Simple J-Factors and D-Factors for Indirect Dark Matter Detection
  J-factors (or D-factors) describe the distribution of dark matter in an
astrophysical system and determine the strength of the signal provided by
annihilating (or decaying) dark matter respectively. We provide simple analytic
formulae to calculate the J-factors for spherical cusps obeying the empirical
relationship between enclosed mass, velocity dispersion and half-light radius.
We extend the calculation to the spherical Navarro-Frenk-White (NFW) model, and
demonstrate that our new formulae give accurate results in comparison to more
elaborate Jeans models driven by Markov Chain Monte Carlo methods. Of the known
ultrafaint dwarf spheroidals, we show that Ursa Major II, Reticulum II, Tucana
II and Horologium I have the largest J-factors and so provide the most
promising candidates for indirect dark matter detection experiments. Amongst
the classical dwarfs, Draco, Sculptor and Ursa Minor have the highest
J-factors. We show that the behaviour of the J-factor as a function of
integration angle can be inferred for general dark halo models with inner slope
$\gamma$ and outer slope $\beta$. The central and asymptotic behaviour of the
J-factor curves are derived as a function of the dark halo properties. Finally,
we show that models obeying the empirical relation on enclosed mass and
velocity dispersion have J-factors that are most robust at the integration
angle equal to the projected half-light radius of the dSph divided by
heliocentric distance. For most of our results, we give the extension to the
D-factor which is appropriate for the decaying dark matter picture.
