Stability of the Solar System - Tremaine

7/29/2019 Stability of the Solar System - Tremaine

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The long-term stability of planetary systems


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Long-term stability of planetary systems

The problem:A point mass is surrounded by N > 1 much smaller masses on nearlycircular, 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 nowbecome more general:

can we explain the properties of extrasolar planetary systems?


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Stability of the solar system

Newton:blind fate could never make all the Planets move one and thesame way in Orbs concentric, some inconsiderableirregularities excepted, which could have arisen from themutual Actions of Planets upon one another, and which will beapt 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 allthings of which the universe consists, would be able tocomprehend the motions of the largest bodies of the world and

those of the smallest atoms in one single formula, provided itwere sufficiently powerful to subject all data to analysis. To it,nothing would be uncertain; both future and past would bepresent before its eyes.


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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 bycomputer, 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


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Small corrections include:

general relativity (


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To a very good approximation, the solar system is an isolatedHamiltonian 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 yrbackwards to formation, or 7109 yr forwards to red-giant stage whenMercury and Venus are swallowed up)

Goal is quantitative accuracy (


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innermost fourplanets

Ito & Tanikawa(2002)

0 - 55 Myr

-55 0 Myr -4.5 Gyr

+4.5 Gyr


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Ito & Tanikawa (2002)


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Plutos 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 Neptunes semimajor axis ( a = 30.1 AU )

How do they avoid colliding?


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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 whenPluto is at perihelion it is

approximately 90o away fromNeptune

Resonant argument:

= 3(longitude of Pluto) 2(longitude of Neptune) (perihelionof Pluto)

librates around with 20,000 yearperiod (Cohen & Hubbard 1965)


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early in the history of the solar systemthere was debris left over between theplanets

ejection of this debris by Neptunecaused its orbit to migrate outwards

if Pluto were initially in a low-eccentricity,low-inclination orbit outside Neptune it isinevitably captured into 3:2 resonancewith Neptune

once Pluto is captured its eccentricityand inclination grow as Neptune

continues to migrate outwardsother objects may be captured in theresonance as well

Malhotra (1993)

Plutos eccentricity

Plutos inclination

resonant argument

PPluto/PNeptune

Plutos semi-major

axis


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Kuiper belt objects

Plutinos (3:2)

Centaurs

comets

as of March 82006 (MinorPlanet Center)


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Two kinds of dynamical system

Regular highly predictable, well-

behaved

small differences growlinearly: x, v t

e.g. baseball, golf, simplependulum, all problems inmechanics 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 isLiapunov time appears regular on

timescales short comparedto Liapunov time lineargrowth on short times,

exponential growth on longtimes e.g. roulette, dice, pinball,

weather, billiards, doublependulum


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Laskar (1989)

linear,

t

expo

nenti

al,

e

xp(t/t

L)

10 Myr

separationi

n

phase

space


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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 timetL is 5-20 Myr factor of at least 10

100 over lifetime of solar system

400 million years 300 million years

Pluto Jupiter

factor of 10,000factor of 1000

saturated


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Integrators:

double-precision (p=53 bits) 2nd order mixed-variable symplecticmethod 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 withh=220 days

saturated

Hayes (astro-ph/0702179)

tL=12My

r

200 Myr


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Chaos in the solar system

orbits of inner planets (Mercury, Venus, Earth, Mars) are chaoticwith e-folding times for growth of small changes (Liapunov times)of5-20 Myr (i.e. 200-1000 e-folds in lifetime of solar system

chaos in orbits of outer planets depends sensitively on initialconditions but usually are chaotic

positions (orbital phases) of planets are not predictable ontimescales 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 Gyror longer


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Laskar (1994)

start finish


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Consequences of chaos

orbits of inner planets (Mercury, Venus, Earth, Mars) are chaoticwith 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 initialconditions but usually are chaotic

positions (orbital phases) of planets are not predictable ontimescales 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 sotrajectory wanders chaotically through large distances in phasespace (Arnold diffusion). Thus solar system is unstable,although probably on verylong timescales

most likely ejection has already happenedone or more times


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Holman (1997)

age of solarsystem

J S U N


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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 potentials1=2=0 so resonances are degenerate

planetary perturbations lead to fine-structure splitting ofresonances 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 enhancechaos

cannot predict lifetimes analytically


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Murray & Holman (1999)


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(198 planets)


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HD 82943

planet 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)


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(6 planets)


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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)


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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


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(4 planets)


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Gravitational

lensing

surface brightness isconserved so

distortion of image ofsource across largerarea of sky impliesmagnification


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Einstein rings around distant galaxies


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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 thefirst three planetsdiscovered bymicrolensing, butthe detection

probability for a low-mass planet of thiskind is 50 timeslower terrestrialplanets are common


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What have we learned?

planets are remarkablycommon, especiallyaround metal-richstars (20% even withcurrent technology)

probability of finding aplanet mass in

metals in the star


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What have welearned?

giant planets like Jupiterand Saturn are found atverysmall orbital radii

if we had expected this,planets could have beenfound 20-30 years ago

OGLE-TR-56b: M = 1.45M

Jupiter

, P = 1.21 days, a =

0.0225 AU


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wide range of masses

up to 15 Jupiter masses

down to 0.02 Jupitermasses = 6 Earth masses

maximum

planetmass

incomplete


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eccentricities aremuch larger thanin the solarsystem

biggesteccentricity e =0.93

tidal circularization


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currentsurveys could(almost) havedetected

Jupiter

Jupiterseccentricity isanomalous

therefore thesolar systemis anomalous


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Some of the big questions:

how do giant planets form at semi-major axes 200 times smallerthan 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?


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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


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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


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Planet formation can be divided into twophases:

Phase 1

protoplanetary gas disk dust disk planetesimals planets

solid bodies grow in mass by45 orders of magnitudethrough at least 6 differentprocesses

lasts 0.01% of lifetime (1Myr)

involves very complicatedphysics (gas, dust,turbulence, etc.)

Phase 2 subsequent dynamical

evolution of planets due togravity

lasts 99.99% of lifetime (10Gyr)

involves very simple physics(only gravity)


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Planet formation can be divided into twophases:

Creation science

protoplanetary gas disk dust disk planetesimals planets

solid bodies grow in mass by45 orders of magnitudethrough at least 6 differentprocesses

lasts 0.01% of lifetime (1Myr)

involves very complicatedphysics (gas, dust,turbulence, etc.)

Theory of evolution subsequent dynamical

evolution of planets due togravity

lasts 99.99% of lifetime (10Gyr)

involves very simple physics(only gravity)


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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


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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 = Rocheradius


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most active systems end up

with an average of only 2-3planets, i.e., 1 planet perdecade

inactive

partially active

active


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all active systemsconverge to acommon spacing(median a in unitsof Hill radii)

solar system is notactive

inactive

partially active

active

solar system (all)

solar system (giants)

extrasolar planets


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a wide variety of active systemsconverge to a commoneccentricity distribution

initial eccentricitydistributions

inactive

partially active

active


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a wide variety of active systemsconverge to a commoneccentricity distribution whichagrees with the observations

see also Chatterjee, Ford &

Rasio (2007)

initial eccentricitydistributions

inactive

partially active

active


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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 fromthe solar system in the past

the solar system is anomalous

currently, planet-formation theory in Phase 1 (first 1 Myr) has virtually nopredictive power

relative roles of Phase 1 and Phase 2 (last 10 Gyr) are poorly understood, butPhase 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

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Stability of the Solar System - Tremaine