We extend the concept of central configurations to the N-body problem in spaces of nonzero constant curvature. Based on the work of Florin Diacu on relative equilib- ria of the curved N-body problem and the work of Smale on general relative equilibria, we find a natural way to define the concept of central configurations with the effective potentials. We characterize the ordinary central configurations as constrained critical points of the cotangent potential, which helps us to establish the existence of ordi- nary central configurations for any given masses. After these fundamental results, we study central configurations on H2, ordinary central configurations in S3, and special central configurations in S3 in three separate chapters. For central configurations on H2, we generalize the theorem of Moulton on geodesic central configurations, the theorem of Shub on the compactness of central configurations, the theorem of Conley on the index of geodesic central configurations, and the theorem of Palmore on the lower bound for the number of central configurations. We show that all three-body central configurations that form equilateral triangles must have three equal masses. For ordinary central configurations in S3, we construct a class of S3 ordinary central configurations. We study the geodesic central configurations of two and three bodies. Three-body non-geodesic ordinary central configurations that form equilateral trian- gles must have three equal masses. We also put into the evidence some other classes of central configurations. For special central configurations, we show that for any N ≥ 3, there are masses that admit at least one special central configuration. We then consider the Dziobek special central configurations and obtain the central con- figuration equation in terms of mutual distances and volumes formed by the position vectors. We end the thesis with results concerning the stability of relative equilibria associated with 3-body special central configurations. We find that these relative equilibria are Lyapunov stable when confined to S1, and that they are linearly stable on S2 if and only if the angular momentum is bigger than a certain value determined by the configuration. / Graduate

The increasing number of discovered extra-solar planets opens a new opportunity for studies of the formation of planetary systems. Their diversity keeps challenging the long-standing theories which were based on data primarily from our own solar system. Resonant planetary systems are of particular interest because their dynamical configuration provides constraints on the otherwise unobservable formation and migration phase. In this thesis, formation scenarios for the planetary systems HD128311 and HD45364 are presented. N-body simulations of two planets and two dimensional hydrodynamical simulations of proto-planetary discs are used to realistically model the convergent migration phase and the capture into resonance. The results indicate that the proto-planetary disc initially has a larger surface density than previously thought. Proto-planets are exposed to stochastic forces, generated by density fluctuations in a turbulent disc. A generic model of both a single planet, and two planets in mean motion resonance, being stochastically forced is presented and applied to the system GJ876. It turns out that GJ876 is stable for reasonable strengths of the stochastic forces, but systems with lighter planets can get disrupted. Even if a resonance is not completely disrupted, stochastic forces create characteristic, observable libration patterns. As a further application, the stochastic migration of small bodies in Saturn’s rings is studied. Analytic predictions of collisional and gravitational interactions of a moonlet with ring particles are compared to realistic three dimensional collisional N-body simulations with up to a million particles. Estimates of both the migration rate and the eccentricity evolution of embedded moonlets are confirmed. The random walk of the moonlet is fast enough to be directly observable by the Cassini spacecraft. Turbulence in the proto-stellar disc also plays an important role during the early phases of the planet formation process. In the core accretion model, small, metre-sized particles are getting concentrated in pressure maxima and will eventually undergo a rapid gravitational collapse to form a gravitationally bound planetesimal. Due to the large separation of scales, this process is very hard to model numerically. A scaled method is presented, that allows for the correct treatment of self-gravity for a marginally collisional system by taking into account the relevant small scale processes. Interestingly, this system is dynamically very similar to Saturn’s rings.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23007号 / 理博第4684号 / 新制||理||1672(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 嶺重 慎, 准教授 前田 啓一, 教授 太田 耕司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This paper describes the IBM Power8 system in comparison to the Intel Xeon processors, widely used in today’s solutions. The performance is not evaluated only on the whole system level but also on the level of threads, cores and a memory. Different metrics are demonstrated on typical optimized algorithms. The benchmarked Power8 processor provides extremely fast memory providing sustainable bandwidth up to 145 GB/s between main memory and processor, which Intel is unable to compete. Computation power is comparable (Matrix multiplication) or worse (N-body simulation, division, more complex algorithms) in comparison with current Intel Haswell-EP. The IBM Power8 is able to compete Intel processors these days and it will be interesting to observe the future generation of Power9 and its performance in comparison to current and future Intel processors.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18795号 / 理博第4053号 / 新制||理||1583(附属図書館) / 31746 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 嶺重 慎, 教授 太田 耕司, 准教授 樽家 篤史 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

In this dissertation, regularization of simultaneous binary collision, existence of a Schubart-like periodic orbit, existence of a planar symmetric periodic orbit with multiple simultaneous binary collisions, and their linear stabilities are studied. The detailed background of those problems is introduced in chapter 1. The singularities of simultaneous binary collision in the collinear four-body problem is regularized in chapter 2. We use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions in both the decoupled case and the coupled case. All the solutions are found and more importantly, we find a crucial first integral which describes the relationship between the decoupled solutions and the coupled solutions. In chapter 3, we show the existence of a Schubart-like orbit, a periodic orbit with singularities in the symmetric collinear four-body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proven by using a "turning point" technique and a continuity argument on differential equations of the regularized Hamiltonian. Analytical methods are used in chapter 4 to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method can be extended to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated. In chapter 5, we apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m , 1, and also in a symmetric planar four-body problem with equal masses. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability.

We investigate several techniques to identify voids in the galaxy distribution of matter in the universe. We utilize galaxy number counts as a function of apparent magnitude and Wolf plots to search a two- or three-dimensional data set in a pencil-beam fashion to locate voids within the field of view. The technique is able to distinguish between voids that represent simply a decrease in density as well as those that show a build up of galaxies on the front or back side of the void. This method turns out to be primarily useable only at relatively short range (out to about 200 Mpc). Beyond this distance, the characteristics indicating a void become increasingly difficult to separate from the statistical background noise. We apply the technique to a very simplified model as well as to the Millennium Run dark matter simulation. We then compare results with those obtained on the Sloan Digital Sky Survey. We also created the Watershed Void Examiner (WaVE) which treats densities in a fashion similar to elevation on a topographical map, and then we allow the "terrain" to flood. The flooded low-lying regions are identified as voids, which are allowed to grow and merge as the level of flooding becomes higher (the overdensity threshold increases). Void statistics can be calculated for each void. We also determine that within the Millennium Run semi-analytic galaxy catalog, the walls that separate the voids are permeable at a scale of 4 Mpc. For each resolution that we tested, there existed a characteristic density at which the walls could be penetrated, allowing a single void to grow to dominate the volume. With WaVE, we are able to get comparable results to those previously published, but often with fewer choices of parameters that could bias the results. We are also able to determine the the density at which the number of voids peaks for different resolutions as well as the expected number of void galaxies. The number of void galaxies is amazingly consistent at an overdensity of −0.600 at all resolutions, indicating that this could be a good choice for comparing models.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24328号 / 情博第812号 / 新制||情||137(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 柴山 允瑠, 教授 矢ヶ崎 一幸, 教授 山下 信雄, 教授 田口 智清 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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