Apsidal Alignment in υ Andromedae

Abstract

One of the parameters fitted by Doppler radial velocity measurements of extrasolar planetary systems is ω, the argument of pericenter of a given planet's orbit referenced to the plane of the sky. Curiously, the ω's of the outer two planets orbiting Upsilon Andromedae are presently nearly identical:This observation is least surprising if planets C and D occupy orbits that are seen close to edge-on (sin i C , sin i D 0.5) and whose mutual inclination Θ does not exceed 20 • . In this case, planets C and D inhabit a secular resonance in which ∆ω librates about 0 • with an amplitude of ∼30 • and a period of ∼4 × 10 3 yr. The resonant configuration spends about one-third of its time with |∆ω| ≤ 10 • . If Θ 40 • , either ∆ω circulates or the system is unstable. This instability is driven by the Kozai mechanism which couples the eccentricity of planet C to Θ to drive the former quantity to values approaching unity. Our expectation that Θ 20 • suggests that planets C and D formed in a flattened, circumstellar disk, and may be tested by upcoming astrometric measurements with the FAME satellite.

A Dynamical Mechanism for Establishing Apsidal Resonance

“…A good candidate among the known exoplanetary systems to apply this theory is the u Andromedae system, which represents perhaps the most (dynamically) constrained of the known exoplanetary systems. A recent orbital solution for this system (as quoted in Chiang et al 2001) gives present eccentricities of planets C and D of 0.25 and 0.34, respectively, and yields secular dynamics for these two planets similar to that shown in Figures 1e and 1f, with an apsidal libration amplitude of 25Њ-35Њ (for relative orbital inclinations Շ20Њ). An independent orbital solution obtained previously by Stepinski, Malhotra, & Black 2000 (from an older and therefore smaller set of observational data) has orbital parameters within ∼2 j uncertainties of the more recent solution; it yields secular dynamics similar to that shown in Figures 1a and 1b After submitting this Letter, we became aware of a recent preprint by Chiang & Murray (2002) who propose an adiabatic eccentricity perturbation to explain the apsidal resonance in the u Andromedae planetary system.…”

“…A recent orbital solution for this system (as quoted in Chiang et al 2001) gives present eccentricities of planets C and D of 0.25 and 0.34, respectively, and yields secular dynamics for these two planets similar to that shown in Figures 1e and 1f, with an apsidal libration amplitude of 25Њ-35Њ (for relative orbital inclinations Շ20Њ). An independent orbital solution obtained previously by Stepinski, Malhotra, & Black 2000 (from an older and therefore smaller set of observational data) has orbital parameters within ∼2 j uncertainties of the more recent solution; it yields secular dynamics similar to that shown in Figures 1a and 1b After submitting this Letter, we became aware of a recent preprint by Chiang & Murray (2002) who propose an adiabatic eccentricity perturbation to explain the apsidal resonance in the u Andromedae planetary system. They invoke torques from an exterior massive primordial disk to provide the adiabatic eccentricity excitation to the outermost planet D. However, the adiabaticity of the perturbation is the result of a specific choice of disk parameters, which are not well constrained; a different, and perhaps equally plausible, choice of disk parameters may instead provide an impulse perturbation (E. Chiang 2002, private communication); then the secular dynamics described here would apply.…”

“…INTRODUCTION Orbital eccentricities in exoplanetary systems 1 discovered thus far are often surprisingly large and have proved to be a major puzzle in understanding these systems. At least two systems with multiple planets in eccentric orbits (u Andromedae and HD 83443) are suspected of exhibiting secular apsidal resonance (Chiang, Tabachnik, & Tremaine 2001;Wu & Goldreich 2002). Apsidal resonance is the phenomenon of phase locking of the apsidal longitudes of two orbits, such that the two planets have a common average rate of apsidal precession and the angular difference of their apsidal longitudes, D p , librates around 0.…”

Orbital migration and the frequency of giant planet formation

“…Additionally, by tidal arguments, υ And, HD 75289, HD 187123, 51 Peg, and HD 217107 are all constrained to have i > ∼ 30 degrees as well, although there are no direct observations as yet to confirm this (Trilling 2000); the Hipparcos data implies an inclination ∼25 degrees for υ And (Mazeh et al 1999). Additionally, from dynamical and astrometric arguments, the multiple-planet systems Gl876, HD 168443, and υ And likely have inclinations that are far from face-on (Laughlin & Chambers 2001;Marcy et al 2001b;Chiang et al 2001). Therefore, most systems for which inclinations are known or suspected appear to have mass factors less than around 2.…”

Excitation of Orbital Eccentricities by Repeated Resonance Crossings: Requirements

“…This is a hybrid mechanism because the required planetary migration is driven by an underlying disk, while each planet's eccentricity is directly excited by the other planet. A different hybrid scenario may have played out for planet C in the system Upsilon Andromedae (Butler et al 1999;Chiang, Tabachnik, & Tremaine 2001b). A primordial disk may have directly excited the eccentricity of the outermost planet, D; secular interactions between D and C could then have siphoned off the eccentricity of the former to grow that of the latter (Chiang & Murray 2002).…”