John E. Chambers

Short-Term Dynamical Interactions among Extrasolar Planets

We show that short-term perturbations among massive planets in multiple planet systems can result in radial velocity variations of the central star that differ substantially from velocity variations derived assuming the planets are executing independent Keplerian motions. We discuss two fitting methods that can lead to an improved dynamical description of multiple planet systems. In the first method, the osculating orbital elements are determined via a Levenberg-Marquardt minimization scheme driving an N-body integrator. The second method is an improved analytic model in which orbital elements are allowed to vary according to a simple model for resonant interactions between the planets. Both of these methods can determine the true masses for the planets by eliminating the sin i degeneracy inherent in fits that assume independent Keplerian motions. We apply our fitting methods to the GJ 876 radial velocity data and argue that the mass factors for the two planets are likely in the 1.25-2.0 range.

Terrestrial Planet Formation around Individual Stars within Binary Star Systems

We calculate herein the late stages of terrestrial planet accumulation around a solar-type star that has a binary companion with semimajor axis larger than the terrestrial planet region. We perform more than 100 simulations to survey binary parameter space and to account for sensitive dependence on initial conditions in these dynamical systems. As expected, sufficiently wide binaries leave the planet formation process largely unaffected. As a rough approximation, binary stars with periastron qB > 10 AU have a minimal effect on terrestrial planet formation within ~2 AU of the primary, whereas binary stars with qB 5 AU restrict terrestrial planet formation to within ~1 AU of the primary star. Given the observed distribution of binary orbital elements for solar-type primaries, we estimate that about 40%-50% of the binary population is wide enough to allow terrestrial planet formation to take place unimpeded. The large number of simulations allows us to determine the distribution of results—the distribution of plausible terrestrial planet systems—for effectively equivalent starting conditions. We present (rough) distributions for the number of planets, their masses, and their orbital elements.

How Eccentric Orbital Solutions Can Hide Planetary Systems in 2:1 Resonant Orbits

The Doppler technique measures the reflex radial motion of a star induced by the presence of companions and is the most successful method to detect exoplanets. If several planets are present, their signals will appear combined in the radial motion of the star, leading to potential misinterpretations of the data. Specifically, two planets in 2:1 resonant orbits can mimic the signal of a single planet in an eccentric orbit. We quantify the implications of this statistical degeneracy for a representative sample of the reported single exoplanets with available data sets, finding that (1) around 35% of the published eccentric one-planet solutions are statistically indistinguishable from planetary systems in 2:1 orbital resonance, (2) another 40% cannot be statistically distinguished from a circular orbital solution, and (3) planets with masses comparable to Earth could be hidden in known orbital solutions of eccentric super-Earths and Neptune mass planets.

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 Jupiters 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.

A Dynamical Analysis of the 47 Ursae Majoris Planetary System

. These mass and period ratios suggesta possible kinship to the Jupiter-Saturn pair in our own solar system. We explore thecurrent dynamical state of this system with numerical integrations, and compare theresults with analytic secular theory. We find that the planets in the system are likelyparticipating in a secular resonance in which the difference in the longitudes of pericenterlibrates around zero. Alternately, it is possible that the system is participating in the7:3 mean motion resonance (in which case apsidal alignment does not occur). Usinga self-consistent fitting procedure in conjunction with numerical integrations, we showthat stability considerations restrict the mutual inclination between the two planets to∼ 40 degrees or less, and that this result is relatively insensitive to the total mass ofthe two planets. We present hydrodynamical simulations which measure the torquesexerted on the planets by a hypothesized external protoplanetary disk. We show thatplanetary migration in response to torques from the disk may have led to capture of thesystem into a 7:3 mean-motion resonance, although it is unclear how the eccentricities ofthe planets would have been damped after capture occured. We show that Earth-massplanets can survive for long periods in some regions of the habitable zone of the nominalco-planar system. A set of planetary accretion calculations, however, shows that it isunlikely that large terrestrial planets can form in the 47 UMa habitable zone.

The stability of the orbits of terrestrial planets in the habitable zones of known exoplanetary systems

We show that terrestrial planets could survive in variously restricted regions of the habitable zones of 47 Ursae Majoris, Epsilon Eridani, and Rho Coronae Borealis, but nowhere in the habitable zones of Gliese 876 and Upsilon Andromedae. The first three systems between them are representative of a large proportion of the 90 or so extrasolar planetary systems discovered by mid-2002, and thus there are many known systems worth searching for terrestrial planets in habitable zones. We reach our conclusions by launching putative Earth-mass planets in various orbits and following their fate with a mixed-variable symplectic integrator.

Planet Formation with Migration

In the core-accretion model, gas-giant planets form solid cores that then accrete gaseous envelopes. Tidal interactions with disk gas cause a core to undergo inward type I migration in 104-105 yr. Cores must form faster than this to survive. Giant planets clear a gap in the disk and undergo inward type II migration in <106 yr if observed disk accretion rates apply to the disk as a whole. Type II migration times exceed typical disk lifetimes if viscous accretion occurs mainly in the surface layers of disks. Low turbulent viscosities near the midplane may allow planetesimals to form by coagulation of dust grains. The radius r of such planetesimals is unknown. If r < 0.5 km, the core formation time is shorter than the type I migration timescale, and cores will survive. Migration is substantial in most cases, leading to a wide range of planetary orbits, consistent with the observed variety of extrasolar systems. When r ~ 100 m and the midplane α ~ 3 × 10-5, giant planets similar to those in the solar system can form.

STELLAR ELEMENTAL ABUNDANCE PATTERNS: IMPLICATIONS FOR PLANET FORMATION

The solar photosphere is depleted in refractory elements compared to most solar twins, with the degree of depletion increasing with an elements condensation temperature. Here, I show that adding 4 Earth masses of Earth-like and carbonaceous-chondrite-like material to the solar convection zone brings the Suns composition into line with the mean value for the solar twins. The observed solar composition could have arisen if the Suns convection zone accreted material from the solar nebula that was depleted in refractory elements due to the formation of the terrestrial planets and ejection of rocky protoplanets from the asteroid belt. Most solar analogs are missing 0-10 Earth masses of rocky material compared to the most refractory-rich stars, providing an upper limit to the mass of rocky terrestrial planets that they possess. The missing mass is correlated with stellar metallicity. This suggests that the efficiency of planetesimal formation increases with stellar metallicity. Stars with and without known giant planets show a similar distribution of abundance trends. If refractory depletion is a signature of the presence of terrestrial planets, this suggests that there is not a strong correlation between the presence of terrestrial and giant planets in the same system.

Symplectic Integrator Algorithms for Modeling Planetary Accretion in Binary Star Systems

We derive and test two new symplectic integrator algorithms, suitable for studying planetary accretion in binary star systems. The algorithms incorporate the hierarchical nature of stable planetary orbits in binary star systems. In one case, planets orbit a single star, perturbed by a distant companion; in the second case, planets orbit both binary members. Each algorithm integrates close encounters between planets symplectically using a hybrid symplectic scheme.

The fragility of the terrestrial planets during a giant-planet instability

Many features of the outer solar system are replicated in numerical simulations if the giant planets undergo an orbital instability that ejects one or more ice giants. During this instability, Jupiter and Saturns orbits diverge, crossing their 2:1 mean motion resonance (MMR), and this resonance-crossing can excite the terrestrial planet orbits. Using a large ensemble of simulations of this giant planet instability, we directly model the evolution of the terrestrial planet orbits during this process, paying special attention to systems that reproduce the basic features of the outer planets. In systems that retain four giant planets and finish with Jupiter and Saturn beyond their 2:1 MMR, we find at least an 85% probability that at least one terrestrial planet is lost. Moreover, systems that manage to retain all four terrestrial planets often finish with terrestrial planet eccentricities and inclinations larger than the observed ones. There is less than a ~5% chance that the terrestrial planet orbits will have a level of excitation comparable to the observed orbits. If we factor in the probability that the outer planetary orbits are well-replicated, we find a probability of 1% or less that the orbital architectures of the inner and outer planets are simultaneously reproduced in the same system. These small probabilities raise the prospect that the giant planet instability occurred before the terrestrial planets had formed. This scenario implies that the giant planet instability is not the source of the Late Heavy Bombardment and that terrestrial planet formation finished with the giant planets in their modern configuration.