  Most proposed dark matter candidates are stable and are produced thermally in
the early Universe. However, there is also the possibility of unstable (but
long-lived) dark matter, produced thermally or otherwise. We propose a strategy
to distinguish between dark matter annihilation and/or decay in the case that a
clear signal is detected in gamma-ray observations of Milky Way dwarf
spheroidal galaxies with gamma-ray experiments. The sole measurement of the
energy spectrum of an indirect signal would render the discrimination between
these cases impossible. We show that by examining the dependence of the
intensity and energy spectrum on the angular distribution of the emission, the
origin could be identified as decay, annihilation, or both. In addition, once
the type of signal is established, we show how these measurements could help to
extract information about the dark matter properties, including mass,
annihilation cross section, lifetime, dominant annihilation and decay channels,
and the presence of substructure. Although an application of the approach
presented here would likely be feasible with current experiments only for very
optimistic dark matter scenarios, the improved sensitivity of upcoming
experiments could enable this technique to be used to study a wider range of
dark matter models.
