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The long-term stability of planetary systems
Long-term stability of 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 1010 orbits)?
Why is this interesting? one of the oldest problems in theoretical physics what is the fate of the Earth? why are there so few planets in the solar system? can we calibrate geological timescale over the last 50 Myr? how do dynamical systems behave over very long times?
Historically, the focus was on the solar system but the context has now become more general:
can we explain the properties of extrasolar planetary systems?
Stability of the solar systemNewton: blind fate could never make all the Planets move one and the
same way in Orbs concentric, some inconsiderable irregularities excepted, which could have arisen from the mutual Actions of Planets upon one another, and which will be apt to increase, until this System wants a reformation
Laplace: An intelligence knowing, at a given instant of time, all forces
acting in nature, as well as the momentary positions of all things of which the universe consists, would be able to comprehend the motions of the largest bodies of the world and those of the smallest atoms in one single formula, provided it were sufficiently powerful to subject all data to analysis. To it, nothing would be uncertain; both future and past would be present before its eyes.
Stability of the solar system
The problem:A point mass is surrounded by N much smaller masses on nearly circular, nearly coplanar orbits. Is the configuration stable over very long times (up to
1010 orbits)?
How can we solve this? many famous mathematicians and physicists have attempted to find analytic
solutions or constraints, with limited success (Newton, Laplace, Lagrange, Gauss, Poincar, Kolmogorov, Arnold, Moser, etc.)
only feasible approach is numerical solution of equations of motion by computer, but:
needs 1012 timesteps so lots of CPU needs sophisticated algorithms to avoid buildup of truncation error
geometric integration algorithms + mixed-variable integrators roundoff error difficult to parallelize; but see
Saha, Stadel & Tremaine (1997) parareal algorithms
Stability of the solar system
Small corrections include:
general relativity (
To a very good approximation, the solar system is an isolated Hamiltonian system described by a known set of equations, with
known initial conditions
All we have to do is integrate them for ~1010 orbits (4.5109 yr backwards to formation, or 7109 yr forwards to red-giant stage when Mercury and Venus are swallowed up)
Goal is quantitative accuracy (
innermost four planets
Ito & Tanikawa (2002)
0 - 55 Myr
-55 0 Myr -4.5 Gyr
+4.5 Gyr
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 )
closest approach to Sun is q = a(1 e) = 29.6 AU, which 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
Resonant argument:
= 3(longitude of Pluto) 2(longitude of Neptune) (perihelion of Pluto)
librates around pi with 20,000 year period (Cohen & Hubbard 1965)
Pluto s peculiar orbit
early in the history of the solar system there was debris 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
Malhotra (1993)
Pluto s eccentricity
Pluto s inclination
resonant argument
PPluto/PNeptune
Pluto s semi-major axis
Kuiper belt objects
Plutinos (3:2)
Centaurs
comets
as of March 8 2006 (Minor Planet Center)
Two kinds of dynamical system
Regular highly predictable, well-
behaved small differences grow
linearly: x, v t e.g. baseball, golf, simple
pendulum, all problems in mechanics textbooks, 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, weather, billiards, double pendulum
Laskar (1989)
linear, t
expo
nent
ial,
exp
(t/t L)
10 Myr
sepa
ratio
n in
pha
se s
pace
The orbit of every planet in the solar system is chaotic (Sussman & Wisdom 1988, 1992)
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
400 million years 300 million years
Pluto Jupiter
factor of 10,000 factor of 1000
saturated
Integrators:
double-precision (p=53 bits) 2nd order mixed-variable symplectic method with h=4 days and h=8 days
double-precision (p=53 bits) 14th order multistep method with h=4 days
extended-precision (p=80 bits) 27th order Taylor series method with h=220 days
saturated
Hayes (astro-ph/0702179)
t L=12
Myr
200 Myr
Chaos in the solar system
orbits of inner planets (Mercury, Venus, Earth, Mars) 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
chaos in orbits of outer planets depends sensitively on initial conditions but usually are chaotic
positions (orbital phases) of planets are not predictable on timescales longer than 100 Myr
the solar system is a poor example of a deterministic universe shapes of some orbits execute random walk on timescales of Gyr
or longer
Laskar (1994)
start finish
Consequences of chaos
orbits of inner planets (Mercury, Venus, Earth, Mars) 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
chaos in orbits of outer planets depends sensitively on initial conditions but usually are chaotic
positions (orbital phases) of planets are not predictable on timescales longer than 100 Myr
the solar system is a poor example of a deterministic universe shapes of some orbits execute random walk on timescales of Gyr
or longer most chaotic systems with many degrees of freedom are unstable
because chaotic regions in phase space are connected so trajectory wanders chaotically through large distances in phase space ( Arnold diffusion ). Thus solar system is unstable, although probably on very long timescales
most likely ejection has already happened one or more times
Holman (1997)
age of solar system
JSUN
Causes of chaos
chaos arises from overlap of resonances orbits with 3 degrees of freedom have three fundamental
frequencies i. In spherical potentials, 1=0. In Kepler potentials 1=2=0 so resonances are degenerate
planetary perturbations lead to fine-structure splitting of resonances by amount O() where mplanet/M*.
two-body resonances have strength O() and width O()1/2. three-body resonances have strength O(2) and width O(),
which is matched to fine-structure splitting. Murray & Holman (1999) show that 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)
small changes in initial conditions can eliminate or enhance chaos
cannot predict lifetimes analytically
Murray & Holman (1999)
(198 planets)
HD 82943planet 1:
m sin I = 1.84mJ
P = 435 d
e = 0.18 0.04
planet 2:
m sin I = 1.85mJ
P = 219 d
e = 0.38 0.01
(Mayor et al. 2003)
(6 planets)
mass = 0.69 Jupiter masses
radius = 1.35 Jupiter radii ( bloated )
orbital period 3.52 days, orbital radius 0.047 AU or 10 stellar radii
stellar obliquity < 10o
T = 1130 150 K
sodium, oxygen, carbon detected from planetary atmosphere
HD 209458Brown et al. (2001),
Deming et al. (2005)
primary (visible) secondary (infrared)
Transit searches
COROT (France) launched December 27 2006
180,000 stars over 2.5 yr
0.01% precision
Kepler (U.S.) launch 10/2008
100,000 stars over 4 yr
0.0001% precision
(4 planets)
Gravitational lensing
surface brightness is conserved so distortion of image of source across larger area of sky implies magnification
Einstein rings around distant galaxies
Beaulieu et al. (2006): 5.5 (+5.5/-2.7) MEarth, 2.6(+1.5/-0.6) AU orbit, 0.22(+0.21/-0.11) MSun, DL=6.61.1 kpc
this is one of the first three planets discovered by microlensing, but the detection probability for a low-mass planet of this kind is 50 times lower terrestrial planets are common
What have we learned?
planets are remarkably common, especially around metal-rich stars (20% even with current technology)
probability of finding a planet mass in metals in the star
What have we learned?
giant planets like Jupiter and Saturn are found at very small orbital radii
if we had expected this, planets could have been found 20-30 years ago
OGLE-TR-56b: M = 1.45 MJupiter, P = 1.21 days, a = 0.0225 AU
What have we learned?
wide range of masses
up to 15 Jupiter masses
down to 0.02 Jupiter masses = 6 Earth masses
maximum planet mass
incomplete
What have we learned?
eccentricities are much larger than in the solar system
biggest eccentricity e = 0.93
tidal circularization
What have we learned?
current surveys could (almost) have detected Jupiter
Jupiter s eccentricity is anomalous
therefore the solar system is anomalous
Some of the big questions:
how do giant planets form at semi-major axes 200 times smaller than any solar-system giant?
why are the eccentricities far larger than in the solar system?
why are there no planets more massive than 15 MJ? why are planets so common? why is the solar system unusual?
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 1 protoplanetary gas disk
dust disk planetesimals planets
solid bodies grow in mass by 45 orders of magnitude through at least 6 different processes
lasts 0.01% of lifetime (1 Myr)
involves very complicated physics (gas, dust, turbulence, etc.)
Phase 2 subsequent dynamical
evolution of planets due to gravity
lasts 99.99% of lifetime (10 Gyr)
involves very simple physics (only gravity)
Planet formation can be divided into two phases:
Creation science protoplanetary gas disk
dust disk planetesimals planets
solid bodies grow in mass by 45 orders of magnitude through at least 6 different processes
lasts 0.01% of lifetime (1 Myr)
involves very complicated physics (gas, dust, turbulence, etc.)
Theory of evolution subsequent dynamical
evolution of planets due to gravity
lasts 99.99% of lifetime (10 Gyr)
involves very simple physics (only gravity)
Modeling phase 2 (M. Juric, Ph.D. thesis)
distribute N planets randomly between a=0.1 AU and 100 AU, uniform in log(a); N=3-50
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 K=(number of planets per system) X (number of orbital periods)
X (number of systems) here K=51012, factor 50 more than before
Crudely, planetary systems can be divided into two kinds:
inactive: large separations or low masses
eccentricities and inclinations remain small preserve state they had at end of phase 1
active: small separations or large masses multiple ejections, collisions, etc. eccentricities and inclinations grow
Modeling phase 2 - results
Hill = tidal = Roche radius
most active systems end up with an average of only 2-3 planets, i.e., 1 planet per decade
inactive
partially active
active
all active systems converge to a common spacing (median a in units of Hill radii)
solar system is not active
inactive
partially active
active
solar system (all)
solar system (giants)
extrasolar planets
a wide variety of active systems converge to a common eccentricity distribution
initial eccentricity distributions
inactive
partially active
active
a wide variety of active systems converge to a common eccentricity distribution which agrees with the observations
see also Chatterjee, Ford & Rasio (2007)
initial eccentricity distributions
inactive
partially active
active
Summary we can integrate the solar system for its lifetime the solar system is not boring on long timescales planet orbits are probably chaotic with e-folding times of 5-20 Myr the orbital phases of the planets are not predictable over timescales > 100 Myr Is the solar system stable? can only be answered statistically it is unlikely 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 the solar system is anomalous currently, planet-formation theory in Phase 1 (first 1 Myr) has virtually no
predictive power relative roles of Phase 1 and Phase 2 (last 10 Gyr) are poorly understood, but
Phase 2 may be important a late phase of dynamical evolution lasting 10-100 Myr can explain two observed
properties of extrasolar planet systems: eccentricity distribution typical separation in multi-planet systems