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The long-term evolution of planetary orbits
The problem:
A point mass is surrounded by N > 1 much smaller masses on nearly
circular, nearly coplanar orbits. Is the configuration stable over
very long times (up to 1012 orbits)?
Why is this interesting?
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one of the oldest and most influential problems in theoretical
physics (perturbation theory, Hamiltonian mechanics, nonlinear
dynamics, resonances, chaos theory, Laplacian determinism,
design of storage rings, etc.)
occupied Newton, Laplace, Lagrange, Gauss, Poincaré, Kolmogorov,
Arnold, Moser, etc.
what is the fate of the Earth?
where do asteroids and comets come from?
why are there so few planets?
calibration of geological timescale over the last 50 Myr
can we explain the properties of extrasolar planetary systems?
The problem:
A point mass is surrounded by N > 1 much smaller masses on nearly
circular, nearly coplanar orbits. Is the configuration stable over
very long times (up to 1012 orbits)?
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many famous mathematicians and physicists have attempted to find
analytic solutions (e.g. KAM theorem), with limited success
only feasible approach is numerical solution of equations of motion by
computer, but:
– needs up to 1014 timesteps so lots of CPU
– inherently serial, so little gain from massively parallel computers
– needs sophisticated algorithms to avoid buildup of errors
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symplectic integration algorithms
mixed-variable methods
warmup
symplectic correctors
optimal floating-point arithmetic
The solar system
d2 x
dt 2
i
= ¡ G
P
m j (x i ¡ x j )
N
j = 1 jx i ¡ x j j3
+ small correct ions
Known small corrections include:
• satellites ( < 10-7)
• general relativity (fractional effect < 10-8)
• solar quadrupole moment (< 10-10 even for Mercury)
• Galactic tidal forces (fractional effect < 10-13)
Unknown small corrections include:
• asteroids (< 10-9) and Kuiper belt (< 10-6 even for outermost planets)
•mass loss from Sun through radiation and solar wind, and drag of solar
wind on planetary magnetospheres (< 10-14)
1 AU = 1 astronomical unit
• passing stars (closest passage about 500 AU)
= Earth-Sun distance
Neptune orbits at 30 AU
Masses mj known to better than 10-9M
Initial conditions known to fractional accuracy better than 10-7
solar mass
The solar system
To a very good approximation, the solar system is an isolated,
conservative dynamical system described by a known set of
equations, with known initial conditions
All we have to do is integrate the equations of motion for ~1010
orbits (4.5109 yr backwards to formation, or 7109 yr forwards
to red-giant stage when Mercury and Venus are swallowed up)
Goal is quantitative accuracy (  1 radian) over 108 yr and
qualitative accuracy over 1010 yr
0 - 55 Myr
+4.5 Gyr
• an integration of
the solar system
(Sun + 9 planets)
for § 4.5£ 109 yr
= 4.5 Gyr
-55 – 0 Myr
-4.5 Gyr
• figures show
innermost four
planets
Ito & Tanikawa
(2002)
Ito & Tanikawa (2002)
Pluto’s peculiar orbit
Pluto has:
• the highest eccentricity of any planet (e = 0.250 )
• the highest inclination of any planet ( i = 17o )
• perihelion distance q = a(1 – e) = 29.6 AU that is smaller than
Neptune’s semimajor axis ( a = 30.1 AU )
How do they avoid colliding?
Pluto’s peculiar orbit
Orbital period of Pluto = 247.7 y
Orbital period of Neptune =
164.8 y
247.7/164.8 = 1.50 = 3/2
Resonance ensures that when
Pluto is at perihelion it is
approximately 90o away from
Neptune
Cohen & Hubbard (1965)
Pluto’s peculiar orbit
period ratio
semi-major axis of Pluto
eccentricity of Pluto
17o
inclination of Pluto
• early in the history of the solar system
there was debris (planetesimals) left over
between the planets
• ejection of this debris by Neptune caused
its orbit to migrate outwards
• if Pluto were initially in a low-eccentricity,
low-inclination orbit outside Neptune it is
inevitably captured into 3:2 resonance with
Neptune
• once Pluto is captured its eccentricity and
inclination grow as Neptune continues to
migrate outwards
• other objects may be captured in the
resonance as well
resonant angle
Malhotra (1993)
Kuiper belt
objects
Plutinos (3:2)
Centaurs
comets
Two kinds of dynamical system
Regular
• highly predictable, “wellbehaved”
• small differences in
position and velocity grow
linearly: x, v  t
• e.g., simple pendulum, all
problems in mechanics
textbooks, baseball, golf,
planetary orbits on short
timescales
Chaotic
• difficult to predict,
“erratic”
• small differences grow
exponentially at large
times: x, v  exp(t/tL)
where tL is Liapunov time
• appears regular on
timescales short compared
to Liapunov time  linear
growth on short times,
exponential growth on long
times
• e.g., roulette, dice, pinball,
billiards, weather
The orbit of every
planet in the solar
system is chaotic
(Laskar 1988,
Sussman & Wisdom
1988, 1992)
ln(distance)
saturated
factor of 1000
Jupiter
separation of
adjacent orbits
grows  exp(t / tL)
where Liapunov time
tL is 5-20 Myr 
factor of at least
10100 over lifetime
of solar system
300 million years
saturated
Hayes (astro-ph/0702179)
Integrators:
200 Myr
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double-precision (p=53 bits) 2nd order mixed-variable symplectic
(Wisdom-Holman) method with h=4 days and h=8 days
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double-precision (p=53 bits) 14th order multistep method with h=4
days
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extended-precision (p=80 bits) 27th order Taylor series method
with h=220 days
Chaos in the solar system
• the orbits of all the planets are chaotic with e-folding times
for growth of small changes (Liapunov times) of 5-20 Myr (i.e.
200-1000 e-folds in lifetime of solar system
• positions (orbital phases) of planets are not predictable on
timescales longer than 100 Myr – future of solar system over
longer times can only be predicted probabilistically
• the solar system is a poor example of a deterministic universe
• dominant chaotic motion for outer planets is in phase, not
shape (eccentricity and inclination) or size (semi-major axis)
• dominant chaotic motion for inner planets is in eccentricity
and inclination, not semi-major axis
Laskar (1994)
birth
death
Laskar (2008)
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1000 integrations of solar
system for 5 Gyr using
simplified but accurate
equations of motion
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all four inner planets exhibit
random walk in eccentricity
(more properly, in e cos ! and e
sin !)
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probability Mercury will reach e
= 0.6 in 5 Gyr is about 2%
1 curve per
0.25 Gyr
Causes of chaos
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orbits with 3 degrees of freedom have three actions Ji, three
conjugate angles µi, and three fundamental frequencies i.
in spherical potentials, 1=0. In Kepler potentials 1=2=0
all resonances involve perturbations to Hamiltonian of the form
©(Ji) cos (m1µ1+m2µ2+m3µ3-!t)
two types of resonance:
– secular resonances have m3=0. If ©» O(®) ¿ 1 the characteristic
oscillation frequency in the resonance is O(®1/2) but width of
resonance is O(1)
– mean-motion resonances have m30. Width of resonance is
O(®1/2) but resonances with different m1,m2 and same m3 overlap
(“fine-structure splitting).
resonance overlap leads to chaos
chaos arises from secular resonances in inner solar system and meanmotion resonances in outer solar system
chaos in outer solar system arises from a 3-body resonance with
critical argument  = 3(longitude of Jupiter) - 5(longitude of
Saturn) - 7(longitude of Uranus) (Murray & Holman 1999)
small changes in initial conditions can eliminate or enhance chaos
cannot predict lifetimes analytically
eccentricity of Mercury
time (Myr - not Gyr!)
Laskar (2008)
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change PPN parameter 2°-¯ from +1 to -2
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precession of Mercury / 2°-¯ so changes from +43 arcsec/100
yr to -86 arcsec/100 yr
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Mercury collides with Venus in < 4 Myr or < 0.1% of solar system
age
J
S
U
N
age of solar
system
Holman (1997)
Summary
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we can integrate the solar system for its lifetime
the solar system is not boring on long timescales
planet orbits are chaotic with e-folding times of 5-20 Myr
the orbits of the planets are not predictable over timescales
> 100 Myr
the shapes (eccentricities, inclinations) of the inner
planetary orbits execute a random walk
the phases of the outer planetary orbits execute a random
walk
“Is the solar system stable?” can only be answered
probabilistically
it is unlikely (2-¾) that any planets will be ejected or collide
before the Sun dies
most of the solar system is “full”, and it is likely that planets
have been lost from the solar system in the past
(261 planets)
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Given the star mass M (known from spectral type),
radial-velocity observations yield:
I ' 0
I ' 90±
orbital period P
semi-major axis a
combination of planet mass m &
inclination I, m sin I
orbit eccentricity e
• current accuracy ¢ v » 3 m/s in best observations
• Jupiter  v ~ 13 m/s, P = 11.9 yr – just detectable
• Earth  v ~ 0.1 m/s, P = 1 yr – not detectable yet
51 Peg:
m sin I = 0.46 MJ
P = 4.23 d
a = 0.05 AU
e = 0.01
1 MJ = 1 Jupiter mass =
0.001 M¯ = 318 M©
70 Vir:
m sin I = 7.44 MJ
P = 116.7 d
a = 0.48 AU
e = 0.40
What have we learned?
• 277 extrasolar planets known (March 2008)
• 261 from radial-velocity surveys
• 36 from transits
• 5 from timing
• 6 by microlensing
• 5 by imaging
(see http://exoplanet.eu)
What have we learned?
• planets are
remarkably
common (~ 5%
even with current
technology)
log10 Z/Z¯
• probability of
finding a planet /
Za, a' 1-2, where Z
= metallicity =
fraction of
elements heavier
than He
(Fischer & Valenti 2005)
What have we
learned?
• giant planets like
Jupiter and Saturn are
found at very small
orbital radii
• record holder is Gliese
876d: M = 0.018 MJupiter,
P = 1.94 days, a =
0.0208 AU
M V E Mars
Jupiter
• maximum radius set by
duration of surveys
(must observe 1-2
periods at least for
good orbit)
What have we learned?
• wide range of masses
• up to 15 Jupiter masses
• down to 0.01 Jupiter
masses = 3 Earth masses
• lower cutoff is
determined entirely by
observational selection
• upper cutoff is real
incomplete
Earth mass
Jupiter
mass
maximum
planet
mass
What have we learned?
• eccentricities
are much
larger than in
the solar
system
tidal
circularization
• biggest
eccentricity
is e = 0.93
What have we learned?
• current
surveys could
(almost) have
detected
Jupiter
• Jupiter’s
eccentricity
is anomalous
• therefore the
solar system
is anomalous
Theory of planet formation:
Theory failed to predict:
– high frequency of planets
– existence of planets much more massive than Jupiter
– sharp upper limit of around 15 MJupiter
– giant planets at very small semi-major axes
– high eccentricities
Theory of planet formation:
Theory failed to predict:
– high frequency of planets
– existence of planets much more massive than Jupiter
– sharp upper limit of around 15 MJupiter
– giant planets at very small semi-major axes
– high eccentricities
Theory reliably predicts:
– planets should not exist
Planet formation can be divided into two
phases:
Phase 2
Phase 1
• protoplanetary gas disk 
dust disk  planetesimals
 planets
• subsequent dynamical
evolution of planets due to
gravity
• solid bodies grow in mass
by 45 orders of magnitude
through at least 6
different processes
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• lasts 0.01% of lifetime (1
Myr)
• involves very complicated
physics (gas, dust,
turbulence, etc.)
• lasts 99.99% of lifetime
(10 Gyr)
involves very simple
physics (only gravity)
Modeling phase 2 (M. Juric, Ph.D. thesis)
• distribute N = 3-50 planets randomly between a = 0.1 AU and
100 AU, uniform in log(a)
• choose masses randomly between 0.1 and 10 Jupiter masses,
uniform in log(m)
• choose small eccentricities and inclinations with specified
e2, i2
• include physical collisions
• repeat 200-1000 times for each parameter set N, e2, i2
• follow for 100 Myr to 1 Gyr
Modeling phase 2 - results
Characteristic radius of influence of a planet of mass m at semi-major
axis a orbiting a star M is
rHill=a(m/M)1/3
(Hill, Roche, or tidal radius)
Crudely, planetary systems can be divided into two kinds:
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inactive:
– large separations or low masses (¢ r >> rHill)
– eccentricities and inclinations remain small
– preserve state they had at end of phase 1
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active:
– small separations or large masses (¢ r < a few times rHill)
– multiple ejections, collisions, etc.
– eccentricities and inclinations grow
partially active
active
inactive
• most active systems end up
with an average of only 2-3
planets, i.e., 1 planet per
decade
solar system (all)
solar system
(giants)
median ¢ r/rHill
inactive
extrasolar planets
partially active
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all active systems
converge to a
common spacing
(median r in units
of Hill radii)
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solar system is not
active
active
108 yr
initial eccentricity
distributions
• a wide variety of
active systems
converge to a
common
eccentricity
distribution
initial eccentricity
distributions
• a wide variety of
active systems
converge to a
common
eccentricity
distribution,
which agrees
with the
observations
Summary:
• late dynamical evolution (Phase II), long after planet
formation is complete, may establish some of the properties
of extrasolar planetary systems, such as the eccentricity
distribution
Questions:
• why are there planets at all?
• why are giant planets (M > 0.1 MJupiter = 30 MEarth) so
common?
• how common are smaller planets (< 30 MEarth )?
• how do giant planets form at semi-major axes 200 times
smaller than any solar-system giant?
• why are there no planets more massive than » 15 MJ?
• what are the relative roles of Phase I and II evolution in
determining the properties of extrasolar planetary systems?
• why is the solar system unusual? (anthropic principle?)
Kozai oscillations
• Kozai (1962); Lidov (1962)
• arise most simply in restricted three-body problem (two massive
bodies on a Kepler orbit + a test particle $ binary star + planet
orbiting one member of binary)
•in Kepler potential ©=-GM/r, eccentric orbits have a fixed
orientation
generic axisymmetric potential
Kepler potential
generic axisymmetric potential
Kepler potential
Kozai oscillations
• in Kepler potential ©=-GM/r, eccentric orbits have a fixed
orientation because of conservation of the Laplace-Runge-Lenz or
eccentricity vector
e ´ v £ ( r £ v ) ¡ ^r
GM
• vector e points towards
pericenter and |e| equals the
eccentricity
e
e
Kozai oscillations
• now subject the orbit to a weak, time-independent external force
• because the orbit orientation is fixed even weak external forces
can act for a long time in a fixed direction relative to the orbit and
therefore change the angular momentum or eccentricity
• if Fexternal / ² then timescale for evolution / 1/² but nature of
evolution is independent of ² so long as all other external forces are
negligible, i.e. ¿ ²
Fexternal
e
Kozai oscillations
Consider a planet orbiting one member of a
binary star system:
• can average over both planetary and binary star
orbits
• both energy (a) and z-component of angular
momentum (GMa)1/2(1-e2)1/2cos I of planet are
conserved
• thus the orbit-averaged problem has only one
degree of freedom (e.g., eccentricity e and angle
in orbit plane from the equator to the
pericenter, ), so ©=©(e,!)
• motion is along level surfaces of ©(e,!)
Kozai oscillations
circular
• initially circular orbits remain circular if and
only if the initial inclination is < 39o = cos-1(3/5)1/2
• for larger initial inclinations the phase plane
contains a separatrix
• ! can either librate around ¼/2 or circulate
• orbits undergo coupled oscillations in
eccentricity and inclination (Kozai oscillations)
• the separatrix includes circular orbits, so
circular orbits
• cannot remain circular and are excited to
high eccentricity (surprise number 1)
• are chaotic (surprise number 2)
radial
Holman, Touma & Tremaine (1997)
circular orbits have inclination 60±
• initially circular orbits remain circular if and
only if the initial inclination is < 39o = cos-1(3/5)1/2
Kozai oscillations
• for larger initial inclinations the phase plane
contains a separatrix
circular
• ! can either librate around ¼/2 or circulate
• orbits undergo coupled oscillations in
eccentricity and inclination (Kozai oscillations)
• the separatrix includes circular orbits, so
circular orbits
• cannot remain circular and are excited to
high eccentricity (surprise number 1)
• are chaotic (surprise number 2)
•as the initial inclination approaches 90±, the
maximum eccentricity achieved in a Kozai
oscillation approaches unity ! collisions (surprise
number 3)
• mass and separation of companion affect period
of Kozai oscillations, but not the amplitude
(surprise number 4)
radial
eccentricity
oscillations of a
planet in a
binary star
system
M3=0.9M¯
M3=0.08 M¯
•
aplanet = 2.5 AU
•
companion has
inclination 75±, semimajor axis 750 AU,
mass 0.08 M¯ (solid) or
0.9 M¯ (dotted)
(Takeda & Rasio 2005)
• initially circular orbits remain circular if and
only if the initial inclination is < 39o = cos-1(3/5)1/2
• for larger initial inclinations the phase plane
contains a separatrix
Kozai oscillations
• ! can either librate around ¼/2 or circulate
circular
• orbits undergo coupled oscillations in
eccentricity and inclination (Kozai oscillations)
• the separatrix includes circular orbits, so
circular orbits
• cannot remain circular and are excited to
high eccentricity (surprise number 1)
• are chaotic (surprise number 2)
•as the initial inclination approaches 90±, the
maximum eccentricity achieved in a Kozai
oscillation approaches unity ! collisions (surprise
number 3)
• mass and separation of companion affect period
of Kozai oscillations, but not the amplitude
(surprise number 4)
• small additional effects such as general
relativity can strongly affect the oscillations
(surprise number 5)
radial
Kozai oscillations
•may excite eccentricities of planets in some binary star systems, but
probably not all planet eccentricities:
•not all have stellar companion
stars (so far as we know)
• suppressed by additional
planets
• suppressed by general
relativity (!)
Desidera & Barbieri (2007)
filled circle  binary star
dot size represents planet mass
Kozai oscillations – applications
• the Moon would crash into the Earth if its inclination were 90±, not near
zero
• distant satellites of the giant planets have inclinations near 0 or 180± but
not near 90±
• may be responsible for the high eccentricities of stars near the Galactic
center
• long-period comets and asteroids
• survival of binary black holes in galactic nuclei
• homework: why do Earth satellites stay up?