UBC Theses and Dissertations
Critical collapse of Newtonian fluids Aguilar-Martinez, Silvestre
This thesis constitutes a numerical study concerning the dynamics of an inviscid fluid subject to Newtonian gravity. Type-II critical phenomena has been previously measured in gravitational collapse simulations of isothermal-gas-spheres in Newtonian gravity. Our first objective was to extend this work by applying the more general polytropic-gas equation-of-state to the spherically symmetric fluid. We showed that under generic conditions of critical collapse, the polytropic gas allows for scale-invariant solutions. These solutions display self-similarity of the first kind with non-linear scaling between the space and time variables. One of these solutions was identified as the critical solution in critical collapse simulations. Such solution was found to have a single unstable mode with a Lyapunov exponent whose value depends on the polytropic index (Γ) from the equation of state. We argued that this behavior constitutes evidence of type-II critical phenomena with a transition from type-II to type-I behavior occurring at Γ ≥ 6/5. Thus, the polytropic gas exhibits both types of critical behavior. These phenomena was investigated dynamically and also from perturbation analysis. In the second phase of the project we extended the hydrodynamic model to treat axi-symmetric gravitational collapse. This allowed us to study the effect of angular momentum on the critical solution. As previously predicted, infinitesimal initial rotation introduces a non-spherical, unstable axial mode into the dynamics. The measured scaling behavior of the specific angular momentum of the collapsed core agrees with the predicted growth rate (Lyapunov exponent) of the axial perturbation. This two-mode linear regime modifies the scaling laws via the introduction of universal functions that depend on the two-parameter family of initial data. The predicted universality of these functions was confirmed through careful measurements of the collapsed mass and its angular momentum near the collapse threshold. A two-parameter space survey reveals a universal behavior of the order-parameters, with no mass-gap even in the presence of finite initial rotation. The behavior changes slightly beyond some initial rotation threshold. The results then, can be interpreted as an intermediate convergence to a non-spherical self-similar critical solution with a single unstable mode.
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