The discovery of over 400 extrasolar planets allows us to statistically test our understanding of formation and dynamics of planetary systems via numerical simulations. Traditional N-body simulations of multiple-planet systems without gas disks have successfully reproduced the eccentricity (e) distribution of the observed systems, by assuming that the planetary systems are relatively closely packed when the gas disk dissipates, so that they become dynamically unstable within the stellar lifetime. However, such studies cannot explain the small semi-major axes (a) of extrasolar planetary systems, if planets are formed, as the standard planet formation theory suggests, beyond the ice line. In this paper, we numerically study the evolution of three-planet systems in dissipating gas disks, and constrain the initial conditions that reproduce the observed semi-major axis and eccentricity distributions simultaneously. We adopt the initial conditions that are motivated by the standard planet formation theory, and self-consistently simulate the disk evolution, and planet migration by using a hybrid N-body and 1D gas disk code. We also take account of eccentricity damping, and investigate the effect of saturation of corotation resonances on the evolution of planetary systems. We find that the semi-major axis distribution is largely determined in a gas disk, while the eccentricity distribution is determined after the disk dissipation. We also find that there may be an optimum disk mass which leads to the observed a-e distribution. Our simulations generate a larger fraction of planetary systems trapped in mean-motion resonances (MMRs) than the observations, indicating that the disk's perturbation to the planetary orbits may be important to explain the observed rate of MMRs. We also find much lower occurrence of planets on retrograde orbits than the current observations of close-in planets suggest.
Deep Dive into Unstable Planetary Systems Emerging Out Of Gas Disks.
The discovery of over 400 extrasolar planets allows us to statistically test our understanding of formation and dynamics of planetary systems via numerical simulations. Traditional N-body simulations of multiple-planet systems without gas disks have successfully reproduced the eccentricity (e) distribution of the observed systems, by assuming that the planetary systems are relatively closely packed when the gas disk dissipates, so that they become dynamically unstable within the stellar lifetime. However, such studies cannot explain the small semi-major axes (a) of extrasolar planetary systems, if planets are formed, as the standard planet formation theory suggests, beyond the ice line. In this paper, we numerically study the evolution of three-planet systems in dissipating gas disks, and constrain the initial conditions that reproduce the observed semi-major axis and eccentricity distributions simultaneously. We adopt the initial conditions that are motivated by the standard planet
Out of over 360 planetary systems discovered so far, about 12.4% are known to be multiplanet systems (http://exoplanet.eu/). Also, recent observations have started revealing that many of the detected planets are accompanied by a planet on a further orbit (e.g. Wittenmyer et al. 2007;Wright et al. 2007). It will become increasingly more important to understand the formation and evolution of multiplanet systems, which can explain the observed properties of extrasolar planetary systems.
Recent numerical N-body simulations of planetary systems without a gas disk demonstrated that dynamical instabilities occurring in the multiplanet systems, which are characterized by orbital crossings, collisions, and ejections of planets, could increase planetary eccentricities (e) efficiently (e.g., Rasio et al. 1996;Weidenschilling & Marzari 1996). These studies successfully reproduced the observed eccentricity distribution of extrasolar planets (Ford & Rasio 2008;Chatterjee et al. 2008;Jurić & Tremaine 2008, from here on C08, and JT08, respectively.)
Such N-body simulations also suggest that the planetplanet interactions alone cannot explain small semimajor axes (a) of the observed planets, if giant planets are formed beyond the ice line as expected from the standard planet formation theory. More specifically, starting with giant planet systems beyond 3 AU, C08 found that it is difficult to scatter planets within ∼ 1 AU. This is because planet-planet interactions are not particularly efficient in shrinking the planetary orbits.
The disk-planet interactions, on the other hand, are known to decrease semi-major axes of planets efficiently (Ward 1997). The overall effect of such interactions on the orbital eccentricity is highly uncertain, and depends on a detailed disk structure, as well as planetary masses. Generally, disk-planet interactions lead to eccentricity damping, but for planets massive enough to open a clean gap in the disk, eccentricity can increase rapidly depending on the level of saturation of corotation resonances (Goldreich & Sari 2003;Moorhead & Adams 2008). However, hydrodynamic simulations show that the disk-planet interactions typically lead to e < 0.2 (e.g., D’ Angelo et al. 2006).
The time to dynamical instability scales with the distance between planets (e.g., C08). Thus, all the Nbody studies on planet-planet scattering assume initially dynamically active planetary systems (i.e., distance between planets being less than about a few Hill radii). However, with the aid of a gas disk, planets which would not easily reach dynamical instability could experience strong interactions. For example, when planets are embedded in the inner cavity of a disk, the surrounding disk would push the outer planet closer to the inner one, triggering the dynamical instability (e.g., Adams & Laughlin 2003;Moorhead & Adams 2005). The orbital eccentricity can also be increased if planets are trapped in mean motion resonances (MMRs) during such a convergent migration (e.g., Snellgrove et al. 2001;Lee & Peale 2002).
Alternatively, when a gas disk annulus is left between planets, the eccentricities of planets could still increase by repeated resonance crossings due to divergent migration (Chiang et al. 2002). Thus, whether the combined effects of disk-planet and planet-planet interactions would lead to eccentricity excitation, or damping should be studied carefully. One of the goals of our study is to verify the initial assumptions of N-body studies (i.e., planets are on nearly circular, and coplanar orbits when the gas disk is around), and figure out whether the eccentricity distribution is largely determined before, or after the disk dissipation.
In this paper, we numerically study the evolution of three-planet systems in a dissipating gas disk, and constrain the “initial” conditions of planetary systems which can reproduce the a and e distributions simultaneously. We calculate the disk-planet interactions directly so that the disk and planetary orbits evolve self-consistently. Also, we take account of the effect of saturation of corotation resonances on eccentricity damping. We introduce the numerical methods in Section 2, and the initial conditions in Section 3. In Section 4, we show that the observed ae scattered plot can be reproduced well for a reasonable range of disk masses. We also discuss the mass distribution, and mean-motion resonances for representative cases. Finally in Section 5, we compare this work with some recent observations, and summarize our results.
To simulate multiplanet systems in gas disks, we use a hybrid code which combines the symplectic N-body integrator SyMBA (Duncan et al. 1998) with a onedimensional gas disk evolution code (Thommes 2005). SyMBA utilizes a variant of the so-called mixed-variable symplectic (MVS) method (Wisdom & Holman 1991), which treats the interaction between planets as a perturbation to the Keplerian motion around the central star, and handles close encounters between b
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