random variables with E(Xj ) = µ
n
and V ar(Xj ) = σ 2 < ∞. Let Sn = n1 k=1 Xk , then as n goes to infinity, the
√
random variables n(Sn − µ) converge in distribution to a normal N (0, σ 2 ),
Pn
√
i=1 Xi
n
− µ → N (0, σ 2 ).
n
Definition 1. A one-dimensional random walk is the random walk on aZ, starting
from 0 and at each step moving +a or −a with equal probability.
In the one-dimensional random walk, the steps can √
be treated as i.i.d. random
n
is in accordance with
variables, Xk = ±a with equal probability. Then nS
a
normal distribution N (0, 1) when n → ∞,
Z az
√
1
−x2
lim P rob( nSn ≤ az) = ψ(z), where ψ(z) = √
exp(
)dx.
n→∞
2
2π −∞
After introducing the one-dimensional random walk and the Central Limit Theorem, together with the numerical exploration, the stochastic behaviors of the orbits
of asteroids near the resonances were considered. The following conjecture was
posed:
Conjecture 1. Eccentricity of an asteroid inside of mean motion resonance sat3 −1
isfies Central Limit Theorem with a ∼ −cp,q ln(µe1 )µ− 2 e1 , where e1 is the eccentricity of Jupiter.
6. Future Works
The following open problems and plans of attacks in the future were brought up.
The main results shown in the talk were based on the situations in which mean
motion resonances are 1 : 7 and 3 : 1. The mechanism should apply to a substantially larger interval of eccentricities, e.g, to include [0.1, 0.375]. However, more
sophisticated numerics are required in order to achieve this. Moreover, estimating
the time of instability remains an open problem. Proving the stochastic conjecture
proposed in the last section would also be a future direction and there is work to
be done for the nonrestricted three-body problem.
References
[1] J. Wisdom, The origin of the Kirkwood Gaps: a mapping for asteroidal motion near 3/1
commensurability. Astronom. J., 87(3):577-593, 1982.
[2] J. Laskar, Large scale chaos in the solar system. Astron. Astrophysics., 287, 1994.
[3] J. Fejoz, M. Guardia, V. Kaloshin, P. Roldan, Kirkwood gaps and diffusion along mean motion
resonances in restricted plannar three-body problem. Preprint, University of Maryland, 2012.