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Understanding how matter is distributed in galaxies is a fundamental problem in astronomy. In particular, cosmological simulations of structure formation within the standard ΛCDM model of the Universe suggest that the stars we see collected together as galaxies are surrounded by much more massive and extended dark matter halosdark matter halos. The simulations outline expectations for the average properties of halos, including their characteristic density profiles (Navarro, Frenk & White 1997) and triaxial shapes and orientations as a function of radius (Jing & Suto 2002). They also suggest that each Milky-Way-sized dark matter halo encompasses a swarm of satellite subhalosdark matter subhalos!satellite in numbers far greater than the number of observed satellite dwarf galaxies in the Milky Way halo (traditionally referred to as the “missing satellites”missing satellites problem, see Moore et al. 1999, Klypin et al. 1999). While it is possible that the observed dwarf galaxies account for all of the largest of these satellite subhalos (though this is far from clear, see Boylan-Kolchin et al. 2011), a multitude of smaller halos (masses of order 107 M or less) are still predicted to exist for which observed counterparts have not been identified (see Section 3 for a more complete discussion).

The expectations for the structure of and substructure within dark matter halos from the cosmological simulations are in general hard to test with great accuracy since we currently observe dark matter only by its gravitational effects on stars, and the dark matter extends well beyond where the majority of the stars in any galaxy lie. Tracer populations such as planetary nebulae and globular clusters have been used to estimate global masses beyond the visible components of galaxies (Côté et al. 2003), while satellite systems are thought to provide some accounting of the substructure — though it remains far from clear how complete and unbiased that accounting is (Tollerud et al. 2008). Some progress on the projected shapes, total masses and largest substructures within dark matter halos has come from gravitational lensing (Vegetti et al. 2010). This chapter outlines why tidal debris is considered a promising and sensitive probe of dark matter halos as well as the subhalos they are expected to contain. In particular, this chapter concentrates on debris around our own Milky Way galaxy since this is the one place in the Universe where we might hope to measure the three-dimensional structure of a dark matter halo, as well as be sensitive to the proposed multitude of lower mass subhalos that may not contain gas or stars.

Historically, the knowledge that stars in disk systems are moving on near-circular orbits has been exploited to sensitively measure their mass distributions — indeed, the first systematic studies of the rotation curves of galaxies promoted the idea that galaxies were dominated by dark matter rather than baryonic matter (Rubin & Ford 1970). Although the nature of the dark matter particles themselves has yet to be determined, the idea that the majority of the mass in the Universe is composed of particles that are so far not observed because they do not interact with light is generally accepted in the field.

Stars in tidal debris structures make excellent probes of the matter distribution around galaxies for analogous reasons to disk stars. Like disk stars, we know more about their orbits than we would know for a purely random population — we know that the stars in tidal streams were once all part of the same parent satellite galaxy, and consequently have a small range of orbital properties about the progenitor satellite's orbit. Moreover, to be detectable in a photometric study, these debris structures must lie well outside the bulk of the stars in the parent galaxy and hence typically probe a very different region of the dark matter halo than the bulk of the stars in a galaxy.

Images of streams of debris in particular suggest that stars lie close to a single orbit, and this gives some simple insight into why they are such sensitive potential probes. If the stars were actually on exactly the same orbit and you could measure the positions vector{x} and velocities vector{v} of each, then the potential (up to the unknown, constant energy E of the orbit) would simply be Φ(vector{x}) = Ev2/2. A more detailed discussion of methods for measuring the global potential is presented in Section 2.

The dynamical coldness of streams, as determined from the small velocity dispersions and narrow spatial cross sections observed for many streams, also provides simple insight into their use as substructure probes. This low temperature means that discontinuities in the debris on much smaller scales than the apparent orbital path can arise due to asymmetries in the potential that are also on much smaller scale than the global potential. This leads to the exciting possibility of using debris to detect substructures within the potential, such as those arising from the thousands of dark matter subhalos that are expected to be orbiting within the main halo. Direct evidence of a large number of dark subhalos would solve the missing satellite problem. The characteristic signatures that substructure may leave in streams are discussed in Section 3.

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