Title: Orbits, Orbitals, and Dark Matter Halos
Description: In this dissertation, we develop two novel methods for studying the nature of the Milky Way's dark matter halo. In both cases, we rely on the relationship between the dark matter halo's gravitational potential and the orbital structure it supports. The first method explores the morphology of stellar streams orbiting in non-spherical gravitational potentials. When globular clusters or dwarf galaxies fall into the Milky Way, tidal forces shred them into long filaments of stars called stellar streams. We show that in non-spherical potentials, stream morphologies are heavily dependent on the characteristics of the progenitor's orbit. Flattened axisymmetric galactic potentials, for example, are known to host minor orbit families surrounding special orbits with commensurable frequencies. The behavior of orbits that belong to these orbit families is fundamentally different from that of typical orbits with non-commensurable frequencies. We show that streams evolving near the boundaries, or separatrices, between orbit families, may become fanned out, develop a bifurcation, or both.

We utilize perturbation theory to estimate the timescale of this effect and the likelihood of a stream evolving close enough to a separatrix to be affected by it. Next, we study the dynamical reasons for stream fanning and bifurcations near resonances, and find that each morphological outcome has a slightly different dynamical cause. Using a novel numerical approach for measuring the libration frequencies of resonant and near-resonant orbits, we reveal that fans come about due to a large spread in the libration frequencies near a separatrix, whereas bifurcations arise when a separatrix splits the orbital distribution of the stellar stream between two (or more) distinct orbit families. We then demonstrate how these features can arise in streams on realistic galactic orbits, in realistic galactic potentials, over timescales as short as 2-3 Gyr, and discuss how this might be used to constrain the global shape of the Milky Way's gravitational potential. The second method studied in this dissertation enables dynamical tests of a dark matter candidate known as Fuzzy (or Ultra-Light) Dark Matter.

Our method relies on a wave generalization of the classic Schwarzschild approach for constructing self-consistent halos -- such a halo consists of a suitable superposition of waves instead of particle orbits, chosen to yield a desired mean density profile. As an illustration, we apply the method to spherically symmetric halos. We derive an analytic relation between the particle distribution function and the wave superposition amplitudes, and show how it simplifies in the high energy (WKB) limit. We verify the stability of such constructed halos by numerically evolving the Schrodinger-Poisson system. The proposed algorithm provides an efficient and accurate way to simulate the time-dependent halo substructures from wave interference, and to test how they will affect dynamical tracers or other observables in a galaxy. The dissertation concludes with a brief discussion of the future prospects of these two methods, especially in the context of upcoming ground- and space-based missions like Rubin LSST and the Roman Space Telescope.