Particle-Particle Particle-Tree: A Direct-Tree Hybrid Scheme for Collisional N-Body Simulations

In this paper, we present a new hybrid algorithm for the time integration of collisional N-body systems. In this algorithm, gravitational force between two particles is divided into short-range and long-range terms, using a distance-dependent cutoff function. The long-range interaction is calculated using the tree algorithm and integrated with the constant-timestep leapfrog integrator. The short-range term is calculated directly and integrated with the high-order Hermite scheme. We can reduce the calculation cost per orbital period from O(N^2) to O(N log N), without significantly increasing the long-term integration error. The results of our test simulations show that close encounters are integrated accurately. Long-term errors of the total energy shows random-walk behaviour, because it is dominated by the error caused by tree approximation.Comment: 22 pages, 15 figure

Ejection of close-in super-Earths around low-mass stars in the giant impact stage

Earth-sized planets were observed in close-in orbits around M dwarfs. While more and more planets are expected to be uncovered around M dwarfs, theories of their formation and dynamical evolution are still in their infancy. We investigate the giant impact growth of protoplanets, which includes strong scattering around low-mass stars. The aim is to clarify whether strong scattering around low-mass stars affects the orbital and mass distributions of the planets. We perform $N$-body simulation of protoplanets by systematically surveying the parameter space of the stellar mass and surface density of protoplanets. We find that protoplanets are often ejected after twice or three times close-scattering around late M dwarfs. The ejection sets the upper limit of the largest planet mass. Adopting the surface density scaling linearly with the stellar mass, we find that as the stellar mass decreases less massive planets are formed in orbits with higher eccentricities and inclinations. Under this scaling, we also find that a few close-in protoplanets are generally ejected. The ejection of protoplanets plays an important role in the mass distribution of super-Earths around late M dwarfs. The mass relation of observed close-in super-Earths and their central star mass is well reproduced by ejection.Comment: accepted for publication in A&

Potassium Abundances in Red Giants of Mildly to Very Metal-Poor Globular Clusters

A non-LTE analysis of K I resonance lines at 7664.91 and 7698.97 A was carried out for 15 red giants belonging to three globular clusters of different metallicity (M 4, M 13, and M 15) along with two reference early-K giants (rho Boo and alpha Boo), in order to check whether the K abundances are uniform within a cluster and to investigate the behavior of [K/Fe] ratio at the relevant metallicity range of -2.5 <[Fe/H] < -1. We confirmed that [K/H] (as well as [Fe/H]) is almost homogeneous within each cluster to a precision of < ~0.1 dex, though dubiously large deviations are exceptionally seen for two peculiar stars showing signs of considerably increased turbulence in the upper atmosphere. The resulting [K/Fe] ratios are mildly supersolar by a few tenths of dex for three clusters, tending to gradually increase from ~+0.1-0.2 at [Fe/H] ~-1 to ~+0.3 at [Fe/H] ~ -2.5. This result connects reasonably well with the [K/Fe] trend of disk stars (-1 < [Fe/H]) and that of extremely metal-poor stars (-4 <[Fe/H] < -2.5). That is, [K/Fe] appears to continue a gradual increase from [Fe/H]~0 toward a lower metallicity regime down to [Fe/H]~-3, where a broad maximum of [K/Fe]~+0.3-0.4 is attained, possibly followed by a slight downturn at [Fe/H]<~-3.Comment: 13 pages, 4 tables, 6 figures, and 1 electronic table (accepted for publication in Publ. Astron. Soc. Japan