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

Kozai cycles Paper: M.E. Bailey, J.E. Chambers, G. Hahn Origin of sungrazers: a frequent cometary end-state Astron. Astrophys. 257, 315-322 (1992)

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

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

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

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

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

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

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

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, 466-469 (2005)

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”

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!

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

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 ~200-900 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

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)

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

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

The Nice Model: Simulation

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. 55-69 (2002)

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?

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:

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

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

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

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

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

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

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