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Planetary System Dynamics
Part II
Resonances: Periodic orbits
• Periodic orbits (equilibria) exist for perihelicand aphelic conjunctions
• The perihelic conjunction case is stable andthe aphelic one is unstable (chaotic)
Example: asteroid in2/1 mean motionresonance with Jupiter
Analogue: theplanar pendulum
Examples of Mean MotionResonance
• Asteroid Main Belt:Kirkwood gaps
• Outside the MB:Hildas and Trojans
• TransneptunianPopulation: Largegroup of “Plutinos”
Stability of the Solar System• The truncated part of the series is a possibly non-
integrable perturbation• Resonances involve both periodic motion and
chaos (Poincaré): Do the “tori” of quasi-periodicmotion in phase space survive in the presence ofperturbations?
• KAM (Kolmogorov-Arnol’d-Moser) theorem:Quasi-periodic motions dominate the phasespace if the perturbations are small enough
• Nekhoroshev theorem: Even if chaos exists, thedeviations from regular motion are boundedduring a finite time
Circular restricted 3-body problem
• A massless body moves in thecombined gravitational field ofthe Sun and one planet, andthe orbit of the planet iscircular
• Lagrange points: Equilibriumpoints for the massless body inthe rotating frame following theplanet
Zero-velocity surfaces
• The force field in therotating system isconservative ⇒existence of anenergy integral
• v = 0: zero-velocitysurfaces
!
C = x 2 + y 2( ) + 2 1"m#1
+m#2
$ % &
' ( ) " v 2
These cannot be crossed!
Tisserand parameter
• Largest closed zero-velocity surface aroundthe planet: Hill sphere
• Orbital energy in the rotating system: Jacobiintegral
• Approximation far from the Sun or the planet:Tisserand parameter
!
T =aJa
+ 2 aaJ(1" e2) cos i
!
"H =m3
#
$ %
&
' ( 1 3
-E Hz
Kozai cycles• Without close encounters: both a and Hz
are constant, but H may change• Coupled (e,ω) and (i,Ω) variations with
constant a: Kozai cycle
Sungrazing Comets
• Comet Ikeya-Seki(1965 S1):groundbasedcoronograph image
• SOHO image of twocomets plunging intothe Sun in 1998
These are results of Kozai cycles
Orbital evolution due to closeencounters
• Many comets have low-inclination orbits andexperience closeencounters with Jupiter
• The Tisserand criterionwas used to identify thesame comet before andafter a large change of theorbit
• It can be used to plotevolutionary curves in the(a,e) or (Q,q) planes
Capture routes into the Jupiter family
The Jupiter Family
T=3.0
T=2.8
T=2.5
Circles: discovered before 1980
Pluses: discovered after 1980
Capture of the Jupiter Family
• Low-inclination routeswith Tp ≈ 3
• “Handing down”process: Neptune-Uranus-Saturn-Jupiter
• Most efficient source:the Scattered Disk
Starts from the TNO populationGoes via Centaur population
scattered disk
JF
Comet Dynamics roadmap
Oort Cloud
Observedcomets
Oort Cloud dynamics
• Comets are orbiting at distances ~ 104 -105 AU from the Sun
• Their orbits are continually perturbed by(1) the Galactic tides; (2) impulses dueto passing stars
• These perturbations are important tofeed new comets into observable orbits
Galactic tides• The disk tide - Gravitational potential of the
Galactic disk - Induces regular oscillations of
q and iG (small enough a forintegrability)
• The radial tide - In-plane, central force field of
the Galaxy - Changes also the semimajor
axis and may eject comets
The Disk tide
• The q oscillations of the disk tide are animportant mechanism to inject comets intothe inner Solar System
• Heisler & Tremaine (1986) developed ananalytic theory of the regular disk tide byorbital averaging
!
"q #q1/ 2 $ a7 / 2 $ sin2%G Tidal change of q per orbitalrevolution due to the disk tide
Gal. latitude of perihelion
Imprint of the disk tide
• Delsemme (1987)recognized a double-peaked distribution ofGalactic latitudes ofperihelia of newcomets, indicating theimportance of the disktide for comet injection
recent orbit sample
Impulse Approximation
• Heliocentric impulse(velocity changeI=Δvc):
Rickman (1976)
Assumptions: the comet and the Sun are at rest; the starpasses at constant speed along a straight line
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