  Terrestrial searches for the conversion of dark matter axions or axion-like
particles into photons inside magnetic fields are sensitive to the phase space
structure of the local Milky Way halo. We simulate signals in a hypothetical
future experiment based on the Axion Dark Matter eXperiment (ADMX) that could
be performed once the axion has been detected and a frequency range containing
the axion mass has been identified. We develop a statistical analysis to
extract astrophysical parameters, such as the halo velocity dispersion and
laboratory velocity, from such data and find that with only a few days
integration time a level of precision can be reached matching that of
astronomical observations. For longer experiments lasting up to a year in
duration we find that exploiting the modulation of the power spectrum in time
allows accurate measurements of the Solar peculiar velocity with an accuracy
that would improve upon astronomical observations. We also simulate signals
based on results from N-body simulations and find that finer substructure in
the form of tidal streams would show up prominently in future data, even if
only a subdominant contribution to the local dark matter distribution. In these
cases it would be possible to reconstruct all the properties of a dark matter
stream using the time and frequency dependence of the signal. Finally we
consider the detection prospects for a network of streams from tidally
disrupted axion miniclusters. These features appear much more prominently in
the resolved spectrum than suggested by calculations based on a scan over a
range of resonant frequencies, making the detection of axion minicluster
streams more viable than previously thought. These results confirm that
haloscope experiments in a post-discovery era are able to perform "axion
astronomy".
