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, Poincaré, 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 (<10-8)

• satellites (<10-7)

Unknowns include:

• asteroids and Kuiper belt (fractional effect < 10-7)

• solar quadrupole moment (fractional effect < 10-10)

• mass loss from Sun through radiation and solar wind, and drag of solar wind on planetary magnetospheres (<10-14)

• Galactic tidal forces (fractional effect <10-13)

• passing stars (closest passage about 500 AU)

Masses mj known to better than 10-9M

Initial conditions known to fractional accuracy better than 10-7

d2 x idt 2 = ¡G

PNj = 1

m j (x i ¡ x j )jx i ¡ x j j 3 + small corrections

1 AU = 1 astronomical unit = Earth-Sun distance

Neptune orbits at 30 AU

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 (4.5×109 yr backwards to formation, or 7×109 yr forwards to red-giant stage when Mercury and Venus are swallowed up)

Goal is quantitative accuracy (∆ϕ << 1 radian) over 108 yr and qualitative accuracy over 1010 yr

Stability of the solar system

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:

Φ= 3×(longitude of Pluto) − 2×(longitude of Neptune) – (perihelion of Pluto)

librates around π 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, ∆ ∝ tex

pone

ntia

l, ∆

∝ e

xp(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

J S U N

Causes of chaos

• chaos arises from overlap of resonances• orbits with 3 degrees of freedom have three fundamental

frequencies Ωi. In spherical potentials, Ω1=0. In Kepler potentials Ω1=Ω2=0 so resonances are degenerate

• planetary perturbations lead to fine-structure splitting of resonances by amount ∼ O(α) where α ∼ mplanet/M*.

• two-body resonances have strength ∼ O(α) and width ∼ O(α)1/2. • three-body resonances have strength ∼ O(α2) and width ∼ O(α),

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 Φ = 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=6.6±1.1 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=5×1012, 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