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A broad summary of the dependencies of debris structures can be drawn from the physical models outlined in the previous section (and by inspection of Figure 2):

Armed with these descriptions we can now go on to interpret observations of debris structures in terms of the history of the progenitor satellite. There is a rich literature with exact models (typically based on N-body simulations) of individual structures (Johnston et al. 1995, Velazquez & White 1995, Helmi & White 2001, Peñarrubia et al. 2005). Here we instead discuss in principle how and why these models are uniquely sensitive to progenitor properties.

Note that the potential of the parent galaxy also affects debris properties, so debris can also be used to constrain the mass distribution in the Milky Way. This will be discussed in Chapter 7.

4.1. Young debris

As debris ages, it spreads further and further apart in orbital phase away from the progenitor along its orbit, overall decreasing in density over many orbits. The debris is considered fully phase-mixed once it fills the configuration-space volume defined by the progenitor's orbit. For example, for an eccentric orbit in a spherical potential, fully phase-mixed debris would be a spread over a (near-planar) annulus with inner and outer radii corresponding to the pericenter and apocenter of the parent satellite's orbit. For a loop orbit in a potential where the orbital plane precesses, the debris would fill a three-dimensional donut shape. For a given orbit, debris becomes fully phase-mixed much more rapidly for more massive progenitor objects.

The term “young debris” is used to refer to structures that have not had time to fully phase-mix and are apparent as distinct spatial overdensities in star-count maps, such as the Sagittarius and Orphan streams, and the GD-1 and Pal 5 globular cluster streams. These have been found at distances of 10 kpc to 100 kpc from the Sun where the background stellar density is low enough for such low surface brightness features to be apparent and orbital timescales sufficiently long for mixing not to have proceeded far during the lifetime of the Galaxy.

As illustrated in Figure 1, streams result from satellite destruction along mildly eccentric orbits. In these cases, the spreading due to differences in turning point precession (Ψl, see equation [10]) is much smaller than the initial angular width of the stream (of order s Rrtide) for many orbits and always less than the spreading along the orbit due to differences in orbital time (Ψe, see equation [9]). Hence the width of a stream is an indication of the progenitor mass. Once the mass is known, the age of the debris (or time since the satellite first started losing stars) can be estimated from the length of the streams.

Shells result from destruction along more eccentric orbits. There have been extensive studies of the properties and interpretation of shell systems seen around external galaxies (Quinn 1984, Sanderson & Helmi 2013), though these have largely concentrated on structures formed along almost radial orbits. The transition between conditions that produce stream-like or shell-like morphologies primarily depends on orbital eccentricity (e.g. Johnston et al. 2008, found this to occur on orbits where L / Lcirc ∼ 0.3−0.5), but also depends on satellite mass and time since disruption. Both stream-like and shell-like morphologies can occur in the same structure in this transition region. For a given mass and orbit, streams are most apparent in the early stages of evolution as the spreading along the orbit that produces them (i.e. as estimated by Ψe in equation [9]) typically occurs more rapidly than the differential precession of the apocenters that gives rise to shells (i.e. as estimated by Ψl in equation [10]). However, along the more eccentric orbits, the rapid passage of debris through pericenter significantly reduces the density at these orbital phases and the apparent angular extent of contiguous streams can be effectively limited by the angular size subtended by the region around the apocenter of the orbit where debris spends most of its time (estimated in Hendel & Johnston 2015, as the angle centered on apocenter where the satellite orbit spends half of its time, α in Figure 3). Under these circumstances, a single disruption event can produce what appear to be distinct structures, centered at two or more orbital apocenters with orbital precession becoming the dominant effect that dictates the apparent angular spread around each one. The time at which these structures start taking on shell-like characteristics can be estimated by finding when Ψl is greater than α. (See also Amorisco 2014, for an independent discussion of these effects).

4.2. Fully phase-mixed debris

Despite the low density and lack of spatial coherence of fully phase-mixed debris, its presence can often still be detected. For example, Liouville's theoremLiouville's theorem states that the flow of points through phase-space is incompressible: that the phase-space density of debris remains constant in time even as it evolves to form streams and shells. An inevitable consquence is that debris must become locally more concentrated in velocity space even as it becomes more diffuse in configuration space (Helmi & White 1999). Hence, signatures of the disruption of satellites can remain apparent in catalogues of stellar velocities even if no spatial structures remain detectable. This idea was first conjectured by Olin Eggen (for a summary see Eggen 1987), who proposed that nearby moving groups of stars could be the remnant of long-dead star clusters. In Chapter 5 the more recent work on velocity substructure in the stellar halo (e.g. Xue et al. 2011, Schlaufman et al. 2009) is discussed in more detail.

Surveys with full phase-space information can exploit the fact that the orbital properties of the debris (e.g. their actions or frequencies) remain constant in a static potential. Helmi et al. (1999) were the first to apply this idea to data using the HipparcosHipparcos mission catalogue of proper motions and parallaxes to derive angular momenta and estimate energies for giant stars within 1kpc of the Sun. They found ∼10% of the stars in their sample to be clumped in orbital-property space and concluded that 10% of the local halo must be formed from an object similar to the dwarf galaxy Sagittarius being disrupted in the distant past.

Once larger samples with more accurate orbital properties are found in near-future surveys, Gómez & Helmi (2010) have shown how the ages of these structures might also be determined by looking for substructure in orbital properties within a group (see also McMillan & Binney 2008). To develop some intuition for this idea, consider the (incorrect) model of debris spreading exactly along a single orbit. In the early stages of mixing the local volume will contain just one wrap of the debris stream at one orbital phase. As the debris spreads, the local volume will gradually fill with more and more wraps. The spreading itself is caused by the debris having a range of orbital properties. Hence, each wrap of the debris within the local volume will have different orbital properties and the degree and spacing of substructure within the orbital properties of the group gives an estimate of the age.

The beauty of using orbital properties to identify debris members is that stars can be connected even if they have no clear association in velocity or configuration space alone. For example, looking to the future, combining a distance indicator with Gaia's assessment of proper motions and radial velocities suggests that satellite remnants throughout the inner 100 kpc of the Milky Way might be identified using this method even if their members are spread on disparate planes and a wide range of radii.

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