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Extrasolar Trojans: The Viability and Detectability of Planets in the 1:1 Resonance

We explore the possibility that extrasolar planets might be found in the 1 : 1 mean motion resonance, in which a pair of planets share a time-averaged orbital period. There are a variety of stable co-orbital configurations, and we specifically examine three different versions of the 1 : 1 resonance. In the first configuration, the two planets and the star participate in tadpole-type librations about the vertices of an equilateral triangle. The dynamics of this situation resemble the orbits of Jupiter's Trojan asteroids. We show analytically that an equilateral configuration consisting of a star and two equal-mass planets is linearly stable for mass ratios μ = 2mpl/(2mpl + M*) < 0.03812. When the equilateral configuration is subjected to larger perturbations, a related 1 : 1 resonance occurs. In this second family of configurations, the planet pair executes horseshoe-type orbits in which the librating motion in the corotating frame is symmetric about a 180° separation. The Saturnian satellites Janus and Epimetheus provide a solar system example of this phenomenon. In the case of equal-mass planets, a numerical survey indicates that horseshoe configurations are stable over long periods for mass ratios μ < 0.0004, indicating that a pair of Saturn-mass planets can exist in this resonance. The third configuration that we examine is more exotic and involves a pair of planets that exchange angular momentum in a manner that allows them to indefinitely avoid close encounters. An illustrative example of this resonance occurs when one planet has a highly eccentric orbit while the other planet moves on a nearly circular orbit; the periapses are in alignment, and conjunctions occur near periapse. All three of these resonant configurations can be stable over timescales comparable to or longer than stellar lifetimes. We show that pairs of planets in 1 : 1 resonance yield characteristic radial velocity signatures that are not prone to the sin i degeneracy. Indeed, Keplerian fits to the radial velocities cannot reveal the presence of two planets in the 1 : 1 resonance. We discuss a dynamical fitting method for such systems and illustrate its use with a simulated data set. Finally, we argue that hydrodynamic simulations and torqued three-body simulations indicate that 1 : 1 resonant pairs might readily form and migrate within protostellar disks.