Planets in Binary Systems

Lucasfilm

In the previous post, I discussed how the oscillating orbits of planets within binary star systems could create extreme climate cycles as a result of the fluctuating energy input.  What exactly is going on in these systems, though? In this post I seek to describe more technically what is referred to as the Kozai-Lidov mechanism by which a second star can alter the shape of an inner planet's orbit.

In binary systems, planets can orbit in two configurations. The first, like that on Tatooine in Star Wars, has the two stars at the center with the planet orbiting around both. For these circumbinary systems, dynamical stability requires that the planet orbit at a distance significantly greater than the separation between the two stars.

The second configuration, which my research focuses on, has the planet orbiting one of the two stars, with the second circling the system further out. Under these circumstances, the gravitational influence of this stellar companion can augment the eccentricity of the planet's orbit  (science speak for making the orbit more elliptical) via the Kozai-Lidov mechanism.  For this to occur, the orbits of the secondary star and the planet must be inclined relative to each other by a sufficiently large angle (approximately 40 degrees or greater).

In the setup above, the planet (p) orbits the primary star (1). The secondary star (2) orbits with a large inclination angle i relative to the orbit of the planet.  The angular momentum of the planet in the direction parallel to that of Star 2 is conserved.

Under this configuration, the angular momentum of the planet's motion in the direction parallel to that of the secondary star is conserved. This quantity, denoted as Lz, depends on the eccentricity of the planet's orbit e along with the inclination i.  As Star 2 tugs on the planet, the decreasing inclination between the two orbits is traded for eccentricity, and the planet's orbit becomes more elliptical.  This dynamical exchange between eccentricity and inclination occurs in cycles with a frequency that depends on the masses of the stars and the radii of the two orbits.  The oscillation occurs more rapidly for a system with a more massive secondary star, a less massive primary star, and a planet which orbits further out.  This means that the effect is most prominent when the secondary star has a substantial gravitational influence relative to the primary.  The maximum eccentricity that the planet can reach depends only on the initial inclination angle.  If the orbit of the secondary star is eccentric, however, Lz is not strictly conserved and arbitrarily high eccentricities can be attained.  The phenomenon was originally studied with reference to asteroids in highly elliptical orbits that were influenced by Jupiter's gravity, but in recent years has been applied to the dynamics of extrasolar planets.  

These cycles can be visualized in the plot below.  An eccentricity of zero corresponds to a circular orbit, while an eccentricity of 1 is the maximum limit.  Higher eccentricity corresponds to a more elliptical orbit. The setup involves a planet orbiting a star 1.4 times as massive as our Sun at a distance twice that of Earth. The secondary star is 0.4 times as massive as the Sun and orbits at a distance 20 times greater than that of Earth's orbital radius.  Under this configuration, the planet's eccentricity cycles from nearly zero to approximately 0.4 over a timescale on the order of 10,000 years. At this eccentricity, the average stellar flux received by the planet would be about 1.1 times greater than if the orbit were circular. If Earth's orbit were this elliptical, its annually averaged temperature would be warmer by about 5 degrees Celsius, although seasonal temperature variation would be substantially greater.  Within one orbit, the planet's energy input would vary by a factor of over five between the nearest and farthest points from the primary star.

So what are the implications for planetary habitability?  One potential research question would be whether or not these cycles could thaw a planet like Earth out of a state of global glaciation. On the other end, these spikes in eccentricity may render an otherwise habitable planet too warm to sustain liquid water.  Under these circumstances, would the planet retain its total water supply in the atmosphere, or would it lose vapor to space and dry out over the course of its lifetime? Such questions pertaining to the habitability of these planets have yet to be investigated.