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 migration-see 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 nodesI, mutual inclination

Celestial Mechanics

• 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 Lp and LB along L. These are called Kozai constants.

Kozai constants

Lp

Ltotal

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

• 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 back into the orbit of the planet

How does Kozai work?

Lp

Ltotal

• 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

How does Kozai work?

Lp

Ltotal

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)