two for one deals?
The Gliese 876 system is remarkable for a number of reasons. It makes a mockery of the notion that the minimum-mass solar nebula has a universal validity. It harbors one of the lowest-mass extrasolar planets known (discovered by our own Eugenio Rivera). And of course, the outer two planets are famously caught in a 2:1 mean motion resonance, in which the inner 0.8 Jupiter-mass planet makes (on average) exactly two trips around the red dwarf for every one trip made by the outer 2.5 Jupiter-mass planet.
As users of the console know, the planet-planet interactions between the Gliese 876 planets are strong enough so that one needs a self-consistent dynamical fit to the system. Even on the timescale of a single outer planet orbit, the failure of the Keplerian model can be seen on a 450-pixel wide .gif image:
The following three frames are from a time-lapse .mpg animation of the Gliese 876 system over a period of roughly one hundred years:
Each frame strobes the orbital motion of the planets at 50 equally spaced intervals which subdivide the P~60 day period of the outer planet. Upon watching the movie, it’s clear that the apsidal lines of the outer two planets are swinging back and forth like a pendulum. This oscillation has an amplitude (or libration width) of 29 degrees, and acts like a fingerprint identifier of the Gliese 876 system.
The derangement of the orbits is reflected in their continual inability to maintain an exact 2:1 orbital commensurability. The first figure up above shows that when planet c has finished exactly two orbits, it has already managed to lap planet b, which was still dawdling down Boardwalk prior to passing GO.
Planet b, however, doesn’t always run slow. The gravitational perturbations between the two planets provide a second pendulum-like restoring action which prevents the bodies from straying from the average period ratio of 2:1, which, over the long term, is maintained exactly. The degree to which the orbits themselves librate, combined with the planets’ abilities to run either ahead or behind exact commensurability is captured by the resonant arguments of the configuration. These can be defined as,
where the lambdas are mean longitudes and the curly pi’s are the longitudes of periastron. The two resonant arguments capture the simultaneous libration of the mean motions and the apsidal lines. The smaller the arguments, the more tightly the system is in resonance.
In the Gliese 876 system, the resonant arguments are both librating with amplitudes of less than 30 degrees. This is evidence that a dissipative mechanism was at work during the formation of the system. Interestingly, however, when one looks at the other extrasolar planetary systems that are thought to be in 2:1 resonance, one finds that the libration amplitudes in every case are much larger. In fact, in the HD 73526 system and in the HD 128311 system, only one of the arguments is librating, while the other is circulating. In this state of affairs, the apsidal lines act like a pendulum that is swinging over the top. In addition, the orbital eccentricites are higher, and the sum of planet-planet activity is strikingly greater (see this animation of the evolution of the HD 128311 system).
A gas disk seems to be the most likely mechanism for pushing a planetary system into mean-motion resonance. Protoplanetary disks are likely, however to experience turbulent density fluctuations. These density fluctuations lead to stochastic gravitational torques, which provide a steady source of orbital perturbations to any planets that are embedded in a disk. For a reasonable spectrum of turbulent fluctuations, it turns out that it’s rather difficult to wind up with a planetary system that is as deeply in resonance as Gliese 876. The conclusion, then, is that Gliese 876-like configurations should be quite rare. Indeed, 2:1 resonances of every stripe should constitute only a minor fraction of planetary systems, and the majority that do exist should either large libration widths or only a single argument in resonance.
If you’re interested in more detail, we’ve submitted a paper that goes into much more detail (Adams, Laughlin & Bloch, ApJ, 2008 Submitted).