Transcript Seminar topics - Studentportalen

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:
T
aJ
a
2
1 e 2 cosi

a
aJ
E
Hz
2
2


1
e
cos
Both
a
and
T
are
constants

  i is constant

Theory, ctd
The third integral (the most complicated one!)
The average of the perturbing gravitational
potential of Jupiter:
This reduces to:
e2 5sin 2 isin 2   2 const.
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 ~200900 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
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
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:
v e2
bE  rE 1 2
v
where ve is the Earth’s escape velocity:

2GME
v 
rE
2
e
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:
U  3T
2
• 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