Celestial Mechanics II

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Celestial Mechanics II
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This Lecture

• N-body problem
• General Equation of Motion
• Example of numerical solutions
• perturbed motion
• Research papers

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Why is this important?

• Accurate ephemerides
• Long-term orbital evolution studies
• Scientifically interesting problems can be
  studied
• Predict future behaviour of PHAs

Bild http//www.cfa.harvard.edu/iau/Animations/An
imations.html
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N-body Equation of Motion

• Coupled system of equations all bodies affect
  each other!
• ? In inertial Cartesian system, arbitrarily
  placed origo
• With respect to the Sun ?

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N-body Equation of Motion
Separate Sun from other bodies and simplify
notations!
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A Note on Calculation Cost

• All bodies influence each other!
• With 103 asteroids and 8 planets, we have 1008
  coupled differential equations.
• But asteroids do not affect the planets or each
  other measurably waste of time.
• Only apply this equation for the planets and
  treat each asteroid as a massless particle in the
  combined planetary gravity field!

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Massless particle
Calculate planetary r(t) and use it as source
function when solving this equation for each
asteroid, individually. Thus we Have 8 coupled,
and 1000 uncoupled differential equations.
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Perturbations
Force function
Perturbation function
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Relativistic Equation of Motion
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Numerical integration
Minor changes in orbital elements during a 50
year integration Short-period perturbations Need
for osculating or mean elements
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Numerical integration
104 yr integration Changes in e, ?, ?
substantial
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Numerical integration
105 yr integration
Long-periodic perturbation
Secular perturbations ? 15000 yr ? 27000 yr
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Tsiganis et al. (2005)

• Eccentricities of Jupiter (0.06), Saturn (0.09)
  and Uranus (0.08) and a 2º difference in
  inclination difficult to explain
• Nice model Gas giants form within 18 AU from
  Sun, surrounded by a 30-50 Earth-mass disk
• Migration due to interaction with disk (angular
  momentum conservation) leads to present
  semi-major axes J,S,U,N5.2, 9.5, 19.2, 30.1
  and eccentricities
• Jupiter and Saturn briefly in 12 resonance

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Tsiganis et al. (2005)
Bild Figur 1, Tsiganis et al (2005)
33 of runs resulted in ice giant ejection, the
rest in (satellite-conserving) close encounters
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Gomes et al. (2005)

• Late Heavy Bombardment 700 million years after
  Solar System formation
• Moon struck by 61018 kg material, forming the
  lunar basins (Maria)
• Various simulations with different initial
  interplanetary distances, disk cut-offs, disk
  mass etc ? 12 resonance reached 0.2-1.1 Gyr
  into the simulation
• Severe depletion of disk, sending large numbers
  of comets towards terrestrial planets

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Gomes et al. (2005)
t100 Myr
t879 Myr
Bild Figur 2, Gomes et al. (2005)
t1082 Myr (3 of disk is left)
t882 Myr
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Gomes et al. (2005)

• About right amount of mass hits moon
• 50 arrives in first 3.7 Myr, and 90 in first 29
  Myr
• Earth hit by 1.81020 kg comets. Water content
  6 of current ocean mass
• Asteroid belt depleted by factor 10, many
  impacting terrestrial planets within 150 Myr
  after 12 resonance

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Levison Duncan (1997)

• Empirical data Physical theory Numerical
  simulations Comparative analysis ? Extraction
  of Knowledge

Initial conditions 1300 test particles, 0.01?e?0.
03 ilt1º, 4 Gyr simulation Neptune-encountering p
articles with e0.05 and ilt15º selected. 2200
clones
Bild Figur 1, Levison Duncan (1997)
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Levison Duncan (1997)

• Sun and four giant planets
• 1 Gyr simulation time
• Removal of egt1 objects
• Removal of agt1000 AU objects
• Removal of object colliding with Sun/planet
• Visible if qlt2.5 AU

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Orbit element distribution
JFC 2lt T lt 3 HTC T lt 2
Bild Figur 2, Levison Duncan (1997)
Edgeworth-Kuiper belt (EKB) likely source of JFCs
but unlikely source of HTCs Why few real
simulated objects with T 2?
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Evolutionary path
Bild Figur 3a, Levison Duncan (1997)

• a39.4 AU, e0.25 ? TN2.86, qgt17.7 AU (handover
  to Uranus)
• TU2.86, qgt9.0 AU (handover to Saturn)
• TS2.94, qgt3.8 AU (handover to Jupiter)
• TJ2.82
• Handover requires T just below 3!

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Statistics

• 31 become visible comets (all but two JFC)
• These later become ejected (97) or impact
• 69 never within 2.5 AU.
• Of non-visible objects
• 25 ejected (egt1)
• 68 removed (agt1000 AU)
• 2 impact Sun/planet
• 5 survives simulation in EKB

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Impact rates