Indirect Dark Matter Detection for Flattened Dwarf Galaxies
  Gamma-ray experiments seeking to detect evidence of dark matter annihilation
in dwarf spheroidal galaxies require knowledge of the distribution of dark
matter within these systems. We analyze the effects of flattening on the
annihilation (J) and decay (D) factors of dwarf spheroidal galaxies with both
analytic and numerical methods. Flattening has two consequences: first, there
is a geometric effect as the squeezing (or stretching) of the dark matter
distribution enhances (or diminishes) the J-factor; second, the line of sight
velocity dispersion of stars must hold up the flattened baryonic component in
the flattened dark matter halo. We provide analytic formulae and a simple
numerical approach to estimate the correction to the J- and D-factors required
over simple spherical modeling. The formulae are validated with a series of
equilibrium models of flattened stellar distributions embedded in flattened
dark-matter distributions. We compute corrections to the J- and D-factors for
the Milky Way dwarf spheroidal galaxies under the assumption that they are all
prolate or all oblate and find that the hierarchy of J-factors for the dwarf
spheroidals is slightly altered (typical correction factors for an ellipticity
of $0.4$ are $0.75$ for the oblate case and $1.6$ for the prolate case). We
demonstrate that spherical estimates of the D-factors are very insensitive to
the flattening and introduce uncertainties significantly less than the
uncertainties in the D-factors from the other observables for all the dwarf
spheroidals (for example, ${}^{+10\mathrm{\,percent}}_{-3\mathrm{\,percent}}$
for a typical ellipticity of $0.4$). We conclude by investigating the spread in
correction factors produced by triaxial figures and provide uncertainties in
the J-factors for the dwarf spheroidals using different physically-motivated
assumptions for their intrinsic shape and axis alignments. (abridged)
