By Steve Lubow, Observatory Scientist at STScI
The orbits of a particle about a central point mass are well known and have analytic solutions that were determined by Newton in 1687. In the case that a second point mass is present, forming a binary system, the particle orbits are much more complicated and simple analytic descriptions are possible for only special cases, such as the famous Lagrange points. This so-called three-body problem has been the subject of many studies for several centuries by Euler, Lagrange, Poincare, and others. The three-body problem has many astronomical applications.
In the early 1960s, Kozai and Lidov independently discovered a fairly general class of orbits in the three-body problem that has some peculiar properties. Kozai was motivated by his studies of asteroids that orbit under the influence of the Sun and Jupiter. Lidov was motivated by the analysis of orbits of artificial satellites at the beginning of the Russian space program. Consider a particle that orbits about a single central object whose motion is disturbed by a distant companion. One might think the effects of the companion do not matter much, since it is far away. However, they found that over time its effects can build up to produce strong changes in the particle’s orbit. They showed that a particle orbit that is initially circular and sufficiently inclined would undergo tilt and eccentricity oscillations. During the oscillations, the particle orbit eccentricities can become quite large for sufficiently inclined initial orbits. For example, for an initial orbit tilt of 60 degrees, an initially circular orbit reaches an eccentricity of about 0.75.
The key to understanding the Kozai-Lidov (KL) effect is that the vertical component (perpendicular to the binary orbital plane) of the particle’s angular momentum is nearly conserved, while the particle’s orbit plane undergoes tilt oscillations. As the orbit tilt evolves to a lower inclination angle over part of the oscillation cycle, its eccentricity must grow to maintain the same vertical angular momentum.
There have been many applications of this KL effect apart from these original motivations. These include triple star systems (Eggleton & Kiseleva-Eggleton 2001; Fabrycky & Tremaine 2007), extrasolar planets with inclined stellar companions (Wu & Murray 2003; Takeda & Rasio 2005), inclined planetary companions (Nagasawa et al. 2008), merging supermassive black holes (Blaes et al. 2002), stellar compact objects (Thompson 2011), and blue straggler stars (Perets & Fabrycky 2009).
Early in 2014, I was working with Rebecca Martin, a Sagan Fellow at the University of Colorado, and other collaborators on explaining some fluid (SPH) simulation results that she had obtained for Be star disks. The disks were taken to orbit about Be stars that have neutron star companions on an eccentric orbits, as suggested by observations. In addition, the simulations considered the disk to be initially circular and inclined with respect to the orbit plane of the binary. The simulations showed that the disk became substantially eccentric. We had initially thought that the disk eccentricity could be explained by some tidal effects that had been found in previous simulations and had been explained analytically. These effects were analyzed for a disk that is coplanar with the binary. For example, an eccentric binary has an eccentric component of its tidal potential that can induce eccentricity in a disk. We thought the latter was the most plausible explanation for the eccentricity, as we discussed in our paper (Martin et al 2014a).
However, I was suspicious of our explanation because I thought this tidal effect would become smaller with higher orbit inclination, while we found that eccentricity was larger at higher inclinations. So I suggested that Rebecca try numerically to determine the orbits of particles, rather than performing the computer-intensive SPH simulations. The result was that the particles underwent KL oscillations — obvious in hindsight. To confirm this result, Rebecca performed simulations with a circular orbit companion, for which the KL effect can operate, while the eccentric tidal effect would not. She found that the eccentricity growth was still present. We found that the disk undergoes coherent tilt oscillations much like a rigid body. The oscillation period agrees well with the expectations of KL theory. This is the first time that the KL effect has been found to operate on a fluid (Martin et al 2014b).
During tilt oscillations, the disk eccentricity gets fairly high, but it is reduced by disk dissipation. After a few KL oscillations, the disk evolves to an eccentric state and at the critical minimum tilt angle for KL oscillations of about 40 degrees with respect to the binary orbit plane (see Figure 1). With Rice University postdoctoral fellow Wen Fu, we are exploring the range of parameters for which the KL effect operates on disks. Misaligned disks are likely common in young wide binary systems and so the KL effect may have an important influence on the evolution of some protoplanetary disks.
Figure 1: Evolution of disk eccentricity and inclination in degrees as a function of time in binary orbital periods for a disk that orbits a member of an equal mass binary binary system that has an initial tilt of 60 degrees from the binary orbit plane and is initially circular.
The video (found at this link) shows the evolution of the disk in Figure 1 that undergoes KL oscillations viewed in different planes. The binary lies in the X-Y plane, while the disk is initially tilted by 60 degrees with respect to this plane. The disk precesses about the Z-axis. The disk eccentricity is apparent from the displacement of the central star from the disk center. Note that some disk mass is transferred to the companion as a consequence of the reduction of tidal forces at high inclination (Lubow et al 2014) and because the disk becomes eccentric.
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