Transcript Kozai Migration

Kozai Migration
Yanqin Wu
Mike Ramsahai
The distribution of orbital
periods
• P(T) increases from
120 to 2000 days
• Incomplete for
longer periods
• Clear excess at 3-4
days
Evidence for tidal evolution
• Maximum e declines
with a: tidal
circularization
• The two highest e
planets are in binary
star systems
– (plot stolen from
former CITA
postdoc Phil
Armitage)
Migration: Disk-Planet
Interactions
• A planet embedded in a gas disk excites
spiral density waves via gravitational
interactions; like the wake from a boat, these
waves exert a torque on the planets
• In type I migration (low mass planets) the gas
occupies orbits coincident with the planet
• In type II migration (Jupiter mass planets) the
gas is pushed away from the planet, leaving a
gap. This slows the migration rate to match
the viscous evolution time scale of the disk
Why Kozai Migration?
• The masses of close in planets tend to be
smaller than those at larger a (disk migrationsee plot)
• However, the pile up of planets at three days,
where tidal effects are strong, is very
suggestive.
• Similarly, the finding that the most massive
planets in 3-4 day orbits are in binary
systems is suggestive, and consistent with
Kozai
• Finally, the fact that the highest e planets are
in binary systems is evidence that Kozai is
operating
Why Kozai Migration?
• The pile up of planets at three days, where tidal
effects are strong, is very suggestive.
• The masses of close in planets tend to be smaller
than those at larger a (see plot)
• Similarly, the finding that the most massive planets in
3-4 day orbits are in binary systems is also
suggestive
• The fact that the highest e planets are in binary
systems is evidence that Kozai is operating, at least
in those (relatively large a) systems
Celestial Mechanics
• a semi major axis; e eccentricity; I inclination
• f ~ nt with n2=GM/a3;  longitude of
periapse;  longitude of node
• r = a(1-e2)/[1+e cos(f-)]
• E/mp = 1/2 v2 - GM/r = GM/2a
• L/mp = [Gma(1 -e2)]1/2
• r2df/dt = L/mp
• rp = a(1-e)
K K1 is the line of nodes
I, mutual inclination
Celestial Mechanics
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a semi-major axis; e eccentricity; I inclination
f ~ nt;  longitude of periapse;  longitude of node
r = a(1-e2)/[1+e cos(f-)]
Ep/mp = 1/2 v2 - GM/r = GM/2a
Lp/mp = [Gma(1 -e2)]1/2
r2df/dt = L/mp
rperi = a(1-e)
How does Kozai work?
• The effect works on times much longer than the orbital
period of either object, so imagine that the mass of both
planet and secondary star is distributed in a ring around
the primary.
• If the rings have a mutual inclination i, they will exert a
torque on each other; T is perpendicular to L, so the
orbits exchange angular momentum, but total L=const.,
as is the component of L p and LB along L. These are
called Kozai constants.
Kozai constants
Ltotal
Lp
How does Kozai work?
• For low i, the apsidal line (from star to periapse)
undergoes a prograde precession, while the nodal
line (where the two orbit planes intersect) undergoes
a retrograde precession with the same frequency
• As a result there are only small oscillations of i and e
How does Kozai work?
• For high enough i, the apsidal line precession
slows and eventually reverses, becoming
prograde. A resonance occurs when the
precession rate of the apsidal line equals that
of the nodal line.
• As noted above, the mutual torque of the two
rings is always along the nodal line, so it
cannot affect the z component of Lp; the
projection Lpz of Lp along L is fixed, but |Lp|
will oscillate, as angular momentum is
drained out of and backLinto the orbit of the
total
planet
Lp
How does Kozai work?
• As Lp = [a(1-e2)]1/2, a decrease in i
corresponds to a decrease in Lp, which
in turn corresponds to an increase in e.
• However, Lpz is constant, so Lp will not
go to zero; when Lp=Lpz the angular
moment will begin to flow from the outer
orbit back into the planet’s orbit
Ltotal
Lp
Tides
• The Kozai mechanism reduces e, but does
not affect the semimajor axis a.
• However, the periapse rp=a(1-e) (the closest
approach to the star) does shrink
• For I large enough, rp can approach the stellar
radius
• When it does, the star raises a substantial
tidal bulge on the planet; since the planet is
no longer spherical, the mutual gravitational
attraction of the planet and star is no longer
given by just 1/r2. The extra force induces a
rapid precesion of the apsidal line, halting the
Kozai-induced reduction of the periapse
Tides
• While the periapse is held at a small
value, tidal dissipation removes energy
but not angular momentum from the
orbit of the planet; hence a is reduced
but rp is held fixed. This is effectively
migration.
Binary Star Model Ingredients
• The model assumes that the mutual
inclination of proto-planetary disk and binary
orbit are random (there is weak evidence for
this)
• P(aB) and P(q) are take from observations
• The frequency of planets in binary systems is
assumed to be similar to that around single
stars (some evidence for this from current
surveys)
Numerical Results
Numerical Results (different Q)
Predictions
• Kozai migration implies that many short period
planets will be in binary star systems; the
frequency of binarity will be higher for short
period systems than for long period systems
• Kozai planets will inhabit dynamically empty
systems
• Kozai migration leaves planets with a
substantial inclination to the spin of the primary
star (Rossiter-McLaughlin effect)
• Transiting Kozai planets will have secondary
stars orbiting near the plane of the sky.
Kozai In Single Star Systems
• The Kozai mechanism does not require
a stellar companion to work: a second
planet can also do the job
• The trick is to get a large mutual
inclination
• Several groups have studied this
Kozai Combined with
Scattering
Why should you believe a
theorist?
• “An observational result should not be
believed until it is confirmed by theory”
• A theoretical result should not be
believed until it is confirmed by
observation
• A numerical result should not be
believed until it is confirmed by both
Rossiter-McLaughlin Effect
Rossiter-McLaughlin Effect
Fabrycky & Winn 2009
HAT-P-7b retrograde orbit
Winn et al. 0908.1672
Conclusions
• The Kozai mechanism must operate in binary star
systems with single planets
• It will produce highly inclined (including retrograde)
orbits
• It can also operate in multiplanet systems, possibly
with mutual inclinations generated by planet-planet
scattering.
• Such highly inclined orbits are now seen (4/14, 2
retrograde)