Yoko Funato

Missing Link Found? The "Runaway" Path to Supermassive Black Holes

Observations of stellar kinematics, gasdynamics, and masers around galactic nuclei have now firmly established that many galaxies host central supermassive black holes (SMBHs) with masses in the range of ~106-109 M☉. However, how these SMBHs formed is not well understood. One reason for this situation is the lack of observations of intermediate-mass BHs (IMBHs), which could bridge the gap between stellar mass BHs and SMBHs. Recently, this missing link (i.e., an IMBH) has been found in observations made by ASCA and Chandra of the central region of the starburst galaxy M82. Subsequent observations by Subaru have revealed that this IMBH apparently coincides with a young compact star cluster. Based on these findings, we suggest a new formation scenario for SMBHs. In this scenario, IMBHs first form in young compact star clusters through runaway merging of massive stars. While these IMBHs are forming, the host star clusters sink toward the galactic nucleus through dynamical friction and upon evaporation deposit their IMBHs near the galactic center. The IMBHs then form binaries and eventually merge via gravitational radiation, forming an SMBH.

TO CIRCULARIZE OR NOT TO CIRCULARIZE?—ORBITAL EVOLUTION OF SATELLITE GALAXIES

We investigate the orbital evolution of satellite galaxies using numerical simulations. It has long been believed that orbits suffer circularization due to dynamical friction from the galactic halo during orbital decay. This circularization was confirmed by numerous simulations in which dynamical friction was added as an external force. However, some of the recent N-body simulations have demonstrated that circularization is much slower than expected from approximate calculations. We find that the dominant reason for this discrepancy is the assumption that the Coulomb logarithm log Λ is constant, which has been used in practically all recent calculations. Since the size of the satellite is relatively large, an accurate determination of the outer cutoff radius is crucial to obtaining a good estimate for the dynamical friction. An excellent agreement between N-body simulations and approximate calculations is observed when the outer cutoff radius is taken to be the distance of the satellite to the center of the galaxy. When the satellite is at the perigalacticon, the distance to the center is smaller, and therefore log Λ becomes smaller. As a result, the dynamical friction becomes less effective. We apply our result to the Large Magellanic Cloud (LMC). We find that the expected lifetime of the LMC is twice as long as that which would be predicted with previous calculations. Previous studies predict that the LMC will merge into the Milky Way after 7 Gyr, while we find that the merging will take place 14 Gyr from now. Our result suggests that generally, satellites formed around a galaxy have longer lifetimes than previous estimates.

The formation of Kuiper-belt binaries through exchange reactions.

Recent observations have revealed that an unexpectedly high fraction—a few per cent—of the trans-Neptunian objects (TNOs) that inhabit the Kuiper belt are binaries. The components have roughly equal masses, with very eccentric orbits that are wider than a hundred times the radius of the primary. Standard theories of binary asteroid formation tend to produce close binaries with circular orbits, so two models have been proposed to explain the unique characteristics of the TNOs. Both models, however, require extreme assumptions regarding the size distribution of the TNOs. Here we report a mechanism that is capable of producing binary TNOs with the observed properties during the early stages of their formation and growth. The only required assumption is that the TNOs were initially formed through gravitational instabilities in the protoplanetary dust disk. The basis of the mechanism is an exchange reaction in which a binary whose primary component is much more massive than the secondary interacts with a third body, whose mass is comparable to that of the primary. The low-mass secondary component is ejected and replaced by the third body in a wide but eccentric orbit.

On the mass distribution of planetesimals in the early runaway stage

We derived the stationary distribution of the mass of planetesimals from the coagulation equation under the assumption of energy equipartition. In the three-dimensional case, we found that the stationary solution is expressed as n(m)∝m−83, where n is the surface number density and m is the mass of the planetesimals. This solution is in excellent agreement with the result of direct N-body simulations and the numerical integration of the coagulation equation, for the range of mass which contains most of the total mass of the planetesimals (1022–1025 g). In the two-dimensional case, the stationary solution exists but cannot be realized in a finite time. This difference is the direct consequence of the fact that runaway growth takes place in three dimensions, but not in two dimensions.

Evolution of Massive Black Hole Triples. I. Equal-Mass Binary-Single Systems

We present the results of N-body simulations of dynamical evolution of triple massive black hole (BH) systems in galactic nuclei. We found that in most cases two of the three BHs merge through gravitational wave (GW) radiation in a timescale much shorter than the Hubble time before ejecting one BH through a slingshot. In order for a binary BH to merge before ejecting the third one, it has to become highly eccentric, since the GW timescale would be much longer than the Hubble time unless the eccentricity is very high. We found that two mechanisms drive the increase of the eccentricity of the binary. One is the strong binary-single BH interaction, which results in the thermalization of the eccentricity. The second is the Kozai mechanism, which drives the cyclic change of the inclination and eccentricity of the inner binary of a stable hierarchical triple system. Our result implies that many supermassive BHs are binaries.

BRIDGE: A Direct-Tree Hybrid N -Body Algorithm for Fully Self-Consistent Simulations of Star Clusters and Their Parent Galaxies

We developed a new direct-tree hybrid N -body algorithm for fully self-consistent N -body simulations of star clusters in their parent galaxies. In such simulations, star clusters need high accuracy, while galaxies need a fast scheme because of the large number of particles required to model it. In our new algorithm, the internal motion of the star cluster is calculated accurately using the direct Hermite scheme with individual timesteps, and all other motions are calculated using the tree code with a second-order leapfrog integrator. The direct and tree schemes are combined using an extension of the mixed variable symplectic (MVS) scheme. Thus, the Hamiltonian corresponding to everything other than the internal motion of the star cluster is integrated with the leapfrog, which is symplectic. Using this algorithm, we performed fully self-consistent N -body simulations of star clusters in their parent galaxy. The internal and orbital evolutions of the star cluster agreed well with those obtained using the direct scheme. We also performed fully self-consistent N -body simulation for large-N models (N = 2 � 10 6 ). In this case, the calculation speed was seven-times faster than what would be if the direct scheme was used.

Time-Symmetrized Kustaanheimo-Stiefel Regularization

In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo--Stiefel (KS) regularization is widely used to remove the singularity in the equations of motion, making it possible to integrate orbits having very high eccentricity. However, even with KS regularization, long-term integration is difficult, simply because the required accuracy is usually very high. We present a new time-integration algorithm which has no secular error in either the binding energy or the eccentricity, while allowing variable stepsize. The basic approach is to take a time-symmetric algorithm, then apply an implicit criterion for the stepsize to ensure strict time reversibility. We describe the algorithm in detail and present the results of numerical tests involving long-term integration of binaries and hierarchical triples. In all cases studied, we found no systematic error in either the energy or the angular momentum. We also found that its calculation cost does not become higher than those of existing algorithms. By contrast, the stabilization technique, which has been widely used in the field of collisional stellar dynamics, conserves energy very well but does not conserve angular momentum.

Dynamical Friction on Satellite Galaxies

For a rigid model satellite, Chandrasekhars dynamical friction formula describes the orbital evolution quite accurately, when the Coulomb logarithm is chosen appropriately. However, it is not known if the orbital evolution of a real satellite with the internal degree of freedom can be described by the dynamical friction formula. We performed N-body simulation of the orbital evolution of a self-consistent satellite galaxy within a self-consistent parent galaxy. We found that the orbital decay of the simulated satellite is significantly faster than the estimate from the dynamical friction formula. The main cause of this discrepancy is that the stars stripped out of the satellite are still close to the satellite, and increase the drag force on the satellite through two mechanisms. One is the direct drag force from particles in the trailing tidal arm, a non-axisymmetric force that slows the satellite down. The other is the indirect effect that is caused by the particles remaining close to the satellite after escape. The force from them enhances the wake caused in the parent galaxy by dynamical friction, and this larger wake in turn slows the satellite down more than expected from the contribution of its bound mass. We found these two have comparable effects, and the combined effect can be as large as 20% of the total drag force on the satellite.

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.

Change in Mass and Energy of Galaxies through Mutual Encounters

We present the result of a systematic study of the evolution of galaxies through their encounters. We performed a series of numerical experiments of encounters of two galaxies and determined how the change of mass and binding energy depends on galaxy models and collision parameters. Using both numerical experiments and analytic theory, we found that the relation between the relative change of energy ?E and that of mass ?M of one galaxy per unit time is expressed as ?E1.5?M if the evolution is solely due to encounters. This result suggests that when galaxies in a cluster evolve mainly through frequent encounters with each other, a relation ?4~M develops between the mass M and the velocity dispersion ? of the galaxies. If we assume that M/L is constant, this relation is equivalent to the Faber-Jackson relation, ?4~L, which is observed for elliptical galaxies in a cluster of galaxies.This agreement with observations suggests that the encounters of galaxies play an important role in the evolution of galaxies in clusters.