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?

• 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)?

• 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

• symplectic integration algorithms• mixed-variable methods• warmup• symplectic correctors• optimal floating-point arithmetic

The solar system

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)

• 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

d2x idt2 = ¡ G

P Nj =1

m j (x i ¡ x j )jx i ¡ x j j3

+ small corrections

1 AU = 1 astronomical unit = Earth-Sun distance

Neptune orbits at 30 AU

solar mass

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 (4.5109 yr backwards to formation, or 7109 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

The solar system

• an integration of the solar system (Sun + 9 planets) for § 4.5£ 109 yr = 4.5 Gyr

• figures show 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 )

• 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

• 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

Malhotra (1993)

period ratio

inclination of Pluto

eccentricity of Pluto

semi-major axis of Pluto

resonant angle

17o

Kuiper belt objects

Plutinos (3:2)

Centaurs

comets

Two kinds of dynamical system

Regular • highly predictable, “well-

behaved”• 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)

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

Jupiter

factor of 1000

saturated

ln(d

ista

nce

)

Integrators:

• double-precision (p=53 bits) 2nd order mixed-variable symplectic (Wisdom-Holman) 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)

Liapunov time t L

=12 Myr

200 Myr

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)

• 1000 integrations of solar system for 5 Gyr using simplified but accurate equations of motion

• all four inner planets exhibit random walk in eccentricity (more properly, in e cos ! and e sin !)

• probability Mercury will reach e = 0.6 in 5 Gyr is about 2%

1 curve per 0.25 Gyr

Causes of chaos

• 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 m30. 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

mean-motion 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

Laskar (2008)

• change PPN parameter 2°-¯ from +1 to -2

• precession of Mercury / 2°-¯ so changes from +43 arcsec/100 yr to -86 arcsec/100 yr

• Mercury collides with Venus in < 4 Myr or < 0.1% of solar system age

ecc

en

tric

ity o

f M

erc

ury

time (Myr - not Gyr!)

Holman (1997)

age of solar

system

J S U N

• 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

Summary

(261 planets)

Given the star mass M (known from spectral type), radial-velocity observations yield:

• 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

I ' 0 I ' 90±

51 Peg: m sin I = 0.46 MJ

P = 4.23 d a = 0.05 AU e = 0.01

70 Vir: m sin I = 7.44 MJ

P = 116.7 d a = 0.48 AU e = 0.40

1 MJ = 1 Jupiter mass = 0.001 M¯ = 318 M©

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)

• probability of finding a planet / Za, a' 1-2, where Z = metallicity = fraction of elements heavier than He

(Fischer & Valenti 2005)

log10 Z/Z¯

What have we learned?

M V E Mars Jupiter

• 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

• 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

maximum planet mass

incomplete

Earth mass

Jupiter mass

What have we learned?

• eccentricities are much larger than in the solar system

• biggest eccentricity is e = 0.93tidal

circularization

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 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)

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

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:

• inactive:– large separations or low masses (¢ r >> rHill)– eccentricities and inclinations remain small– preserve state they had at end of phase 1

• active:– small separations or large masses (¢ r < a few times rHill)– multiple ejections, collisions, etc.– eccentricities and inclinations grow

Modeling phase 2 - results

• 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 r in units of Hill radii)

• solar system is not active

inactive

partially active

active

solar system (all)

solar system (giants)

extrasolar planets

med

ian

¢

r/r H

ill

108 yr

• 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

initial eccentricity distributions

• 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?)

Summary:

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

• vector e points towards pericenter and |e| equals the eccentricity

e ´ v£ (r£ v)GM

¡ r̂

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

• 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)

circular

radial

Holman, Touma & Tremaine (1997)

circular orbits have inclination 60±

Kozai oscillations• 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)

•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)

circular

radial

eccentricity oscillations of a planet in a binary star system

• aplanet = 2.5 AU

• companion has inclination 75±, semi-major axis 750 AU, mass 0.08 M¯ (solid) or 0.9 M¯ (dotted)

(Takeda & Rasio 2005)

M3=0.9M¯

M3=0.08 M¯

Kozai oscillations• 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)

•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)

circular

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?