We perform numerical simulations on the merger of multiple black holes (BHs) in primordial gas at early cosmic epochs. We consider two cases of BH mass: $M_{BH} = 30 M_{odot}$ and $M_{BH} = 10^4 M_{odot}$. Attention is concentrated on the effect of t
he dynamical friction by gas in a host object. The simulations incorporate such general relativistic effects as the pericentre shift and gravitational wave emission. As a result, we find that multiple BHs are able to merge into one BH within 100 Myr in a wide range of BH density. The merger mechanism is revealed to be categorized into three types: gas-drag-driven merger (type A), interplay-driven merger (type B), and three-body-driven merger (type C). We find the relation between the merger mechanism and the ratio of the gas mass within the initial BH orbit ($M_{gas}$) to the total BH mass (${Sigma}M_{BH}$). Type A merger occurs if $M_{gas} gtrsim 10^5 {Sigma}M_{BH}$, type B if $M_{gas} lesssim 10^5 {Sigma}M_{BH}$, and type C if $M_{gas} ll 10^5 {Sigma}M_{BH}$. Supposing the gas and BH density based on the recent numerical simulations on first stars, all the BH remnants from first stars are likely to merge into one BH through the type B or C mechanism. Also, we find that multiple massive BHs ($M_{BH} = 10^4 M_{odot}$) distributed over several parsec can merge into one BH through the type B mechanism, if the gas density is higher than $5times 10^6$ cm$^{-3}$. The present results imply that the BH merger may contribute significantly to the formation of supermassive BHs at high redshift epochs.
A phase-space distribution function of the steady state in galaxy models that admits regular orbits overall in the phase-space can be represented by a function of three action variables. This type of distribution function in Galactic models is often
constructed theoretically for comparison of the Galactic models with observational data as a test of the models. On the other hand, observations give Cartesian phase-space coordinates of stars. Therefore it is necessary to relate action variables and Cartesian coordinates in investigating whether the distribution function constructed in galaxy models can explain observational data. Generating functions are very useful in practice for this purpose, because calculations of relations between action variables and Cartesian coordinates by generating functions do not require a lot of computational time or computer memory in comparison with direct numerical integration calculations of stellar orbits. Here, we propose a new method called a torus-fitting method, by which a generating function is derived numerically for models of the Galactic potential in which almost all orbits are regular. We confirmed the torus-fitting method can be applied to major orbit families (box and loop orbits) in some two-dimensional potentials. Furthermore, the torus-fitting method is still applicable to resonant orbit families, besides major orbit families. Hence the torus-fitting method is useful for analyzing real Galactic systems in which a lot of resonant orbit families might exist.
To determine the local dark matter density (LDMD) of the solar system is a classical problem in astronomy. Recently, a novel method of determining the LDMD from stellar distribution and vertical velocity dispersion profiles perpendicular to the Galac
tic plane was devised. This method has the advantage of abolishing conventional approximations and using only a few assumptions. Our aims are to carefully scrutinize this method and to examine influences by uncertainties of astrometric observations. We discuss how the determinations of the LDMD vary with observational precisions on parallax, proper motion, and line-of-sight velocity measurements. To examine the influences by the observational imprecision, we created mock observation data for stars that are dynamical tracers based on an analytical galaxy model and applied parametrized observational errors to the mock data. We evaluated the accuracy of determining the LDMD by applying the method to the mock data. In addition, we estimated a sample size and observational precision required to determine the LDMD with accuracy. We find that the method is capable of determining the LDMD with accuracy if the sample size and observational precisions are satisfactory. The random errors of parallaxes and proper motions can cause systematic overestimation of the LDMD. We estimate the required precisions of the parallax measurements to be approximately 0.1-0.3 milliarcseconds at 1 kpc away from the Sun; the proper motion precisions do not seem to be as important as the parallaxes. From these results, we expect that using the Hipparcos catalog would overestimate the LDMD because of the imprecise parallax measurements if this method is applied; however, we emphasize the capability of the method. We expect that Gaia will provide data precise enough to determine the LDMD.
We investigate the properties of HI-rich galaxies detected in blind radio surveys within the hierarchical structure formation scenario using a semi-analytic model of galaxy formation. By drawing a detailed comparison between the properties of HI-sele
cted galaxies and HI absorption systems, we argue a link between the local galaxy population and quasar absorption systems, particularly for Damped Ly-alpha absorption (DLA) systems and sub-DLA systems. First, we evaluate how many HI-selected galaxies exhibit HI column densities as high as those of DLA systems. We find that HI-selected galaxies with HI masses M(HI) > 10^8 solar masses have gaseous disks that produce HI column densities comparable to those of DLA systems. We conclude that DLA galaxies where the HI column densities are as high as those of DLA systems, contribute significantly to the population of HI-selected galaxies at M(HI) > 10^8 solar masses. Second, we find that star formation rates (SFRs) correlate tightly with HI masses rather than B- (and J-) band luminosities. In the low-mass range M(HI) < 10^8 solar masses, sub-DLA galaxies replace DLA galaxies as the dominant population. The number fraction of sub-DLA galaxies relative to galaxies reaches 40%-60% at HI masses 10^8 solar masses and 30%-80% at 10^7 solar masses. The HI-selected galaxies at 10^7 solar masses are a strong probe of sub-DLA systems that place stringent constraints on galaxy formation and evolution.
