Rosemary Mardling

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School of Mathematical Sciences
Rosemary Mardling
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Planetary Dynamics

• Planet formation
• assembling terrestrial planets and giant cores
• Close encounters and collisions between
  planetesimals (unstable orbits)
• Gravitational influence of existing giant planets
• planet-disk interaction (tidal interaction) (hot
  Jupiters - migration)
• secular and dynamical stability (resonance)
• resonance capture (eg. 21 extrasolar planets)
• moon formation (in situ and capture/collision)
  (web animation-Kokubo)
• effect of binary companion

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Planetary Dynamics

• Small bodies in the Solar System
• asteroid belt
• resonant structure (Kirkwood gaps)
• stability

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Planetary Dynamics

• Small bodies in the Solar System
• Kuiper belt (old and scattered)
• resonant structure (32v 21?)
• stability
• edge
• Comets (high e, unstable)
• Dust (debris disks)

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Planetary Dynamics

• Long-term stability
• Is the Solar System stable?
• Secular resonance
• resonance and chaos

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Planetary Dynamics

• non-point mass dynamics
• Tidal and spin-orbit coupling
• Planet-moon interactions
• Star-planet interactions (hot Jupiter companion
  planet)
• Planet-disk interactions
• Planet-ring-moon interactions (Saturns rings and
  shepherds)
• Star-planet-moon (Sun-Earth-Moon)

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Planetary Dynamics

• other phenomena
• relativistic potential of star (post-Newtonian)
• Apsidal advance of Mercury
• Survival of short-period terrestrial planets
• Kozai cycles
• Large oscillations in eccentricity (up to unity)
• and inclination caused by inclined binary
  companion

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• NUMERICAL METHODS
• symplectic integration (energy conserved)
• Hermite with regularization for close encounters
  (N-body)
• hydrodynamical codes
• rubble piles of sticky particles (N-body)
• direct integration with additional accelerations
  (tides, spin, GR)
• orbit-averaging with additional accelerations
• Lyapunov exponents (chaosinstability)

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• ANALYTICAL METHODS
• Circular restricted planar three-body problem
• One body massless
• Other body small compared to primary
• Perturbation methods
• Small eccentricities and inclinations
• Small masses compared to primary
• New formalism which drops these restrictions

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• Extrasolar planets many hot Jupiters
• many planets with high eccentricities
• 9 systems with two or more planets
• 2 with three planets
• 2 systems in 21
  resonance
• 1 system in 31 resonance
• Planet formation questions
• What kind of planetary systems is the star
  formation process
• capable of producing?
• What can observed systems tell us about the
  planet formation process?
• What kind of structures can we expect from
  disk-like systems
• prograde motion, small inclinations
• Lots of opportunity for resonance

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• Planet formation questions
• Do non-coplanar systems exist?
• how does the cluster environment affect
  configuration?
• How common are systems like the Solar System?
• small eccentricites, inclinations
• Terrestrials inner, giants outer
• not much migration
• Is the edge of the Solar System due to a
  passing star?
• Are we certain planets cannot form close to star?
• migration
• scattering

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Question
Why is resonance associated with stability AND
instability???

• Asteroid belt gaps associated with resonances
• Kuiper belt bodies safe in resonances with
  Neptune
• (example Pluto)

A resonance can be a safe harbour or a stormy
sea. WHY?
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Linear resonance (damped)
response curve
Resonance width DEFINED as full width half max
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The three-body problem
Easy to write down but hard to extract secrets!
Lots of symmetry to take advantage of
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Jacobi coordinates (hierarchies)
--- responsible for all interesting behaviour!
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Spherical harmonic expansion to identify
resonance angles
In practice we keep l2,3 quadrupole and
octopole terms
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Fourier analysis
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Identify resonance angle
resonance angle
Away from resonance, short-period terms
time-average to zero. nn0 terms responsible
for secular evolution.
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Secular evolution Upsilon Andromedae
no energy exchanged
semi-major axes
eccentricities
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short-period variations
orbital variation of inner planet over 3 years
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resonance
2 planets 21 resonance Tout 2 Tin
Energy exchange tends to be in same direction at
conjunction
THEY ARE CAPABLE OF EXCHANGING ENERGY
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resonance
non-resonant system
resonant system
libration
circulation
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resonance
resonance width has precise meaning
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resonance
Resonance width twice separatrix amplitude
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resonance overlap
KAM, Chirikov
chaos
dots translate to stability boundary
n1 resonances
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n1 resonances
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Smaller masses skinnier resonances
But n2 resonances important for extreme mass
ratios these are missing from this plot
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Thus a resonance can be a safe haven if there are
no other resonances overlapping
There are many resonant orbits in the Solar
System many were formed when the protoplanetary
disk was present -- dissipation nonlinear
fixed points RESONANCE CAPTURE
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