Constraints on the Shape of the Milky Way Dark Matter Halo from Jeans Equations Applied to SDSS Data

We search for evidence of dark matter in the Milky Way by utilizing the stellar number density distribution and kinematics measured by the Sloan Digital Sky Survey (SDSS) to heliocentric distances exceeding ~10 kpc. We employ the cylindrically symmetric form of Jeans equations and focus on the morphology of the resulting acceleration maps, rather than the normalization of the total mass as done in previous, mostly local, studies. Jeans equations are first applied to a mock catalog based on a cosmologically derived N-body + SPH simulation, and the known acceleration (gradient of gravitational potential) is successfully recovered. The same simulation is also used to quantify the impact of dark matter on the total acceleration. We use Galfast, a code designed to quantitatively reproduce SDSS measurements and selection effects, to generate a synthetic stellar catalog. We apply Jeans equations to this catalog and produce two-dimensional maps of stellar acceleration. These maps reveal that in a Newtonian framework, the implied gravitational potential cannot be explained by visible matter alone. The acceleration experienced by stars at galactocentric distances of ~20 kpc is three times larger than what can be explained by purely visible matter. The application of an analytic method for estimating the dark matter halo axis ratio to SDSS data implies an oblate halo with q_DM = 0.47 +/- 0.14 within the same distance range. These techniques can be used to map the dark matter halo to much larger distances from the Galactic center using upcoming deep optical surveys, such as LSST.