Chaotic diffusion of small bodies in the Solar System
Kleomenis Tsiganis ([email protected]) and Alessandro Morbidelli
Observatoire de la Côte d’ Azur - Nice, CNRS, B.P. 4229, 06304 Nice Cedex 4, France
Abstract
Near Earth Asteroids (NEAs) are small bodies that orbit
around the Sun, in the vicinity of the Earth. Since the orbits of NEAs are chaotic, catastrophic collisions with our
planet can occur. Most NEAs originate from the main asteroid belt. Resonances, between the orbital frequencies
of an asteroid and those of the major planets, are responsible for generating chaotic motion and transporting asteroids
towards the near-Earth space. Assessing the long-term effects of different types of resonance on the replenishment
of the NEAs population is the main goal of our research
project. In this paper we focus on one class of resonances:
medium/high-order mean motion resonances with Jupiter.
1. INTRODUCTION
During the last 20 years, our perception of the dynamical
evolution of our Solar System changed drastically; from a
well-tuned stable system, envisioned by Newton and Laplace,
to a dynamically vivid chaotic world sculpted by collisions,
migration, and ejection of bodies. It is now widely accepted
that chaotic, rather than regular, motion prevails in the Solar System. The planets themselves follow weakly chaotic
orbits, as was recently shown [1-2]. This complicated dynamical behavior sets important constraints on our theories
of formation and long-term evolution of planetary systems,
as well as on the habitability of terrestrial-like planets.
Our planetary system consists of much more bodies
than just the major planets and their satellites. A huge
number of small bodies,asteroids or comets, exhibit highly
irregular motion, owing to the gravitational perturbations
exerted on their orbits by the major planets. Collisions
between the proto-Earth and water-rich small bodies may
be responsible for the delivery of water to our planet [3].
The Moon itself is thought to be the by-product of an early
collision between the Earth and a Martian-sized body; a
much felicitous incident, if we take into account that the
Moon prevents the Earth’s spin axis from suffering chaotic
episodes that would not have allowed our planet to develop
stable climate (or life...) [4].
Such violent events, with catastrophic consequences for
life, can still occur. The biggest threat comes from Near
Earth Asteroids (NEAs). These objects follow chaotic or
bits, whose dynamical lifetime is short (
years) [5]
compared to the age of the Solar System ( years),
but long enough to perform millions of revolutions in the
vicinity of the Earth. Given these short dynamical lifetimes, it is evident that NEAs must originate from another
region and their population is somehow sustained in a sort
of steady-state [6].
Figure 1: Distribution of main-belt asteroids in the plane. Several “gaps” are visible, at the location of the
main mean motion resonances. Mars-crossing objects are
located above the solid line.
Most asteroids in the Solar System are concentrated in a
big reservoir, known as the main asteroid belt. Their orbits
are (to a first approximation) elliptic and lie between those
of Mars and Jupiter, with semi-major axes in the range
(Fig. 1). The eccentricities, and
inclinations, ! , of their orbital planes (with respect to the
ecliptic) are generally small. Their orbits are perturbed
by the planets, mostly Jupiter which has by far the largest
#%$'&
mass ( "
Solar masses). Chaotic motion is born in the
vicinity of resonances, between the orbital frequencies of
an asteroid and the planets. Different types of resonance
occur, the main ones being (i) mean motion resonances
with Jupiter, (ii) secular resonances between the frequencies of precession of the asteroid orbit and those of the
planetary orbits, and (iii) three-body mean motion resonance between the orbital frequencies of an asteroid and
two planets (see [7] for a detailed description). We differentiate between strong (low-order mean motion or secular)
resonances and weak (high-order mean motion or threebody) resonances. Low-order resonances are responsible
for shaping the main belt [7-8] (see also Fig. 1). They can
increase the eccentricity of an asteroid’s orbit up to Marscrossing or even Earth-crossing values on a short time scale
(
(
years). Mars-crossers can in turn evolve to NEAs,
due to a complicated interplay between resonances and close
encounters with Mars, which drives them slowly towards
the Earth [9]. Weak resonances on the other hand generate a slow chaotic diffusion [10], whose long-term effects
on the production of NEAs, as well as on the spreading of
2. RESONANT MOTION
We study the dynamics of a massless asteroid (test-particle)
in the Newtonian gravitational field of the Sun and Jupiter.
We assume that Jupiter moves on a fixed elliptic orbit around
the Sun and that the particle moves in the plane defined
by Jupiter’s orbit. In a heliocentric reference frame, the
Hamiltonian of the elliptic restricted three-body problem
has the form
&!
(1)
where ( ) are the position vectors of the asteroid and
) the masses of the Sun and
Jupiter, respectively, (
Jupiter, respectively, the magnitude of the heliocentric
velocity vector of the asteroid and the gravitational constant. In celestial mechanics we use the osculating orbital
elements to characterize the perturbed elliptic orbits of asteroids: the semi-major axis, , eccentricity, and longitude of pericenter and " (the orientation angle of the ellipse). The position of the body on its orbit is given by the
mean longitude, # 1 . We use functions of the elements, e.g.
the modified Delaunay elements
1 In a 3-D configuration
space we also need the inclination of the orbital
plane, , and the longitude of the node,
$
%
0.3
8/3
0.2
e cos(w)
0.1
0
-0.1
-0.2
-0.3
0.515
0.516
0.517
0.518
0.519
0.52
0.521
0.522
0.523
0.524
0.482
0.484
0.486
0.488
0.49
a/a_j
0.3
3/1
0.2
0.1
e cos(w)
asteroid families [11], have yet to be demonstrated.
In this paper we concentrate on high-order mean motion resonances between an asteroid and Jupiter, which occur when the mean revolution frequency of an asteroid,
(called mean motion), and thatof
a rela
Jupiter,
, satisfy
tion of the form , where are integers
and is called the order of the resonance. These resonances
cover the whole range of semi-major axes. We limit our
study to
, since low-order three-body resonances are
more
numerous
and stronger than simple resonances with
#
[12]. Mean motion resonances exist even in the
simplest version of the three-body problem (Sun - Jupiter
- asteroid), where Jupiter is assumed to move on a circular
orbit. However, the eccentricity of Jupiter’s orbit has to be
taken into account. In this 2-D elliptic (2DE) or 3-D elliptic
(3DE) problem (depending on whether we study asteroids
that have orbits co-planar to Jupiter’s orbit or not), each resonance splits into a multiplet of near-by resonances which
overlap with each other and generate chaotic motion [7].
We note that the presence of other planets in the model can
have significant consequences for resonant dynamics. In a
first approximation one can incorporate the secular effects
by assuming Jupiter’s elements to be varying according to
a secular planetary theory (e.g. [13]). These variations will
force mean motion resonances to pulsate, by generating additional near-by resonant terms. We use the 2-D and 3-D
secularly precessing problems (2DP and 3DP) in our numerical studies described below.
In the following section we present some analytical and
numerical results, in the framework of the 2DE model. In
Section 3, we present the results of numerical experiments
in the framework of the 2DE, 3DE, 2DP and 3DP models.
A discussion on these results and our conclusions are given
in Section 4.
0
-0.1
-0.2
-0.3
0.472
0.474
0.476
0.478
0.48
a/a_j
Figure 2: The representative plane of the 8/3 (top) and 3/1
(bottom) resonances. The thick “V”-shaped
(
curves
'
show
the maximal width of the resonance for &
) .
Several level curves of are shown in each case.
1 2"
#
+*
+3
-, .0/
, . /
(2)
which are canonical variables and transform the Hamiltonian to
.
/ .54 6 # 1 7* 83:98# 8* 1 83 (3)
* As usual, primed elements refer to Jupiter.
system of
; and The
units is chosen such that ,
are all equal
to 1. In
these units, the revolution period
of Jupiter is equal
to <
. . The ratio of
) and its mean motion is
>=% ?
Jupiter’s
mass
to
the
total
mass
of
the
system
is
"
. / .
and
@C.BESince
D8FHG Jupiter
@KBLD8FHG on a fixed
A
CI is
assumed
I
to
Jmove
ellipse, "
,#
and .
Using canonical perturbation theory we can construct
a simpler averaged Hamiltonian, suitable for studying motion in the vicinity
4 of a resonance [7]. We start by taking
the expansion of (see [14]). The lowest-degree term for
each combination of angles takes the form
M
6N >OCP
ORQ
@CBFS T /
WV
T /
# # ; 1 U 1
(4)
for the 2DE problem, or
M
6N O P O QX>YZP[X Y\Q2]
@CBF[S T[/ T U / 1 U 1 ; /_^ ; ^ V
]
#
#
(5)
M
for the 3DE problem,
6N F is
of the semi
= where
D a = function
major axis), N
, and X
!
. The permissible
combinations
obey the d’Alembert
T which
/ of angles
T[/ are those
/ rules, T[i.e.
(for
case,
/ T / A and / the
3-D
even). When
# # " the
asteroid is
= resonance with Jupiter. Kepler’s
in the 3rd law
/
gives
the
nominal
location
of
the
resonance,
i.e.
S = U V & . / &
Y
O
. In the
[ 2DE problem, there appear
combinations for every order resonance, which are
all nearly-resonant when " vary much slower
O Y and
than all other combinations of # , # . Thus, we can aver# and keep only the resonant terms
age over the fast angle
4
in the expansion. also contains secular terms, which do
not depend on the mean longitudes. We keep only the leading secular terms, of degree and respectively. Thus,
the averaged Hamiltonian takes the form
. /
. . , C@ BF 1 1 * 3
3
. Y M $ Y
3
]
T
V
@CBFKS ]
# 8# 1 1 1 (6)
M
= where , and are functions of . For resonant orbits, the semi-major axis has small-amplitude oscillations
about O Y . Thus, apart from the Keplerian part, the semimajor axis is assumed to be constant, when changing from
the orbital
elements to the canonical variables, while ,
M
and s are substituted by their numerical values. The averaged Hamiltonian may be further simplified, by applying a
series of canonical transformations (see also [15]), leading
to the final form
/ . Y M @KBEF
T ' (7)
the new variables being
. /
6 * , O Y * "
'
!
'
# # ; "
(8)
where is the opposite of the free longitude
of perihelion,
/2
= ,
" is the free eccentricity (see [14]),
$#
. O Y
M
= and the s are functions of and . The free
elements are a set of action-angle variables for the secular
problem, i.e. a model where
4 only the 2nd-degree secular
terms of the expansion of are retained. Then, is the
'
first approximation of the frequency of the perihelion, .
Note
that the Hamiltonian is similar
. to that of a pendulum
('' ) coupled to a slow (i.e. ) harmonic oscillator
( ), in a non-linear fashion.
The unperturbed location, , the width % , and the
T
'
frequency, & T , of small-amplitude
oscilla= resonant
tions of each resonance of the multiplet, are given by
Figure 3: Surface of section for the 8/3 resonance. The
borders of the narrow chaotic zone are well reproduced by
the trace of the separatrix (thick curve) of the modulated
pendulum.
T '
/
(%
(. M
*) ++ - ,/.
0
+ P
+
+
+ (& 21 4/ 3 (9)
and
is the value of
where at the
nominal resonance location.
Thus,
the
separation
between
6 = /
the resonances is 5 . The amount of chaos
generated in the vicinity of the resonant multiplet
is con = 5 . We caltrolled by the mutual overlap
ratios,
i.e.
%
M
culated the values of the s and (at O7 Y ), using
a code written by M. Sidlichovsky and kindly provided to
us by D. Nesvorný. Our results show that, for all resonant
multiplets studied (
), the widths of the resonances
'
are much larger than their mutual separations and 9
.
8:8
Therefore, the resonances are lying almost on top of each
other, the dynamics of the mean motion resonant multiplet
becoming similar to those of a slowly modulated pendulum
[7]. We may treat this problem in the adiabatic
' approximation, where the slow degree of freedom ( ) is “frozen”
and the characteristics of the fast' pendulum
( ' ) depend
parametrically on the values of ( ). In this approximation the separatrix of the pendulum (i.e. the curve that separates librations from circulations of the resonant angle, )
expands and contracts at a rate approximately equal to ,
causing orbits in the separatrix-swept zone to change their
oscillation mode erratically (from libration to circulation),
thus becoming chaotic.
We calculated the maximum width of the separatrix (i.e.
at the stable equilibrium of the resonance) and constructed
the portrait of each resonance in the so-called “representative
@CBF all' orbits
plane” [16], i.e. the plane to which almost
of '
come arbitrarily close. This is the ( ' ) plane
;
for
and ) , and the resonant angle
being
equal to or ) , depending on whether is odd or even,
respectively. Figure (2) shows the results for the 8/3 and
3/1 resonances. On the same
we have superimposed
; plane
several level curves of
. Note that, depending on ,
each level curve may have one or more branches. It is par3
Figure 4: Surface of section for the 3/1 resonance. The
homoclinic orbit crosses the separatrix of the pendulum and
the chaotic zone extends to Mars-crossing values of .
ticularly interesting that, for resonances with
# , there
exist level curves which are closed (see Fig. 2). Even if
such a level curve intersects the separatrix, chaotic motion
will be restricted to small eccentricities, due to energy conservation. This means that low-eccentricity chaotic orbits
cannot access the high-eccentricity regions and are therefore confined at low values of . In the non-averaged problem, chaotic orbits cannot be eternally confined. However,
the jumps between different energy levels are controlled by
the magnitude of the remainder that is disregarded when
constructing the Hamiltonian (6). Apart from resonances
which are close to other, lower-order, resonances, the effects of the remainder should be miniscule for times comparable to the age of the solar system.
The equations of motion resulting from Eq. (6) (for
different resonances) were numerically integrated, using a
2nd-order symplectic implicit scheme [17].
, Surfaces
of
@KBEF'
section
for
the
8/3
and
3/1
resonances
(
, F D2'
' H) ) are shown in Figs. (3)-,
, at (4). Regular orbits appear as smooth curves on a surface
of section, while chaotic orbits appear as a cloud of points.
We have also superimposed the trace of the separatrix of the
pendulum, as calculated
' in the adiabatic approximation, for
different values of ( ). The 8/3 and 3/1 resonances are
representative of the two types of mean motion resonance
that exist in the main belt. In the 8/3 case, a narrow chaotic
zone at moderate values of is found. Chaotic orbits are confined in this narrow zone and low-eccentricity
orbits cannot reach the high-eccentricity region. The borders of the chaotic zone are almost tangent to the analytically calculated separatrix.
In the 3/1 case, however, the topology of the surface of
section is very different. Low-order resonances can force
the frequency of perihelion to become
This leads to
' zero.
a corotation resonance, when , which appears
as a set of fixed points on the surface of section.
In Fig.
' (4)
the fixed points are located at ' " ? ,
(
,
stable) and
,
(
" ) , unstable). In the
integrable approximation there exists a homoclinic curve,
composed of two knots which join smoothly at the unstable point. Each knot encircles one of the two stable islands
(like the ones shown in Fig. 4). Chaotic motion is generated in the vicinity of the homoclinic orbit. If the homoclinic orbit intersects the separatrix of the pendulum, the
two chaotic regions merge and large-scale chaos sets in. We
note though that the adiabatic approximation cannot well
reproduce the inner borders of the chaotic zone. Note that
now almost circular orbits can random-walk to the higheccentricity region. In [16] it was shown that asteroids in
the low-order mean motion resonances, associated with the
Kirkwood gaps, can escape from the main belt because of
this mechanism.
It is of course necessary to check now (i) the behavior
of orbits in the non-averaged 2-D elliptic problem and (ii)
how the above results may change by including additional
perturbations in our model, such as the third spatial dimension and the secular perturbations of Jupiter’s orbit due to
its interaction with Saturn.
3. ADDITIONAL PERTURBATIONS
We numerically integrated the orbits of a carefully selected
sample of fictitious asteroids, within all four models described above. All integrations were performed with a mixed
variable symplectic integrator [18], as it is implemented in
the SWIFT package [19].
3.1. Numerical experiments
The initial conditions were selected, by considering particles initially placed in the vicinity of several mean motion
resonances
resonances up
of the 2DE model. We studied
# )
in
the
central
belt
(
to order
for the outer belt (# # ); 22
and
resonances are studied in total. For each resonance we integrated
orbits of 60 particles, 30 with free eccentricity
the
and 30 with " # . We set " " ) , so
"
that the resonance’s width is at maximum. The mean
lon
gitude was selected so that the critical angle " corresponded to the stable equilibrium. Finally, the initial value
of was varied, with respect to the nominal resonance location, within a range that agrees with our calculations for
the width of the resonance. A short term ( yrs) integration was performed, in order to select 10 chaotic orbits
in each resonance for a 1 Gyrs integration. The selection
was based on the behavior of the critical angle. For highorder inner-belt resonances it was not always easy to find
chaotic orbits; the width of the chaotic domain can be very
small. In such cases, slowly circulating or librating (but
with large libration amplitude) orbits were selected. After
selecting our sample of 220 fictitious asteroids we verified,
by means of Lyapunov exponent2 estimates, that more than
of our selected particles indeed follow chaotic orbits.
The orbits of these 220 fictitious asteroids were first
#
integrated, for a time corresponding to
years, in the
framework of the
2DE model. The
orbital
elements
of Jupiter
% AU, E?
(present epoch),
were
set
to
I "
and #
(at
). The same particles were
2 The maximal Lyapunov exponent is the mean rate of exponential divergence of near-by orbits, in a chaotic domain.
6
10
Model
2DE
2DP
3DE
3DP
5
TL (yrs)
10
4
10
2-D E
2-D P
3-D E
3-D P
2
2.0
MC ( )
2
6
2
6
JC ( )
7
21
19
16
JC ( )
36
51
88
85
Table 1: The percentage of particles in each final state (MC,
SG or JC). Columns 2-4 are for inner-belt and column 5 for
outer-belt particles. The numbers are given with respect
to the total number of particles in the inner (140) or outer
belt (80), respectively. The complement (not shown) corresponds to orbits that are not strongly excited.
3
10
10
SG ( )
1
14
22
2.5
3.0
3.5
4.0
a (AU)
0.6
initial
maximum
0.5 Earth-crossers
0.4
e
Figure 5: The Lyapunov time of the test-particles’ orbits
in all models. For most particles, the value of < is already dictated by the 2DE perturbations. As Jupiter is
approached, < drops significantly, i.e. the orbits become
more chaotic. Particles above the solid line most probably
follow regular orbits.
0.3
Mars-crossers
7/3
0.2
integrated in the framework of the 2DP model, the frequencies and amplitudes of Jupiter’s precession taken from
[20].
Then, the particles were given an inclination of !
with
respect to Jupiter’s orbital plane and the integration was repeated, both in the 3DE and 3DP models. In all integrations
the particles were stopped if they became Jupiter-crossers
(JCs) or Sun-grazers (SGs). From the particles surviving
the 1 Gyrs run, we recorded those that reached perihelion
distances smaller than 1.5 AU, i.e. became Mars-crossers
(MCs).
0.1
0.0
3/1
4/1
2.00
2.25
2/1
2.50
5/2
2.75
3.00
3.25
3.50
3.75
a (AU)
3.2. Results
Figure 6: The initial (open circles) and maximum values
(filled circles) of the time-averaged elements of the
surviving particles. Note the 7/2 particles, which become
Earth-crossers. The rest of the inner-belt particles are not
excited enough to cross the orbit of Mars (within 1 Gyr).
Our results show that the degree of stochasticity of the orbits is already determined by the least sophisticated model
(2DE). The value of the Lyapunov exponent does not change
significantly, as additional perturbations are taken into account (Fig. 5). The global transport properties, however,
are significantly modified. The results of all the runs are
summarized in Table 1.
In the 2DE model, the inner-belt particles that reach
planet-crossing eccentricities within the integration timespan are those that start from the vicinity of the lowestorder resonances, which are associated to the Kirkwood
gaps. Note that most particles end up encountering Jupiter,
after spending a considerable amount of time as MCs. The
rest of inner-belt resonances do not contribute to the Marscrossing population. In the outer belt, the large fraction
of escapes can be attributed to the overlap between neigh boring mean motion resonances, for "
. For the
remaining particles the numerically averaged elements (as
computed using a running-window averaging) change by
very small amounts. These numerical results confirm what
was suggested by our analytical approach, concerning the
existence of chaotic orbits semi-confined in high-order resonances, for times comparable to the age of the Solar System.
In what concerns the outer belt, the only way asteroids
can escape is by encountering Jupiter. The percentage of
JCs ranges from #
to ?E? , depending on the model.
It is easy to note that including the secular precession of
Jupiter enhances chaotic transport in 2-D space, but it is
the 3rd spatial degree of freedom that greatly increases the
percentage of JCs. Only 1-2 particles seem to follow stable orbits, so we expect the rest of the particles to escape
as well, in the frame of (3DE) and (3DP). For this population, the escape time is comparable to the age of the Solar
System.
For the inner-belt resonances, the percentage of escap (2DE) to (3DP). Chaotic
ing particles goes from
diffusion is again much more effective in 3-D space. Two
snapshots of the evolution of the surviving inner-belt particles in the (3DP) model are shown in Fig. (6). Each particle
I is projected on the ( ) plane of averaged elements at
Myrs (termed “initial”) and at the moment when the
eccentricity is at maximum. In this model the (MC)-(SG)
end state becomes the most probable ( ? ). High-order
resonances, like the 7/2 at
AU,
drive asteroids
I can
to the Mars-crossing region within 8 Myrs and even
AU). However,
lead to the Earth-crossing region (
for AU, the 1 Gyr time-scale is short to produce
Mars-crossers.
The fact that the 3rd spatial degree of freedom seems to
be more important for large-scale transport than the secular
precession of Jupiter’s orbit may seem surprising. However, we remind the reader that the expansion of the disturbing function has resonant harmonics whose strength is
proportional to
V whose strength is proF D Y , as well
S F asD terms
or
portional to ! Y or ! Y (for
F D even values of
= , respectively). For ) all
! (i.e. !
harmonics have comparable strength. Thus, when going to
a 3-D space we superimpose harmonics of equal strength
that couple the ! degrees of freedom. This leads to an
effective transport mechanism for small- and small-! orbits.
4. CONCLUSIONS
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We presented the results of a study of chaotic diffusion of
asteroids, initially placed in mean motion resonances with
Jupiter. Analytical results were given, in the framework
of the 2-D elliptic three-body problem. Numerical results
on the long-term evolution of small-eccentricity chaotic orbits were also presented. Based on these results we can
draw some important conclusions, concerning the mechanisms that generate chaotic diffusion inside mean motion
resonances and the long-term effects of this process.
If the secular precession of Jupiter is not taken into account, an important fraction of chaotic orbits with small
eccentricities is “quasi-confined” in the resonances. This
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constant-energy surfaces and (ii) the absence of corotation
resonances for high-order resonant multiplets. However,
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orbits to the Mars-crossing limit. When the secular precession of Jupiter is added in the model, chaotic diffusion is
enhanced. However, for small eccentricities, the inclination terms still dictate the motion.
In our most sophisticated model (3DP) the final distribution of fictitious asteroids shows that (i) in the outer belt,
) could
only a small fraction of resonant orbits (
survive for times comparable to the age of the Solar Sys
tem, and (ii) of the resonant orbits reach the terrestrial planets’ region, the planet-crossers coming mainly
from low-order resonances. The high-order resonances do
not seem to be so important for the production of NEAs.
However, a comprehensive analysis of the diffusion process and an estimate of the diffusion rate is still required, in
order to explain the spreading of asteroid families and the
evolution of dust particles produced in the family-forming
events. We expect that diffusion will be enhanced if we include in our models the short-term variations of Jupiter’s
orbit due to Saturn. The evolution of the present work towards more sophisticated models and more extensive simulations is currently under way.
5. ACKNOWLEDGEMENTS
The work of K. Tsiganis is supported by an EC Marie Curie
Individual Fellowship (contract N HPMF-CT-2002-01972).
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