The Milky Way Galaxy serves as a laboratory for testing models of galaxy formation. Discovering the nature of dark matter is often cited as the second most important problem in astrophysics, preceded only by dark energy. Mapping the structure and dynamics of the Milky Way Galaxy can tell us how galaxies form, and place constraints on the properties of dark matter.

We can map the distribution of dark matter in the Milky Way using tidal streams, collections of stars that have been gravitationally stripped from satellite dwarf galaxies and globular clusters. By knowing the positions and velocities of these stars, and assuming they came from a compact source, we can follow them back in time and constrain the shape of the Milky Way dark matter halo.

This Thesis presents a method that allows us to constrain the parameters of a static Galactic gravitational potential using the data from any number of tidal debris streams. The method is tested on simulated tidal streams, and successfully recovers the original model parameters in most cases. The importance of simultaneously fitting the measured rotation curve of the Milky Way is explored, and the strengths and weaknesses of the algorithm are discussed.

The orbit fitting algorithm is applied independently to the Stream of Grillmair and Dionatos (GD-1), the Orphan Stream, and the Cetus Polar Stream (CPS). We show that no known globular cluster or dwarf galaxy in the Milky Way has kinematics consistent with being the progenitor of the GD-1 stream. The Orphan Stream constrains the Milky Way dark matter halo as having a mass at the low end of previous measurements, giving a best fit halo speed of v_halo = 73 ± 24 km s^-1, compared to typical values of v_halo ≈ 115 km s ^-1. A lower halo speed implies a less massive halo.

The GD-1 and Orphan streams are then fit simultaneously with the Sagittarius Dwarf Tidal Stream (Sgr), within a triaxial dark matter halo. Results for restricted triaxial cases are shown to be consistent with previous authors. Simultaneous fits within an unrestricted triaxial halo (free to rotate in any direction) give flattenings q_x = 1.33 ± 0.16, q_z = 1.52 ± 0.14 and XYZ pitch-roll-yaw Euler orientation angles of (&thgr;, &phis;, ψ) = (-50° ± 18°, 86° ± 11°, 1° ± 6°). The best fit halo speed and scale length are v_halo,t = 126 ± 9 km s^-1 and d_halo,t = 22.2 ± 3.3 kpc, respectively. The &phis; Euler angle is broadly consistent with that found for the stellar halo by Newberg et al. (2006). The significance of these orientation angles within the context of Galaxy formation and evolution are discussed.

Utilizing the orbit fits to the Orphan Stream, a novel technique is presented to fit the density of F-turnoff stars along the stream utilizing semi-analytic N-body simulations. Baseline estimates of the mass and scale radius of the Orphan Stream progenitor are obtained. We discuss the development of a stellar density fitting algorithm, which is implemented on the Milkyway@home volunteer computing platform.