Mon. Not. R. Astron. Soc.000, 000–000 (0000) Printed 13 April 2013 (MN LATEX style file v2.2)

The satellites of the Milky Way - Insights from semi-analyticmodelling in a ΛCDM cosmology

Else Starkenburg1,2,3, Amina Helmi1, Gabriella De Lucia4, Yang-Shyang Li5,Julio F. Navarro2, Andreea S. Font6, Carlos S. Frenk7, Volker Springel8,9,Carlos A. Vera-Ciro1, Simon D. M. White10

1 Kapteyn Astronomical Institute, University of Groningen,P.O. Box 800, 9700 AV Groningen, the Netherlands2 Dept. of Physics and Astronomy, University of Victoria, P.O. Box 3055, STN CSC, Victoria BC V8W 3P6, Canada3 CIfAR Junior Fellow and CITA National Fellow4 INAF - Astronomical Observatory of Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy5 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China6 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA7 Institute of Computational Cosmology, University of Durham, Science Laboratories, South Road, Durham DH13LE8 Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany9 Zentrum fur Astronomie Universitat Heidelberg, Astronomisches Recheninstitut, Monchhofstr. 12-14, 69120 Heidelberg, Germany10 Max Planck Institut fur Astrophysik Karl-Schwarzschild-Str. 1 85741 Garching, Germany

13 April 2013

ABSTRACTWe combine the six high-resolutionAquarius dark matter simulationswith a semi-analyticgalaxy formationmodel to investigate the properties of the satellites of Milky Way-likegalaxies. We find good correspondence with the observedluminosity function, luminosity-metallicity relation and radial distribution of theMilky Way satellites. The star formationhistoriesof thedwarf galaxiesin our model vary widely, in accordance with what is seen ob-servationally. Some systems are dominated by old populations, whereas others are dominatedby intermediate populations;star formation historiescan either be continuous or more bursty.Ram-pressurestripping of hot gas from the satellites leaves a clear imprint of the environmenton the characteristics of adwarf galaxy. We find that the fraction of satellites dominated by oldpopulations ofstarsmatches observations well. However, the internal metallicity distributionsof the model satellites appear to be narrower than observed.This may indicate limitations inour treatment ofchemical enrichment, which is based on the instantaneous recycling approxi-mation. We find a strong correlation between the number of satellites and thedark mattermassof the host halo. Our model works best if thedark matter haloof theMilky Way has a mass of∼ 8×10

11M⊙, in agreement with the lower estimates from observations, but about a factor of

two lower than estimates based on theLocal Grouptiming argument orabundance matchingtechniques. Thegalaxythat resembles theMilky Way the most also has the best matchingsatellite luminosity function, although it does not contain an object as bright as theSMC orLMC. Compared to other semi-analytic models andabundance matchingrelations we findthat centralgalaxiesreside in less massive haloes, but the halo mass-stellar mass relation inour model is consistent both withhydrodynamical simulationsand with recent observations.

Key words: galaxies: evolution –galaxies: formation –galaxies: dwarf – galaxies: abun-dances –galaxies: stellar content –Galaxy: halo

1 INTRODUCTION

There is much to be learned aboutgalaxy formationand evolu-tion from our own ‘backyard’, theMilky Way galaxyand its satel-lite system. Resolved stellar spectroscopy of theMilky Way stellarhalo and thesatellite galaxiesprovides ‘archaeological’ evidenceof thechemical enrichmentof the interstellar mediumback to theearliest times. The details with which theMilky Way and its satel-

lites can be studied make them a useful testbed of thecosmologicalparadigm.

Thestar formation historyof theMilky Way satellitescan bederived from the study ofstellar populationsidentified in colourmagnitude diagrams (CMD) (seeTolstoy, Hill & Tosi 2009, andreferences therein). These studies have revealed large variations inthe star formation historiesof Local Group dwarf galaxies, evenfor those of similarstellar mass. These range from solely old- to

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2 E. Starkenburg et al.

predominantly intermediate-age to even significantly young stellarpopulations. Somestar formation historiesmay be bursty, such asfor the Carina dSph(Hurley-Keller, Mateo & Nemec 1998). It iscurrently not completely understood what physical mechanisms areresponsible for the exactstar formation historyand how the envi-ronment of thegalaxyinfluences thestar formationprocess.

With additional spectroscopic observations, several teamshave investigated the dynamical and chemical properties ofbothclassical (e.g.Tolstoy et al. 2006; Battaglia et al. 2006; Koch et al.2007b; Walker et al. 2009; Kirby et al. 2010) and the recentlydiscoveredultra-faint dwarf spheroidal galaxies(e.g.,Kirby et al.2008; Aden et al. 2009; Norris et al. 2010). The discovery of thisnew, very faint class of satellites (Willman et al. 2005; Zucker et al.2006a,b; Belokurov et al. 2007; Walsh, Jerjen & Willman 2007; Ir-win et al. 2007) has revived the interest in the so-called ‘missingsatellites’ problem (Klypin et al. 1999; Moore et al. 1999), whichcontrasts the huge number ofdark mattersatellites predicted to or-bit in Milky Way-sized haloes with the relatively modest numberof luminous satellites observed.

In parallel to observational efforts, largecosmologicalN-bodydark matter only simulations, like the AquariusProject (Springelet al. 2008a), Via Lactea II (Diemand et al. 2008) and GHALO(Stadel et al. 2009), have greatly improved mass resolution andhave now reached a regime in which the formation and evolutionof (satellite)galaxiescan be studied in exquisite detail down to thisultra-faint regime in aΛCDM universe.

Following the early suggestion byEfstathiou (1992) andKauffmann, White & Guiderdoni(1993) that thereionizationoftheintergalactic mediumat high redshift could suppress the forma-tion of faint galaxies, Bullock, Kravtsov & Weinberg(2000) wereable to show, usingdark matter halomerger trees and a 1-D gassimulation, that the effects ofreionizationcould indeed help to rec-oncile the distribution of subhalocircular velocities expected inthe CDM cosmologywith inferences from satellite data.Bensonet al. (2002) then developed a detailed treatment ofreionizationand, using asemi-analytic model of galaxy formation, showed thatthe combined effects ofreionizationandsupernova feedbackcouldaccount for the observedluminosity functionof satellites in theLocal Group. They also predicted the existence of a large popu-lation of ultra-faint satellites. Several recent semi-analytical andhydrodynamicalstudies have made use of the new generation ofN-body simulations(e.g.,Munoz et al. 2009; Okamoto & Frenk2009; Okamoto et al. 2010; Busha et al. 2011; Cooper et al. 2010;Maccio et al. 2010; Li, De Lucia & Helmi 2010; Wadepuhl &Springel 2011; Font et al. 2011; Sawala, Scannapieco & White2012; Guo et al. 2011b) to confirm the importance ofreionization,and also feedback mechanisms, to suppress the formation of smallgalaxieswithin all haloes and reproduce the observed number ofdwarf satellites down to the ultra-faint regime.

However, many issues remain. Relevant questions to be askedare for instance: ‘How manysatellite galaxiesare still undiscoveredin theMilky Way stellar halo?’ (e.g.Koposov et al. 2008), ‘Whatwas their time of infall?’ (e.g.,Li, De Lucia & Helmi 2010; Rocha,Peter & Bullock 2012), ‘What is themass of the Milky Way darkmatter halo?’ (e.g Battaglia et al. 2005; Sales et al. 2007; Smithet al. 2007; Li & White 2008; Xue et al. 2008; Guo et al. 2010), ‘Aretheluminosity functionsof satellites linked to the properties of theirhost in any way?’ (e.g.McConnachie & Irwin 2006; McConnachieet al. 2009; Guo et al. 2011a; Lares, Lambas & Domınguez 2011;Wang & White 2012a).

In this work, we study the formation and evolution ofdwarfgalaxiesin and around Milky Way-likegalaxiesusing theN-body

simulationsof the Aquarius Project (Springel et al. 2008a). Wecombine these with semi-analytical modelling to study the physicalprocesses associated with the baryonic components of thegalaxies.We use the model described byLi, De Lucia & Helmi (2010),which has been extended to include new prescriptions to followthe stellar stripping andtidal disruptionof satellites.

Font et al.(2011) also combined theAquarius simulationswith a semi-analytical code to study the properties ofsatellitegalaxies, but their focus was in particular on a more sophisticatedtreatment ofreionization, while our interest is mainly in thestarformation historiesof the satellites and of isolateddwarf galaxies.Because the two codes were developed independently, it is instruc-tive to compare their results on general properties for the satel-lite population, such asluminosity function, metallicity distribu-tion and radial profile. A similar semi-analytical model to that ofFont et al.(2011) was used byCooper et al.(2010) to study thestripping ofsatellite galaxiesand the formation ofGalactic haloesin theAquarius simulations. Additionally, Guo et al.(2011b) haveused an adapted version of the semi-analytical code ofDe Lucia &Blaizot (2007) to studygalaxiesand satellites in theMillennium IIsimulationand show their results to be consistent with thesatelliteluminosity functionover the (lower) resolution range in that simu-lation.

This paper is structured as follows. In Sections2.1 and 2.2we describe theAquarius simulationsand the particular semi-analytical model we use. Some additional prescriptions have beenimplemented to account for thetidal strippingand disruption ofsatellites, and these are described in detail in AppendixA. In Sec-tion 3, we investigate the properties of the modelled maingalaxiesas well as theluminosity function, the luminosity-metallicity rela-tion, and the radial and spatial distributions of their satellites. Sec-tion 4 is devoted to a more in-depth analysis of thestar formationhistoriesof the modelleddwarf galaxies, both satellites and isolatedgalaxies, whereas in Section5 we investigate the closest modelanalogs to thedwarf spheroidal galaxiesSculptor,Carinaand For-nax. In Section6 we discuss our findings and compare them withother semi-analytical andhydrodynamicalwork. We summariseour results in Section7.

2 THE MODEL

2.1 The Aquarius simulations

The six Milky Way-like haloes (Aq-A to Aq-F) of theAquariusProject were selected from a lower resolution version ofthe Millennium-II Simulation (Boylan-Kolchin et al. 2009), acosmological N-body simulationof a cubic region125h−1Mpc ona side with parametersΩm = 0.25, ΩΛ = 0.75, σ8 = 0.9, ns = 1,h = 0.73 andH0 = 100h km s−1Mpc−1. We refer the readerto Springel et al.(2008a,b) for further information. The parametersare the same as those of theMillennium Simulation(Springel et al.2005) and were based on the first-year results from theWMAPsatellite. They are no longer consistent with the latestWMAP anal-ysis (Komatsu et al. 2011), but we do not expect this to affect ourresults significantly (seeWang et al. 2008, for a comparison of firstand third year parameters). The simulated Milky Way-like haloeshavevirial masses(M200, defined as the mass enclosed in a spherewith mean density200 times thecritical value) in the range 0.8–1.9×1012M⊙, broadly consistent with the mass estimated for theMilky Way (e.g.Battaglia et al. 2005; Smith et al. 2007; Li & White2008; Xue et al. 2008; Guo et al. 2010). One halo, Aq-A, was sim-ulated at five different numerical resolution levels (summarised in

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The satellites of theMilky Way 3

Name mp M200 r200 Nr. snapshots(M⊙) (M⊙) (kpc)

Aq-A-5 3.14 × 106 1.85 × 1012 246.37 128Aq-A-4 3.93 × 105 1.84 × 1012 245.70 1024Aq-A-3 4.91 × 104 1.84 × 1012 245.64 512

Aq-A-2 1.37 × 104 1.84 × 1012 245.88 1024Aq-B-2 6.45 × 10

38.19 × 10

11 187.70 128Aq-C-2 1.40 × 10

41.77 × 10

12 242.82 128Aq-D-2 1.40 × 10

41.77 × 10

12 242.85 128Aq-E-2 9.59 × 10

31.19 × 10

12 212.28 128Aq-F-2 6.78 × 10

31.14 × 10

12 209.21 112

Table 1. Some basic parameters for the Aquarius haloes fromSpringel et al.(2008a) (see the original paper for more information). The columnscorre-spond to the simulation name, the particle mass (mp), the virial mass ofthe halo (M200), the corresponding virial radius (r200) and the number ofsnapshots we use for each simulation.

Table1). We focus on the high-resolution level 2 (common to allsix haloes) and use the lower resolution series Aq-5 throughAq-2to test the numerical convergence of our model.

Dark matter haloesare identified in the simulations using afriends-of-friends algorithm(Davis et al. 1985) and the codeSUB-FIND (Springel et al. 2001) which identifies self-bound structureswithin larger structures. Following previous work, we haveonlyconsideredsubhaloesthat retain at least 20 particles.

2.2 The semi-analytical code

We use theAquarius simulationsas a backbone for modelling bary-onic processes ingalaxies. Subhalocatalogues are used to constructmerger (history) trees for all self-bound haloes andsubhaloesintheAquarius simulations(Springel et al. 2005; De Lucia & Blaizot2007) by determining one unique descendant for each (sub)halo.These merger trees are combined with semi-analytical modellingto study thegalaxiesthat reside in suchsubhaloes. The semi-analytical modelling technique follows the relevant physical pro-cesses using simple but observationally and astrophysically mo-tivated ‘prescriptions’. One advantage of this method is that itis applicable to largecosmological simulationsand provides rel-atively fast predictions ofgalaxy properties. The method, how-ever, does not follow explicitly the gas dynamics (as is doneinhydrodynamical simulations), and does not usually provide spa-tially resolved information about the baryonic components.

The specific model we use in this work is the ‘ejection model’described inLi, De Lucia & Helmi (2010), with new prescriptionsto follow the stellar stripping andtidal disruptionof dwarf galaxiesas they become satellites. These new prescriptions are described inAppendixA. The model builds upon the methodology introducedby Kauffmann et al.(1999), Springel et al.(2001) andDe Lucia,Kauffmann & White(2004) and subsequently updated byCrotonet al.(2006) andDe Lucia & Blaizot(2007). The model ofDe Lu-cia & Blaizot (2007) has since been modified to follow more accu-rately processes on the scale of theMilky Way and its satellites byDe Lucia & Helmi(2008); Li et al. (2009); Li, De Lucia & Helmi(2010). In particular, bothLi, De Lucia & Helmi (2010) and thispaper use the ‘ejection’ feedback scheme ofDe Lucia, Kauffmann& White (2004), which is different from the (default) feedbackscheme adopted in most previous work (e.g.,Croton et al. 2006; DeLucia & Blaizot 2007; De Lucia & Helmi 2008), a somewhat ear-

lier reionization epochand additionally suppress cooling in smallhaloes (Tvir < 104 K). Guo et al.(2011b) show that the model ofDe Lucia & Blaizot(2007) significantly overpredicts the number ofgalaxieswith stellar massesbetween107 and1010M⊙. Our modelis based onDe Lucia & Blaizot(2007), but as outlined above sev-eral changes have been made since that paper. Most changes arefocused on improving the modelling on low mass scales and areshown to mostly affect thedwarf galaxyscale and preserve theproperties ofgalaxieswith a Milky Way-like mass (see Table 2.of Li, De Lucia & Helmi 2010, for a comparison). However,DeLucia & Blaizot (2007) show that the choice of feedback schemewill also affect theluminosityand evolutionary rate of thebrightestcluster galaxies. In particular, the feedback model ofDe Lucia,Kauffmann & White(2004), as used in this work, will result ina more prolongedstar formationactivity and a higherluminosityfor thesemassive galaxiescompared to the feedback model usedin Croton et al.(2006). A careful analysis of the combined impactof all the model changes on moremassive galaxieswill be part offuture work.

A schematic diagram of the main processes modelled, isshown in Figure1. Below, we outline the main ingredients of themodel, but for a full description and the analytic expressions werefer the reader toLi, De Lucia & Helmi (2010) and referencestherein.

• Reionizationis modelled followingGnedin(2000), andCro-ton et al.(2006). Reionization causes the baryonic content of a haloto decrease in haloes with mass comparable or smaller than a so-called filtering mass, which evolves with redshift. The reionizationepoch is assumed to last fromz0 = 15 to zr = 11.5. Our reion-ization prescription results in a stronger effect than suggested forthe global reionization inOkamoto, Gao & Theuns(2008). How-ever,Font et al.(2011) show using a detailed treatment of reion-ization for the Aquarius simulations that the proto-Galactic regionis completely photo-ionized byz = 10 due to the contribution oflocal sources. In the end, this results in a comparable reionizationhistory on the scale of the Milky Way to that in our model (see Ap-pendix A ofFont et al. 2011, and references therein for a completediscussion).• Coolingof the hot gas is dependent on its metallicity and tem-

perature. BelowTvir = 104 K (the atomic hydrogen cooling limit)cooling is forbidden since we are assuming that cooling via molec-ular hydrogen is prevented by photo-dissociation caused byUV ra-diation from the (first) stars in most cases (in contrast to other semi-analytical work where star formation in haloes withTvir < 104 Kis allowed through molecular hydrogen cooling; see for instance,Madau et al. 2008; Salvadori, Ferrara & Schneider 2008).• Star Formationtransforms cold gas, which is assumed to be

in an exponential thin disc with properties given by the formalismof Mo, Mao & White(1998), into stars. Stars form in the gas of thedisc that is above a critical density threshold. The star forming discradius, rdisc, is assumed to be three scale-lengths. Assuming thatthe disc has a flat rotation curve with a rotational velocity equal tothe circular velocity of the halo (V200, see alsoKauffmann 1996)and the gas velocity dispersion is 6 km s−1, the critical densitythreshold is described by Equation1 (Kennicutt 1989).

Σcrit

M⊙pc−2= 0.59

V200

kms−1/rdisc

kpc(1)

The star formation rate is then proportional to the amount ofgas inthis state. Star formation can also happen in bursts during minor ormajor mergers, when (part of) the cold gas of the merging galax-

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4 E. Starkenburg et al.

Figure 1. A schematic diagram of the semi-analytical modelling scheme for the central galaxy within a dark matter halo and a satellite that has just beenaccreted. The yellow boxes linked to the galaxies representall different ‘phases’ of the baryons (these are all modelled analytically) and the red arrowsrepresent all modelled physical prescriptions that affectthem.

ies is turned into stars. A Chabrier initial mass function (Chabrier2003) and instantaneous recycling approximation are assumed andaccordingly 43% of the mass in stars formed at each time step is(instantaneously) recycled back into the gas phase, representing su-pernovae Type II events.

• Heating and ejection of gasis due to supernova feedback pro-cesses. We use the feedback prescription from the ejection modeldescribed inDe Lucia, Kauffmann & White(2004). The mass ofgas reheated by supernovae depends on the depth of the halo poten-tial well (i.e.,∝ 1/V 2

200), which implies that smaller haloes (withshallow potential wells) are more sensitive to the effects of feed-back. The material reheated is put in an ejected component that canbe reincorporated into the hot gas at later times.

• Metalsare created in star formation events and follow the flowof the mass between different components. We assume an instanta-neous recycling approximation that is appropriate only forelementsproduced by Type II supernovae.

• If a galaxy becomes a satellite(i.e., its halo becomes a sub-halo) we assume that its hot gas and ejected gas componentsare stripped and transferred to the central galaxy. This processcrudely models the physical process of ram-pressure stripping ofhot gaseous components. In this paper we have also added twophysical mechanisms that only operate on satellite galaxies dueto interactions with the host halo,stellar stripping and tidal dis-ruption. The implementation of these physical processes and theirresults are described and shown in AppendixA. We find that ourimplementation of stellar stripping does not have a significant ef-fect on the luminosity functions, it affects very few galaxies. Thetidal disruption prescription allows us to deal with galaxies whichhave ‘lost’ their dark matter subhalo in the simulation (when it isstripped down to fewer than theSUBFIND resolution of 20 particles)and decide whether the galaxies themselves should survive the tidesof the main galaxy. We refer to these galaxies as ‘orphans’, in con-trast to satellites which still live in dark matter subhaloes of morethan 20 particles. Our implementation of tidal disruption has a sig-nificant influence on the shape of the satellite luminosity function,as we will show in Section3.2.

3 COMPARISON TO THE MILKY WAY AND ITSSATELLITES: GENERAL PROPERTIES

3.1 Milky Way properties

In Figure2 we show the various properties of the central (MilkyWay-like) galaxiesresiding in the mainAquariushaloes comparedto the values observed for theMilky Way. We show the results ofour basic model (Li, De Lucia & Helmi 2010, filled circles) and thesame model with our additional prescriptions for stellar strippingandtidal disruption(see AppendixA, open squares). Our treatmentof satellite stripping and disruption changes only slightly the prop-erties of the maingalaxies. The scatter from maingalaxyto maingalaxyin the differentAquariushaloes is clearly visible.

Aquarius galaxiesB and E have very similarstellar massestotheMilky Way, estimated to be in the range of4.9−5.5×1010 M⊙

(Flynn et al. 2006), one of the most robust and important constraintsfor which the comparison can be made.

In our models the spheroid refers to both the bulge andstellarhaloesof thegalaxies. Since in the case of theMilky Way thestellarhalo contains a very small mass (in comparison to the bulge), wecompare the spheroidal component in the model with the bulgecomponent of theMilky Way. As explained in more detail byDeLucia & Helmi (2008), spheroid formation can occur in our modelthrough both mergers and discinstabilities. The treatment of discinstabilitiesis one of the least constrained physical processes inthe model. It is very sensitive to small variations of the other pre-scriptions, but nevertheless is a key channel for bulge formation inMilky Way sizegalaxies(seeParry, Eke & Frenk 2009; De Luciaet al. 2011, for a comprehensive discussion). The disc and bulgecomponents of theMilky Way are estimated to have amass ratiobetween 0.2 and 0.3 (Bissantz, Debattista & Gerhard 2004) whichimplies a bulge mass of0.8 − 1.3 × 1010 M⊙. This bulge-to-discratio is well reproduced by theAquarius galaxies, except in the caseof galaxiesE and F, which have a very dominant bulge component.

The slight increase of the total spheroid mass (and thus alsoofthe bulge/disc ratio) in the models with stripping andtidal disrup-tion is expected since thestarsstripped from the satellites are addedto this component. Additionally, these processes can affect the de-tails of the discinstability phenomenon through an increase in thecold gas mass of the disc. However, we find that for all haloes themajor source of the bulge mass increase is the addition ofstellarmassfrom disrupted and stripped satellites. For instance, the rel-

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Figure 2. Properties of the central galaxies of the haloes Aq-A to F bothfor the ‘satellite-model ejection’ fromLi, De Lucia & Helmi (2010) (filledcircles) and our fiducial model (i.e., extended to include stellar stripping andtidal disruption, open squares). The solid lines show observed Milky Wayvalues (see text for references). The properties of the Milky Way bulge arecompared to the spheroid component in our models.

atively large difference found in Aq-F is almost completelydueto one quite luminoussatellite galaxythat istidally disrupted(seeCooper et al. 2011, which includes a movie of the evolution of thisobject).

In all models, except Aq-E which shows almost nostar forma-tion at the present day, we find that the currentstar formation rateisabove the corresponding value for theMilky Way. This is also truefor the total amount of present-day cold gas, which is also abovetheMilky Way value for most models (with the exception of Aq-F)(using a total HI+H2 mass value for theMilky Way of ∼ 6 × 109

M⊙, seeBlitz 1997, and references therein). Among the modelsshowing significant ongoingstar formation, halo B and F give theclosest match to theMilky Way.

The two bottom panels of Figure2 show the average metallic-ities obtained for all thestarsand just the spheroid component. Theoverall metallicities lie systematically above theMilky Way values,but despite the crude approximations made to follow the evolutionof metals in our model, the mismatch is only on average∼0.25dex, or a factor∼1.8 in total mass of metals.GalaxyB, E, and Fshow the best match to the measured [Fe/H] for the disc (0.1-0.2dex above the measured value), and halo B a very close match tothe bulgestars(Freeman & Bland-Hawthorn 2002).

Figure 3. Cumulative luminosity functions of all satellite galaxieswithin280 kpc of the main galaxy for all different Aquarius haloes (solid lines), forthe Milky Way satellites (black filled circles), with corresponding Poissonerror bars, and for the Milky Way satellites as derived and corrected forincompleteness byKoposov et al.(2008) (thick dashed black line). In thetop panel surviving orphan satellite galaxies are also included, in the bottompanel they are not shown.

Altogether, Figure2 suggests that thegalaxymodelled withinhalo B is the closest analogue of theMilky Way galaxy.

3.2 Satellite luminosity functions

Figure3 shows theluminosity functionof all satellites of the mainAquarius galaxies(solid lines of different colours). Black circlesshow the cumulative number ofMilky Way satellites. Given thatthe Sloan Digital Sky Survey(SDSS) footprint covers roughly aquarter of the sky, we may expect many moregalaxieshiding in theMilky Way that are either outside theSDSSfootprint or too faintand/or too far away to be identified inSDSSdata. TheKoposovet al. (2008) (thick dashed black line) relation takes these uncer-tainties into account and attempts to correct for them. However, itshould be noted that for the brightest end (MV < −11) Koposovet al.(2008) use an averageluminosity functionof theMilky WayandM31 for their fit. SinceM31 has more bright satellites than theMilky Way, the relation overpredicts satellites in this regime.

In the top panel of Figure3 both the satellites stillembedded

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6 E. Starkenburg et al.

within a dark matter subhaloand the orphan satellites (for whichtheir dark matter subhalohas been stripped below the resolutionlimit of the simulation) which survive according to ourtidal dis-ruption prescription (see AppendixA) are plotted. In the bottompanel these orphan satellites are not shown. Both the data and themodel counts ofsatellite galaxiesare restricted to a distance of 280kpc from the centre of the maingalaxy, as assumed byKoposovet al.(2008). This radius is comparable to, but generally a bit largerthan, the virial radii (r200) of the mainAquariushaloes (see Table1).

The shape of theluminosity functionroughly agrees with theMilky Way data down to MV = −5 where resolution effects arestarting to play a role (see AppendixB for the discussion of thislimit, the vertical dashed line in Figure3) for all Aquariushaloes.Halo B shows clearly the best quantitative correspondence with theluminosity functionderived byKoposov et al.(2008) in our fiducialmodel (top panel). It is interesting thatAquariusB has a similarsatellite luminosity functionas well as the centralgalaxythat mostclosely resembles theMilky Way galaxy, as shown in the previoussubsection. However, one shouldbearin mind that the number ofsatellites formed around any halo is very sensitive to the choice offeedback scheme andreionizationphysics (e.g.,Guo et al. 2011b;Font et al. 2011).

Most Aquariushaloes do not contain satellites as bright as theLMC. This has been investigated byBoylan-Kolchin et al.(2010)and Busha et al.(2011), who estimate fromdark matter simula-tionsthat the probability of finding both anLMC andSMCarounda Milky Way-sized halo is∼ 10 per cent (up to 25 per cent witha dependence on the exactMilky Way dark matter halomass andenvironment).Liu et al.(2011) find from an analysis of Milky-Way-like hosts in theSDSSDR7 catalogue that only 3.5 per cent of themhave two such bright satellites within 150 kpc of their host.A fur-ther warning against making too strong a statement about thebrightend of theluminosity functioncomes fromGuo et al.(2011a), whofind that isolatedhost galaxiesof luminosity comparable to theMilky Way and toM31 contain∼2 times fewer satellites brighterthanMV = −14 (see alsoLares, Lambas & Domınguez 2011). Inour models, 4 out of 6 haloes have one satellite as luminous asorslightly more luminous than theSMC (the exceptions are haloes Band C). Aq-F hosts asatellite galaxyslightly brighter than theLMCand one ofluminositysimilar to theSMC. Halo D hosts three veryluminous satellites, but all fainter than theLMC.

Although all Aquarius haloes have masses consistent withthat of theMilky Way halo, they still span a factor 2.25 in mass.AquariusB is the least massive withM200 = 8×1011 M⊙. Figure3 shows that this range of masses is reflected in the number of satel-lites. This was noted also byMaccio et al.(2010), who remark thatthe trend betweenhalo massandsatellite luminosity functiondoesnot depend on the particular semi-analytical model used. For oursemi-analytical model this correlation is shown in Figure4, wherethe number of satellites brighter than MV=-5 within thevirial ra-dius is plotted against thevirial massof the host halo. The trendis very clear and the scatter is small, although based on justa fewpoints. We plan to investigate these results further using the muchlarger volume of theMillennium II simulation, where over 7000Milky Way-sized haloes are found (Boylan-Kolchin et al. 2010).

Since our model for halo B best resembles both the propertiesof theMilky Way and theluminosity function of satellite galaxies,this would favour adark mattermass estimate close to 8×1011 M⊙

for theMilky Way galaxyin agreement with the work ofBattagliaet al.(2005); Smith et al.(2007); Sales et al.(2007) andXue et al.(2008), but a factor two less massive than the best estimate ofLi &

Figure 4. The virial masses of the six different Aquarius dark matter haloesversus the number of satellite galaxies of the main galaxy inour model.Open black squares show the number of satellites brighter than MV=-5within the corresponding virial radius.

White (2008) andGuo et al.(2010). In terms of formation history,halo B forms relatively late, its massaccretionis slower than thatof otherAquariushaloes, in particular atz > 2 (Boylan-Kolchinet al. 2010). In other properties, like spin or concentration, halo Bis not special (Boylan-Kolchin et al. 2010).

3.3 Luminosity-metallicity relation

The Milky Way satellites (including the ultra-faint dwarfspheroidals; Kirby et al. 2008) show a strong correlation betweenluminosity and metallicity. It is not straightforward to comparethe observed abundances to the metallicities in our model, sincethe model adopts an instantaneous recycling approximationthat isvalid for the majority of alpha elements formed like O and Mg,but not for Fe (mainly produced by Type IaSN) which is the mostcommonly measured element in stellar spectra.

For our comparison we therefore use Mg. Correcting the ob-served average [Fe/H] to [Mg/H] requires knowledge of [Mg/Fe].Observations ofred giant branch starsin dwarf galaxiesand theMilky Way stellar haloshow clear trends of [Mg/Fe] with [Fe/H],which vary slightly from satellite to satellite and are distinct fromthose in theMilky Way stellar halo, especially at [Fe/H]> −1.5.Nonetheless, we adopt the following function based on observedsatellite data compiled byTolstoy, Hill & Tosi (2009) for [Mg/Fe]with [Fe/H]:

[Mg/Fe] = +0.4 for [Fe/H] < −2

[Mg/Fe] = −0.4[Fe/H]− 0.4 for − 2 < [Fe/H] < 0 (2)

[Mg/Fe] = −0.4 for [Fe/H] > 0

The metallicity given by our model is mass-weighted, i.e. itis the logarithm of the ratio of mass in metals over the total massin stars. The average observed metallicity to compare with shouldtherefore also be obtained by taking the logarithm of an averageover the ratio of metals to hydrogen, which will be differentfromthe average of [Fe/H]. We have tested the offset between the two es-

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The satellites of theMilky Way 7

Figure 5. Luminosity and metallicity for the satellite galaxies (grey filledcircles) and those present only in halo B (black filled circles). Grey opencircles represent satellite metallicities we do not trust,mainly due to in-complete modelling of first star physics. Overplotted as redasterisks arethe average values for the Milky Way satellites, corrected to approximatea mass-weighted average of [Mg/H] for a better comparison tothe models(see text for details and references). The error bars indicate the metallicityscatter found inside the galaxies.

timates of the mean metallicity on the data set of theDwarf Abun-dances andRadial velocityTeam (DART) (Tolstoy et al. 2004),which contains (Ca II triplet derived) metallicities for the classicaldwarf spheroidalsFornax, Sculptor,Carinaand Sextans. We foundthat the average [Fe/H] is on average 0.23 dex lower than the loga-rithm of the average over the ratio of Fe to H, although ranging from0.15 to 0.4 dex. In Figure5 we therefore show [Fe/H] corrected by0.23 dex and subsequently transformed to [Mg/H] values. ForFor-nax, Sculptor, Sextans andCarinawe do not use the 0.23 correction,but use the averages directly from the DART data (seeTolstoy et al.(2006), Battaglia et al.(2008) andStarkenburg et al.(2010) for adescription of the data sets and methods) by taking the logarithmof the average ratio of Fe to H for allstars.

Figure5 shows a comparison for thoseMilky Way satelliteswhich have an average [Fe/H] available from the literature (see, forinstance, the compilation inLi 2009). The mean iron abundancesand their dispersions (error bars) in the plot are taken fromWest-erlund(1997) for theLMC andSMC, Cole(2001) for Sagittarius,the DART survey (Helmi et al. 2006; Starkenburg et al. 2010, andreferences therein) for Fornax, Sculptor,Carinaand Sextans,Har-beck et al.(2001) for Draco andUrsa Minor, Koch et al.(2007b)for Leo I andKoch et al.(2007a) for Leo II, Kirby et al. (2008) formost of the ultra-faints; Bootes I and Segue I are fromNorris et al.(2010) andBootes IIfrom Koch et al.(2009).

The filled and empty grey circles in Figure5 show the averagemetallicity for the modelsatellite galaxieswithin 280 kpc of theirhosts. The larger black filled circles represent the satellites in Aq-B, to highlight the number of satellites and dispersion in metallicityfound in that simulation (which most resembles theMilky Way).

All model galaxieswith an average metallicity below [Fe/H]=−3 are shown as open grey circles. All of them have experiencedvery few star formationevents (typically less than 4). Because of

their few star formationepisodes these satellites will enrich verylittle and not compensate for the first generation ofstarsformedwith no metals at all, in contrast to higher mass objects which cansustainstar formationfor more extended periods. Their very lowmetallicities could therefore result mainly from our neglect of anykind of pre-enrichment, which could be driven by a top-heavyIMFfor the first stars. These events are likely to enrich thegalaxy, oreven theintergalactic mediumto a metallicity floor of [Fe/H]∼ −3(e.g.Salvadori, Ferrara & Schneider 2008).

The metallicities forSMC and LMC-like satellites are over-predicted, although they are found to be consistent within the rangeof the measured scatter inside thesegalaxies. A more thoroughmodeling of the chemical processes will have to be conductedhow-ever, before enabling any conclusions on the possibility that ourmodel retains too much of its produced metals for the moremassivegalaxies. The overall slope and normalisation of the metallicity-luminosity relation are reproduced well in our models when con-sidering thosegalaxieswhich have sufficientstar formationeventsto enrich above [Fe/H]=-3, as shown in the grey filled circles.

3.4 Radial and spatial distribution

In Figure6 we show the radial distributions of all satellites brighterthan MV=−8.5 within theMilky Way and in each of theAquariushaloes. For the (surviving) orphan satellites included in the toppanel, the position is that of the most bound particle withinthehost subhalobefore disruption. For theMilky Way satellites, thedistances are taken fromMateo(1998), except forCanes Venatici I(Martin et al. 2008). We have assumed a distance from theSuntotheGalactic centreof 8.5 kpc. The uncertainty in the radial distri-bution due toPoisson noiseis indicated by the grey area.

The distribution of satellites is broadly consistent with thatobserved in theMilky Way for mostAquariushaloes, except forthe inner regions. If orphan satellites are removed, the profiles arein general slightly less centrally concentrated, since theorphangalaxiesare mainly found in the inner regions. The bottom panelof Figure6 shows that the radial distributions ofAquariusC, D, E,and F match that of theMilky Way if orphangalaxiesare excludedfrom the analysis. Such an exclusion could be justified, since or-phangalaxiesare almost certainly affected bytidesand could havevery low surface brightness, which would hinder their detectability.

Note that the radial distribution of luminous satellites isdiffer-ent from that of the total population ofdark matter substructures,which has a less centrally concentrated profile as shown as dashedlines in the top panel of Figure6 (see alsoGao et al. 2004; Springelet al. 2008a).

One interesting property of theMilky Way satellitesis thatthey seem to lie close to a plane, rather than being distributedisotropically on the sky (e.g.Kunkel & Demers 1976; Lynden-Bell1976; Majewski 1994; Lynden-Bell & Lynden-Bell 1995; Hartwick2000; Palma, Majewski & Johnston 2002; Kroupa, Theis & Boily2005; Metz, Kroupa & Jerjen 2007). In Figure7 we investigate theanisotropy of the spatial distribution of satellites with MV < −8.5in each of theAquariushaloes. We calculate theflatteningusingthe normalised inertia tensor. The short-to-long axis ratio (c/a) iscomputed from theeigenvaluesof the diagonalized inertia tensor.In each panel theflatteningof the Milky Way satellitesystem isindicated by a dashed black vertical line, theflatteningof the totalsystem ofsubhaloesin eachAquariushalo as a dashed grey line,whereas that of all bright satellites is shown as a full coloured line.

All Aquarius haloes show a fairly spherical distribution oftheir total system of (dark)subhaloeswithin 280 kpc, although

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8 E. Starkenburg et al.

Figure 6. Radial distribution of the bright model satellites (MV < −8.5,solid lines). The top panel shows all bright satellites (solid lines) as well asthe total population of all subhaloes (dashed lines). The grey area indicatesthe distribution of Milky Way satellites including a Poissonian error barin both panels. In the bottom panel additionally the radial distribution ofclassical Milky Way satellites is overplotted (dotted black line) and orphansatellites are excluded from the solid lines.

some variations can be seen from halo to halo. The spatial distribu-tion of their bright satellites is in allAquariushaloes lessflattenedthan theMilky Way satellite system. However, all host a largernumber of bright satellites as well, as shown in Figure3. To in-vestigate the effect of the number of satellites on this comparison,we have overplotted in all panels in Figure7 the distribution ofc/a when instead of all bright satellites a random subsampleof 12satellites is taken, equal to the number of classical satellites in theMilky Way. The restriction to a smaller number of satellites greatlyenhances the chances of selecting a moreflatteneddistribution, ascan also be concluded from the bottom panel where 12 points areselected randomly distributed on a sphere. From a purely spheri-cal distribution one expects aflatteningcomparable to that seen intheMilky Way in ∼1 per cent of the cases if 12 points are drawnrandomly.

The chance of getting such a highlyflatteneddistribution asseen in theMilky Way from 12 satellites within theAquariusmod-

Figure 7. Flattening (c/a) of the twelve classical satellites (MV < -8.5) inthe Milky Way (black dashed vertical line) and the distribution of c/a valuesthat can be reached with random selections of twelve MV < -8.5 modelsatellites for the Aquarius haloes A-F. For each Aquarius halo also the flat-tening ofall satellites with MV < -8.5 is overplotted as a full coloured line.We also show the flattening of all subhaloes within 280 kpc of each mainhalo as a dashed grey line. The distribution of c/a obtained by taking twelverandom points from a spherical distribution is shown in the bottom panelfor comparison.

els is low, but can also not be completely ruled out. In particular,AquariusB and E show distributions close to the spherical casewhen 12 satellites only are selected. It is extremely unlikely forAquariusB to host a Milky Way-likeflattenedsystem (but noticethat Aq-B has significantly fewer bright satellites in totalthan theother haloes).AquariusA, C and F are on average moreflattened.In some of these cases theflatteningof the satellite system fol-lows the shape of the present-day hostdark matter haloand/orlargescale structure, most notably inAquariusA as illustrated in Figure8. AquariusA is found to have a long, thinfilamentwhich is co-herent in time, whereas in some of the otherAquariushaloes thefilamentis either less well-defined or broader (such that it encom-passes the whole halo) or changes itsorientationover time (for afull discussion of the shapes of theAquarius dark matter haloesand theirfilamentswe refer the reader toVera-Ciro et al. 2011).

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The satellites of theMilky Way 9

Figure 8. The spatial distribution of galaxies of−19 <MV < −8.5 inAquarius A. Symbols and colours denote the age of the dominant stellarpopulation: old (red asterisks), or intermediate (blue diamonds). The greymap shows the underlying density of dark matter in the same simulation.The frame is rotated such that the major axis of the main halo is verticaland the minor axis horizontal. The grey errors show the relative sizes of themajor and minor axis.

Other studies have also indicated that aflattening similarto that of theMilky Way satellitescan be reached in aΛCDMcosmology, although it is not very common (e.g.Libeskind et al.2005; Kang et al. 2005; Zentner et al. 2005; Li & Helmi 2008;Libeskind et al. 2009; Deason et al. 2011).

4 STAR FORMATION HISTORIES

4.1 A comparison to Local Group dwarf galaxies

The resolution for which thestellar agescan be determined dependson the populations that can be used as tracers. Overall, the resolu-tion decreases with increasing age and ranges from a few Myr atthe youngest end (ages up to 1 Gyr), to several Gyrs forstarsolderthan a few Gyrs. In this work we consider three different age binsthat can be well separated in a CMD analysis: an old population(> 10 Gyr), an intermediate population (1-10 Gyr) and a youngpopulation,<1 Gyr (see alsoTolstoy, Hill & Tosi 2009).

In the top left panel of Figure9 we show the percentage ofsatellite galaxiesthat contain observable populations (defined as>1 per cent of the total mass) in each of the age bins in the simu-lations Aq-A to F, where red asterisks correspond to the old,bluediamonds to the intermediate and green triangles to the young pop-ulations ofstars.

From this panel we see that∼9 per cent of all modelled satel-lites do not have an old population, whereas allMilky Way satel-litesand all isolateddwarf galaxiesobserved with sufficiently deepcolour-magnitude diagrams do containold stars. We have checkedexplicitly that all of thesesubhaloeshave appeared in the simula-tion well before a lookbacktime of 10 Gyrs, therefore the lack of oldpopulations is probably related to the implementation of the semi-

analytic prescriptions. There are at least two possible explanationsfor this difference associated to how we modelstar formationandcooling.

Firstly, we do not allow cooling indark matter haloeswithTvir < 104K. If, however, werelax this assumption and also al-low cooling in such haloes with the same efficiency as a halo ofTvir = 104K, we find that a larger percentage of allgalaxies(95 percent of alldwarf galaxies, and 97.5 per cent of those that are satel-lites presently) does form an old population. But as a consequencethe faint end of thesatellite luminosity functionis also much en-hanced (most notably at MV > −5). A proper implementation ofthe physical processes playing a role in the formation of thefirststarswould require many extra assumptions, for instance on theIMF and the interplay between H2 cooling and H2 dissociation inthe host haloes which we prefer not to include in our models atthisstage.

Another limitation of our semi-analytical model is that we donot represent the stochasticity ofstar forming regions. In our mod-elling, forstarsto form in a disc, the total (global) surface density ofthe disc has to be above thestarforming density threshold. How-ever, in realitystar formationprocesses can be much more local,i.e. onemolecular cloudcan have the required density while itssurroundings might not.

In the left middle panel we plot the percentage ofgalaxiesfor which a given population is dominant (i.e. more than 50 percent of thestarsbelong to it) as a function ofMV. Table2 sum-marizes the available data for theMilky Way satellites(the first 5columns are taken from Table 2 ofTolstoy, Hill & Tosi 2009). Adirect comparison can be made forgalaxiesin the luminosity bin−12 <MV < −8.5. Most modeled satellites in this bin are dom-inated by an old population, while a significant minority (∼20 percent) is dominated by an intermediate population ofstars. This isvery comparable to the results for thesatellite galaxiesaround theMilky Way, as can be directly seen from their percentages over-plotted in the left middle panel of Figure9 as filled circles, whereout of the sevengalaxiesin this luminositybin only two are clearlydominated by an intermediate population.

The left bottom panel of Figure9 shows the relative contribu-tions of old, intermediate and young populations averaged for allgalaxiesin a particularluminositybin.

While all left panels show thesatellite galaxiesof the mainAquarius galaxy, the right panels of Figure9 show the contribu-tions of the various populations fordwarf galaxiesoutside the mainhalo, i.e., from 400 kpc to 2 Mpc. We focus here on slightly brighterdwarf galaxies, with −19 < MV < −8.5, as these may be ob-servable with current instrumentation. Most of these systems arethe maingalaxywithin their dark matter halo, which are the onlygalaxiesfed by cooling. The isolateddwarfsare mostly dominatedby intermediate age instead of old populations. Also, a young pop-ulation ofstarsis present much more frequently (compare the twotop panels). Qualitatively, it is clear that our model produces anage-density relation in agreement with the observations ofLocalGroup satellites. Gas-poor and oldergalaxiesare found in over-dense regions near biggergalaxies, while thestar forming galaxiesare found in isolation. This result is more clearly illustrated in Fig-ure8, where we show the location of−19 < MV < −8.5 galaxiesoverlaid on thedark matterdistribution for theAquarius A simula-tion. (We have checked that the results shown in Figure8 are notinfluenced by projection effects.) Thegalaxiesdominated by an oldstellar populationshow a very different distribution as function ofthe density of their environment than the intermediate agegalaxies.

Somegalaxiesoutside the main haloes are dominated by old

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10 E. Starkenburg et al.

Figure 9. The percentage of galaxies in each luminosity bin which have>1 per cent (top panels) old (red asterisks), intermediate (blue diamonds)and/or young (green triangles) populations. In the middle panels the lineswith similar symbols indicate whether the old, intermediate or young pop-ulations are the dominant one in the galaxy (that is, more than 50 per centof their stars originate from this epoch). In the bottom panels we show thefractions of the old, intermediate and young stars averagedover all galax-ies in the luminosity bin. In all left panels the satellite galaxies, within 280kpc of the main halo are displayed, in all right panels the isolated galaxies.Overplotted in the left middle panel are the percentages of these Milky Waysatellites with−12 <MV < −8.5 dominated by either old or intermediatepopulation as filled red and blue circles respectively for a direct comparison.

populations. In all of these somestarsare formed at intermediateages as well, but there are fewer of these thanold stars. All showa burstystar formationat intermediate age. Some have never beensatellites in their life, so this behaviour is entirely due to their owninternal feedback processes. In theLocal Groupwe also find a fewexamples of old and passively evolving smallgalaxies, like the Ce-tus andTucana dwarfspheroidals, with no clear association to theMilky Way or M31, although a past association can not be ruledout. The existence, or non-existence, of truly isolatedquencheddwarf galaxiesin observational data will provide important con-straints on the modelling of thestar formationthreshold fordwarfgalaxies.

A more quantitative comparison between isolateddwarfs inour models and in the nearby Universe is difficult. Many of thewellstudiedgalaxiesoutside theMilky Way are still associated with itsnearest neighbour, Andromeda, or with theLocal Groupenviron-ment as a whole. Beyond theLocal Group, at∼ 1.3 Mpc, the ob-

Figure 10. Mass-weighted age as a function of their distance of the satel-lite to the main galaxy. The black circles show all Aquarius satellites withMV < −5 and the grey line represents the mean in this sample. Red aster-isks represent mass-weighted ages for the Milky Way satellites fromOrbanet al. (2008). Note that these are derived from HST observations and, be-cause of the limited field of view and population gradients known in dwarfgalaxies, might be biased towards lower mass-weighted ages. The two bluetriangles are the Magellanic Clouds. The smaller red asterisk representing alower limit is the Sagittarius galaxy, which has been severely stripped andtherefore will have lost preferentially part of its older population.

servational CMDs are much harder to interpret, since only thegiantbranchis bright enough to be resolved. We have made a tentativecomparison with the ACSNearby GalaxySurvey (ANGST) survey,a project measuring thestar formation historiesof galaxiesoutsidetheLocal Groupout to 4 Mpc in a systematic manner (Dalcantonet al. 2009; Weisz et al. 2011). Only threegalaxieshave reliablemeasurements of the dominant population in their sample from 1.3to 2 Mpc. Of these threegalaxies, two are dominated by an inter-mediate population and just one by an old population (Weisz et al.2011).

Figure10 shows that there is a clear trend in mass-weightedage with distance from the host. Systems dominated by intermedi-ate populations are found preferentially in the outskirts of the hosthalo while older systems are found closer in. We see a similartrendin theMilky Way satellites, where thegalaxiesdominated by inter-mediate populations are located at greater distances than∼ 100 kpcas indicated in Table2 (with the exception of theMagellanic Cloudsand the heavily disruptedSagittarius galaxy). Note that the mass-weighted ages for theMilky Way satellitesplotted here are derivedfrom HST observations, with a relative smallfield of view. Thesemass-weighted ages will therefore be biased towards the more con-centrated younger population if a radial age gradient is present inthegalaxyand metallicity and age gradients have been observed inmanydwarf spheroidalsystems (e.g.,Harbeck et al. 2001; Tolstoyet al. 2004; Battaglia et al. 2006; Bernard et al. 2008; de Boer et al.2012) as well as along theSagittarius stream(Bellazzini et al. 2006,e.g.).

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The satellites of theMilky Way 11

Name MV Detected Pops Dom. Pop refold im yng

LMC -18.1 yes yes yes old [1]SMC -16.2 yes yes yes im [2]

Sagittarius -13.4 yes yes no ? [3][4]Fornax -13.2 yes yes no im [5][6]

Leo I -11.9 yes yes no im [3][7][8]Sculptor -11.1 yes yes no old [3][9]Leo II -9.6 yes yes no ? [3][7][10]Sextans -9.5 yes yes no old [11][12][13]Carina -9.3 yes yes no im [3][7][14]Ursa Minor -8.9 yes yes no old [3][7][15]Draco -8.8 yes yes no old [3][16]CVn I -8.6 yes yes no old [17]

Hercules -6.6 ? ? ? ?Bootes -6.3 yes yes no old [17]UmA I -5.5 ? ? ? ?Leo IV -5.0 ? ? ? ?

CVn II -4.9 ? ? ? ?Leo V -4.3 ? ? ? ?UmA II -4.2 yes yes no old [17]Com Ber -4.1 ? ? ? ?Boo II -2.7 ? ? ? ?Willman 1 -2.7 ? ? ? ?Segue 1 -1.5 ? ? ? ?

Table 2. Populations in Milky Way satellite galaxies. References are: [1]Harris & Zaritsky(2009) (but note that the intermediate and old stars are al-most contributing equally in the LMC SFH), [2]Harris & Zaritsky(2004),[3] Dolphin (2002), [4] Bellazzini, Ferraro & Buonanno(1999), [5] Cole-man & de Jong(2008), [6] Gallart et al.(2005), [7] Hernandez, Gilmore& Valls-Gabaud(2000), [8] Gallart et al.(1999), [9] (de Boer et al. 2012),[10] Gullieuszik et al.(2008), [11] Mateo et al.(1992), [12] Bellazzini,Ferraro & Pancino(2001), [13] Lee et al.(2003), [14] Hurley-Keller, Ma-teo & Nemec(1998), [15] Carrera et al.(2002), [16] Aparicio, Carrera &Martınez-Delgado(2001), [17] de Jong et al.(2008).

4.2 The physics shaping the modeled star formation histories

Figure11 illustrates the variety ofstar formation historiesfoundfor the different satellites inAquariushalo B. The 18 most lumi-nous satellites are shown down to a magnitude of MV=-7.9. Thestar formation rateshave been normalized in this figure to the high-est peaks, but the absolute values range approximately from0.2M⊙/yr in the top row to 0.005 M⊙/yr in the bottom row. As canbe seen from the figure, moststarsin our satellites are made af-ter reionization(at z = 11.5 in our models), but before the timethat thegalaxy fell into the main halo (and thus became a satel-lite) as indicated by the vertical dashed blue lines. Takingall thestarsin satellites brighter than MV = −5 and within 280 kpc inAquariushaloes A-F, we find that 99.7 per cent of thestarsformedafterreionization, and 96.8 per cent before the satellite fell into thehalo. The small satellites make a larger percentage of theirstarsbe-fore reionization. This is because they have just a few small burstsof star formationeach of which contributes a significant fractionof the stars. In the largergalaxiesthe star formation rateis muchhigher and it extends over longer periods so that the initialepisodeof star formation(i.e. beforereionization) is not important in a rel-ative sense. On the other hand thequenchingof star formationafterinfall ensures that a relatively small amount ofstarswill be madeafterwards.

Infall onto a host system in our modelquenches star forma-

tion, because we assume the hot halo of agalaxy is stripped assoon as thegalaxy becomes a satellite and the hot component isadded to that of the host. Some models followram pressurestrip-ping processes more gradually (e.g.Font et al. 2008; Weinmannet al. 2010; Guo et al. 2011b; Nichols & Bland-Hawthorn 2011).Font et al.(2008) and Guo et al.(2011b) show that their imple-mentation does not make a significant difference for small systemsas considered here. Some fullyhydrodynamical simulationshavealso shown that the remaining gas in thesatellite galaxiesis (al-most) completely stripped once agalaxy falls into the main halo(e.g.Okamoto & Frenk 2009; Okamoto et al. 2010) and that thecooling of gas onto satellites seems to be small and important onlyfor massive systems (Saro et al. 2010). In our model, a satellite canstill make somestarsafter infall, but only as far as its cold gas reser-voir allows it. We find that the amount of cold gas still present inthe modeledsatellite galaxies, is in almost all cases larger than isobserved withinsatellite galaxiesin theMilky Way halo(e.g.Grce-vich & Putman 2009). However, because this gas is below thestarformationthreshold, it does not form anystars. It is quite possiblethat ram-pressurestripping of cold gas might have removed someof this remaining gas from the observedsatellite galaxies. Addi-tionally, a more realistic model ofstar formation, not based on aglobal density threshold, but following the local density of the gas,might also alter the amount of gas left in thesegalaxies. The factthat all the classicaldwarf spheroidalshave a detected intermedi-ate population, whereas this population is not always present in themodels (see top left panel of Figure9) hints that our model mightshut offstar formationtoo soon after infall.

A change in the value of the threshold density forstar for-mation, equivalent to a more stochastic implementation to reflectmolecular cloudphysics, will not lead to continuousstar forma-tion historiesfor most satellites, since the density of cold gas istoo far below the threshold most of the time. However, there aresome cases where thestar formationthreshold is barely not met,and a small change in the value of the threshold, could significantlychange thestar formation history. The most burstystar formationhistoriesare found in thegalaxieswhich are constantly close, butmostly below, thestar formationthreshold.

5 MILKY WAY DWARF ANALOGS

5.1 Star formation histories

In Figure12 we show some modelsatellite galaxiesthat are com-parable in either brightness, average metallicity andstar formationhistories, or several of these properties, to theCarina, Sculptor andFornax dwarfspheroidals. For eachgalaxywe show three examplesto illustrate the scatter in these properties.

In the bottom panels of Figure12, we plot the observedstarformation historyfor eachdwarf spheroidal. Even though it is pos-sible to observe all threegalaxiesdown to their main sequenceturnoff magnitudes, large differences are found in derivedstar for-mation ratesdue to incomplete spatial coverage or the set of modelisochrones used (see for exampleGallart et al. 2005). We have cho-sen to show here thestar formation historiesas obtained byCole-man & de Jong(2008) for Fornax and byde Boer et al.(2012) forSculptor, because these are the most recent analyses, derived fromphotometrywhich covers (nearly) the entiregalaxy. For Fornax,we also show, with a dashed line, thestar formationas compiledby Grebel (1998). The star formation historyof Carinamatchesthe one derived byHurley-Keller, Mateo & Nemec(1998), with

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12 E. Starkenburg et al.

Figure 11. The relative SFRs as a function of lookback time for the most luminous satellite galaxies in Aquarius B (black solid lines). We have indicated theinfall time (blue vertical dotted line). In the top left corner of each panel, the absolute V-band magnitude of that particular galaxy is given and the scaling factorby which the star formation rates have been multiplied to obtain the normalized values.

its characteristic three peaks. Other studies claim slightly differ-entstar formation histories(Dolphin 2002; Hernandez, Gilmore &Valls-Gabaud 2000). Of course, one needs tobearin mind that theerror bars on age are quite large for the observations (generally inthe order of Gyrs for the regime we are most interested in), sothestudies are not necessarily inconsistent.

ForCarina, we show several candidates selected onluminosity(−10.5 <MV < −8.5) and their distinct burstystar formationwitha majority of thestars formed atintermediate age. Amongst all ourAquarius simulations, we only find six candidates that have a dom-inant intermediate population in the correspondingluminositybin.In Figure12 we show three candidates which have a very burstystar formation history. It is interesting that also bursty Carina-likegalaxiesare produced. In our modelsCarina’svery burstystar for-mation historyoccurs before it becomes a satellite and is the resultof an interplay between gas density andstar formationthreshold,and is clearly not due totidal interactionswith the host (such assuggested byPasetto et al. 2011). All of these Carina-analogs fallinto the main halo quite late. As discussed before in Section4.2,most of these modelgalaxiesfind themselves close to the thresholdof cold gas needed to makestars, and are only above the thresholdoccasionally. This explains the bursty nature of thesestar formationhistoriesin our model. We have explicitly checked their merginghistory, but found no significant merging events causing or preced-ing any of the bursts. TheCarinamodel shown in the bottom panelof Figure12, depicts astar formation historythat matches best theobserved general properties ofCarina, i.e. also matches the metal-licity. Also this galaxyhas a strong intermediate age burst ofstarformation, but a lower peak at a young age.

For Sculptor we found many analogs for thestar formationhistory, since a great majority of our modelsatellite galaxiesaredominated by an old population. However, most candidates ofcom-parableluminosityhave a slighter higher metallicity than Sculptor.Three typical examples of comparableluminosity, and also averagemetallicity (the largest offset is 0.5 dex), are shown in themiddlecolumn of Figure12.

The Fornax candidate model shown on the bottom panel ofthe right column of Figure12 matches very well the observedstarformation historyfrom Coleman & de Jong(2008) and the gen-eral properties of thegalaxy. On the other hand the models shownin the top and middle panels havestar formation historieswhichmatch better older observations of Fornax in which the peak of starformationoccurs at an older age, such as that compiled byGrebel(1998). All examples shown here were chosen to have a comparableluminosityand average metallicity (the largest offset in metallicityis 0.2 dex).

5.2 Metallicity distributions

Figure 13 shows the metallicity distributions for the same set ofmodelgalaxieschosen to be relatively close analogs of theMilkyWay dwarf spheroidals Carina, Sculptor and Fornax shown in Fig-ure 12. The bottom panels show the observed metallicity distri-butions for thesedwarf spheroidalsas taken from the DART datasets. The black histograms correspond to the observed [Fe/H] dis-tribution as derived from the Ca II triplet calibration ofStarkenburget al.(2010). The grey histograms give the metallicity distributionscorrected for [Mg/H] using the simplified relation of [Mg/Fe] dis-

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The satellites of theMilky Way 13

Figure 12. Star formation histories for models which resemble the threeclassical dwarf galaxies Carina, Sculptor and Fornax (top panels). The starformation histories are shown both in the original binning of the simulation(grey lines) and rebinned to bins of 1 Gyr, to provide an easier comparisonto the observations. In both cases the star formation histories are normal-ized to their peak value. Overwritten within the panels are the metallicity,luminosity and the scaling factor by which the model star formation rateshave been multiplied to get the normalized values shown here. Division bythis scaling factor thus returns the model star formation rate values, for thenon-rebinned model (grey lines) in M⊙/yr. We have also indicated the infalltime of the satellite as a blue vertical dotted line. In the bottom panels weshow observational star formation histories for the same galaxies (see textfor references).

cussed in Section3.3. We regard this as a more direct comparisonto our model, which assumed instantaneous recycling.

In general, themetallicity distribution functionsof the modeldwarfsare narrower, although a typical error on the observed metal-licities is 0.2 dex, which will broaden the distribution. There is alsoa lack of extremelymetal-poor starsin most metallicity distribu-tions of our models. In the case of theCarinamodel shown in thetop left panel however, too many verymetal-poor starsare pro-duced and the global metallicity is too low. This also is probablydriven by our crude modelling of the first generations ofstars, asdiscussed in Section4. Since the first peak ofstar formationis verysmall, it will only enrich thegalaxyby a small amount leading tothe formation of very low-metallicitystarsin the second burst.

Overall, Figure13 shows that the form of the distribution canvary significantly, depending on the exactstar formation historyofthe system.

Figure 13. Metallicity distributions for the models displayed in Figure 12which resemble the three classical dwarf galaxies: Carina,Sculptor and For-nax (top three panels). In the bottom panels we show observational metal-licity distributions for the same galaxies (see text for references) in [Fe/H]from CaT samples (black solid lines), or corrected for a global relation to[Mg/H] values (grey solid line).

6 DISCUSSION

6.1 Comparison to the GALFORM code

The Durhamsemi-analytic model of galaxy formation, GALFORM

(e.g. Cole et al. 2000; Bower et al. 2006), has also been run onthe same sixAquariushaloes byCooper et al.(2010) and byFontet al.(2011). The latter implemented a novel and detailed treatmentof both local and globalreionizationand compared theluminosityfunctionsof their model and the satellite ejection model ofLi, DeLucia & Helmi (2010). This is identical to the one presented herein the bottom panel of Figure3, since in the comparison no or-phangalaxiesare considered and our stellar stripping mechanismdoes not significantly alter its shape.Font et al.(2011) concludedthat both models are quite similar in methodology as well as re-sults, including our treatment ofreionizationwhich is much lesssophisticated as theirs. However, we note that a detailed treatmentof local reionizationcould possibly affect also other observablessuch as the radial distribution of satellites, as shown byOcvirk &Aubert(2011).

Our model and those resulting fromGALFORM show some in-teresting differences however. For example the totalstellar massin the maingalaxiesare generally lower in theGALFORM mod-els, ranging from7 × 109 M⊙ to 1 × 1011 M⊙ (compare Ta-

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14 E. Starkenburg et al.

ble 1 from Font et al.(2011) and Figure2 in this work). Thenumber of satellites around the main halo are very similar, al-though there clearly are fewer very luminous satellites (brighterthan MV = −15) present in theGALFORM models. This seemsto be due to their slightly stronger feedback in this regime.In gen-eral theGALFORM satellite luminosity functionis slightly steeper,but consistent withKoposov et al.(2008) and our results, in par-ticular for luminositiesfainter than MV = −15. Their luminosity-metallicity relationflattensout significantly at the fainter end (seetheir Figure 5, right panel), which is an interesting resultdirectlyrelated to their saturated feedback scheme, which is clearly distinctfrom the feedback scheme used in this work. Unfortunately, the er-rors on the available data are too large to test this prediction.

6.2 A stellar mass-halo mass relation: comparison toabundance matching and hydrodynamical simulations

An interesting issue in the context of theΛCDM paradigm is whichgalaxiesreside in which haloes. Several groups have quantified therelationship betweenstellar massandhalo massusingN-body sim-ulationscombined with (semi-analytical) models or observations(e.g.Frenk et al. 1988; Navarro & Steinmetz 2000; Dekel & Woo2003; Yang, Mo & van den Bosch 2003; Wang et al. 2006; Shankaret al. 2006; Yang, Mo & van den Bosch 2008; Conroy & Wech-sler 2009; Moster et al. 2010; Guo et al. 2010). This comparison ismade under the assumption that thedark matter halomass and thestellar massof a centralgalaxy follow a monotonic relation, tak-ing into account some amount of scatter. For exampleGuo et al.(2010) combined the abundance ofgalaxieswith stellar massesinthe range108M⊙ < M∗ < 1012M⊙ from SDSS(Li & White2009), and thedark matter haloabundances from theMillenniumSimulations(Springel et al. 2005; Boylan-Kolchin et al. 2009). Therelation derived from this analysis predicts adark mattermass fora Milky Way-typegalaxyof ∼ 2 × 1012M⊙, which is a factor oftwo larger than the favoured result from our semi-analytic model,but within the current observational constraints.

Sawala et al.(2011) noticed that currenthydrodynamical sim-ulationsof isolateddwarf galaxiesproduce an order of magnitudelarger stellar massesthan the expected relation fromGuo et al.(2010) extrapolated to the lowerhalo massend. In Figure14 weshow the relation fromGuo et al.(2010) andMoster et al.(2010)as well as the results predicted by our model for all the centralgalaxiesin theAquarius simulations1 and thehydrodynamical sim-ulationsfrom Pelupessy, van der Werf & Icke(2004); Stinson et al.(2007); Governato et al.(2010); Sawala et al.(2011), as compiledby Sawala et al.(2011, see also their Figure 4.).

We see a strong correlation between totalstellar massanddarkmatter halomass (taken as theirvirial massatz = 0, as inGuo et al.2010) for the centralgalaxiesin our model, as shown in Figure14. However, the relation predicted from our model and the onepredicted byGuo et al.(2010) andMoster et al.(2010) are clearlyoffset: at a givenluminosity our modelgalaxiesreside in smallerdark matter haloes. The difference is greatest for the lower massobjects, for which theGuo et al.(2010) andMoster et al.(2010)relations are extrapolations (i.e. these scales are poorlyconstrainedby the data used by these authors; and are indicated by the dashedand dotted lines in Figure14), but also at the regime of the maincentralgalaxies(our ‘Milky Way ’s) there is a significant difference

1 We have used a distance limit of 2.5 Mpc from the centre of the box tostay well within the high-resolution region of our simulations.

Figure 14. The relation between stellar masses and dark matter halo massesfor all central haloes within 2.5 Mpc from the main halo in theAquariussimulations (black filled circles). The red symbols are results from hydro-dynamical simulations, as compiled inSawala et al.(2011), fromPelupessy,van der Werf & Icke(2004); Stinson et al.(2007); Governato et al.(2010);Sawala et al.(2011). The blue asterisks are galaxies from an SDSS samplefor which stellar masses were derived from spectroscopy anddark mattervirial halo masses from weak gravitational lensing byMandelbaum et al.(2006); the error bars give the 95 per cent confidence intervals. The blacksolid line represents the stellar mass-halo mass relation as derived byGuoet al. (2010), constrained by SDSS DR7 data. The extrapolation into thelow-mass regime is indicated by the black dashed line, whereas the lightgrey area shows the maximum dispersion,σlogM∗

= 0.2 dex. The stellarmass-halo mass relation derived byMoster et al.(2010), including a scatterof σlogM∗

= 0.15 dex (dark grey area), is shown as a black dashed-dottedline and its extrapolation into the low-mass regime is givenby the dottedblack line.

(with the exception of the maingalaxyin AquariusE). Generally,our results are in agreement withMandelbaum et al.(2006), whousedSDSSto derivestellar massesfrom spectroscopy and virialhalo massesfrom weak gravitational lensing. Interestingly, whilebeing offset from the results ofGuo et al.(2011b) and the relationfrom Guo et al.(2010) at the lower masses, our model predictionsare also consistent with mosthydrodynamical simulationsshown.

Both Moster et al.(2010) andGuo et al.(2010) demonstratethat the relation for the most likelystellar masswithin a givendarkmatter halochanges with different assumptions about scatter. Thescatter within theAquariushaloes is larger than the predictionsfrom eitherGuo et al.(2010) and Moster et al.(2010), certainlyat the lowest masses, raising the question of whether environmentalstochastic effects (such as those associated to the time of formation,the mass atreionization, the cold gas density and thestar formationthreshold) may play a more important role for low mass systems.

Recent work ofVera-Ciro et al.(2013) andWang et al.(2012b)show that the assumption of a lower mass for theMilky Wayaround∼ 8 × 1011 will alleviate significantly the problem raisedby Boylan-Kolchin, Bullock & Kaplinghat(2011) that satellite sys-tems around Milky Way-sized haloes would be too dense to hosttheknown Milky Way dwarf galaxysatellites. Using the model pre-sented in this paperVera-Ciro et al.(2013) subsequently show thatgood agreement can be found for the relation betweenluminosityto total mass of the modeled satellites in comparison with observa-tional constraints (in disagreement with the Vmax-MV dichotomybetween modeled and observed satellites as shown in Figure 6. ofBoylan-Kolchin, Bullock & Kaplinghat 2012, based onabundancematching).

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The satellites of theMilky Way 15

Using an extension of theDe Lucia & Blaizot(2007) model,Guo et al.(2011b) obtain a good correspondence with the relationderived byGuo et al.(2010) for SDSS galaxiesand consistencywith galaxy luminosity functionsover a large range in magnitudesin various bands and with the extension of theGuo et al.(2010) re-lation for lower massgalaxies. They found that in order to reach thisgood correspondance they needed to adopt asupernova feedbackejection efficiency which depends on the circular velocity of theunderlyingdark matter halowith an exponentβ1 = −3.5, whereasin our modelβ1 = −2. A careful comparison of both models onlargercosmologicalscales is beyond the scope of this paper andalso not feasible with the set of simulations used, but we plan toaddress this issue in future work.

7 CONCLUSIONS

In this work we have confirmed that asemi-analytic model ofgalaxy formationapplied to a high resolutioncosmological N-bodysimulationis able to match observed relations on the scale of theMilky Way and its satellites simultaneously. We have compared theresults for theluminosity function, luminosity-metallicity relationand radial and spatial distribution to observations ofdwarf galaxiesaround theMilky Way and investigated the age-density relation andstar formation historieswithin the model.

We find that the sameAquariushalo in which we found theclosest Milky Way-likegalaxyanalogue, halo B, also provides uswith the best matching satellite system in terms of itsluminosityfunction. This galaxydoes not resemble theMilky Way in everyrespect. For example the radial distribution of satelliteswithin thisparticular simulation is more centrally concentrated thanthe ob-served distribution of satellites around theMilky Way and it doesnot show the same degree of spatialflattening. It also does nothosta galaxywith a luminosity comparable to theLMC or SMC. Ofcourse, theMilky Way, although in many ways the best studiedgalaxy we know, is just one example and it might be unusual inseveral respects (e.g.,Flynn et al. 2006; Guo et al. 2011a).

We find a clear relation in our models between the numberof bright satellites and the hostdark halomass. With our currentfeedback andreionizationprescriptions, our bestMilky Way ana-logue has adark haloclose to 8×1011M⊙, in agreement withthe lower estimates from observations (Battaglia et al. 2005; Smithet al. 2007; Xue et al. 2008), but a factor of two lower than the bestestimates fromLi & White (2008); Guo et al.(2010). Additionally,over the mass range probed by theAquarius simulations, we finda different relation betweendark matter halomass andstellar massof the centralgalaxiesthan that derived byGuo et al.(2010) andMoster et al.(2010), and reproduced byGuo et al.(2011b).

Based on theirstar formation histories, we find modelsatellites analogous to the Sculptor, Fornax andCarina dwarfspheroidals, although none of the modelgalaxiesprovides a matchof all observable properties. However, the metallicity distributionsfor thesegalaxiesare generally too narrow compared to the ob-servations and they lack an (extremely) metal-poor population. Itis unclear at the moment whether this can be completely ascribedto the lack of a detailed prescription of the chemical evolution ofdifferent elements and to the adoption of the instantaneousrecy-cling approximation. Thistopicwill be the subject of further work.Also our model does not allow cooling via H2 in haloes below theHydrogen atomiccooling limit, and hence does not provide a fullyphysical model for the formation of thefirst stars. Another short-coming of our current model is that our modeledsatellite galaxies

have too much cold gas compared to observations. This could per-haps be (partly) solved by includingram-pressurestripping of thecold gas when an object becomes a satellite.

However, various predictions can be made from our model thatare expected to be independent of these shortcomings. We predictthe ratio betweengalaxiesdominated by old- or intermediate popu-lations ofstarsto be close to 1:2 beyond theLocal Group. We alsoexpect a large majority of thesatellite galaxiesthat are dominatedby intermediate agestellar populations, to have fallen into the mainhalo relatively late, includinggalaxieswith a burstystar formationlike Carina. Generally, a very small percentage ofstarsin a satel-lite is formed after its infall-time, due to stripping of thesatellite’shot halo which prevents further gas from cooling and formingstars.However, somegalaxieswithin the model have stoppedstar forma-tion several Gyrs before their infall, due to internal processes, suchassupernova feedback, which have expelled their gas. Our modelthus predicts the last majorstar formationevent within a (classical)satellite to rather provide a lower limit on the time elapsedsinceinfall.

The brightergalaxiesamongst the ultra-faint satellites (e.g−8.5 <MV < −5), which could be well resolved in our model,have formed a larger percentage of theirstarsin a single burst,because feedback has a larger impact and prevents a continuousmode ofstar formation. These model ultra-faintgalaxiesare gener-ally older than the more luminous counterparts, and hence containa higher fraction ofstarsformed around, and even before, theepochof reionization.

ACKNOWLEDGMENTS

We thank Eline Tolstoy, Stefania Salvadori and Andrew Cooper andthe referee for very helpful suggestions that helped improve the pa-per. E.S., A.H. and C.V.-C. thank the Netherlands Foundation forScietific Research (NWO) and the Netherlands Research School forAstronomy (NOVA) for financial support. E.S. also gratefully ac-knowledges the Canadian Institute for Advanced Research (CIfAR)Junior Academy and the Canadian Institute for Theoretical Astro-physics (CITA) National Fellowship for partial support, and the or-ganisers of the KITP programme First Galaxies and Faint Dwarfs:Clues to the Small Scale Structure of Cold Dark Matter for a stim-ulating work environment where much progress on this article hasbeen made. As such, this research was supported in part by theNational Science Foundation under Grant No. NSF PHY11-25915.A.H. was supported by the ERC-StG Galactica 240271. G.D.L. ac-knowledges financial support from the European Research Councilunder the European Community’s Seventh Framework Programma(FP7/2007-2013)/ERC grant agreement n. 202781. C.S.F. acknowl-edges a Royal Society Wolfson Research Merit award and is sup-ported in part by ERC Advanced Investigator grant COSMIWAY.This work was also supported in part by an STFC rolling grant tothe ICC. V.S. acknowledges support through SFB 881, ‘The MilkyWay system’, of the DFG.

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APPENDIX A: IMPLEMENTATION OF STELLARSTRIPPING AND TIDAL DISRUPTION

A1 Stellar stripping

In our simulations, the dark matter subhalo of a satellite may beso heavily tidally stripped that also its stellar componentshouldbe affected. Here we implement an approach to ensure that novery extended satellite galaxies reside within much smaller, heavily

c© 0000 RAS, MNRAS000, 000–000

18 E. Starkenburg et al.

stripped dark matter subhaloes. This approach is thus only used onsatellites which still have a dark matter component.

The half-mass radius of an exponential disc is related to itsscale-length throughR∗

1

2

= 1.67 × Rd. At each time-step we now

compare the half-mass radius of the stars (and cold gas),R∗1

2

, to the

half-mass radius of the dark matter subhalo,RDM1

2

. The value of

RDM1

2

is measured by theSUBFIND algorithm (Springel et al. 2001)

directly using the dark matter particles belonging to the satellite. Ifthrough tidal stripping in the simulationsRDM

1

2

< R∗1

2

, we remove

the stars (and corresponding cold gas) up toRDM1

2

. This implies

that after stripping the disc now has a mass given by

Md−new = 2Md

[

1 −(

1 +RDM

1

2

Rd

)

e−RDM

1

2

/Rd]

. (A1)

The updated exponential disc, consisting of the leftover cold gas,has a half-mass radius set byR∗

1

2−new

= RDM1

2

and the new scale

radius for the disc is thereforeRd−new = RDM1

2

/1.67.

The cold gas and stars stripped from the satellite, are trans-ferred respectively to the cold gas disc and spheroidal stellar com-ponent, which includes the bulge and stellar halo, of the maingalaxy.

Although the assumption that 50 per cent of the stars shouldat least be contained within the dark matter half-mass radius seemslike a natural first guess, there are very little direct observationalconstraints on the exact ratio. However there is general agreementthat the dark matter halo indeed has to be severely stripped beforestars are affected. For instance, the scale length of the Milky Waystellar thin disc is∼ 3 kpc, whereas its dark matter half-mass ra-dius is thought to be in the order of 100 kpc. For a more in deptdiscussion of tidal stripping in disc galaxies, and the influence ofparameters as galaxy mass, orbit and disc inclination from numeri-cal simulations we refer the reader toVillalobos et al.(2012).

A2 Tidal disruption

In our tree files a dark matter subhalo is ‘lost’ when it is strippeddown to less than 20 particles (corresponding to a dark matter massof ∼ 2.7 × 105 M⊙ in the Aquarius A halo simulated at level 2Springel et al. 2008a). We refer to these galaxies as ‘orphan’. Thequestion arises: when their dark matter subhalo disappearswhatshould be their fate.

We assume that the satellite galaxies residing at the centres ofthe fully (below the resolution limit) stripped subhaloes will mergewith the central galaxy after a dynamical friction time scale, as ex-plained byDe Lucia & Helmi(2008). However, for low mass galax-ies this time scale will be longer than the Hubble time. In some pre-vious work on semi-analytical models of Milky Way haloes (e.g.Li et al. 2009; Li, De Lucia & Helmi 2010; Font et al. 2011) theseorphan satellites galaxies were not taken into account, since theywere believed to get completely stripped. However this needs to bechecked explicitly and in some cases we can expect the galaxies tosurvive the tidal forces.

Here we improve further the code, by comparing the averagemass density of orphan satellites to that of their host system anddetermine whether they will survive. We do thisonly for those or-phan galaxies that would otherwise survive, so will not merge on adynamical friction time-scale with the central galaxy beforez = 0.This means we treat the more and less massive orphan galaxiesvia different mechanisms. The more massive orphans are dragged

in and merge with the central at the dynamical friction time-scale,the less massive might disrupt within the halo (or not, depending ontheir density). Differences between the two implementations appearfirstly in the time-scale of merging (which is set at the dynamicalfriction time-scale for the massive orphans and just taken as thefirst snapshot at which they become orphans for the less massivegalaxies) and secondly their impact on the host galaxy: onlywithinthe merging prescription black hole growth and the impact ofmajormergers are modeled (which can induce starbursts and morpholog-ical changes). Tests showed that if all orphans be treated such todisrupt within the halo, the final properties of the host galaxies canbe affected. In most cases, the changes are small and mainly affectthe bulge-to-disc ratio and the mass of the central black hole.

For the disruption mechanism we compare the density of thesesatellites, at the time they become a orphan galaxy, to the densityof the host galaxy at pericenter. In the case that the densityof thesatellite is not sufficiently high compared to the environment, theyare (assumed to be) completely disrupted. A similar tidal disrup-tion mechanism was already followed in several semi-analyticalcodes (e.g.Monaco, Fontanot & Taffoni 2007; Henriques, Bertone& Thomas 2008; Guo et al. 2011b). The approach described hereis in essence most comparable to the implementation ofGuo et al.(2011b), but there are some significant differences, most impor-tantly we use a Navarro, Frenk and White profile (NFW;Navarro,Frenk & White 1996) for the host halo whereasGuo et al.(2011b)assume an isothermal sphere.

To establish whether a orphan satellites will survive or not,we need to estimate its density at the pericenter of its orbit. Thisdistance may be derived numerically from:

Etot =1

2

L2

R2peri

+ ΦNFW(Rperi), (A2)

here the total energy,Etot, and angular momentum,L, are thosefrom the last recorded time.

Subsequently the density of the satellite,〈ρsat〉 is compared tothe average density of the host,〈ρhost〉, at the pericenter distance:

〈ρhost〉 =Mhalo(Rperi) + M∗ + Mgas

R3peri

. (A3)

Since the satellite’s dark matter content has fallen below the reso-lution limit of the N-body simulation it is not possible to evaluateits density precisely. Here we adopt two approaches to bracket thetrue density of the satellite.

• To obtain an upper limit to the satellite’s density (〈ρsat〉), wemeasure the average dark matter density withinRDM

1

2

in the last

snapshot the satellite was still detected within the simulation. Tocompute the satellite’s average density, we also take into account afraction of the stellar and cold gas mass of the satellite galaxy. Thisfraction of baryonic mass withinRDM

1

2

is determined as:

f =Md(RDM

1

2

)

Md

=[

1 −(

1 +RDM

1

2

Rd

)

e−RDM

1

2

/Rd]

, (A4)

〈ρsat〉 =MDM

1

2,sat

+ fM∗sat + fMgas

sat

(RDM1

2,sat

)3. (A5)

• To obtain a lower limit to the satellite’s density, we assumealldark matter is stripped and we take as the average satellite densitythat given by the full stellar and cold gas mass of the galaxy withinfive disc scale lengths (within which 95 per cent of the stars shouldbe contained). In this case:

c© 0000 RAS, MNRAS000, 000–000

The satellites of theMilky Way 19

〈ρsat〉 =M∗,gas

sat

(5Rd,sat)3. (A6)

We consider the satellite galaxy to be disrupted when〈ρsat〉 <〈ρhost〉, where〈ρsat〉 is given by either Eq. (A5) or (A6). The coldgas component of the disrupted satellites is added to that ofthemain galaxy. We tested that the addition of this gas to eitherthecold or the hot gas component of the main galaxy does not sig-nificantly affect the properties of the main galaxies, sinceit mightnot be completely physical to assume that all the cold gas from thesatellite flows into the cold gas reservoir of the host. The stars fromthe disrupted satellites are added to the spheroid component (whichincludes the stellar halo of the main galaxy).

A3 The effects of the stripping and disruption prescriptions

In FigureA1 we show the mass functions of present-day massesfor the systems of subhaloes, and the mass- and luminosity func-tions of luminous satellites in Aq-A-2 using different modelingprescriptions. From the middle panel of the figure, where both thetotal subhalo and luminous subhalo mass functions are overplot-ted for Aq-A-2, it is clear that all massive halos at present day(Mbound >∼109M⊙) will have formed stars, but that at the lowermass end subhalos with similar present-day masses (which can beaffected also by stripping processes) might or might not have a lu-minous component.

Figure A1 also shows the effect of the stellar stripping andtidal disruption prescriptions, introduced in SectionA1 and A2,on both the mass function and the luminosity function in Aq-A-2.Overall, the stripping mechanism implemented in the model has asmall effect on the total luminosity function, only affecting a smallpercentage of the bigger satellite galaxies (MV ∼ −10). This isstill true if we relax our assumption on the percentage of stars to becontained within the dark matter half-mass radius, and change thisfrom 50 per cent to 20 or 80 per cent. In Aq-A-2, just 3 present-daygalaxies with MV < −5 are affected by the stripping mechanism.Nonetheless, the stellar stripping criterion is an important additionfor the study of individual galaxies since it prevents the unphysicalpresence of too massive galaxies within very heavily stripped darkmatter subhaloes in the model. It also strips down galaxies whichwill become orphans eventually, making the comparison of theirdensities to that of the halo more realistic.

The implementation of tidal disruption affects the orphansatellites, but not all of them, as shown in FigureA1.

APPENDIX B: THE EFFECTS OF NUMERICAL ANDTIME RESOLUTION

B1 Numerical resolution

The Aquarius Project haloes have been run at different numericalresolution levels and these runs show remarkable good convergenceon the properties (mass, position and kinematics) of the simulatedhaloes (Springel et al. 2008a). This enables us to explore the ef-fects of numerical resolution on our models very directly. FigureB1 shows the effect of numerical resolution for both tidal disruptionapproaches on the luminosity functions. The luminosity functionsof Aq-A-2 and Aq-A-3 start to diverge significantly for MV > −5.This justifies the choice used throughout the article to use this mag-nitude limit in our comparisons to the Milky Way satellites,whichincludes many of the (brighter) ultra-faint satellites. Lower level

Figure B1. Luminosity function for Aquarius A levels 2-5 for the two waysof estimating the average density of a satellite for tidal disruption (see Sec-tion A1). In the top panel the density estimate includes the contribution ofthe dark matter, while in the bottom panel it does not. The dashed verticalline and grey area in the bottom panels indicate the luminosity of satellitegalaxies at which Aq-A-2 and Aq-A-3 start to deviate and thusthe numeri-cal resolution be considered to affect the results.

resolution merger trees are only available for halo A, we canthere-fore not perform similar convergence tests for the other simulations.

We also choose to use the disruption prescription without darkmatter (i.e., based on the lower limit estimate of the density of thesatellite) in the rest of this work. Either choice would be anapproxi-mation, due to the intrinsic uncertainties in the modellingof orphangalaxies. However FigureA1 shows that the tidal disruption with-out dark matter shows a convergence between Aq-A-2 and Aq-A-3down to fainter magnitudes. At the level of MV = −5 the lowerlimit approach predicts the disruption of 20 additional galaxies (seeFigureA1), while in Aq-B the difference is only 7 systems, sincethis halo has a much smaller population of orphan galaxies.

APPENDIX C: RESOLUTION OF TIMESTEPS

Additionally, we test the dependence of the star formation histo-ries obtained in the model for their dependence on the resolution oftimesteps taken. We use Aq-A-4 for this purpose, which has a verylarge number of snapshots and can thus be rebinned into different

c© 0000 RAS, MNRAS000, 000–000

20 E. Starkenburg et al.

Figure A1. Leftmost panel: Mass functions for all systems of subhaloeswithin 280 kpc of all six Aquarius haloes. Right panels: Mass(left) and luminosity(right) functions for all luminous satellite galaxies within 280 kpc of the main galaxy in Aq-A-2. Different models are used: the default model (‘ejection’model ofLi, De Lucia & Helmi 2010, black solid line), only the galaxies with a dark matter subhalo within the default model (grey solid line), default modelnow including stellar stripping (red solid line), and default model including both stripping and tidal disruption withor without dark matter (providing an upperand lower limit to the luminosity function, blue and green solid lines respectively). Additionally, the mass function of the system of all subhaloes is overplottedas a dashed black line in the left panel. The effect of stellarstripping alone, shown here as the difference between the black and red solid line, is not visiblein the mass function (as one would expect) and has only a smalleffect on the luminosity function. Notice that the masses shown are the present-day massesobtained by adding all bound particles of a subhalo. This value can be affected by stripping processes and therefore be significantly reduced from the mass atinfall. For the orphan galaxies the mass is used from the lastsnapshot in which they were found.

time resolutions. We find that the less luminous galaxies, which of-ten have the more bursty star formation histories, are the most vul-nerable to a change in the number of time-steps taken in the simu-lation. By changing the time resolution in the simulation bya factorof two and rebinning the final output to the lowest time resolution,typically∼30 per cent of the stars in a dwarf galaxy will be formedin a different snapshot. For the galaxies with MV < −8.5 only, thisshrinks to 25 per cent. In all cases, this has hardly any effect on thecharacterization of the star formation history in old, intermediate,or young bins as used in this paper. Only typically∼1 per cent ofstars ever will change between these bins in the tests. We have alsochecked the time resolution within the semi-analytical model itselfand found that the resolution used is sufficient; i.e. by increasingthe number of time steps by a factor 2, changes to the results asused in the paper are negligible.

c© 0000 RAS, MNRAS000, 000–000