N-body simulations of planetary accretion around M dwarf stars

Over 300 extrasolar planets have been discovered. Target stars for exoplanet search were mostly solar-type stars (F, G, K dwarfs), although M dwarfs make up 70-80% of all stars in the galactic disk. The low luminosity of M dwarfs is disadvantageous for high-dispersion spectroscopic observation, so that radial velocity surveys have not discovered large number of planets around M dwarfs. However, as improvement of spectroscopic observations, ground-based radial velocity surveys are revealing planetary systems around M dwarfs. Due to the low luminosity of M dwarfs, habitable zones (HZ), in which a planet with sufficient amount of atmosphere can sustain liquid water on its surface, are close to the host stars (Kasting et al. 1993). The proximity of the HZs to the host stars allow for detection of planets in HZs by current radial velocity observation. In fact, two planets with minimum masses below 10M ⊕ were discovered near the HZ in a triple planet system around an M star, Gliese 581, with stellar mass M * = 0.31M ⊙ (Udry et al. 2007). The habitability of these planets (Gl 581c,d) is vigorously under discussion theoretically. In addition, gravitational microlensing survey is suited for detection of M dwarf planets, because its detection efficiency is independent of stellar luminosity. Most of the planets detected by microlensing are orbiting M dwarfs. Recent radial velocity and microlensing observations show that Jupiter-mass gas giants are generally rare (e.g., Endl et al. 2006, Johnson et al. 2007), but Neptune-mass planets are rather abundant (e.g., Beaulieu et al. 2006), compared with solar-type stars. GJ 436b is the planet that was discovered first among Neptune-mass planets (Butler et al. 2004). Transit observations revealed a planet's radius, and its combination with radial velocity measurements permits a determination of the planet's density. The evaluated internal density suggests that GJ 436b can be composed mainly of ice (Gillon et al. 2007, Deming et al. 2007), in spite of proximity to the host star. On-going and upcoming transit surveys using space telescopes such as Corot, Kepler and TESS, besides ground-based transit surveys, are expected to reveal lower mass exoplanets around M dwarfs.

“Core accretion” model (e.g., Hayashi et al. 1985) naturally accounts for the low abundance of gas giants around M dwarfs, because observationally inferred low disk mass around M dwarfs inhibits formation of cores large enough for runaway gas accretion (Laughlin et al. 2004b;Ida & Lin 2005). The relatively high abundance of Neptune-mass planets could be accounted for by truncation of gas accretion at smaller planetary mass due to lower disk temperature (Ida & Lin 2005). Thus, the Monte Carlo calculation by Ida & Lin (2005) (at least qualitatively) explained the observed properties in the population of gas giants and Neptune-mass planets around M dwarfs. However, they did not predict abundance of habitable planets, because it is affected by type-I migration and detailed orbital configurations of close-in planets, which were not taken into account in their calculation. Ida & Lin (2008a) included the effect of type-I migration in the similar calculation, but treatment of close-in planets was still too simple to discuss the abundance of habitable planets at ∼ 0.1AU around M dwarfs.

N -body simulation is an efficient tool to address this issue. Since physical sizes of planetesimals occupy a larger fraction of their Hill radii in the terrestrial planet regions (∼ 0.1 AU) around M dwarfs than around solar-type stars, strong gravitational scattering is suppressed, which reduces computational cost. Moreover, we can neglect perturbations from gas giants because they are rare in the M-dwarf planetary systems, which also makes N -body simulation simple. Raymond et al. (2007) performed a first N -body simulation of terrestrial planet formation from planetary embryos around low-mass stars. They found that the planets in a HZ may be too small to retain ocean because they assumed that disk surface density is proportional to the stellar mass and the disk model for 1M ⊙ is the minimum mass solar nebula (MMSN) model (Hayashi 1981). Under this assumption, the isolation mass of the planets is proportional to stellar mass (section 2.3). In HZs, icy grains do not condense in the protoplanetary disk in which gas pressure is much smaller than the planetary atmospheric pressure. One of available sources for the water on the planets is delivery of icy planetesimals from the regions beyond the ice line (Morbidelli et al. 2000), although the possibility of forming H 2 O through chemical interaction between the planetary magma ocean and primitive H 2 atmosphere is also pointed out (Ikoma & Genda 2006). Assuming the delivery hypothesis of origin of H 2 O, Raymond et al. (2007) suggested that the planets in HZs around M dwarf stars are likely to be dry, since radial mixing and therefore water delivery are inefficient in the lower-mass disks. L

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