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