Chaos in the Solar System
J.Laskar
Astronomie et Syst`emes Dynamiques,
CNRS-UMR8028, IMCCE-Observatoire de Paris,
77 Av Denfert-Rochereau, 75014 Paris
email: laskar@imcce.fr
June 5, 2003
Abstract
The implications of the chaotic evolution of the Solar System are briefly
reviewed, both for the orbital and rotational motion of the planets. In partic-
ular, Why Venus spins backward ? can be now understood while considering
the transition through a highly chaotic state during its history; chaotic state
that the Earth itself would have experienced in absence of the Moon, while
the large variations of Mars’ obliquity were probably at the origin of con-
siderable climate variations that may have left some geological traces on its
surface.
The limits of predictability for a precise solution of the planetary orbits
is an obstruction to the use of the astronomical insolation computations
as an absolute geological time scale through paleoclimates reconstructions
beyond a few tens of millions of years. On the opposite, as the paleoclimate
geological records increase in duration and quality, they may provide an
ultimate constraint for the dynamical model of the Solar system.
1
The chaotic motion of the Solar System
The orbital motion of the planets in the Solar System is chaotic. After Pluto
(Sussman and Wisdom, 1988), the evidence that the motion of the whole Solar
system is chaotic was established with the averaged equations of motion (Laskar,
1989, 1990), and conï¬rmed later on by direct numerical integration (Sussman
and Wisdom, 1992). For a review, see (Laskar, 1996), and for some historical
considerations (Laskar, 1992a). The most immediate expression of this chaotic
behavior is the exponential divergence of trajectories with close initial conditions.
Indeed, the distance of two planetary solutions, starting in the phase space with
a distance
d
(0) =
d
0
, evolves approximately as
d
(
T
)
≈
d
0
e
T /
5
or in a way which
is even closer to the true value,
d
(
T
)
≈
d
0
10
T /
10
(1)
1
Figure 1: Error in the eccentricity of the Earth resulting from an initial change of
10
8
n
radian in the perihelion of the Earth at the origin. After about
n
×
10 millions
of years, the exponential divergence of the orbits dominates, and the solutions are
no longer valid. Error in eccentricity is plotted versus time (in Ma). (Laskar, 1999)
where
T
is expressed in million of years. Thus, an initial error of 10
8
10
leads to
an indeterminacy of
≈
10
8
9
after 10 millions of years, but reaches the order of
1 after 100 millions of years. When speciï¬c planets are considered, these results
may vary, as the coupling between some of the planets is small, and the outer
planets (Jupiter, Saturn, Uranus, Neptune) are more regular than the inner ones
(Mercury, Venus, Earth, Mars). For the Earth the above formula is a very good
approximation of the divergence of the orbits (Fig.1) (Laskar, 1999a).
1.1
Secular resonances
In (Laskar, 1990, 1992b), it was demonstrated that this chaotic behavior arise from
multiple secular resonances in the inner solar system. In particular, the critical
2
argument associated to
(
s
4
−
s
3
)
−
2(
g
4
−
g
3
)
(2)
where
g
3
, g
4
are related to the precession of the perihelion of the Earth and Mars,
s
3
s
4
are related to the precession of the node of the same planets, is presently in in
a librational state, but can evolve in a rotational state, and even move to libration
in a new resonance, namely
(
s
4
−
s
3
)
−
(
g
4
−
g
3
) = 0
,
(3)
showing that a region of relatively strong instability extend over these two res-
onances. This result was questioned by Sussman and Wisdom (1992) which did
not recover such a large variation for the secular frequencies, but reached only
2(
s
4
−
s
3
)
−
3(
g
4
−
g
3
) = 0 instead of (
??
). Recently, in our group, we performed
precise new numerical integrations of the complete equations of motion of the Solar
System, including all 9 main planets, the Moon as a separate object, Earth and so-
lar oblateness, tidal dissipation in the Earth Moon system, and the effect of general
relativity. This new solution was also adjusted and compared to the JPL ephemeris
DE406 (Standish, 1998). Different numerical integrations were conducted, with in-
clusion or not of the oblateness of the Sun (
J
2
= 10
8
7
) (num2002 and num2001 in
Fig.2). The secular equations were also slightly readjusted and integrated over the
same time (sec2001 in Fig.2). In all cases, the resonant argument corresponding
to (
s
4
−
s
3
)
−
2(
g
4
−
g
3
) was in libration over 30 Myr. The transition to circulation
occurred at 30 Myr and about 40 Myr for sec2001 and num2002, while no transi-
tion was observed for num2001 (over 100 Myr). In both cases when the transition
occurs, the orbit reaches the resonance (
s
4
−
s
3
)
−
(
g
4
−
g
3
) = 0 before 100 Myr,
comforting the conclusions of (Laskar, 1990, 1992b). The other important resonant
argument ((
g
1
−
g
5
)
−
(
s
1
−
s
2
)) that is at the origin of the chaotic behavior of the
inner planets (Laskar, 1990, 1992) presents a similar behavior.
The chaotic motion of the solar system is mainly due to the interactions
of the secular resonances in the precessing motion of the inner planets, while the
secular motion of the outer planets is very regular. In order to evaluate the possible
chaotic diffusion of the planetary orbits over the age of the Solar system, numerous
numerical integrations have been conducted over several billions of years (Gyr),
extending even the age of the Solar system. From these integrations, it appears
that Venus and the Earth present some moderate chaotic diffusion, while the
lightest planets, Mars and Mercury, can experience very large changes in their
orbit eccentricity, allowing even for collision between Mercury and Venus in less
than 5 Gyr (Laskar, 1994). From these integrations, it appears that the chaotic
diffusion of the orbits in the earlier stages of the solar system formation could
provide some clues for the planetary distribution in semi major axis (Laskar, 1997,
2000).
3
0
10
Ï€
20
Ï€
30
Ï€
-100
-80
-60
-40
-20
0
−π
0
Ï€
-50
-40
-30
-20
-10
0
-80
Ï€
-60
Ï€
-40
Ï€
-20
Ï€
0
-100
-80
-60
-40
-20
0
time (Ma)
2(
g
4
−
g
3
)
−
(
s
4
−
s
3
)
(
g
4
−
g
3
)
−
(
s
4
−
s
3
)
sec2001
num2001
num2002
Figure 2: Evolution of he critical angle related to 2(
g
4
−
g
3
)
−
(
s
4
−
s
3
) (top
and middle) and (
g
4
−
g
3
)
−
(
s
4
−
s
3
) (bottom) for three different solutions for
the Solar system over 100 Myr. sec2001 is obtained with an averaged system,
while num2001 and num2002 are two direct numerical integrations with very close
dynamical models.
1.2
The outer planets
The secular motion of the outer planets is very regular. Nevertheless, there exists
also some instabilities among these planets, due to resonances of high order in
their orbital motion around the Sun (mean motion), although these instabilities
do not lead to signiï¬cative changes in the orbits (Murray and Holman, 1999).
A global view of the short period dynamics of the outer planets is given in
ï¬gure 3 using frequency analysis (Laskar, 1999b, Laskar and Robutel, 2001, Robu-
tel, 2003). In each plot, three of the planets initial conditions are ï¬xed, and the
initial conditions of the remaining one ( respectively Saturn, Uranus, or Neptune)
are changed in a large area of the phase space, spanning semi-major axis and
excentricity in a regular mesh around their current position. In each case, an inte-
gration is conducted over 1 Myr, and a stability index, obtained as the variation
4
-8
-6
-4
-2
0
20
25
30
35
40
0.0
0.2
0.4
10
15
20
25
0.0
0.2
0.4
6
8
10
12
14
16
18
0.0
0.2
0.4
saturn
Uranus
Neptune
Figure 3: Global dynamics of the giant planets. The abscissa and ordinate corre-
spond to the initial semi major axis and eccentricity of respectively Saturn (top),
Uranus (middle) and Neptune (bottom), while in each case the initial conditions
of the other planets are taken at their actual values. The current position of each
planet is given by a circle. The colors correspond to a stability index obtained by
frequency analysis on two consecutive time interval of 1 Myr, and the black regions
the mean motion resonances locations (Robutel, 2003).
of the mean motion frequency with time (Laskar, 1990, Laskar and Robutel, 2001)
is reported, from very stable (blue) to very unstable (red), while close encounter
or ejection is denoted by white dots. The resonances are identiï¬ed and reported
in black. It is thus very clear that although the motion of the outer planets are
essentially stable, they are numerous mean motion resonances and small chaotic
zones that in the vicinity of the present solar system solution (see also Michtchenko
and Ferraz-Mello, 2001).
2
Obliquity of the planets
The planetary perturbations induce also some effect on the tilt (obliquity) of their
equatorial plane over their orbital plane . For the Earth, as the precession frequency
of the axis is far from the main orbital secular frequencies of the precession motion
of the ecliptic, the obliquity presents only small variations of about 1.3 degrees
around the mean value of 23.3 degrees (Laskar
et al.
, 1993a). In the absence of
the Moon, the situation would be very different, as multiple resonances then occur
between the precession of the axis and the precession of the orbital plane, and a
5
very large chaotic zone would exist, ranging from 0 to about 85 degrees (Laskar
et al.
, 1993b). This is also what will happen as the Moon recedes from the Earth
due to tidal dissipation (N´
eron de Surgy and Laskar, 1997). This situation is very
similar for all inner planets (Laskar and Robutel, 1993, Laskar, 1995).
Figure 4: Two different solutions for Mars obliquity over 100 Myrs, obtained with
direct integration of the planetary orbits, and with initial conditions of the planet
spin within the uncertainty of the most recent determinations (Folkner
et al.
, 1997,
Laskar
et al.
, 2002)
2.1
Mars obliquity
Mars presents a very large chaotic zone for its obliquity ranging from 0 to more
than 60 degrees (Laskar and Robutel, 1993, see also Touma and Wisdom, 1993). On
Figure 4 are plotted two possible solutions for the past evolution of the obliquity
of Mars, obtained using our latest numerical integration for the orbital motion
of all the planets, and initial conditions and parameters within the uncertainty
of the best known values for Mars rotational parameters (Folkner
et al.
, 1997).
It is clear that the climatic history of the planet would be very different in the
two situations. It is presently particularly important to understand what are the
possible past evolutions of the obliquity (and thus climate) of Mars in the past, as
the recent Martian spacecrafts provide very detailed observations of the Martian
surface (Malin and Edgett, 2001), and give some accurate account of the weather
on the planet. In particular, the polar caps of Mars present crevasses that reveal the
layered features of the ice (Fig. 5). This layers are thought to be induced by climate
variations on Mars surface, and the succession of layers can indeed be related to
the insolation variations on its surface (Laskar
et al.
, 2002), although the possible
relation from insolation to surface feature on Mars remains very uncertain.
6
Figure 5: Mars North polar ice cap, and detailed view of a crevasse, showing the
layered nature of the ice. The picture was taken by the Mars Orbital Camera
(MOC) on april 13, 1999 (Malin and Edgett, 2001)
2.2
Venus ï¬nal states
In 1962, using radar measurements, the slow retrograde rotation of Venus was
discovered. Since, the understanding of this particular state becomes a challenge
as many uncertainties remain in the dissipative models of Venus’ rotation. Various
hypothesis were proposed for its evolution, aiming to search wether Venus was
born with a direct or retrograde rotation. The most favored scenario assumes that
its axis was actually tilted down during its past evolution as a result of core mantle
friction and atmospheric tides (for a detailed review, see Correia
et al.
, 2003 and
references therein). Nevertheless, this requires high values of the initial obliquity,
and it was proposed that Venus was strongly hit by massive bodies which would
have tilted it signiï¬cantly or started its rotation backward (Dones and Tremaine,
1993).
The discovery of a very large chaotic zone in the obliquity evolution of Venus
allowed new possible scenario for driving Venus to 180 degrees obliquity (Laskar
and Robutel, 1993, Yoder,1997), but some difficulties remained. Recently, we have
shown that due to the dissipative effects, there are only 4 possible ï¬nal states
for Venus’ rotation, and only 3 of them are really reachable. When the planetary
perturbations are added, most of the initial conditions lead to the two states
corresponding to the present conï¬guration of Venus, one with period
−
243
.
02
days and nearly 0 degree obliquity, and the other with opposite period and nearly
7
180 degree obliquity. We thus demonstrate that a large impact is not necessary
to have a satisfying scenario for the reverse rotation of Venus (Fig. 5) (Correia
and Laskar, 2001, 2003, Correia
et al.
, 2003), and that most initial conditions can
lead to the present reverse rotation. On the opposite, we demonstrate that the
present observed retrograde spin state of Venus can be attained by two different
processes that cannot be discriminated by the observation of the present rotational
state of Venus. In the ï¬rst scenario, the axis is tilted towards 180 degrees while
its rotation rate slows down, while in the second one, the axis is driven towards
0 degree obliquity and the rotation rate decreases, stops, and increases again in
the reverse direction, being accelerated by the atmospheric tides, until a ï¬nal
equilibrium value.
Figure 6: examples of the possible evolution of Venus spin axis during its history.
The precession frequency (in arcsec/years) is plotted versus obliquity (in degrees).
The initial obliquity is 1 degree, and initial period 3 days. The initial precession
frequency is about 16 arcsec/yr, but due to tidal dissipation and core-mantle fric-
tion, the planet slows down and the precession frequency decreases. The obliquity
then enters a zone of larger chaos (in grey) where the variations of the obliquity
increases, until the dissipations drives the obliquity out of the chaotic zone, for a
high obliquity. The various dissipative effects can then drive the axis towards 180
degrees (Correia and Laskar, 2003).
8
3
Earth paleoclimates and chaos
For Mars, the chaotic variations of the obliquity are large, and certainly induce
some dramatic effects on its climate. For example, it is probable that at large
obliquity (
>
45 degrees), the polar caps entirely sublimate, the ice been eventually
redisposed in the equatorial regions (Levrard, 2003). For the Earth, the situation
is very different. The variation of the obliquity is also chaotic, but mainly because
it is driven by the chaotic orbital motion of the planet that acts as a forcing
term on the obliquity. The variations of the obliquity and orbital parameters are
small, but they induce signiï¬cative changes of the insolation on Earth at a given
latitude, that are thought to be at the origin of large climatic variations in the
past (Hays
et al.
, 1976). Even more, the astronomical solution is used to provide
an absolute time scale for the geological paleoclimate records, over a few tens of
millions of years, when the effect of the chaotic behavior of the solution is not yet
sensitive (e.g. Schackleton
et al.
, 1990, 1999). The quality of the geological data is
now becoming sufficient to allow on the other hand to constrain some geophysical
parameters of the Earth that are not well known, by comparison of the computed
astronomical evolution of the insolation with the geological data (Lourens
et al.
,
2001, P¨
alike and Shackleton, 2001). As the age of good quality geological records,
obtained over long period of time, is now exceeding 30 to 40 millions of years, it
becomes interesting to search whether it would be possible to use these geological
data to trace back the chaotic diffusion of the solar system in the past.
In order to tackle such a challenging problem it is important, as the quality
of geological records decreases with age, to search in the solution some features
that would have an important implication on the observed data. Such factor could
actually be the resonant argument
(
s
4
−
s
3
)
−
2(
g
4
−
g
3
)
,
(4)
and its evolution with time. It would be indeed fascinating to be able to retrieve in
the geological data the transitions from libration to circulation of this argument,
as well as its eventual transition to the resonance
(
s
4
−
s
3
)
−
(
g
4
−
g
3
) = 0
.
(5)
The direct observation of the individual arguments related to
g
3
, g
4
, s
3
, s
4
is
certainly out of reach. But it may be possible to follow the evolution in time of
the differences
g
3
−
g
4
and
s
3
−
s
4
. Indeed,
s
3
−
s
4
appears as a beat of about 1.2
million of years in the solution of obliquity, as the result of the beat between the
p
+
s
3
and
p
+
s
4
components of the obliquity, where
p
is the precession frequency
of the axis (see Laskar, 1999a) (Fig. 7). In a similar way,
g
3
−
g
4
appears as a beat
with period of about 2.4 Myr in the climatic precession (Fig. 7). One understands
that because of the occurrence of these beats, the detection of the resonant state
in the geological data can be possible, as one has to search now for phenomena
of large amplitude in the geological signal. Indeed, in the newly collected data
9
Obliquity
p
+
s
3
−→
41 000 yrs
p
+
s
4
−→
39 600 yrs
s
3
−
s
4
−→
1
.
2 Ma
Ma
Ma
Climatic precession
p
+
g
3
−→
19 097 yrs
p
+
g
4
−→
18 900 yrs
g
3
−
g
4
−→
2
.
4 Ma
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
-20
-15
-10
-5
0
21.5
22
22.5
23
23.5
24
24.5
25
-20
-15
-10
-5
0
Figure 7: The main components of the obliquity of the Earth are given by two terms
of period close to 40 000 years (
p
+
s
3
and
p
+
s
4
) than induce a beat of about 1.2
Myr period. For the climatic precession (
e
sin
ω
, where
e
is the excentricity and
ω
the longitude of perihelion from the moving equinox), the main terms are of period
close to 20 000 years, and the beat
g
3
−
g
4
has a period of about 2.4 Myrs.
from Ocean Drilling Program Site 926, the modulation of 1.2 Myr of the obliquity
appears clearly in the spectral analysis of the paleoclimate record (Zachos
et al.
,
2001).
The search for the determination of these resonant angles in the geological
data is just starting, but we may expect that a careful analysis, and new data
spanning the last 60 to 100 Myr, will allow to determine the possible succession
of resonant states in the past, allowing, for example, to discriminate between the
solutions displayed in ï¬gure 2. Of particular importance would be the detection of
the ï¬rst transition from libration to circulation of the resonant argument (
s
4
−
s
3
)
−
2(
g
4
−
g
3
). This program, if completed, will provide some extreme constraint for the
gravitational model of the Solar System. Indeed, the observation of a characteristic
feature of the solution at 40 to 100 Myr in the past, because of the exponential
divergence of the solutions, will provide a constraint of 10
8
4
to 10
8
10
on the
10
dynamical model of the Solar System.
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