Kozai Cycles with Tidal Friction

The meatiest paper of my PhD thesis focused on this process. (Read the details there.)

Suppose you have a gas-giant planet orbiting a star at 5 AU, and way out at 1000 AU a stellar binary companion lurks in a high-inclination orbit.
Here's what can happen...

This movie is an attempt to visualize 3-dimensional orbital and spin evolution. The inertial frame is indicated by the blue coordinates, centered on the star HD 80606, with the binary companion HD 80607 in the X-Y plane. The planet's orbit is shown in black, with an arrowhead indicating the position of distance of closest approach (periastron) and the sense of the orbital trajectory. The black stick is the vector normal to the orbital plane. The red stick is the spin axis of the host star at the center. The combatants in the center of the figure have shadows on the three walls of the arena, to aid in visualizing the 3-D motion.

At first, the movie plays 80 Myr of evolution slowly, during which the planetary orbit exhibits 3 Kozai cycles: oscillations in eccentricity and inclination. These are nearly periodic, so the movie is sped up to see the long-term effect of tidal friction during the eccentricity maxima. The orbit is slowly drawn in. The eccentricity cycles are first quenched, and we see it (today) in a monotonically damping state. In the future the eccentricity damps out completely, forming a hot Jupiter.

As the planetary orbit undergoes Kozai oscillations and migrates through tidal friction, at first its host star maintains its original orientation. Its distant planet simply supplies no net torque to its rotational bulge. As the migration plays out, eventually the spin takes note, and starts precessing very quickly around the planetary orbit normal. The final result is an angle of 50 degrees between the planetary orbit and the stellar spin (the angle between the green and red sticks), the sky projection of which has recently been measured via spectroscopy during transits.


This process also works for triple stars, and observations indicate that it has delivering a large fraction of close binaries to their close orbits. The time history of how close-binaries are built up is relevant to the formation of blue stragglers, so I have computed the e-logP diagram as a function of time, and studied whether the proper inital cut-off to period is, say, 0.2 days, 6 days (preferred), or 30 days.


This process makes two predictions regarding angles in the final systems.

First, the outer binary and the inner orbit should prefer the values of ~40 degrees and ~140 degrees; this is a clear hallmark of Kozai cycles at work. Some of the first citations to this paper were optical interferometry observations that confirmed this expectation for two systems. A satisfactory observational study and analysis testing this prediction may require new telescopes (and therefore require many many years).

Second, planetary systems are usually assumed to form in a disk, which itself determines the spin direction of the star. Therefore alignment between the planetary orbits and the host star's spin is expected, and it is observed in the solar system. However, Kozai cycles from the binary companion can torque the planetary orbit through many many cycles before it settles as a hot Jupiter. Therefore spin-orbit mis-alignment may be induced. Surprisingly, even retrograde (inclination > 90 degrees) misalignment angles would be rather common in this process, which might be guessed from the movie above. Once the planet-star torques dominate the planet-companion torques, the inclination is sealed in. The planet cannot torque up (synchronize or parallelize) its host star spin, via tidal friction in the star. If it could, the planet would be shortly consumed. This prediction is interesting because the Rossiter-McLaughlin effect, a spectroscopic distortion observable during planetary transits, can probe the statistics of (mis)alignment, and it is a promising way for deciding how exo-Jupiters migrate to orbits of just a few days.

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