Mathematical Modeling of Lopsided Structures in Self-Gravitating Systems

Abstract

In recent years, an exceptional progress has exposed a great deal of information about the formation and evolution of large-scale struc- tures in this stunning star-spangled Universe. But, with more infor- mation comes many thought-provoking questions for theorists. The images obtained by the Hubble Space Telescope (HST ) has revealed that basic large-scale structures are shaped at the non-stationary non- linear stage of their evolution; therefore modern extragalactic astron- omy is compelled to study early non-linear stages of evolution of self- gravitational systems. A great role is played by global pulsations in different stages of the formation of galaxies. Incidentally though, reliable mechanisms of development of their sub-structures, as well as possible various non- linear effects are not yet fully revealed. Similarly, the physics of the formation of large-scale structures in the non-stationary universe is not completely available. Many authors have put forward various specific models of the system that gravitate. Binney and Tremaine (1987) have obtained a large number of results. The basis of the most of these results are on the linearisation of the Euler-Poison and Vlasov Poison systems around a stationary solution. Kalnajs (1972) has covered milestones in station- ary models of self-gravitating systems. Although the stationary mod- els of gravitating systems are abundance in the research, the presence of non-stationary models is very conspicuous among various models for study of dynamical development of large-scale structures. There- fore it seemed necessary to develop a new non-linear model which is viinon-stationary in nature and discuss its stability, so that our model will be more accurate. Gravitational instability with respect to lopsided oscillation mode is examined in this dissertation. A phase model of non-stationary self- gravitating disks with isotropic and anisotropic diagrams has been constructed. We used well-known generalization of the Bisnovatyi- Kogan-Zel’dovich model is used in order to find out the formation criteria of galaxies whose nucleus is away from their center (lopsided galaxies). Non-stationary dispersion relations are obtained for both isotropic and anisotropic models of lopsided mode. ) calculations ( The 2T show the relationship between initial virial ratio |U and degree of | ◦ rotation Ω. A comparative analysis of increment (growth rate) of lop- sided mode with other oscillatory modes is made and concluded that lopsided mode has a clear lead over other oscillatory modes. A radial instability always occurs if total kinetic energy is no more than 12.4% of the initial potential energy, in non-stationary isotropic model for lopsided mode. Also, it has been shown that instability is aperiodic when Ω = 0 and oscillatory when Ω ̸ = 0. This ratio of total kinetic energy and total potential energy becomes 30.6% for an anisotropic model of lopsided structure. In this thesis, a multi-parameter composite model by the method of linear superposition has also been constructed and analyzed the stabil- ity of lopsided mode for this model. This new composite model inves- tigates intermediate stages between isotropic and anisotropic models. In the end, the application of lopsidedness in our solar system is dis- cussed. Here, we suggested that G. Darwin’s theory of origin of moon would be acceptable if he had calculated his model in the background of non-stationary and non-equilibrium theory. It has been shown that if Nuritdinov’s non-stationary spherical model is applied on the earth- moon system and calculated that at the initial moment of collapse, the kinetic energy will be lesser than 22.3% of the potential energy viiiwhere instability occurred and the earth became lopsided and then split into two parts and hence the moon came into existence.