1. Seminar topics Kozai cycles in comet motions
The Nice Model and the LHB
NEO close encounters

2. Kozai cycles Paper: M.E. Bailey, J.E. Chambers, G. Hahn
Origin of sungrazers: a frequent cometary end-state
Astron. Astrophys. 257, (1992)

3. Sun-grazing comets Example: Comet C/1965 S1 Ikeya-Seki q = 0.0078 AU
More recently, the SOHO spacecraft has discovered > 1000 faint sun-grazers

4. Origin of sun-grazers Why is the perihelion distance so small?
The orbits do not pass close to Jupiter’s orbit – it’s not close encounter perturbations
We need a secular mechanism that can drain angular momentum from a cometary orbit – the Kozai mechanism

5. Kozai cycles
Long-term integrations of cometary motions often show large-scale oscillations of eccentricity and inclination due to a resonance between  and  so that  librates around 90o or 270o:
Kozai resonance
 libration

6. Theory Consider the circular restricted 3-body problem
Average the motion over the orbital periods to get the long-term (secular) behaviour
Then the semi-major axis is constant
We also have the Jacobi integral, which gives rise to the Tisserand relation: E Hz
Both a and T are constants 
is constant

7. Theory, ctd The third integral (the most complicated one!)
The average of the perturbing gravitational potential of Jupiter:
This reduces to:
One can plot curves in a (e,) diagram

8. Kozai diagrams  librations around 90o are seen for several comets
Large amplitudes in e and i are seen near the separatrix  e can get very close to 1

9. Numerical results
The amplitude in q is not constant because of intervening effects like a mean motion resonance
The comet finally falls into the Sun
Conclusion: many comets are subject to this fate, since their Hz values are rather small

10. The Nice Model
Paper:
R. Gomes, H.F.Levison, K. Tsiganis, A. Morbidelli
Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets
Nature 435, (2005)

11. Planetesimal scattering
Hyperbolic deflection changes the orbits of small bodies passing close to major planets – especially if the velocities are small
Energy and angular momentum are exchanged
After interacting with a mass of planetesimals similar to the planet’s own mass, its orbit may be significantly affected  “migration”

12. Shaping of the Solar System
The outward migration of Uranus and Neptune may explain how they could be formed rapidly enough to capture gas from the Solar Nebula
The concentration of resonant TNOs (“Plutinos”) can be explained by trapping induced by Neptune’s outward migration
But the giant planets may have started in dangerously close proximity to each other!

13. The “Nice Model”
- Distribute the planets from 5.5 to 14.2 AU (in a typical case) with PSat < 2PJup
Measure the lifetime of stray planetesimals  dispersal before the gas disk disappears
Integrate the system of planets and an external planetesimal disk of ~30 MEarth
- Migration causes crossing of the Jupiter/Saturn 2:1 mean motion resonance

14. Nice model results (1)
• When the gas disk was blown away, the planetesimal disk started ~1 AU beyond Neptune
Subsequent migration led to 2:1 resonance crossing after ~ Myr
Jupiter’s and Saturn’s eccentricities were excited
Uranus’ and Neptune’s eccentricities increased by secular resonances  close encounters  U+N crossed the disk and migrated outward, removing the planetesimals
Gomes et al. (2005)
The spikes indicate the
lifetimes of individual
planetesimals

15. Nice model results (2)
Rapid clearing of the outer disk  heavy cometary + asteroidal bombardment of the Moon
The clearing also caused migration of Jupiter and Saturn, and sweeping of secular resonances through the Main Belt  heavy asteroidal bombardment
This episodic bombardment fits with lunar crater statistics and may explain the “Late Heavy Bombardment”
Planet
orbits
Lunar
impacts
Gomes et al. (2005)

16. The Late Heavy Bombardment
Lunar stratigraphic units were sampled by the Apollo & Luna missions and radioactively dated
These units are typically associated with impact basins
Their ages correlate with the corresponding crater densities
Near 4 Gyr of age, there is a dramatic upturn in the crater density plot, indicating a very large flux of impacts

17. Nice model results (3)
The initial planetesimal disk must have ended at ~30 AU; thus the TNOs have been emplaced during the gravitational clearing of the disk
Any pre-existing Trojans would have been expelled during the 2:1 resonance crossing but new objects (icy planetesimals) were captured
The same holds for the irregular satellites of the giant planets

18. The Nice Model: Simulation

19. NEO close encounters Paper:
A. Milani, S.R. Chesley, P.W. Chodas, G.B. Valsecchi
Asteroid Close Approaches: Analysis and Potential Impact Detection
Asteroids III (eds. Bottke et al.), pp (2002)

20. Target plane
The plane containing the Earth that is perpendicular to the incoming asymptote of the osculating geocentric hyperbola (also called b-plane)
The plane normal to the geocentric velocity at closest approach is called the Modified Target Plane (MTP)
Question: for a predicted encounter, when the asteroid passes the target plane, is it inside or outside the collisional cross-section of the Earth?

21. Gravitational focussing
If rE is the Earth’s physical radius and bE is the radius of the Earth’s collisional cross-section:
where ve is the Earth’s escape velocity:

22. Encounter prediction
Suppose the asteroid has been observed around a certain time, and the encounter is predicted for several decades later
We need to determine the confidence region of the orbital elements from the scatter of the residuals of the best-fit solution
Then this needs to be mapped onto an uncertainty ellipse on the target plane

23. Encounter prediction, ctd
This mapping is most sensitive to the uncertainty in the semi-major axis or mean motion  uncertain timing of the future encounter  the ellipse is very elongated (“stretching”) and narrow
If it crosses the Earth, the risk of collision is calculated using the probability distribution along the ellipse

24. Target plane coordinates
MOID = Minimum Orbit Intersection Distance
This is the smallest approach distance (minimum distance between the two orbits in space)
If the timing is not “perfect”, the actual miss distance may be larger

25. Case of 1997 XF11 Discovered in late 1997
An 88-days orbital arc observed until March 1998 indicated a very close approach in 2028
Hot debate among astronomers
Impact is practically excluded, but the MOID is very small

26. Resonant returns
If the timing of the 2028 encounter with 1997 XF11 is near the MOID configuration, gravitational perturbation by the Earth may put the object into mean motion resonance, and impact may occur at a later “resonant return”
This is a common feature, and most impacts are likely due to resonant returns

27. Close encounter model
Approximate treatment as hyperbolic deflections (scattering problem)
The approach velocity U is conserved:
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

28. Keyholes
Constant values of ’ after an encounter are found on circles in the b-plane
Resonant returns correspond to special circles
If the uncertainty ellipse cuts such a circle, a resonant return is possible
The intersection of the ellipse and the circle is called a “keyhole”
Keyholes are very small due to the stretching that occurs until the return
1999 AN10