Dynamics of planetary systems

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Dynamics of planetary systems
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Keplers laws

• The orbits of planets are ellipses with the Sun
  in one focus
• In equal times, the radius vector of a planet
  sweeps out equal areas
• The square of the period of revolution is
  proportional to the cube of the semimajor axis

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Newtons law of gravity

• Conservative force field ? Energy integral
• Central force field ? Angular momentum integral
• Solution Conic section satisfying Keplers laws

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Keplers third law

• Exact form
• Approximate form
• The mean orbital velocity of a planet

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Exoplanet detectionexample of the Solar System

• Both the Sun and the planet are moving around the
  center of mass
• Suns orbital semimajor axis ? amp
• Suns orbital mean velocity ? vpmp

APPROXIMATELY!
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Solar System Planets
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Exoplanet mass determination

• Measure period P and radial velocity half
  amplitude v? sin i
• Identifying v? with v? sin i yields a lower limit
  for Mp
• Detection is easiest if P is short, vp is large
  or M? is small

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Solar System planet mass determination

• For planets with natural satellites, classically
• Modern values rely on tracking of artificial
  satellites or space probes, using accurate
  distance and radial velocity from radio
  communication

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Masses of small Solar System bodies

• Asteroids or TNOs with natural satellites
• Mutual perturbations of asteroid orbits
• Space probe rendezvous/orbiter missions to
  asteroids
• Nongravitational perturbations on comets
• Space probe impact experiment (Deep Impact)

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Orbital Elements

• a - semimajor axis
• e - eccentricity
• i - inclination w.r.t. the ecliptic
• ? - longitude of the ascending node
• ? - argument of perihelion
• T - time of perihelion passage

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Useful Relations

• Perihelion distance
• Aphelion distance
• Binding energy
• Speed of motion
• Angular momentum

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Planetary Perturbations (1)

• Small departures from Keplerian motion
• Direct perturbations caused by the acceleration
  of the test body due to the perturbing planet
• Indirect perturbations caused by the
  acceleration exerted by the perturbing planet on
  the Sun

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Planetary Perturbations (2)

• Perturbing function
• If M1 and R1 were zero, the orbital elements
  would be constant
• When M1 and R1 are small, the orbital elements
  will vary slowly
• Set up differential equations for the time
  derivatives of (a,e,i,?,?,T), seek solutions that
  are valid over as long time as possible

DIRECT
INDIRECT
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Stability of the Solar System

• Not guaranteed over infinite time (resonances,
  chaos vs quasi-periodic motion)
• KAM (Kolmogorov-Arnold-Moser) theorem
  Quasi-periodic motions dominate the phase space
  if the perturbations are small enough
• Nekhoroshev theorem Even if chaos exists, the
  deviations from regular motion are bounded during
  a finite time

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Proper Elements

• Over time spans of 105 yr
• a remains on the average constant
• e and i show periodic oscillations, coupled to
  variations of ? and ?, respectively
• (ep,ip) are proper elements

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Resonances

• Secular resonances Period of circulation of
  (e,?) or (i,?) equals the corresponding for
  Jupiter or Saturn
• Mean motion resonances The mean motion 2?/P is
  commensurable with that of Jupiter (e.g., 3/1)
• Resonance overlapping causes rapid chaos

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Examples of Mean Motion Resonance

• Asteroid Main Belt Kirkwood gaps
• Outside the MB Hildas and Trojans
• Transneptunian Population Large group of
  Plutinos

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Circular restricted 3-body problem (1)

• A massless body moves in the combined
  gravitational field of the Sun and one planet,
  and the orbit of the planet is circular
• Lagrange points Equilibrium points for the
  massless body in the rotating frame following the
  planet

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Circular restricted 3-body problem (2)

• The force field in the rotating system is
  conservative ? existence of an energy integral
• v 0 zero-velocity surfaces

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Circular restricted 3-body problem (3)

• Largest closed zero-velocity surface around the
  planet Hill sphere
• Orbital energy in the rotating system Jacobi
  integral
• Approximation far from the Sun or the planet
  Tisserand parameter

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Orbital evolutions at close encounters

• Many comets have low-inclination orbits and
  experience close encounters with Jupiter
• The Tisserand criterion was used to identify the
  same comet before and after a large change of the
  orbit
• It can be used to plot evolutionary curves in the
  (a,e) or (Q,q) planes

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Circular restricted 3-body problem (4)

• Without close encounters both a and Hz are
  constant, but H may change
• ? Coupled (e,?) and (i,?) variations with
  constant a Kozai cycle

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Sungrazing Comets

• Comet Ikeya-Seki (1965 S1) groundbased
  coronograph image
• SOHO image of two comets plunging into the Sun in
  1998

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Close encounters (1)

• A massive body has a sphere of influence, where
  its gravitational influence exceeds that of the
  Sun (e.g., the Hill sphere)
• This can be defined in terms of the ratio of
  central to perturbing force in the planetocentric
  or heliocentric frame

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Close encounters (2)

• Approximate treatment as hyperbolic deflections
  (scattering problem)
• The approach velocity U is conserved
• U2 3 - T
• As the direction of the velocity vector is
  changed, the heliocentric motion can be either
  accelerated or decelerated

• controls the values of
• E and Hz

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Nongravitational forces (1)
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Nongravitational forces (2)

• Radiation pressure
• acts to scale down the gravity by a factor
  (1-?) particles with negative net force are
  called ? particles
• Poynting-Robertson drag
• aberration effect of absorbed light causes
  the particles to get circular orbits and spiral
  into the center

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Cometary outgassing

• Jet force due to net momentum of asymmetric
  outgassing
• Causes measurable perturbations of
• (1) orbital period (2) perihelion longitude

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Yarkovsky effect

• Jet effect due to asymmetric emission of thermal
  photons with transverse component
• Diurnal effect may cause a drift inward or
  outward
• Seasonal effect only causes a drift inward