
1 :

The stability of the solar system TexPoint fonts used in EMF.
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Stability of the solar system 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?
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 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.” 

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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 solutions, 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 errors
inherently serial so little gain from massively parallel computers 

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Stability of the solar system “Small corrections” include:
general relativity (fractional effect <108)
satellites (<107)
Unknowns include:
asteroids (< 109) and Kuiper belt (< 106 even for outermost planets)
solar quadrupole moment (< 1010)
mass loss from Sun through radiation and solar wind, and drag of solar wind on planetary magnetospheres (< 1014)
Galactic tidal forces (fractional effect < 1013)
passing stars (closest passage about 500 AU)
Masses mj known to better than 109M?
Initial conditions known to fractional accuracy better than 107 1 AU = 1 astronomical unit = EarthSun distance
Neptune orbits at 30 AU solar mass 

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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 (4.5?109 yr backwards to formation, or 7?109 yr forwards to redgiant stage when Mercury and Venus are swallowed up)
Goal is quantitative accuracy (?? ?? 1 radian) over 108 yr and qualitative accuracy over 1010 yr Stability of the solar system 

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Consider integrating an orbit in the Kepler potential V(r) = 1/r.
Equations of motion for particle of unit mass read A primer on orbit integration Examine three integration methods with timestep h: Euler’s method
Modified Euler method
RungeKutta method Error after integrating for unit time is O(h) for Euler methods and O(h4) for RungeKutta (firstorder versus fourthorder)
100 function evaluations per orbit for each method 


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Modified Euler method comes in two flavors: A primer on orbit integration “driftkick”
“kickdrift” An even better method is to combine half a driftkick step and half a kickdrift step: This is the classic leapfrog or Verlet method. 


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Why does modified Euler work so well? Because it:
preserves volume in phase space, just like Newton’s laws (Liouville’s theorem): A primer on orbit integration generates a symplectic or canonical transformation
Modified Euler method is the prototype of geometric integration algorithms Why does leapfrog work even better than modified Euler? Because:
like modified Euler, it generates a symplectic transformation
it has error O(h2), one order higher than modified Euler
is timereversible, just like Newton’s laws (a second geometric property) 

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A primer on orbit integration The system we are examining is described by the Hamiltonian and the equations of motion Any map r(0), p(0) ? r(t), p(t) generated by a Hamiltonian is a canonical or symplectic transformation. So consider Integrating this from t=0+ to t=h+ gives which is the modified Euler method. Leapfrog is simply 

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A primer on orbit integration To follow motion in the potential V(r) we use the Hamiltonian To carry out numerical integration we replace this with Motion of a test particle in a planetary system is described by the Hamiltonian 

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A primer on orbit integration The workhorse for long orbit integrations is the mixedvariable symplectic integrator (Wisdom & Holman 1991) a geometric integrator (symplectic and timereversible)
errors smaller than leapfrog by of order mplanet/M* ? 104
all of the work is in converting back and forth from actionangle variables to Cartesian coordinates once per step
numerical analysis ? dynamical perturbation theory
longterm errors reduced to O(mplanet/M*)2 by techniques such as warmup 

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A primer on roundoff error Roundoff error can cause big problems:
in 1991 Gulf War, Patriot missile defense system converted clock steps of 0.1 sec to decimal by multiplying by a 22bit binary number; after 100 hours the accumulated roundoff error was 0.3 sec, which led to failure to intercept a Scud missile, resulting in 28 deaths
Floatingpoint numbers are stored in the computer as p bits plus an exponent. Typically p=53, corresponding to accuracy ?=2p'1016
Simplest model is that energy error grows like a random walk. After N integration steps the fractional error is ? E/E? ? N1/2 ? t1/2. Phase error is then ?? ? (t/Porbit) ? E/E ? t3/2 (“good” roundoff)
Many numerical integrations exhibit ?? ? t2 (“bad” roundoff”)
Over lifetime of solar system, ?? ? 1 for good roundoff and ? 105 radians for bad roundoff 

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A primer on roundoff error Most important step in managing roundoff is to ensure “good” roundoff behavior rather than “bad” behavior.
To manage roundoff error:
use machines with optimal and unbiased arithmetic, e.g., IEEE 754 standard (to check it out, use “paranoia” programs at www.netlib.org)
carry out selected operations in extended or quadruple precision
beware of any mathematical constants that are not representable (?, 1/3, etc.). Replace (2.0/3.0)*x by 2.0*x/3.0 

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A primer on orbit integration Lessons:
when integrating ordinary differential equations, shortterm quantitative accuracy is not the same asand is often less important thanlongterm qualitative accuracy
use geometric integrators, which preserve the qualitative features of the physical systems they are describing (symplecticity, timereversibility, energy and angular momentum conservation, flux conservation, unitarity, etc.)
if the physical system is close to one that can be integrated exactly, choose the integration algorithm so that it is exact for the integrable system
Hamiltonian mechanics is actually useful 

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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 

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Ito & Tanikawa (2002) 

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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? 

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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 ? with 20,000 year period (Cohen & Hubbard 1965) 

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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 loweccentricity, lowinclination 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 semimajor axis 

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Kuiper belt objects
Plutinos (3:2)
Centaurs
comets
from Minor Planet Center 

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Two kinds of dynamical system Regular
highly predictable, “wellbehaved”
small differences in position and velocity 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 

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Laskar (1989) linear, ? ? t exponential, ? ? exp(t/tL) 10 Myr separation in 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 time tL is 520 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 

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Integrators:
doubleprecision (p=53 bits) 2nd order mixedvariable symplectic (WisdomHolman) method with h=4 days and h=8 days
doubleprecision (p=53 bits) 14th order multistep method with h=4 days
extendedprecision (p=80 bits) 27th order Taylor series method with h=220 days saturated
Hayes (astroph/0702179) Liapunov time tL=12 Myr 200 Myr 

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Chaos in the solar system orbits of inner planets (Mercury, Venus, Earth, Mars) are chaotic with efolding times for growth of small changes (Liapunov times) of 520 Myr (i.e. 2001000 efolds 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 – future of solar system over longer times can only be predicted probabilistically
the solar system is a poor example of a deterministic universe
shapes of some orbits execute random walk on timescales of Gyr or longer 

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Laskar (1994) start finish 

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Causes of chaos dynamical behavior of nonlinear systems is largely determined by resonance structure
a resonance of strength ² in linear theory has an effective width in phase space »²1/2
chaos in nonlinear systems 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 strongly degenerate
planetary perturbations lead to finestructure splitting of resonances by amount ? O(?) where ? ? mplanet/M*.
twobody resonances have strength ? O(?) and width ? O(?)1/2 so width of each resonance is much larger than width of multiplet
more numerous threebody resonances have strength ? O(?2) and width ? O(?).
Murray & Holman (1999) show that chaos in outer solar system arises from overlap in the multiplet of 3body 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 

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Murray & Holman (1999) 

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Consequences of chaos orbits of inner planets (Mercury, Venus, Earth, Mars) are chaotic with efolding times for growth of small changes (Liapunov times) of 520 Myr (i.e. 2001000 efolds 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 

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Holman (1997) age of solar system J S U N 

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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 efolding times of 520 Myr
the orbital phases of the planets are not predictable over timescales > 100 Myr
“Is the solar system stable?” can only be answered probabilistically
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 

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Sunday, January 13, 14:30, Lewiner seminar room
New worlds: the properties and evolution of extrasolar planetary systems
Monday, January 14, 16:30, Lidow 502
Kozai oscillations in binary stars and planetary systems



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Bode’s law planets suffer no close encounters and are spaced fairly regularly (Bode’s law: an=0.4 + 0.3?2n) *predicted
+exceptions 

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Brown et al. (2005)
diameter 2400 § 100 km or 5% bigger than Pluto
has a moon
albedo 8090% 
