The above discussion of local halo streams focused on those found using three-dimensional kinematics. However, once we move beyond the solar neighbourhood, uncertainties in proper motion prohibit the use of tangential velocities. Tangential errors scale linearly with distance; for example, a proper motion error of 1 mas/yr at 1 kpc corresponds to a tangential error of 4.7 km/s, but at 10 kpc this error will grow to 47 km/s. As a consequence, any detailed structure in phase-space is washed out. Fortunately radial velocity uncertainties do not scale with distance and so these can be used to probe large volumes of the halo, with the caveat that it is now harder to interpret any identified substructures, as two components of the phase-space are missing. Also working in our favour is that in the outer halo the mixing times are much longer, which allows ancient structures to remain coherent in configuration space for many Gyr.
The most spectacular and important discovery of halo substructure through radial velocities is that of the Sagittarius dwarf galaxy. For many years observers had found dwarf galaxies around the Milky Way, beginning with the Magellanic Clouds, but one lay hidden behind the bulge of the Milky Way and was only discovered in 1994 (Ibata et al., 1994). This was uncovered serendipitously during a radial velocity survey of stars towards the Galactic bulge. When analysing the radial velocity distributions in certain fields, instead of the expected Gaussian distribution, a secondary peak was identified. This peak corresponded to the Sagittarius dwarf galaxy, whose systemic radial velocity is offset from the bulge by around 150 km/s. Subsequent works have revealed that this dwarf is in the process of being devoured by our Galaxy, with tidal streams being discovered encircling the entire Milky Way (see Chapter 2).
At higher latitudes other streams have been identified by their coherent radial velocities, including the Cetus polar stream (Newberg et al., 2009), the cold metal-poor stream of Harrigan et al. (2010) and the high-velocity stream of Frebel et al. (2013).
One of the most impressive studies of radial-velocity selected substructures in the halo was led by Kevin Schlaufman in the ''ECHOS'' series of papers (Schlaufman et al., 2009, 2011, 2012). Their approach was to take each SDSS/SEGUE spectroscopic plate and determine, using robust statistics, whether the radial velocity distribution of the main-sequence stars matches what one expects for a smooth halo. This was done using two statistical tests, as described in Section 3.2 of Schlaufman et al. (2009) and illustrated in Fig. 9. The first test compared the radial velocity histogram to a similar histogram (with the same number of radial velocity measurements) drawn from a smooth model halo. This realization of the smooth halo was repeatedly resampled in order to test the significance of any peaks in the observed distribution. As can be seen from the upper panel of Fig 9, the grey shaded region shows the 95 per cent confidence interval from these realisations of the smooth halo model; the fact that the observation (black histogram) and its error bar do not overlap the grey shaded region implies that this is a robust detection. The second test is based on the cumulative distribution of velocities, which retains more information than the previous approach (i.e. unlike the previous approach, it avoids any binning of the data). Again the observed distribution is compared to one drawn from a smooth model, but this time they compare the steepness of the cumulative distribution function. This is shown in the lower panel of Fig. 9, where one can see that the observed slope (given by the black curve) reaches into the high-significance regions (shown by the dark grey region).
Figure 9. Example of the search for an ECHOS from Schlaufman et al. (2009). The upper panel shows the radial velocity distribution (black histogram with error bars) and the 95 per cent confidence interval from the multiple realisations of the smooth halo model (grey shaded histogram). The middle panel shows the cumulative distribution function from the observed data (black) and mean from the realisations of the smooth halo model (grey). The lower panel shows their measure of the statistical significance Theta (black line), with the various significance levels denoted in light and dark grey. The wide Gaussian in this panel shows an envelope around the most significant region, which is included to remove any spurious detections due to small-scale fluctuations.
Schlaufman et al. applied these techniques to observations of main-sequence halo stars out to 17.5 kpc on 137 individual spectroscopic plates of SEGUE data. In these 137 lines of sight, they identified a total of 10 strong candidates (where the number of false positive detections is estimated to be less than 1) and a further 21 weaker candidates (estimated to have less than 3 false positives). A number of these are likely to be detections of existing substructures, such as the Monoceros stream, but 7 of the strong candidates are new detections. Note that this does not translate to 7 new independent halo substructures, as some streams could intersect multiple lines of sight, but it does show that the halo of the Milky Way is (at least in terms of its kinematics) lumpy, even in the inner halo where phase mixing should occur on relatively small time-scales. Schlaufman et al. quantify this ''lumpiness,'' concluding that around 34 per cent of the inner halo is in the form of elements of cold halo substructures (ECHOS) and estimate that there could be as many as 1e3 individual kinematic groups in the entire inner halo.
The chemical composition of these ECHOS was investigated in the series' second paper (Schlaufman et al., 2011). They found that these ECHOS were more iron-rich and less alpha-enhanced than the smooth halo, concluding that the most-likely origin is that they were formed from the tidally disrupted debris of relatively massive dwarf galaxies (Mtot > 1e9 M⊙). In the final paper of this series (Schlaufman et al., 2012) the authors investigated the spatial coherence in [Fe/H] as a function of Galactocentric radius, concentrating only on main sequence turnoff stars from the kinematically smooth halo (namely SEGUE fields in which no ECHOS were detected). Although these fields are phase mixed and show no kinematic substructures, the chemistry of these stars can illuminate their origins. By studying the distributions of stars in the [Fe/H]-[α/Fe] plane they found that the accreted halo becomes dominant beyond around 15 kpc from the Galactic centre, arguing that at smaller radii the halo is probably formed from a combination of in-situ star formation and dissipative major mergers at high redshift.
As the careful work of Schlaufman et al. has shown, statistical studies are important if we are to dissect the halo of the Milky Way beyond the solar neighbourhood, especially when the sky coverage is not contiguous and the sampling of stars with spectra is sparse. Various tools have been developed, including the 4distance measure introduced by Starkenburg et al. (2009; see also the similar approach of Clewley & Kinman 2006). This technique, which is based around the separation of pairs of stars in four dimensions (angular position on the sky, distance and radial velocity), has proved influential for subsequent works (e.g. Cooper et al., 2011, Xue et al., 2011) and will undoubtedly continue to be used on surveys of distant halo stars where proper motions are unavailable.