Stability of the Solar System - Tremaine
-
Upload
parag-mahajani -
Category
Documents
-
view
216 -
download
0
Transcript of Stability of the Solar System - Tremaine
-
7/29/2019 Stability of the Solar System - Tremaine
1/48
The long-term stability of planetary systems
-
7/29/2019 Stability of the Solar System - Tremaine
2/48
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?
-
7/29/2019 Stability of the Solar System - Tremaine
3/48
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.
-
7/29/2019 Stability of the Solar System - Tremaine
4/48
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 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
-
7/29/2019 Stability of the Solar System - Tremaine
5/48
Stability of the solar system
Small corrections include:
general relativity (
-
7/29/2019 Stability of the Solar System - Tremaine
6/48
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 (
-
7/29/2019 Stability of the Solar System - Tremaine
7/48
innermost fourplanets
Ito & Tanikawa(2002)
0 - 55 Myr
-55 0 Myr -4.5 Gyr
+4.5 Gyr
-
7/29/2019 Stability of the Solar System - Tremaine
8/48
Ito & Tanikawa (2002)
-
7/29/2019 Stability of the Solar System - Tremaine
9/48
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?
-
7/29/2019 Stability of the Solar System - Tremaine
10/48
Plutos 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 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)
-
7/29/2019 Stability of the Solar System - Tremaine
11/48
Plutos peculiar orbit
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
-
7/29/2019 Stability of the Solar System - Tremaine
12/48
Kuiper belt objects
Plutinos (3:2)
Centaurs
comets
as of March 82006 (MinorPlanet Center)
-
7/29/2019 Stability of the Solar System - Tremaine
13/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
14/48
Laskar (1989)
linear,
t
expo
nenti
al,
e
xp(t/t
L)
10 Myr
separationi
n
phase
space
-
7/29/2019 Stability of the Solar System - Tremaine
15/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
16/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
17/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
18/48
Laskar (1994)
start finish
-
7/29/2019 Stability of the Solar System - Tremaine
19/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
20/48
Holman (1997)
age of solarsystem
J S U N
-
7/29/2019 Stability of the Solar System - Tremaine
21/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
22/48
Murray & Holman (1999)
-
7/29/2019 Stability of the Solar System - Tremaine
23/48
(198 planets)
-
7/29/2019 Stability of the Solar System - Tremaine
24/48
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)
-
7/29/2019 Stability of the Solar System - Tremaine
25/48
(6 planets)
-
7/29/2019 Stability of the Solar System - Tremaine
26/48
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)
-
7/29/2019 Stability of the Solar System - Tremaine
27/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
28/48
(4 planets)
-
7/29/2019 Stability of the Solar System - Tremaine
29/48
Gravitational
lensing
surface brightness isconserved so
distortion of image ofsource across largerarea of sky impliesmagnification
-
7/29/2019 Stability of the Solar System - Tremaine
30/48
Einstein rings around distant galaxies
-
7/29/2019 Stability of the Solar System - Tremaine
31/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
32/48
What have we learned?
planets are remarkablycommon, especiallyaround metal-richstars (20% even withcurrent technology)
probability of finding aplanet mass in
metals in the star
-
7/29/2019 Stability of the Solar System - Tremaine
33/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
34/48
What have we learned?
wide range of masses
up to 15 Jupiter masses
down to 0.02 Jupitermasses = 6 Earth masses
maximum
planetmass
incomplete
-
7/29/2019 Stability of the Solar System - Tremaine
35/48
What have we learned?
eccentricities aremuch larger thanin the solarsystem
biggesteccentricity e =0.93
tidal circularization
-
7/29/2019 Stability of the Solar System - Tremaine
36/48
What have we learned?
currentsurveys could(almost) havedetected
Jupiter
Jupiterseccentricity isanomalous
therefore thesolar systemis anomalous
-
7/29/2019 Stability of the Solar System - Tremaine
37/48
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?
-
7/29/2019 Stability of the Solar System - Tremaine
38/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
39/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
40/48
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)
-
7/29/2019 Stability of the Solar System - Tremaine
41/48
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)
-
7/29/2019 Stability of the Solar System - Tremaine
42/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
43/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
44/48
most active systems end up
with an average of only 2-3planets, i.e., 1 planet perdecade
inactive
partially active
active
-
7/29/2019 Stability of the Solar System - Tremaine
45/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
46/48
a wide variety of active systemsconverge to a commoneccentricity distribution
initial eccentricitydistributions
inactive
partially active
active
-
7/29/2019 Stability of the Solar System - Tremaine
47/48
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
-
7/29/2019 Stability of the Solar System - Tremaine
48/48
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