Stellar Dynamics around Black Holes in Galactic Nuclei

We classify orbits of stars that are bound to central black holes in galactic nuclei. The stars move under the combined gravitational influences of the black hole and the central star cluster. Within the sphere of influence of the black hole, the orbital periods of the stars are much shorter than the periods of precession. We average over the orbital motion and end up with a simpler problem and an extra integral of motion: the product of the black hole mass and the semimajor axis of the orbit. Thus the black hole enforces some degree of regularity in its neighborhood. Well within the sphere of influence, (i) planar, as well as three dimensional, axisymmetric configurations-both of which could be lopsided-are integrable, (ii) fully three dimensional clusters with no spatial symmetry whatsover must have semi-regular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a family of planar, non-axisymmetric potential perturbations, and study the orbital structure for the harmonic case in some detail. In the centered potentials there are essentially two main families of orbits: the familiar loops and lenses, which were discussed in Sridhar and Touma (1997, MNRAS, 287, L1-L4). We study the effect of lopsidedness, and identify a family of loop orbits, whose orientation reinforces the lopsidedness, an encouraging sign for the construction of self-consistent models of eccentric, discs around black holes, such as in M31 and NGC 4486B.Comment: to appear in MNRAS, 10 pages, latex, 20 POstScript figure

Elliptical Galaxy Dynamics

A review of elliptical galaxy dynamics, with a focus on nonintegrable models. Topics covered include torus construction; modelling axisymmetric galaxies; triaxiality; collisionless relaxation; and collective instabilities.Comment: 97 Latex pages, 14 Postscript figures, uses aastex. To appear in Publications of the Astronomical Society of the Pacific, February 199

Capture of planets into mean-motion resonances and the origins of extrasolar orbital architectures

The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean-motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this work, we employ an integrable Hamiltonian formalism for first order planetary resonances that allows both secondary bodies to have finite masses and eccentricities, and construct a comprehensive theory for resonant capture. Particularly, we derive conditions under which orbital evolution lies within the adiabatic regime, and provide a generalized criterion for guaranteed resonant locking as well as a procedure for calculating capture probabilities when capture is not certain. Subsequently, we utilize the developed analytical model to examine the evolution of Jupiter and Saturn within the protosolar nebula, and investigate the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets. Our calculations show that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean-motion commensurabilities on slightly eccentric (e ~ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary discs

Gauge Theories, Toric Varieties and Machine Learning

We discuss some recent study on quiver gauge theories in the setting of toric geometry. After mentioning some basic geometric and topological properties, we consider some mathematical concepts, namely the Mahler measure and the dessins d’enfants, in this context. We then focus on the quiver BPS algebras and their connections to diferent aspects in physics. We also have a discussion on the stability of chiral rings for more general geometry. Besides, we make some comments on the applications of machine learning to relevant topics. This thesis is based on the works [1–16]