  Upcoming high spectral resolution telescopes, particularly Astro-H, are
expected to finally deliver firm quantitative constraints on turbulence in the
intra-cluster medium (ICM). We develop a new spectral analysis technique which
exploits not just the line width but the entire line shape, and show how the
excellent spectral resolution of Astro-H can overcome its relatively poor
spatial resolution in making detailed infer- ences about the velocity field.
The spectrum is decomposed into distinct components, which can be
quantitatively analyzed using Gaussian mixture models. For instance, bulk flows
and sloshing produce components with offset means, while partial volume-
filling turbulence from AGN or galaxy stirring leads to components with
different widths. The offset between components allows us to measure gas bulk
motions and separate them from small-scale turbulence, while component
fractions and widths con- strain the emission weighted volume and turbulent
energy density in each component. We apply mixture modeling to a series of
analytic toy models as well as numerical simu- lations of clusters with cold
fronts and AGN feedback respectively. From Markov Chain Monte Carlo and Fisher
matrix estimates which include line blending and continuum contamination, we
show that the mixture parameters can be accurately constrained with Astro-H
spectra: at a \sim 10% level when components differ significantly in width, and
a \sim 1% level when they differ significantly in mean value. We also study
error scalings and use information criteria to determine when a mixture model
is preferred. Mixture modeling of spectra is a powerful technique which is
potentially applicable to other astrophysical scenarios.
