The SEGUE K Giant Survey. III. Quantifying Galactic Halo Substructure
William Janesh, Heather L. Morrison, Zhibo Ma, Constance Rockosi, Else Starkenburg, Xiang Xiang Xue, Hans-Walter Rix, Paul Harding, Timothy C. Beers, Jennifer Johnson, Young Sun Lee, Donald P. Schneider
rrevised version, 2015 Nov 10
The SEGUE K Giant Survey. III. Quantifying Galactic Halo Substructure
William Janesh , ∗ , Heather L. Morrison , Zhibo Ma , Constance Rockosi , Else Starkenburg , , ,Xiang Xiang Xue , Hans-Walter Rix , Paul Harding , Timothy C. Beers , Jennifer Johnson , ,Young Sun Lee , Donald P. Schneider , ABSTRACT
We statistically quantify the amount of substructure in the Milky Way stellar halousing a sample of 4568 halo K giant stars at Galactocentric distances ranging over 5-125kpc. These stars have been selected photometrically and confirmed spectroscopically asK giants from the Sloan Digital Sky Survey’s SEGUE project. Using a position-velocityclustering estimator (the 4distance) and a model of a smooth stellar halo, we quantifythe amount of substructure in the halo, divided by distance and metallicity. Overall,we find that the halo as a whole is highly structured. We also confirm earlier workusing BHB stars which showed that there is an increasing amount of substructure withincreasing Galactocentric radius, and additionally find that the amount of substructurein the halo increases with increasing metallicity. Comparing to resampled BHB stars, wefind that K giants and BHBs have similar amounts of substructure over equivalent rangesof Galactocentric radius. Using a friends-of-friends algorithm to identify members of Department of Astronomy, Case Western Reserve University, Cleveland, OH 44106, USA * Current Address: Department of Astronomy, Indiana University, Bloomington, IN 47405, USA UCO/Lick Observatory, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA Department of Physics and Astronomy, University of Victoria, PO Box 1700, STN CSC, Victoria BC V8W 3P6,Canada CIFAR Global Scholar Leibniz-Institut f¨ur Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117 Heidelberg, Germany Department of Physics and JINA Center for the Evolution of the Elements, University of Notre Dame, NotreDame, IN 46556, USA Department of Astronomy, Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA Department of Astronomy and Space Science, Chungnam National University, Daejeon 34134, Republic of Korea Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802 Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802 a r X i v : . [ a s t r o - ph . GA ] N ov ∼ Subject headings:
Galaxy: evolution - Galaxy: formation - Galaxy: halo - Galaxy:kinematics and dynamics
1. Introduction
Current cosmological models predict that structure forms through hierarchical processes. Forgalaxies, the hierarchical assembly model implies that satellite galaxies will be tidally disrupted,leaving stellar debris (Bullock & Johnston 2005; Cooper et al. 2010). Large–scale surveys like theSloan Digital Sky Survey (SDSS; York et al. 2000) and the Two Micron All Sky Survey (2MASS;Skrutskie et al. 2006) have the capability to find these substructures around the Milky Way. Kine-matic and chemical information derived from these observations then allow for more detailed anal-ysis of the history of the buildup of the Galaxy.The search for stellar substructure in and around the Milky Way has been remarkably successful(cf. Belokurov et al. 2006). In addition to the the Sagittarius dwarf spheroidal galaxy (Sgr; Ibata etal. 1994), and its associated tidal stream (Mateo et al. 1996; Majewski et al. 2003), about a dozenpresumably distinct streams or overdensities have been discovered photometrically (Duffau et al.2006; Grillmair & Dionatos 2006; Belokurov et al. 2007; Newberg et al. 2007, 2009, 2010; Grillmair2014). These streams have a variety of morphologies, ranging from thin, kinematically cold streamsto ‘clouds’, and may comprise the majority of all halo stars beyond ∼
15 kpc (Bell et al. 2008).By collecting the spatial and kinematic properties of Galactic stars, a statistical analysis ofsubstructure in the Galactic stellar halo becomes possible. Precedent for this kind of analysis iswidespread. Maps of stellar density are a classical method of finding substructure. Substructureregions will appear to be more dense on the sky than a smooth background of stars. Large–scalephotometric surveys are well suited to this method. For example, Belokurov et al. (2006) useda simple color cut on a sample of SDSS stars which highlighted a number of structures in theMilky Way halo, including the Sgr tidal streams, the Orphan Stream, and Monoceros ring. Withmulti-color photometric data, this analysis can also be extended to population information. Bellet al. (2008) quantified the variation in star counts in the color region dominated by halo mainsequence stars, and Deason et al. (2011) made a similar calculation using blue horizontal branch(BHB hereafter) stars, and found a smaller variation in star counts at the same distances. Thisresult is intriguing because, while all stars start their lives on the main sequence, only those with 3 –a certain range of age and metallicity evolve to become BHB stars, which are typical of old, metalpoor populations. Thus the smaller amount of substructure seen in BHB stars may be due to apopulation difference rather than simply indicating a smoother halo than previously thought, asclaimed by Deason et al. (2011).Use of kinematic data enhances a metric’s ability to distinguish substructure from its sur-rounding smooth distribution. Gorski et al. (1989)’s method is similar to the classic two-pointcorrelation function, in that it uses a random smooth distribution for significance testing, but themetric instead finds the probability that two objects separated by a given distance will have thesame velocity, rather than the number of pairs at a given separation in distance. This metric hasbeen largely used in cosmological clustering studies, but could be easily adapted to stellar kinematicdata. Schlaufman et al. (2009) identify substructure by exploring the radial velocity distributionsof metal-poor main-sequence turnoff stars on individual SDSS plug-plates (hereafter, plates). Thistracer limits them to the inner halo: distances of 17.5 kpc and less. This method requires a fairlydense sample to be effective, since it makes use of velocity distributions. Schlaufman et al. (2009)used nearby SDSS dwarfs in their analysis, but were still able to identify ten stellar structures in thenearby halo and make predictions about the overall membership of stars in substructure, findingthat 1 / x, y, z, v x , v y , v z )to identify substructure. Given the very accurate observations that are required to measure all sixof these dimensions, and the current capabilities of large–scale ground based surveys, only four ofthese dimensions are readily available. Starkenburg et al. (2009) used a sample of 101 K giantsfrom the Spaghetti survey (Morrison et al. 2000), with distances up to ∼
100 kpc and line-of-sightvelocities to determine the overall amount of substructure in the Milky Way stellar halo, findingthat it is indeed highly structured, but less so than simulated halos derived from Harding et al.(2001). This difference is due to the sparse spatial sampling of the Spaghetti survey being unableto resolve narrow streams in the halo, an issue which Starkenburg et al. (2009) suggest could bealleviated by a much larger survey. Additionally, the authors were able to identify a number ofstars associated with the Sagittarius stream and Virgo overdensity using a friends-of-friends groupfinding algorithm.Analysis of a much larger sample of SDSS BHB stars with 5 kpc < R gc <
60 kpc by Xue et al.(2011) (hereafter X11) demonstrated with very high significance that position–velocity substructureexists throughout the Milky Way halo. With this sample, X11 showed that the outer halo was morestructured than the inner halo, and that these observations were consistent with, but less prominentthan, levels of substructure in mock catalogs derived from the Bullock & Johnston (2005) models.X11 attributed the difference with the models to the strong representation of BHB stars in olderstellar populations, showing that the older model particles also exhibited less substructure. Further,Cooper et al. (2011) used ( x, y, z ) and line-of-sight velocity to quantify substructure in mock catalogsfrom the Aquarius stellar halo models (Springel et al. 2008; Cooper et al. 2010), comparing thiswith a smaller sample of SDSS BHB stars from Xue et al. (2008). Interestingly, they find that 4 –the clustering signal for stars 20-60 kpc from the Galactic center is consistent with the range ofclustering in the Aquarius models, but for inner halo stars the signal is lower than all of the modelhalos. They suggest that this lack of substructure may be due to a smooth component of the innerhalo.Studying the amount of halo substructure in red giants is preferable to solely using BHBstars, as all intermediate-age and old stars, regardless of metallicity, turn into red giants. Wenote that K giants are drawn from a larger range of age and metallicity than BHB stars, andso comparisons between substructure in the two groups could be very helpful. In this study, wepresent a considerably larger set of red giants than in Starkenburg et al. (2009) to overcome theproblems of statistical sampling and provide a more robust comparison with simulations. We usea statistical method called the 4distance (Starkenburg et al. 2009, described fully in Section 3) toquantify the substructure in the largest and most spatially diverse spectroscopic K giant sampleto date. In Section 2, we describe our sample and the methods used to construct it. In Section3 we also discuss other methods used in this paper, including normalization techniques and thefriends-of-friends group finding algorithm. In Sections 4 and 5, we present our results.
2. The sample
The Sloan Extension for Galactic Understanding and Exploration (SEGUE) obtained nearly350,000 spectra of 21 different target types in its four years of observations (Yanny et al. 2009a, C.Rockosi et al. in preparation, York et al. 2000; Gunn et al. 2006; Eisenstein et al. 2011; Smee etal. 2013). The definition, calibration and verification of our K giant sample is described in detailin H. Morrison et al. (in preparation), and the Bayesian distance estimation technique we employis described in Xue et al. (2014). Here we provide an overview only.SEGUE used a pencil-beam sampling of the sky, which can be seen in Figure 3. The SDSS ugriz system (Fukugita et al. 1996) was not designed for studying stars, so targeting possible giantswas not as easy as it would have been with, for example, specially designed filters such as the DDO51 filter (Geisler 1984). Three target types aimed to identify K giants; all three used regions ofthe u − g / g − r color-color diagram. The first target type ( l -color K giants), designed to identifymetal-poor giants of the halo, used the metallicity sensitivity of the u − g color at the bluer endof the K giant region to identify halo giants. However, this method does not work for the redderstars: as discussed in Yanny et al. (2009a) and illustrated in their Figure 10, the giant sequencecrosses the foreground dwarf locus around ( g − r ) =0.8, and appears above the foreground locusfor even redder g − r . SEGUE used two target types to identify K giants here: the red K giant andproper motion K giant categories.We identified our K giants by first finding all the stars in SDSS data release 9 (DR9; Ahnet al. 2012) with spectra which also satisfied the ugr and other requirements for target selectionof the three types discussed above. We limited our consideration to stars with E ( B − V ) from 5 –Schlegel et al. (1998) less than 0.25 mag. We then used the SEGUE spectra to measure each star’sluminosity by first taking a log g value from the SEGUE Stellar Parameters Pipeline (SSPP; Lee etal. 2008) as an initial cut (log g < all of the following criteria: [Fe/H] > − . | z | <
10 kpc, the distance from the Galactic center inthe plane
R <
20 kpc, and the star is located inside the region of the longitude-velocity plot shownin blue in Figure 2.We chose a metallicity cut as disk stars are on average much more metal-rich than halo stars.Since there are a few halo stars with [Fe/H] greater than –1.0 even in the solar neighborhood (see,for example Carney et al. 1996), and there may be some thick disk stars with [Fe/H] less than–1.0 (the metal-weak thick disk stars discussed below), we used a cut of [Fe/H] > − . z height( | z | <
10 kpc) might seem high for a thick disk scale height of order 1 kpc (Majewski 1993; Juri´c etal. 2008). However, given the expectation that the numbers of a stellar population in a pencil-beamsurvey peak at 2-3 times its scale height due to the volume element growing with distance, we havechosen a conservative cut in order to exclude as many thick disk stars as possible.Our last criterion employs a longitude-velocity plot (often used in studies of gas dynamicsbecause a distance estimate is not readily available) in order to require that the star has kinematicsconsistent with the disk/thick disk. Figure 1 both highlights the kinematics of our giant sampleto show how the disk/thick disk appears in these plots, and also shows how bright streams such asthose from the disruption of the Sgr dwarf galaxy will appear. The Figure shows the radial velocity(corrected to account for the LSR velocity around the Galactic center) versus Galactic longitudefor our enlarged K giant sample ( ∼ l < − , and the other which shows negative velocities and is visible for almost theentire range of longitude.The second panel shows the l − v distribution of more metal-rich stars ([Fe/H] > − .
5) at lowlatitude ( | b | < ◦ ). One would expect these criteria to isolate stars from the Galaxy’s disk andthick disk. Since Galactic rotation is defined to be positive in the direction l = 90 ◦ (Binney &Merrifield 1998), we see that the chosen stars have the correct sign of velocity for disk objects. Therange of v gsr seen for disk stars is a function of both their position along the line-of-sight and of theirpopulation’s velocity dispersion. Since V φ is the only component of the velocity of a disk/thick diskstar with non-zero mean, a star whose position places the direction of V φ close to the line-of-sightwill show a significant velocity in the direction of rotation of the Galaxy’s disk. Conversely, starsat the anticenter will have velocity close to zero because V φ is orthogonal to the line-of-sight there,as can be seen in this panel. As a star’s latitude increases, the component of V φ along the line ofsight will also decrease because of projection. The velocity dispersion of the population along theline-of-sight will then ‘jiggle’ the star’s position away from its geometric expectation.The third panel shows stars from the same latitude range ( b < ◦ ), chosen to have [Fe/H] lessthan –1.0. Here we see a completely different velocity distribution, which is basically symmetricaround zero velocity, except for some substructure likely to be stellar streams. This symmetry invelocity suggests that the stars belong to the Galactic halo, which has a near-zero mean velocity(eg Zinn 1985; Fermani & Sch¨onrich 2013). We also see little contribution from the metal-weakthick disk (Norris et al. 1985; Morrison et al. 1990; Carollo et al. 2010) in our sample.Lastly, the bottom panel shows only giants with distances available in X14, which are chosento have R gc >
20 kpc. Since the edge of the Galaxy’s disk is at approximately this distance, wewould expect this criterion to exclude most disk stars and all stars from the inner halo. The featurewhich we can see stretching across values of longitude from 60 to 360 is caused by the various wrapsof the Sgr dSph, and will be discussed in more detail in Section 3.4. Since the pericenter of theorbit of the Sgr dwarf is at R gc ∼
17 kpc (Law & Majewski 2010) and the current position, closeto pericenter, is in the Southern Hemisphere, and thus not accessible to SEGUE, we should seealmost all Sgr members in our sample in this panel.Figure 2 illustrates the velocity cut we apply to remove disk stars from our sample. We showall stars with
R <
20 kpc and | z | <
10 kpc, highlighting those with [Fe/H] greater than –0.8. Itcan be seen that almost all of the red symbols are inside the blue lines which indicate the velocitycut. In summary, we exclude stars as possible disk or thick disk members if they satisfy two spatial 7 – |b|<35 [Fe/H]>-0.5|b|<35 [Fe/H]<-1R>20 kpc
Fig. 1.— The signature of the disk and of the Sgr dSph stream in a plot of Galactic longitude l against the line of sight velocity of the star, corrected for the projection of the LSR velocity alongthe line-of-sight. The top three panels are drawn from our extended K giant sample of 15,081giants, while the bottom panel comes from the smaller sample of K giants with distances fromX14. The top panel shows the entire extended K giant sample. The next lower panel shows starsat low latitude ( | b | < ◦ ) with high metallicity ([Fe/H] > − . < − .
0, and in contrast, displays a symmetry between negative and positive velocitiestypical of the halo. The last panel includes giants with distances greater than 20 kpc from theGalactic center. The signature of the Sgr dSph stream is clearly seen here.criteria (in R and z) and a kinematical criterion. When these cuts are applied, we end up with4568 stars in our halo sample. 8 –Fig. 2.— Galactic longitude vs radial velocity, corrected for the motion of the LSR projected onthe line-of-sight for all stars with
R <
20 kpc and | z | <
10 kpc. Stars with [Fe/H] > − . | b | < ◦ ) shown as having no giants were excluded fromour giant sample because of their high reddening or because of our disk star exclusion algorithm.We adopt the SSPP [Fe/H] measurement without applying any further corrections, as H. Mor-rison et al. (in preparation) have shown that these [Fe/H] values closely correspond with literaturevalues for our calibration sample of star clusters and stars with high-resolution spectroscopy. Typ-ical errors in [Fe/H] are 0.15 to 0.20 dex. 10 –Fig. 4.— [Fe/H] distribution for the spectroscopically confirmed K giant sample with 4568 giants.The vertical dashed and dot-dash lines show the ranges of [Fe/H] we use to create metallicitysubsamples below. The dot-dash line represents the median value ([Fe/H] ≈ − .
3. The 4distance
There have been many approaches to measuring substructure: one of the simplest being simplyto count the number of halo turnoff stars. This technique highlights spatial overdensities in the halo, 12 –which are particularly apparent in the early stages of a stream’s disruption (Bullock & Johnston2005). Our Figure 3 shows suggestions of such spatial overdensities, but would require a propercorrection for our complex SEGUE observational strategy to be compared with data such as the‘Field of Streams’. Another simple approach is to study the velocity distribution in different fieldsof a pencil-beam survey, as modeled by Harding et al. (2001) and implemented by Schlaufman et al.(2009): deviations from a smooth shape in halo fields are likely to be caused by disrupted streams,and this has the advantage of retaining a signal for later stages of disruption. In Figure 6 we showsome typical velocity histograms from giants in SEGUE fields. It can be seen that few of thesehistograms have smooth or near-Gaussian velocity fields, suggesting a large degree of substructurein our halo sample.The method that we have chosen to study substructure combines several different dimensionsof data: position on the sky, distance, and velocity. As in X11, we use the 4distance methodof Starkenburg et al. (2009, S09 hereafter). This metric takes advantage of SEGUE’s capabilityto measure several dimensions in phase space, specifically latitude, longitude, line-of-sight radialvelocity, and distance ( l, b, v los , d sun ). The 4distance calculates the separation of two stars in eachof these four quantities, applies a weighting factor dependent on the measurement range and error,and then produces a dimensionless distance in phase space, δ d = w φ φ ij + w v ( v i − v j ) + w d ( d i − d j ) , (1)where the angular separation φ ij is defined by:cos φ ij = cos b i cos b j cos( l i − l j ) + sin b i sin b j . (2)The three weighting factors ( w φ , w d , w v ) in the 4distance have a two-fold function: to scale theresult to physical quantities and to take into account the measurement errors of the data in themetric itself. These weights are defined as w φ = 1 π , (3) w d = 1130 ( d err ( i ) d ( i ) ) + ( d err ( j ) d ( j ) ) < d err d > , (4)and w v = 1500 v err ( i ) + v err ( j )2 < v err > . (5)Since the error in angular position is negligible, it is not included here. However, velocity anddistance both often have a significant error component. In our case, distance error scales withdistance, but velocity error does not scale with velocity. This is reflected in the weighting factors.The choice of the constant in the weighting factor is determined by the range of the observedquantities and represents the maximum separation between two stars in the 4distance metric. 13 –Fig. 6.— Examples of velocity distributions (showing v gsr , with the projection of the LSR and solarvelocity on the line-of-sight removed) for K giants on individual plates. The label in each quadrantshows the plate number and ( l, b ) coordinates.Table 1 shows some examples of physical sizes corresponding to a 4distance of 0.03. Forexample, if the 4distance was calculated for two stars with identical values of l, b and distance,then a difference of 15 km s − in velocity would produce a 4distance of 0.03. For stars on the sameSEGUE plate, the range in 4distance can be quite large for plates with large numbers of ( >
20) 14 –stars. An extreme example is plate 3245, which has 36 stars and has a minimum 4distance of0.006, median 4distance of 0.185, and maximum 4distance of 1.084. For stars separated by a platediameter (2.98 deg), the minimum 4distance (equal velocity and distance) is 0.016.We define a pair of stars as being two stars separated by no more than a certain 4distance.We can, in effect, derive a two-point correlation function in a sample by comparing the number ofpairs we observe to the number we would expect from a smooth distribution. The exact value wemeasure is the ratio of the cumulative number of pairs in the data with δ d less than a certain valueto the number of pairs in a “smooth” sample (we describe the characteristics of a smooth sample inSection 3.1), which we will refer to as the 4distance ratio. As δ d (or the 4distance bin) increases,we expect the 4distance ratio to approach one as the cumulative number of pairs in the data andsmooth sample become equal, meaning that at larger separations, there are no additional pairsbeing added to the cumulative sum. Error bars on our measurements are simply due to Poissonstatistics, modeled using a Monte Carlo approach.When small values ( δ d < .
10) of the 4distance have a relatively large signal, we can inferthe existence of streams and other cold substructure in the sample. The 4distance is a robustmeasurement of the amount of substructure in a population, but is not always an ideal indicatorof the presence of streams. The spectrographic plate design of the SDSS forces objects on thesame plate to be observed at relatively small angular separations, which causes the bulk of thecontribution to the 4distance between two stars on the same plate to be from velocity and distance.The number of pairs will also vary as a function of distance. Whether through a magnitude limitor the halo density distribution function, the number density of stars decreases with increasingdistance. This relationship causes a distant star to be less likely to be paired with another, butemphasizes the significance of finding pairs at large distances.Table 1. Maximum physical component size for a selected 4distance (4 δ = 0 . φ ( l, b ) 5 . ◦ distance (heliocentric) 6 kpcvelocity (Galactocentric standard of rest) 15 km s −
15 –Fig. 7.— 4distance measurements for our K giants (red), those of Starkenburg et al. (2009) (greendashed), and a resampled set of BHB stars from X11 (blue) using the shuffle method of normaliza-tion used by Starkenburg et al. (2009)
In order to normalize the global indicator of substructure (the 4 δ ratio) we need to find thenumber of 4distance pairs we would expect to see in a smooth halo. S09, X11, and Cooper etal. (2011) solved the problem of normalization by randomly shuffling their data. By holding ( l, b )constant and independently shuffling v gsr and d sun , it was possible to approximate a smooth halowhile respecting a survey footprint. We compared the pair counting results for our K giant sampleto a shuffle normalization; the results are shown in Figure 7. The measurement for our order ofmagnitude larger sample is consistent with the S09 measurement. We see that, for a given valueof 4distance, the K giants have between 1.5 and 2 times more pairs than the BHB stars that 16 –Fig. 8.— 4distance measurements for K giants in the Sgr plane ( | B | <
12; magenta) and out of theSgr plane (
B > | | ; cyan) using the shuffle method of normalization used by Starkenburg et al.(2009). As expected, the stars in the Sgr plane show a much higher substructure signal than starsout of the Sgr plane.have been resampled to select those only inside the SEGUE footprint: they show signficantly moresubstructure. However, their radial distribution is different, with the K giants stretching to twicethe distance of the BHB stars because of their larger luminosity. X11 have shown that the amountof substructure in their BHB sample increases with distance from the galactic center, and we willshow that the K giants have a similar property in Section 4.1. We will further investigate the Kgiant/BHB differences in Section 4.3.To show one way the 4distance metric can be used to quantify substructure in the halo, wehave divided the K giant sample based on its position in the Sgr coordinate system (Λ, B ) describedby Majewski et al. (2003). Stars with | B | <
12 are classified as being in the Sgr plane, while allother stars are outside of the plane. We show the results of 4distance pair counting for these two 17 –subsamples in Figure 8. We obtain the expected result, which is that stars in the Sgr plane showa significantly larger amount of substructure than stars out of the Sgr plane.However, we note that shuffling normalization method does not account for all aspects ofspatial substructure since it leaves the number of stars at each ( l, b ) constant. We know thatspatial substructure is quite obvious in large samples (see Belokurov et al. (2006)), so we wishto include spatial position in the metric. SEGUE’s pencil-beam spectroscopic footprint has anuneven spatial sampling, so to understand the effect of spatial distribution on the measurementof the 4distance ratio we must “observe” a smooth halo model. Further, spatially concentratedstructures are more likely to be identified with the 4distance metric, but will be unaccounted for inthe pair ratio when using a shuffle normalization. Thus we have chosen to use a smooth halo modelfor our normalization, instead of following S09 and shuffling velocity and distance in the surveyfootprint. We will see below that adding the additional spatial information by using a smooth halofor normalization increases the substructure signal by a factor of ∼ Our choice of a smooth halo model for normalization, while adding sensitivity to substruc-ture, requires assumptions about the properties of a smooth halo, the first of which is the densitydistribution of stars. We use a power-law density distribution, ρ ∝ r − . (see Zinn 1985; Prestonet al. 1991; Vivas & Zinn 2006). It is important to use a reasonable estimate for stellar densitybecause the number of observed pairs depends on the density itself (more dense regions will havemore pairs), and we do not want to over-count pairs in the smooth halo, as this would decreasethe overall substructure signal. We will use a ρ ∝ r − . halo model in Section 3.3 to test thiseffect. Next, we assume a functional form for the velocity distribution of the smooth halo. Thischoice is less critical than the choice of the density distribution because the velocity component ofthe 4distance is the one with the smallest bin size compared to the overall range. We choose thevelocity ellipsoid from Morrison et al. (1990), ( σ x , σ y , σ z ) = (133 , ,
94) km s − in an attempt torealistically model the halo, but make comparisons to a uniform velocity ellipsoid in Section 3.3.Our next set of assumptions concern the observational constraints from our survey. Thefirst two relate to the distance distribution of our sample, but the most important of these is theluminosity function. K giants in SEGUE are a subset of target types, and different target types havedifferent distributions in absolute magnitude. This matters because it affects the K giant detectionlimits in the halo: some target types are redder stars, which can be intrinsically more luminous.Therefore we create an absolute magnitude distribution for the stars to which we attempt to matchthe absolute magnitude distribution of the model points. We also need to take into account theapparent magnitude and R gc distributions. The reason for this is to be sure that our model pointshave a similar, but not explicitly identical, distribution to our data. The apparent magnitudedistribution is also not flat, so we treat that the same way as the luminosity function, but alsoensure that the R gc distribution is similar to that of the data. The simplest of the observational 18 –constraints is the SEGUE plate footprint. Here, we only require a model point to be within 1.49degrees of a SEGUE plate center. Finally, we require the overall number of points to be the sameas observed, and the number of each target type the same as observed. Because the four target categories have different apparent magnitude and luminosity distribu-tions, we generate points for our smooth halo model for each target category separately. Since thesmooth halo needs the same number of points as the actual dataset we are testing for substructure,we count the number of stars from each category, and produce a smooth model with the same num-ber of stars. We produce a random sample with density distribution ρ ∝ r − . within the observedrange of R gc for each target category. We then generate an absolute magnitude (uniformly in therange [1 , − R gc bin for that combination of ( l, b ) position, apparent magnitude, and absolute magnitude. Thisprocedure has the effect of allowing us to generate an R gc distribution similar to that of the data,but drawn from a smooth underlying population. Also, we only keep points which are within theSEGUE footprint on the sky. Each point is then assigned velocities from the Morrison et al. (1990)velocity ellipsoid with ( σ x , σ y , σ z ) = (133 , ,
94) km s − . We add errors of 20% in distance and 5km s − in velocity to mimic observational errors. The 4distance ratio is a measure of global substructure in a sample of stars. As it is a cumulativedistribution, the ratio will change on a bin-by-bin basis depending on the physical size of thesubstructure in the sample. If the structures are more concentrated, the ratio curve will be steeper,while a smoother distribution will have a flatter curve. As 4 δ goes to one (which is defined by theweighting scheme as the maximum possible separation between two points in one dimension), theratio should also go to one. A ratio of one (meaning that the number of pairs is equal in the dataand smooth halo) indicates that the data is indistinguishable from a smooth halo. At the maximumseparation, all pairs in both the data and the smooth sample (which are the same size) should bein the pair ratio. It is possible for the ratio to drop below 1; this is due to the smooth sample beingmore structured than the data at that scale , which is to be expected given the statistical variationof randomly distributed points.Figure 9 shows the amount of substructure in a model of Sgr disruption created by LM10 whencomparing to a ρ ∝ r − . smooth halo. These models are labeled so that stars lost on each successivepericentric passage of the progenitor are assigned to a different ‘wrap’. By measuring each wrap 19 –Fig. 9.— 4distance measurements for the Law & Majewski (2010) (LM10) model, where debris lostat each pericentric passage is shown in different colors. The ratio at each 4distance bin is calculatedby dividing the number of pairs (determined by the 4distance metric, see Equation 1) in the databy the number of pairs in a smooth halo model (see Section 3.1 for details). The x − z plot for eachof LM10’s nine wraps are shown in the right hand panels. Starting at the top, these are: currentlybound stars, stars lost on the most recent pericentric passage, stars lost on the next most recentpassage, and so on.separately with the 4distance ratio, we can determine what different kinds of substructure looklike in the metric, since the wraps very in their morphology. Figure 9 clearly shows the differencebetween the progenitor (in red) and the stars that have had the longest time to mix in the halo(those lost on the first pericentric passage, black). The very dense core of the progenitor has almosta factor of 10 more pairs and a much steeper ratio curve than the more well-mixed stars from thefirst pericentric passage.Figure 10 shows substructure in the LM10 model observed with SEGUE’s pencil-beam geom- 20 –Fig. 10.— 4distance measurements for the LM10 model, but observed with the SEGUE footprint.The right hand panel shows the complete model, the middle panel the object in the SEGUEfootprint, and the left hand panel the number of pairs compared to a smooth model for the objectsin the SEGUE footprint.etry, again compared to a ρ ∝ r − . smooth halo. The 4distance measurements follow the samegeneral trend in the observed model as they do without the SEGUE footprint imposed, with onlyslight differences, indicating that while the survey footprint does add a complication to 4distancemeasurements, it does not do so in a drastic way. For this reason, however, we caution that wecan only directly compare substructure measurements between samples observed with the samesurvey footprint. Bound stars indicate the highest possible level of substructure, while the lowestobservable signal arises from the first wrap, which has had the most time to mix. 21 –Fig. 11.— Substructure measurements for the full K giant sample, compared to two differentmodel halos ( ρ ∝ r − . , red; ρ ∝ r − . , blue). The purple, green, and orange lines show the effectof variations in assumptions on the overall 4distance measurement. The purple line shows a model ρ ∝ r − . halo compared to a ρ ∝ r − . halo, the green line shows the effect of using the observedluminosity function compared to a theoretical luminosity function, and the orange line shows thesignal effect of using a Morrison et al. (1990) velocity ellipsoid with ( σ x , σ y , σ z ) = (133 , ,
94) kms − compared to a uniform velocity ellipsoid with ( σ x , σ y , σ z ) = (100 , , − . Figure 11 shows the result of our 4distance ratio measurement on the full K giant sample,against a number of smooth halo variations. We have measured a stronger substructure signal in Kgiants using our method with a smooth halo than using the same sample with S09’s shuffle method,because the smooth halo method allows for spatial clustering to contribute to the signal.Tests of the various assumptions in our smooth halo model are depicted in Figure 11. The 22 –purple line shows the effect discussed above, where the centrally denser ρ ∝ r − . halo has aamount of substructure not found in the ρ ∝ r − . halo, which is roughly constant over the range of4distance. Again, this effect is simply due to the fact that increased density leads to more measuredpairs. The green line shows the effect caused by the choice of luminosity function. We have chosento model the luminosity function (LF) by sampling the observed LF for each of our K giant targettypes, as opposed to using a theoretical LF for the relevant color range for each target type. Thischoice ensures that our smooth halo model has the same observational properties as the K giantsample. The green line shows that at small 4distance, there is no distinguishable difference betweenthe two, but at larger 4distance, the observational LF counts more pairs than a theoretical LF.Finally, we show that the velocity ellipsoid, shown in orange, has a small effect as well. The causeof this effect is unclear, but could be due to the slightly narrower distributions in σ y and σ z in theMorrison et al. (1990) velocity ellipsoid. Fortunately the change in the 4distance ratio by each ofthese effects is much smaller than the total signal level in our real sample, as shown by the red andblue lines. We therefore disregard the effect of these assumptions in our final measurements.Measuring a relatively strong signal using two different normalization methods confirms thepresence of substructure in the K giant sample. Furthermore, a strong signal is produced whenmeasured against two separate smooth halo models ( ρ ∝ r − . , ρ ∝ r − . ). As expected, the ρ ∝ r − . measurement has a slightly weaker signal than the ρ ∝ r − . measurement, due to thehigher central density in the ρ ∝ r − . halo, which increases the number of smooth pairs, drivingdown the overall measurement. Determining a measurement of overall substructure in the halo is useful, but for certain pur-poses we would like to identify specific structures. Following S09, we use the 4distance in com-bination with a friends-of-friends (FoF) algorithm to obtain a local measure of substructure bylinking stars into groups. These groups are stars with similar characteristics. In a metric system,two stars are associated if they are within a certain linking length, the maximum separation thattwo stars can have to be identified associated in a given metric. Recalling that a 4distance sizecontains information about sky position, velocity, and distance, it is useful to review what thatmeans physically. Our chosen linking length of 4 δ = 0 .
03 corresponds to physical sizes in eachdimension that can be found in Table 1.FoF works by drawing a circle around each point, with a radius of one linking length. If thecircle contains any other points, those points are in the group. Once a point is included in thegroup, a circle is drawn around it, and points inside that circle can now be added. The group iscomplete when no more points can be added. Since the 4distance involves more than just physicalposition, our circles need to be drawn in more than just spatial coordinates, but this highlightsthe power of the 4distance metric–the fact that it uses velocity information to find stars that aremoving in the same direction. 23 –Of additional concern when choosing a linking length is the pencil-beam geometry of theSEGUE survey. We want groups to extend across more than one plate pointing, but at the sametime, we do not want the FoF algorithm to be too generous in linking spatial positions. In otherwords, while we desire the ability to connect nearby plate pointings, the deciding membership factorshould be distance or velocity. Our choice of linking length of 4 δ = 0 .
03 takes these criteria intoaccount. The maximum angular separation of 5.4 ◦ allows for stars on nearby plate pointings to beincluded as possible group members, but will require them to have similar distances and velocitiesto be members of the same group. We will see below (Figure 16) that each Sgr stream wrap ismapped into a number of groups.
4. Results on Measuring Substructure in K giants
We have confirmed that the Milky Way halo is highly structured in K giants, but wish to makemore meaningful conclusions about the nature of the halo. Given our large sample size compared tothe previous K giant sample of S09, we are able to create subsamples that will allow us to exploreboth the inner and outer halo, as well as ranges in metallicity.
We might expect the inner halo to show less substructure than the outer halo for two reasons.First, any smooth component of the halo (formed, for example, by violent relaxation in the earlystages of halo formation) would be found in the inner regions. Second, accreted stars will likelyphase-mix faster in the inner halo due to the shorter dynamical times there. This expectation islargely borne out in the simulations of Bullock & Johnston (2005) and Cooper et al. (2010). Inboth papers the majority (but not all) of the realizations show more substructure in the outer thanthe inner halo (Xue et al. 2011; Cooper et al. 2011).In our data we see substructure increasing monotonically with distance (see Figure 12), withthe most distant stars (with R gc from 30 to 120 kpc) showing the strongest signal, ∼ R gc between 15 and 30 kpc. While stars at these intermediatedistances still show significant substructure, this is not the case for stars with R gc less than 15 kpc.However, there is abundant evidence for substructure in this region from other studies using eithervelocities (Schlaufman et al. 2009) or the full 6-D phase space information (e.g. Helmi et al. 1999).This highlights the fact that the 4distance is a relatively rough measurement of substructure. 24 –Fig. 12.— 4distance measurements for SEGUE K giants, divided into distance ranges in Galacto-centric radius. The innermost subsample (shown in green), from R gc = 0 −
15 kpc, exhibits theleast amount of substructure, while the outermost subsample (shown in blue), R gc >
30 kpc, showsthe most. Red shows the subsample with R gc = 15 −
30 kpc. We also present two separate smoothhalo normalizations in this figure ( R − . , R − . ) in dashed and solid lines, respectively. The use of K giants as tracers allows us to check explicitly for a dependence of substructureon metallicity, since it is considerably easier to obtain accurate [Fe/H] values for K giants thanfor BHB stars due to the much stronger metal lines in K giants. (Morrison et al, in preparation,demonstrate this for the SSPP metallicity measures.) This is of particular interest in the contextof the disruption of accreted satellites into the field halo because of the mass-metallicity relation(Lee et al. 2006) which allows us to infer that if a low-mass object is accreted, its stars are likelyto be metal-poor. 25 –Fig. 13.— 4distance measurements for SEGUE K giants, divided into abundance ranges. Themedian metallicity of our K giant sample is [Fe/H] ∼ − .
5. The red and blue lines represent thetop and bottom 25% of the metallicity range, respectively. More metal-rich K giants show a highlevel of substructure, while the two metal-poor categories show similar, but lower, levels.Figure 13 shows the global substructure measurement for the sample divided into [Fe/H] ranges.The most metal-rich group [Fe/H] > − .
2) shows a very strong substructure signal, with ∼ ∼ < − .
9) shows the most subtle signal, with ∼
50% more pairs than the smooth halo. 26 –
We can use a comparison between BHB stars and K giants to investigate the properties of stellarpopulations of different age, because globular clusters with similar metallicity but blue horizontalbranches are 1.5-2 Gyr older than those with red horizontal branches (Dotter et al. 2010). Becausestars of all ages traverse the red giant branch, while only older stars of the same metallicity willbecome blue horizontal branch stars, any differences in substructure between the two samples islikely due to age.Two photometric surveys have led to claims that BHB stars show less spatial substructurethan the overall halo population. The first (Bell et al. 2008) counted numbers of main sequenceturnoff stars in SDSS photometry covering distances from 7–35 kpc, and showed that there wassignificant substructure. (Deason et al. 2011) used a photometric technique to identify BHB stars inSDSS photometry, covering a distance range from 10- 45 kpc, but using a different method and overspatial scales different from those in Bell et al. (2008). Each photometric method has weaknesseswhich may lead to them underestimating substructure: the distance measures to turnoff stars arenot very accurate (Bell et al. estimate 0.9 mag. scatter) and this distance error will smooth out the“lumpiness” of spatial substructure along the line of sight. The BHB star sample suffers from somecontamination by foreground blue stragglers of the halo, and since X11 show that the substructurein BHB stars increases with galactocentric distance, this too will have a smoothing effect on theamount of substructure. Thus, although it is clear that the photometric BHB sample of Deasonet al. (2011) shows less substructure than the Bell et al. (2008) main sequence sample, it is notclear whether this is due to contamination of the BHB sample by inner halo blue stragglers or toan actual difference in substructure between the two tracers.There is also some indirect evidence for a difference in substructure signal which is arrived atvia a comparison with the models of Bullock & Johnston (2005). Bell et al. (2008) compared theirspatial substructure method with the BJ05 models and found good overall agreement, while X11compared the BHB substructure signal (in both velocity and position) with the same models andfound less substructure than predicted by the models. However, since the two comparisons weremade of a different signal (spatial vs spatial plus velocity) and over a different distance range (outto 35 vs 60 kpc), and we have seen that substructure in both BHB stars and K giants varies withdistance, we feel that this result is not conclusive.Clearly the spectroscopic samples of both BHB stars and K giants provide a safer measure ofsubstructure, without the smoothing effects discussed above. Our original calculation of the amountof substructure in the K giant and BHB samples using the “shuffling” method of normalization(shown in Figure 7) showed significantly more substructure in the K giants. However, since sub-structure in both samples increases with galactocentric radius, a correct comparison will limit theK giant sample to the smaller distance range probed by the BHB stars. Figure 14 shows the resultof this comparison: we see that the difference between the two stellar types has largely disappeared,and within the errors we see no significant difference in substructure using this method. 27 –Fig. 14.— 4distance measurements for SEGUE K giants (magenta) and X11 BHB stars (blue),resampled to have matching distributions in R gc . We also show the 4distance measurement for theintact K giant sample for reference (red). When resampled in this manner, K giants and BHBsshow no significant evidence for differing amounts of kinematic substructure.We thus see no conclusive evidence so far of different substructure signals in K giants and BHBstar samples. It would be interesting to see a full analysis of both BHB and K giants using a smoothhalo normalization as this would be more sensitive, because it also takes into account the spatialvariation of numbers across the sky. In addition, more exploration of the relative contribution ofBHB stars and K giants in the Sgr stream would be illuminating, but both are outside the scopeof this work. 28 –Fig. 15.— [Fe/H] vs distance for our K giant sample. The effects of SEGUE targeting and mag-nitude limits can be seen: there are no stars with [Fe/H] > − . < − .
4. Stars which are part of groups classified as ‘definitely’belonging to Sgr (see Section 5 below) are shown with blue points, and stars belonging to a possibledistant Sgr group are shown with red points.
Since the optical luminosity of a K giant increases as its metallicity decreases, a survey such asours with a limiting magnitude will find only metal poor stars in the most distant regions, and onlyrelatively metal-rich stars in its most nearby ones. We see in Figure 15 how the survey selectioneffects play into the distribution of distance and metallicity in our sample: the most metal richstars cover a smaller distance range than the most metal poor ones.If we saw more substructure in metal-poor and more distant stars we would need to worry aboutthe above degeneracy, but in fact our results show the opposite behavior: the most metal-rich and 29 –the most distant stars show the most substructure.We find that more metal rich stars have more substructure, but what does this imply for haloformation? Though its disruption is ongoing, the Sgr dwarf is a quite massive satellite (a recentestimate gives the current core mass at ∼ × M sun and intial virial mass at ∼ × M sun ,placing Sgr among the most massive Local Group dwarf galaxies (see Mateo 1998; (cid:32)Lokas et al. 2010)and has also been observed to be among the most metal-rich of the MW satellites (Mateo 1998).Given that massive satellites are more metal-rich (see Lee et al. 2006), it seems likely that the largeamount of metal-rich substructure in the sample can be attributed to Sgr, especially since we havetaken pains to remove the disk from the sample and that Sgr is such a visible contributor to spatialsubstructure in the Field of Streams. It should be noted, however, that the other metallicity rangesdo show a significant amount of substructure. We will attempt to quantify the contribution of Sgrand other streams to this substructure below.
5. FoF results
S09 find eight total groups, and only one group larger than two stars, which they suggest isassociated with Sgr. Our results with a larger sample (4950 to their 101), yield a much largernumber of groups; our data contain well over 100 groups larger than three members, and morethan 20 groups larger than 10 members in size. About 38% of our K giant sample is in a group offour or more members, ∼
4% is in a group with three members, and ∼
8% is in a group with twomembers (a pair). This then leaves ∼
50% not in a pair or a group. Of S09’s 101 stars, ∼ ∼
5% are found in a group of 4 or more stars, clearly illustrating that thepower of the 4distance and FoF algorithm increases with sample size.This large number of groups allows us to more deeply examine the properties of structuresin the Milky Way halo. Significant group detections found via the FoF method can be found inTable 2. Figures 16 (groups with ten or more members) and 17 (groups with between four andnine members) show these groups in context in four dimensions of phase-space. The largest groupsappear to be primarily associated with two large structures, one each in the Northern and SouthernGalactic hemispheres. These are, respectively, the Sgr leading and trailing streams, which will bediscussed in the next section. The diversity of substructure in the halo becomes more apparent inFigure 17, which shows smaller groups. Additionally, most of the groups in Figure 17 are locatedin the northern Galactic hemisphere.Unfortunately, the FoF technique suffers from the same problem that afflicts the overall pairratio measurement: there are more groups in regions with ( r (cid:12) ≤∼
20 kpc), due to the higherstar density coupled with fixed linking length. In this case, this means that the regions that arerelatively close to the Sun and Galactic center will appear to have more groups than more remoteregions. 30 –Fig. 16.— The FoF (friends-of-friends) groups greater than or equal to 10 members in size, shownwith the line-of-sight velocity (corrected for the solar and LSR motion) v gsr , against Galacticlatitude l (lower left), Galactic latitude b (lower right) and with longitude plotted against Galacto-centric radius R gc (upper left). The upper right panel shows the x − z plane, which is close to Sgr’sorbital plane. Dashed lines are every 10 kpc in R gc . K giants are shown as colored crosses, witheach color representing a different group (color repeats between groups should not be construed asan indication of group membership, but are merely caused by a limited number of colors). The fullK giant sample is represented by gray dots in the background. Nearly all of these larger groups areassociated with Sgr. By comparing to models and observations, we can identify groups associated with knownsubstuctures. The most important of these substructures is the Sgr stream. In fact, the Sgr streamcan be seen clearly in a simple longitude-velocity plot even without using the FoF algorithm. SinceSgr streams dominate substructure in the Field of Streams and are the best-studied streams in the 31 –Fig. 17.— The FoF groups between 4 and 9 (inclusive) members in size, shown in 4 differentsections of phase space as described in the caption for Figure 16. K giants are shown in coloredcrosses, with each color representing a different group (color repeats between groups should not beconstrued as an indication of group membership, but are merely caused by a limited number ofcolors). The full K giant sample is in greyscale in the background. The majority of these smallgroups fall in the northern Galactic hemisphere. Circled are the four notable non-Sgr substructuregroups discussed in Section 5.2: gold, Orphan stream; purple and teal, Cetus Polar Stream; cyan,possible Sgr stream members at 90 kpc.halo, we begin with quantifying their contribution to our sample. We need a reproducible techniqueto identify substructure belonging to Sgr, so we adopt the LM10 model as a way to define regionswhere Sgr streams should be observable, under the assumption that the LM10 models are anaccurate representation of the spatial and kinematic properties of the Sgr streams. By observingthe models using the SEGUE footprint (see Figure 18), we see the expected appearance of Sgrin a large–scale survey. In addition to LM10’s recommendation that only the five most recent 32 –pericentric passages are used, distance and velocity errors drawn from Gaussian distributions withsigma values of 20% and ∼ − , respectively, have been added to each model point that fallswithin 1 . ◦ of a plate center. By constructing boxes around the observed model, we create theregions used to identify members of Sgr. We use a simple system to determine membership. Thereare boxes in four separate dimensions ( l − v gsr , b − v gsr , l − z gc , x gc − z gc ), shown in Figure 19. If60% or more of a group’s stars fall into the box in all four plots, then the group is assigned toSgr (referred to hereafter as “definitely Sgr”). A significant number of groups are identified as Sgrgroups, and the majority of them have a large number of members, which means that the mostobvious substructures in the halo are associated with Sgr. The width of the potential observedstreams in the model is quite large, on the order of 10 kpc or 100 km s − , because Sgr is a massivedSph.Figure 20 shows groups that are classified as “definitely Sgr,” which appear to have a narrowervelocity distribution than the LM10 model, as well as a slightly different distance distribution, whereobserved stars appear to be on average closer to the Sun in some regions (notably in Figure 20 alongthe northern leading stream, where the average R gc differs by as much as 10 kpc, and the widthof the velocity distribution is 50 km s − narrower in some places). These groups are also largelyconsistent with the positions of the Sgr streams using SDSS K/M-giants in Yanny et al. (2009b),and using multiple target types by Koposov et al. (2012). The LM10 model largely reproduces theobserved distribution of Sgr stars in both 2MASS (Majewski et al. 2003) and SDSS (Ruhland et al.2011), pointing to a generally well constructed model. However, LM10 themselves acknowledge thattheir model does not produce features observed by SDSS (notably the bifurcation in the leadingarm), so it is possible that other small deviations exist. Further observations are required to confirmSgr membership and resolve these inconsistencies.One of our smaller groups (six members; marked in cyan in Figure 17) has a large Galactocen-tric radius that is consistent with Newberg et al. (2003)’s detection of potential Sgr debris 90 kpcfrom the Galactic center at ( l, b ) = (190 ◦ , ◦ ). Debris is also found in this location by Ruhlandet al. (2011), using a large sample of SDSS blue horizontal branch stars, as well as evidence ofan extension to the trailing Sgr stream toward 90 kpc, for which we do not find FoF evidence.Further evidence for Sgr debris near this position was presented in Drake et al. (2013). UsingRR Lyrae stars, Drake et al. (2013) find a structure, which they call the Gemini stream, locatednear ( l, b ) = (195 ◦ , ◦ ) and extending to a distance of ∼
100 kpc. All of these structures arelocated in or near the Sgr plane, though their radial velocities are unmeasured. The distant trail-ing stream hypothesis is further supported by observations presented in Belokurov et al. (2014),which found new Sgr trailing stream detections using photometry and kinematics in the NorthernGalactic hemisphere. The results of Koposov et al. (2015) also support the distant trailing streamwith spectroscopic observations of M giants in the area of the Galactic anticenter, consistent withthe results of Drake et al. (2013) and Belokurov et al. (2014).It has been proposed that NGC2419 (with ( l, b ) = (180 ◦ , ◦ ) and v gsr = −
14 km s − ) isassociated with this structure (Newberg et al. 2003), though the Gemini stream from Drake et al. 33 –Fig. 18.— The LM10 model observed with SEGUE plate centers, shown in the x gc − z gc plane,which is close to the Sgr orbital plane. In the left panel, each color represents stars lost on adifferent pericentric passage in the model (black shows bound stars; red, the most recent passage;blue, the next most recent passage; green, the second most recent; magenta, the third most recent;orange, the fourth most recent). The right panel shows the observed model, with the same colors.In both panels, dashed lines are every 10 kpc in R gc .(2013) is inconclusively linked to this cluster. Our FoF group has a position of ( l, b ) = (187 ◦ , ◦ )and a v gsr of −
64 km s − , making it possibly associated with either the Gemini stream or theNewberg et al. (2003) structure. The LM10 model does not predict debris in this position. Streamsat large Galactocentric radius are likely lost on early passages, and are particularly important toconstrain both Sgr’s accretion and the mass of the Milky Way. 34 –Fig. 19.— The regions defined as Sgr by use of the observed Law & Majewski (2010) model,shown with the line-of-sight velocity (corrected for the solar and LSR motion) v gsr , against Galac-tic longitude l (lower left), Galactic latitude b (lower right) and with longitude plotted againstGalactocentric height from the plane z gc (upper left). The upper right panel shows the x − z plane,which is close to Sgr’s orbital plane. Dashed lines are every 10 kpc in R gc . The observed modelpoints are shown in black. The red boxes enclose streams identified as trailing in LM10, and thepurple boxes enclose leading streams. While Sgr is a significant component of the Milky Way halo, it is by no means the only visiblesubstructure. Among the more interesting of other known substructures are the Orphan Stream(Belokurov et al. 2006), the Virgo Overdensity (Juri´c et al. 2008), and Monoceros Ring (Belokurovet al. 2006); the Grillmair-Dionatos stream (Grillmair & Dionatos 2006); and the Cetus Polarstream (Newberg et al. 2009). Our group classification catalog contains additional classificationsbeyond those for general Sgr/non-Sgr groups. These classifications are designed to identify groups 35 –Fig. 20.— All stars classified as part of groups ‘definitely’ belonging to Sgr, shown in blue crosses.The panels are the same as those described in the caption for Figure 16. This classification is madeby using the method described in section 5.1. The observed K giant substructure does not coverthe extent of the LM10 model’s velocity prediction for the leading stream between l = 220 ◦ and l = 360 ◦ .potentially associated with the Orphan Stream and Cetus Polar Stream. Our choice of FoF linkinglength is not well suited to detecting some substructure; we are unlikely to find a group that coversthe extent of the Virgo Overdensity due to its large spatial size. The Grillmair-Dionatos stream isquite narrow and relatively close; a linking length chosen to find substructure at a given distancewill create larger groups at a closer distance and “wash out” narrow streams. Finally, our criteriafor removing the disk from our sample makes it difficult for us to detect the Monoceros ring. Ourselection boxes are designed in a similar manner to those for Sgr, but instead of using a model, weuse observational data of the streams from SDSS detections (Newberg et al. (2009), Newberg et al.(2010)). 36 –Fig. 21.— All groups classified as ‘definitely not’ belonging to Sgr. This classification is made byusing the method described in section 5.1.We identify several relatively large groups that are candidates for Orphan Stream and CetusPolar Stream membership, with velocities and distances consistent with observations in Newberget al. (2010), and Newberg et al. (2009) and Koposov et al. (2012), respectively. These groupsrepresent a small fraction of our overall sample, but are likely to be members of the Orphan andCetus Polar streams. Aside from their matching kinematic data, the groups have [Fe/H] consistentwith observations for each stream. Interestingly, both the Orphan and Cetus Polar streams arespatially coincident with Sgr in portions of the sky, so finding these groups is an illustration ofthe power of the FoF algorithm combined with 4-dimensional spatial and kinematic data. Thesegroups are marked in Figure 17.We find one high latitude distant group that is spatially coincident with Sgr (group 397), andat roughly the same distance, but which has a strikingly different v gsr (∆ v ≈
200 km s − ). Whilethis group has ( l, b ) and v gsr consistent with observed substructure in Virgo (Newberg et al. 2007), 37 –it has a much greater distance. That this group may be associated with debris lost on an earlypericentric passage of Sgr. In a future paper in this series (Z. Ma et al., in preparation), we furtherinvestigate this group.Other groups listed in Table 2 may belong to known substructures. Two groups with l ≈ ◦ and b ≈ ◦ (group 220 and group 211) have stars with v gsr consistent with the Virgo StellarStream (Duffau et al. 2006), but are over 10 degrees from the observed stream, and could bepossible extensions to the Virgo Stellar Stream. We also detect a number of possible members ofthe Virgo Overdensity. In particular, groups 175 and 293 share the spatial and kinematic propertiesof the Virgo Overdensity (Bonaca et al. 2012). We will also further examine these stars, and inparticular their [ α /Fe] properties, in Z. Ma et al. (in preparation).Three groups with l < ◦ are located near the Galactic center with a large range of v gsr .Their origin is unclear, but they could be associated with the Hercules-Aquila cloud (Belokurov etal. 2007).Grillmair (2014) recently reported the discovery of two new halo streams, named Hermus andHyllus. We find a group of five stars near the location of these streams (group 483). Furtherinvestigation of these streams and the members of the group are needed to conclusively determinetheir membership.Koposov et al. (2014) discovered a new metal poor stream in the ATLAS survey at a distance of20 kpc, though it is mostly outside the SDSS footprint. Finally, Martin et al. (2014) find a numberof new streams in the vicinity of the Andromeda and Triangulum Galaxies using the PAndASsurvey. Both of these locations are near the edge of our Milky Way disk star exclusion cuts, so wefind no groups related to these streams.We provide the full catalog of stars in groups with 4 members or larger in Table 3. A number of false positive groups are expected because our choice of linking length does notchange with R gc . These false positive groups occur especially in the inner halo due to its higherdensity. In a smooth halo, we know that any identified groups are simply chance groupings of starswith similar positions, velocities, and distances. With a sample of stars, however, we expect thereto be larger groups, consistent with the clustering of stars in substructure. For a data sample anda smooth halo model of equivalent size, we expect to find both a smaller number of total starsin groups and a smaller average group size in the smooth halo model than in the strucutred datasample.We have computed the expected number of groups in a smooth halo by generating ten model 38 –smooth halos, as described in section 3.1 above. We find that averaged over the ten halos, we expectto find ∼ ∼ Figure 23 shows a histogram of the number of groups with a given group size and their classifi-cations. While the largest groups (larger than 10 members) are predominantly classified as Sgr, thesmallest groups (10 or fewer members) are mostly unlikely to be Sgr, and only 13% of all groupsare likely to be members of any previously known substructure.We can, therefore, give limits on the fraction of halo stars residing in substructure. At least 50%of K giants in our sample show no detection of substructure. Additionally, at least 13% of stars inour sample are members of Sgr, and roughly 1% of stars are members of other known substructure.The remaining ∼
36% of stars in the sample comprise previously undiscovered substructure.Figure 24 shows the fraction of stars in groups that belong to Sgr groups as a function of R gc in different samples of K giants. For the full K giant sample, we see that about 1/3 of stars ingroups belong to a Sgr group, and at least 50% of stars with R gc >
30 kpc are Sgr members. Thistrend also helps to explain our earlier finding of an increase in substructure with R gc , as Sgr is 39 – groups ≥ smooth halo modelsaverage of smooth halo modelsK giants nu m b e r o f s t a r s i n g r o up s groups ≥ R gc (kpc)10 groups ≥
15 members
Fig. 22.— Number of stars in groups as a function of Galactocentric radius, for both the K giantsample and for model smooth halos. Red lines show the number of stars in groups for the K giantsample, black lines show the number of stars in groups for each of ten individual smooth halomodels. The blue line shows the average number of stars in groups for the smooth halo models.Top panel: All stars in groups of any size (two or more members). Middle panel: Stars in groupsof four or more members. Bottom panel: Stars in groups of ten or more members. In all cases,there are more total groups in the K giant sample than on average in the smooth halo models. 40 –Fig. 23.— Number of groups at a given group size, classified by their associated structure. Blueare groups within Sgr regions, green and red are groups within non-Sgr regions. The rightmost binis the combination of all groups larger than 20 members: the largest group contains 255 stars. Forgroups larger than 20 members, 80% of groups are associated with Sgr. Overall, approximately 700stars are associated with Sgr.a greater percentage of the total stellar population as R gc increases. The Figure also shows fourmetallicity selected samples. It is clear from this diagram that in the more metal poor samples,there is a smaller fraction of Sgr stars than in the more metal rich samples. In fact, for some R gc bins, Sgr represents more than 75% of the stars. Since the Sgr stream is by far the dominantstructure found in our analysis, this result readily explains the trend of increasing substructurewith metallicity, and much of the inner/outer halo differences found in Section 4. This conclusionalso suggests an alternate explanation for the lower level of substructure found by X11 and Cooperet al. (2011) in BHB stars compared to the simulations: not age, but metallicity. Metallicity playsinto substructure detection via the mass-metallicity relation: more massive satellites contain more 41 –Fig. 24.— Fraction of stars in groups associated with Sgr as a function of Galactocentric radius.The colored lines show different metallicity samples discussed in Section 4.2. Dashed verticallines indicate the distance ranges over which we divide our sample into R gc bins in Figure 12.Approximately 1/3 of stars in groups are associated with Sgr, and the most metal rich sample islargely composed of Sgr stars.metal-rich stars and are also easier to detect in pencil-beam surveys such as SEGUE because of thehigher stellar density in their streams. Stars from low-mass satellites will be metal-poor and hardto detect because of their lower stellar density and the wide spacing of the SEGUE fields.
6. Conclusions
We use a sample of 4568 halo K giants with distances up to 125 kpc to measure substructurein the halo using a metric sensitive to both spatial and kinematical substructure. This sample 42 –complements and extends the work of X11 and Cooper et al. (2011) on substructure in largesamples of BHB stars because K giants are not restricted to old, metal poor populations.Outer halo K giants show more substructure than inner halo ones, in agreement with theresults of X11 and Cooper et al. (2011). In addition, we find that the most metal-rich K giantsin our sample (with [Fe/H] ≥ − .
2) show the most substructure of all the K giants. In addition,when directly comparing the amount of substructure in the BHB sample from X11 and the K giantspresented in this work, we find no significant difference between the two samples when selectingstars over equivalent distance ranges. Further work is needed to fully understand the contributionof BHBs and K giants to the Galactic stellar halo.Since Sgr stream stars are on average more metal rich than the rest of the halo and theSgr stream is not found in the inner halo, we investigated the possibility that the Sgr stream isresponsible for both trends in substructure (metallicity and distance). We find that for stars with[Fe/H] greater than –1.9, most of the groups with R gc a FoF groups found at linking length 4 δ = 0 . l b b b v gsr b R gc b [Fe/H] b Notesdeg deg km s − kpc dex3 212 251.48 68.18 -85.77 31.98 -1.39 Sgr11 174 162.10 -53.62 -123.26 30.55 -1.19 Sgr41 59 143.72 -70.97 -80.86 27.38 -1.30 Sgr28 54 300.51 74.36 -56.40 36.15 -1.34 Sgr40 50 88.08 -67.66 -34.97 22.65 -1.21 Sgr210 34 208.45 60.65 -115.48 27.93 -1.48 Sgr369 27 217.83 52.33 -109.82 27.68 -1.18 Sgr9 24 340.01 49.86 49.85 46.70 -1.49 unlikely Sgr5 23 323.22 62.38 -7.61 43.89 -1.18 Sgr110 21 153.98 -51.47 -133.08 31.16 -1.24 Sgr107 20 209.58 43.17 -98.98 25.61 -1.34 likely Sgr376 17 166.29 -31.85 -149.72 39.24 -1.26 Sgr220 16 264.95 56.62 24.56 20.24 -1.79 unlikely Sgr332 10 71.63 -50.49 25.61 27.75 -1.41 not Sgr124 10 135.13 -54.99 -105.92 25.55 -1.39 unlikely Sgr211 10 172.49 65.68 64.95 20.36 -1.28 unlikely Sgr71 10 42.12 35.69 18.20 20.34 -1.59 possible Hercules-Aquila Cloud c
222 10 258.33 55.49 -35.96 17.24 -1.39 unlikely Sgr26 9 43.57 35.84 -148.32 12.81 -1.55 possible Hercules-Aquila Cloud c
175 8 288.81 60.93 103.14 17.72 -1.34 possible Virgo Overdensity32 8 145.52 -46.47 -54.29 35.85 -2.13 Cetus Polar Stream408 8 183.93 48.88 130.70 49.07 -2.11 Orphan Stream275 7 328.32 82.11 -63.96 36.37 -1.34 Sgr261 6 41.88 78.69 -101.72 12.14 -1.57 possible Hercules-Aquila Cloud c
388 6 187.08 14.80 -63.43 90.80 -1.34 possible distant Sgr d
224 6 138.39 -71.76 -21.53 42.94 -2.19 Cetus Polar Stream8 6 355.33 51.05 5.53 43.92 -1.82 Sgr397 5 256.07 69.40 148.04 20.05 -1.22 possible older Sgr e
483 5 77.21 45.34 -4.32 23.80 -1.66 possible Hermus/Hyllus f
93 4 304.25 73.51 153.35 15.10 -1.59 possible Virgo Overdensity 45 – a Larger than 10 members or identified as notable structure b Mean value for all stars in group c see Belokurov et al. (2007); well-matched in l , b , r g c , [Fe/H], but not v gsr d see Newberg et al. (2003) e denoted as SgrP in ? f see Grillmair (2014) 46 – REFERENCES
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