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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2) The importance of tides for the Local Group dwarf spheroidals J. I. Read 1⋆ M. I. Wilkinson 1 N. Wyn Evans 1 G. Gilmore 1 & Jan T. Kleyna 2 1Institute of Astronomy, Cambridge University, Madingley Road, Cambridge, CB3 OHA, England 6 2Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822 0 0 2 5February2008 n a J ABSTRACT There are two main tidal effects which can act on the Local Group dwarf spheroidals 6 (dSphs): tidal stripping and tidal shocking. Using N-body simulations, we show that 2 tidal stripping always leads to flat or rising projected velocity dispersions beyond a v critical radius; it is ∼5 times more likely, when averagingover all possible projection 9 angles, that the cylindrically averaged projected dispersion will rise, rather than be 5 flat. In contrast, the Local Group dSphs, as a class, show flat or falling projected 7 velocity dispersions interior to ∼ 1kpc. This argues for tidal stripping being unim- 1 portant interior to ∼ 1kpc for most of the Local Group dSphs observed so far. We 1 showthattidalshockingmaystillbeimportant,however,evenwhentidalstrippingis 5 not. This could explain the observed correlation for the Local Group dSphs between 0 central surface brightness and distance from the nearest large galaxy. / h These results have important implications for the formation of the dSphs and for p cosmology. As a result of the existence of cold stars at large radii in several dSphs, - a tidal origin for the formation of these Local Group dSphs (in which they contain o r no dark matter) is strongly disfavoured.In the cosmologicalcontext, a naive solution t to the missing satellites problem is to allow only the most massive substructure dark s a matter halos around the Milky Way to form stars. It is possible for dSphs to reside v: within these halos (∼ 1010M⊙) and have their velocity dispersions lowered through the action of tidal shocks, but only if they have a central density core in their dark i X matter, rather than a cusp. A central density cusp persists even after unrealistically r extreme tidal shocking and leads to central velocity dispersions which are too high to a be consistent with data from the Local Group dSphs. dSphs can reside within cuspy darkmatterhalosiftheirhalosarelessmassive(∼109M⊙)andthereforehavesmaller central velocity dispersions initially. Key words: galaxies: dwarf, galaxies: kinematics and dynamics, Local Group 1 INTRODUCTION younger stellar populations, irregular morphology, lie fur- theraway from theirhost galaxy and contain significant HI The Local Group of galaxies provides a unique test-bed gas; and the transition galaxies, which are in between the for galaxy formation models and cosmology. The abun- dSphand dIrrtypes(Mateo 1998). dance, spatial distribution and internal mass distribution, Since the dSph galaxies lie, in general, closer to of satellites within the Local Group can provide sensitive their host galaxy (there are notable exceptions - see e.g. testsofcosmological predictions(seee.g.Moore et al.1999, Grebel et al. 2003) it has often been argued that the tidal Klypin et al. 1999,Kravtsov et al. 2004,Mayer et al.2001a field of the host galaxy has an important role to play in andMayeret al.2001b).However,centraltosuchstudiesis the formation and evolution of these galaxies. Mayer et al. an understanding of what the Local Group satellite galax- (2001a)and Mayer et al. (2001b)havesuggested thatall of ies are. Observationally, they are usually split into three the Local Group satellite galaxies started out looking more types:dSphgalaxies, which typically haveold stellar populations, are spheroidal in morphology, lie close to theirhost galaxy1and are devoid of HI gas; dIrr galaxies, which have 1 Weusetheterminology‘hostgalaxy’throughoutthispaperto refertoeithertheMilkyWayorM31dependingonwhichofthese ⋆ Email:[email protected] isclosertothesatellitebeingdiscussed. 2 Read et. al. like the dIrrs, the dSphs then forming through tidal trans- require an extensive search through parameter space and formations. Kuhn & Miller (1989) and Kroupa (1997) have probably not be relevant when taken out of a cosmological arguedthatthedSphsmighthaveformedinthetidaltailof context.Rather,wewishtoinvestigatethegenericeffectsof a previous major merger; in their model, dSphs contain no tides and how they reshape a dSph. dark matter and are nearly unbound. Stoehr et al. (2002) This paper is organised as follows: in section 2, we have argued that the dSphs inhabit the most massive sub- briefly outline some key analytic results for tidal stripping structure dark matter halos, of mass 1010M⊙, predicted and tidal shocking - the two dominant tidal effects which ∼ by cosmological N-body simulations (see e.g. Klypin et al. act on a satellite orbiting in a host galaxy potential. The 1999 and Moore et al. 1999); in such a scenario, tides are analytic results and definitions therein will be referred to unlikelytohavesignificantlyalteredthevisibledistribution throughoutthispaperandareusefulinbothunderstanding of stars in these galaxies. and testing thesimulation results. In section 3, we describe InthispaperwecompareasuiteofN-bodysimulations our numerical method for setting up the initial conditions of dSph galaxies orbiting in a tidal field, with the latest andintegratingtheorbitofthesatelliteinthefixedpotential data from the Local Group dSphs (see Figure 1). In this of the Milky Way. In section 4, we present the results from way, we can constrain the importance of tidal effects for a set of representative N-body simulations simulations. We thesegalaxiesandbreakthedegeneraciesbetweentheabove demonstratethattidalstrippingwillalwaysleadtoaflator formation scenarios. rising projected velocity dispersion. In section 5 we discuss TheeffectsoftidesontheLocalGroupdSphshasbeen ourresultsinthecontextofcosmology;weplacemasslimits investigated extensively in the literature, both dynamically on the Local Group dSphs. Finally, in section 6 we present (Aguilar & White 1986, Oh et al. 1995, Piatek & Pryor ourconclusions. 1995, Hayashi et al. 2003, Kazantzidis et al. 2004 and Mashchenko et al. 2005), and within a cosmological frame- work(Kravtsov et al.2004).Thispresentworkcomplements 2 THEORETICAL BACKGROUND that of previous authors. Like Hayashiet al. (2003) and Kazantzidis et al.(2004),weperform simulationswhichare A satellite in orbit around a host galaxy will experience not fully cosmologically consistent, but which have the ad- two main tidal effects which will reshape the stellar and vantage that the simulated satellites have very high reso- dark matter distributions: tidal stripping and tidal shock- lution. Unlike these previous authors, however, our dSph ing. Since both of the mechanisms will be referred to many galaxies contain bothstars anddarkmatter. Thisallows us times throughout thispaper, it is instructiveto briefly out- to compare our results more easily with the observational line thesalient properties of each in this section. data from the Local Group dSphs. We consider three possible scenarios for the tidal evolution of the Local Group dSphs. In the first (model A), 2.1 Tidal stripping we revisit the hypothesis that the Local Group dSphs con- Wediscussthetheoryoftidalstrippinginmuchmoredetail tainnodarkmatterandthattheirhighvelocitydispersions in a companion paper, Read et al. (2005). Here we briefly arise instead from the action of tides2. In the second sce- summarise the main results of that work. Tidal stripping nario (models B and C), we investigate thehypothesis that refers to thecapture of stars by a host galaxy from a satel- the dSphs are dark matter dominated, but that their outer lite galaxy beyond a critical radius: the tidal radius, r . In t regionshavebeenshapedbytidaleffects.Inthiscaseweask general, thetidal radius dependsupon four factors: thepo- whethertidescoulddirectlycausethedropinprojectedve- tential of the host galaxy, the potential of the satellite, the locity dispersion in theouter regions of thedSphs, recently orbitofthesatelliteand,whichisnewtothecalculationpre- observed by Wilkinson et al. (2004); and see also Figure 1. sentedin Read et al. (2005),the orbit of the star within the In the third scenario (model D), we investigate a scenario satellite.InRead et al.(2005),wedemonstratethatthislast where dSphs were much more massive in the past. In this point is critical and suggest using three tidal radii to cover case, tidal shocks cause their central density and velocity the range of orbits of stars within the satellite. In this way dispersions to fall significantly over a Hubble time, becom- weshowexplicitlythatprogradestarorbitswillbemoreeas- ing consistent with current data at the present epoch. It ily stripped than radial orbits; while radial orbits are more is important to state explicitly here that we do not wish easily stripped than retrograde ones. to model any one specific Local Group dSph, which would Ingeneral,thesethreetidalradiimustbecalculatednu- mericallyasoutlinedinRead et al.(2005).Wewillusethese more accurate numerical calculations in thepresentation of 2 Adark-matterfreemodelfortheformationoftheLocalGroup tidal radii later on in this paper. However, for the special dSphs is often referred to in the literature as the ‘tidal model’. case of point mass potentials, r reduces to the following t Previous tidal models, such as Kroupa(1997), tried to use tides simple form: toexplaineventhecentralvelocitydispersionoftheLocalGroup dSphs. As a result, in their models, after a Hubble time, the 2/3 duthSseephuhsnigrbheeaccliaesmntitecraafllulylvleylalourcgnietbycoeudnnitsdrp.aelIrnmsiotahnssestmoofoltidghehelstLrwoaetciapolsreGisnreonoutrpdheedrrSetp,ohwfiset. rt ≃xp(cid:18)MMgs(cid:19)1/3(cid:16)1+1 e(cid:17)1/3qα2+1+1+1+21e+2e −α (1) Thismeans that tides need notworkquitesohardto reproduce   thelargeoutervelocitydispersions.Ifourtidalmodelisruledout, where xp and e are the pericentre and eccentricity of the more extreme tidal models such as that investigated by Kroupa satellite’s orbit, Ms is the total mass of the satellite, Mg is (1997)arealsoruledout. the mass of the host galaxy and α= 1,0,1 parameterises − The importance of tides for dSphs 3 Figure 1. Surface brightness profiles (left) and projected velocity dispersions (right) forthree Local Group dSphs observed withgood stellarkinematics.DatatakenfromWilkinsonetal.(2004),Kleynaetal.(2004),Irwin&Hatzidimitriou(1995)andPalmaetal.(2003); butseealsoMart´ınez-Delgadoetal.(2001)fordeepersurfacebrightnessdataforUMi.NoticethesimilaritiesbetweeneachofthedSphs: thesurfacebrightnessprofilesareroughlyexponential,whiletheprojectedvelocitydispersionsareapproximatelyflatwithasharpfall-off atlargeradii(∼1kpc). the orbit of the star within the satellite. From equation 1, standardintuitionthatthetidalradiusdependsonly(within we can see that stars on prograde orbits (α = 1) are more a factor) on the densities of the host galaxy and satellite easilystrippedthanthoseonradialorbits(α=0),whichare galaxyisonlyvalidforstarswithinthesatellitewhichareon more easily stripped than those on retrograde orbits (α = radial orbits. For general star orbits, their stripping radius 1). Notice that α = 0 recovers the standard King (1962) will depend upon the mass distribution within the satellite − tidal radius. Other analytic solutions also exist for power- and not just the enclosed mass. law andarestricted class of split power-law densityprofiles (Read et al. 2005). 2.2 Tidal shocking Over long times, orbital transformations cause a con- vergenceofthesethreetidalradiiontheprogradestripping A second important effect of tides is tidal shocking. Tidal radius and leads to the onset of tangential anisotropy be- shocks can be caused either as a satellite plunges through yond thispoint. the disc of the host galaxy (see e.g. Ostrikeret al. 1972) A final key result from Read et al. (2005) is that the or moves on a highly eccentric orbit through the galactic 4 Read et. al. Model ρ∗ ρDM N∗ ξ∗(kpc) NDM ξDM(kpc) Orbit Time(Gyrs) γ A P:5,0.23 None 105 0.01 - - 9kpc,84kpc,-34.7o 10.43 0.0011 B P:0.072,0.23 P:14,0.5 105 0.01 106 0.01 23kpc,85kpc,-34.7o 4.5 0.00014 C SP:0.072,0.1,0 SP:90,1.956,1 105 0.01 106 0.01 23kpc,85kpc,-34.7o 4.5 0.15 D SP:0.072,0.1,0 SP:103,1.956,1 105 0.01 106 0.01 6.5kpc,80kpc,7.25o 8.85 0.83 Table 1.Simulationinitial conditions for models A-D.The columns fromlefttorightshow the model (labelled byaletter inorder of discussion), the stellar density profile, ρ∗, the dark matter density profile, ρDM, the number of stars, N∗, the force softening for the stars,ξ∗,thenumber of darkmatter particles,NDM,theforcesofteningforthedarkmatter, ξDM,thedwarfgalaxy orbit,theoutput time in Gyrs at which we show the results from the model and the strength of tidal shocks, γ (see section 2.2). The density profiles areeither(P)lummerwithparameters:mass(107M⊙),scalelength(kpc)or(S)plit(P)owerlawwithparameters:mass(107M⊙),scale length(kpc)andcentrallogslope,α(seeequations11and13formoredetails).Theorbitforthedwarfgalaxyisgivenbyitspericentre, apocentreandinclinationangletotheMilkyWaydisc. Figure 2. Equilibriumtests for model A: The upper panels are for the stars, while the lower panels are for the dark matter. The left panelsshowthecumulativemassprofile,themiddlepanelsshowthevelocityanisotropyandtherightpanelsshowthefractionalchange inenergyasafunctionoftime.Thebluelinesarefortheanalyticinitialconditions,theblacklinesareforthesimulatedinitialconditions with106 darkmatterand105 starparticles,theredlinesareforthesimulatedinitialconditionsevolvedfor5Gyrs,whilethegreenlines areforasimulationsetupusingtheMaxwellianapproximation(Hernquist1993)andevolvedfor5Gyrs. centre (see e.g. Spitzer 1987). Both scenarios are discussed where:g isthemaximaldiscforcealongthezdirection(in m indetailbyGnedin & Ostriker(1997),Gnedin et al.(1999) allofthecalculationspresentedinthispaperitisaassumed and Gnedin et al. (1999). that the plane of the Milky Way disc lies perpendicular to In the impulsive limit, the mean energy injected into thez-axis);Vz isthez componentofthesatellitevelocityat the satellite at the r.m.s. radius, r, for disc shocks is given thepointitpassesthroughthedisc;x=ωτ (ωistheangular by (Gnedin et al. 1999): velocity ofa starwithin thesatellite at r;τ 2H/Vz is the ∼ typicalshocktimeforadiscofscaleheight,H);andαisthe 2g2r2 ∆E = m (1+x2)−α (2) disc 3V2 z The importance of tides for dSphs 5 adiabaticcorrectionexponentwhichis 5/2forfastshocks halo and some stars. Since this procedure is non-trivial we ∼ (Gnedin et al. 1999). briefly outline themajor steps involved here: Similarly, themean energy injected for an eccentricor- bit in a spherical system (here an isothermal potential for (i) The particle positions for each component are re- thehostgalaxyisassumed)isgivenby(Gnedin et al.1999): alised using standard accept/reject techniques with numerically calculated comparison functions and analytic ∆E = 1 v02πr 2(1+x2)−α (3) density profiles (Press et al. 1992). sph 6 R V p p (cid:18) (cid:19) (ii) We use isotropic, spherically symmetric, distribution wherev0 istheasymptoticcircularspeedoftestparticlesat functions (Binney & Tremaine 1987). For each separate largeradiiinthehostgalaxypotential,R isthepericentric p component, i, with density, ρ , in the total relative3 grav- radius, V is the satellite velocity at R and x is as above, i p p itational potential of all components, ψ, the distribution but theshock time is now given by,τ πR /V . ∼ p p function is given by theEddington formula: Wemakethesomewhatcrudeassumptionthatthetotal energy injected into the satellite due to a single shock at r 1 ǫ d2ρ dψ 1 dρ is then given bythe sum of theabove two terms. f(ǫ)= i + i (6) Assuming that the satellite is in virial equilibrium be- √8π2 "Z0 dψ2 √ǫ−ψ √ǫ(cid:18)dψ(cid:19)ψ=0# fore the tidal shock and that it has an isotropic velocity whereǫisthespecificenergy.Notethat,foranychoiceofρ i distribution, the total specific energy of the satellite, E, is and ψ which are well behaved as r , the second term given by: → ∞ in the brackets in equation 6 vanishes (Binney & Tremaine 1 1 ∞ GM (r) 1987). E =−T ∼ 2v2(r)= 2ρs(r)Zr ρs(r) r2s dr (4) culatFeodrnaumgeenreicraalllyt.wIoncKoamzpanontzeindtissyetstaelm.(2f0(0ǫ)4)m,wushterbeeocnally- where T is the satellite’s total specific kinetic energy, single component systems (dark matter only) were consid- v2(r)1/2 is the velocity dispersion, ρs(r) and Ms(r) are the ered, theterm dd2ψρ2i could be calculated analytically4. Here, mass and density profile of the satellite and the right hand whereingeneralψ=ψ1+ψ2+...+ψn foranncomponent side follows from the Jeans equations (Binney & Tremaine system, this must be calculated numerically. We consider 1987). the special case where ψ and ρ are known analytically for i i Weassumethattidalshocksareimportantforthesatel- each component.In this case we can write: lite if the energy injected is comparable to the total energy of the satellite initially. This gives: d2ρi = dr 2 d2ρi + dr dρi d dr (7) dψ2 dψ dr2 dψ dr dr dψ ∆E +∆E (cid:18) (cid:19) (cid:18) (cid:19) γ =2 sph disc (5) 2 (cid:20) v2(r) (cid:21) This means that we can calculate ddψρ2i in general analytically for a given value of r. This avoids the need where γ parameterises theimportance of tidal shocks: tidal for a noisy double numerical differential. However, we do shocksareimportant for γ 1andunimportant forγ 1. ∼ ≪ still need to generate a look-up table to solve the inversion Notice from equations 2, 3 and 5 that the effectiveness of for r(ψ). Once this is done, however, we may perform tidalshocksdependsonthemassdensityofthesatellite,the the integral in equation 6 and create a log look-up table potentialofthehostgalaxyandontheorbitofthesatellite. forf(ǫ)overawiderangeofǫasinKazantzidis et al.(2004). The strength of such shocks is very sensitive to the value ofR .FortheMilkyWaypotentialemployedinthispaper, p (iii) Once the distribution function, f(ǫ), is known for bothdiscshocksandsphericaltidalshocksarenegligiblefor each component, we can set up particle velocities using orbits which nevercome closer than 20kpc. ∼ standard accept/reject techniques (Press et al. 1992). We The valueof γ for each of thefourmodels presentedin take advantage of the fact that the maximum value of the thispaper(labelledA-D)aregiveninTable1.ModelDisa distribution function at a given point will be given by the caseofparticularinterest.Forthismodel,γ 1andsotidal ∼ maximum valueof ǫ; that is for v=0 (Kuijken & Dubinski shocksareveryimportant.Yetthetidalradius(asobtained 1995). using equation 1) lies at the edge of the light distribution. ThusitispossiblethatsomeoftheLocalGroupdSphscould This may seem like an unnecessarily large amount of be in a regime where their visible light is strongly affected effort to go to, when simple numerical techniques already by tidal shocks but only very weakly perturbed by tidal exist for setting up multi-component galaxies (Hernquist stripping. 1993). However, as pointed out by Kazantzidis et al. (2004),using approximatemethodssuch as theMaxwellian approximation will lead to a small evolution away from 3 THE NUMERICAL TECHNIQUE the initial conditions. This can lead to significant radial velocity anisotropies being introduced at large radii. In a 3.1 Setting up the initial conditions 3.1.1 The dSph galaxy 3 The relative gravitation potential, ψ=−Φ+Φ0, used here is The initial conditions for the dSph galaxy were set up asdefinedinBinney&Tremaine(1987). 2 in a similar fashion to Kazantzidis et al. (2004), but for 4 Infact,evenforsinglecomponentmodels, ddψρ2i isonlyanalytic two-componentsphericalgalaxiescomprisingadarkmatter forafewspecialcases. 6 Read et. al. Figure 3. Initial conditions for models A-D. The blue data points are for the Draco dSph galaxy taken from Wilkinsonetal. (2004). Theblack,red,green,purplelinesareformodelsA,B,C,Drespectively.RecallthatinmodelAwehavenodarkmatter,hencethereisno blacklineintherightpanelplotofthedarkmatterdistribution. companion paper (Read et al. 2005) we show that such radii (see green line, bottom right panel). In contrast, the anisotropies lead to significantly increased tidal stripping. distribution function generated initial conditions are very This is undesirable when we wish to perform controlled stable overthe whole simulation time. numerical simulations of tidal stripping. Noticethattheenergybothinthestarsanddarkmatter In Figure 2 we show an equilibrium test for our two- is conserved to better than 2% over the whole simulation component model B (the other models showed similar re- time. sults). The upper panels are for the stars, while the lower panelsareforthedarkmatter.Theleft panelsshowthecu- mulative mass profile, the middle panels show the velocity 3.1.2 The host galaxy potential anisotropy and the right panels show the fractional change We used a host galaxy potential chosen to provide a inenergyasafunctionoftime.Weuseamodifiedanisotropy good fit to the Milky Way (Law et al. 2005), with a parameter, β∗, given by: Miyamoto-NagaipotentialfortheMilkyWaydiscandbulge v2 v2 (Nagai & Miyamoto 1976), and a logarithmic potential for β∗ = r − t (8) theMilkyWaydarkmatterhalo.Thesearegivenbyrespec- v2+v2 r t tively (Binney & Tremaine 1987): pwehresrioenvsr2reasnpdecvtt2ivaerlye.tThheisraddeifianliatniodntgainvgesenβt∗ia=l v−el1ocfoitrypduirse- Φmn(R,z)= R2+(−aG+M√dz2+b2)2 (9) circular orbits, β∗ = 0 for isotropic orbits, and β∗ = 1 for pureradial orbits5. where M = 5p 1010M⊙ is the disc mass, a = 4kpc is the The blue lines in Figure 2 are for the analytic initial disc scale leng×th and b = 0.5kpc is the disc scale height, conditions, the black lines are for the simulated initial con- and: ditionswith 106 darkmatterand 105 starparticles, thered 1 lines are for the simulated initial conditions evolved in iso- Φlog(r)= 2v02ln Rc2+r2 +constant (10) lation (i.e.withnoexternaltidalfield)for5Gyrs,whilethe greenlinesareforasimulation set upusingtheMaxwellian where Rc = 4.1(cid:0)kpc is (cid:1)the halo scale length and v0 = approximation (Hernquist 1993) and evolved for 5Gyrs. 220km/s is the asymptotic value of the circular speed of ThemostimportantpointtotakeawayfromFigure2is test particles at large radii in thehalo. thatwecanconfirmthefindingsofKazantzidis et al.(2004) Note that with the choice b = 0.5kpc, we do not con- for our two-component model. In the Maxwellian approxi- sider the case of maximal disk shocking which would occur mation the dark matter evolves away rapidly and signifi- forb 0(seeequation 2).However,given thattheamount → cantly from the initial conditions. In particular, significant ofdiscshockingisalsodegeneratewiththeorbitofthedSph radial anisotropy is introduced to the dark matter at large which is poorly constrained, our simulations span a range fromtypicaltoextremelevelsofshocking.Inaddition,none ofourconclusionswouldbealteredbystrongerdiskshocks. 5 The standard anisotropy parameter, β, is usually defined as β=1−v2/v2(Binney&Tremaine1987).Thisdefinitionisuseful θ r forobtaininganalyticsolutionstotheJeansequations.However, 3.2 Force integration and analysis it can be readily seen that for circular orbits β → −∞. In this paperitisusefultohaveadefinitionofanisotropywhichiswell- The initial conditions were evolved using a version of the behavedforallorbitsandwhichissymmetricalaroundisotropic GADGET N-body code (Springel et al. 2001) modified to orbits. includea fixed potential to model the host galaxy. The importance of tides for dSphs 7 A wide range of code tests were performed using dif- panel), projected velocity (middle panel) and projected ve- ferent softening criteria, force resolution and timestep cri- locity dispersion6 (right panel) for the stars. For the pro- teria and the results were found to be in excellent agree- jected velocity dispersion, the errors are determined from ment with each other. We also explicitly checked the code Poissonstatistics.ModelsA-Darepresenteddownthepage. againstoutputfromtwootherN-bodycodes:Dehnen(2000) InFigure5weshowtheprojected surfacebrightnessprofile and NBODY6 (Aarseth 1999), and found excellent agree- for thestars (left panel), the true velocity dispersion of the ment. Our force softening was chosen using the criteria of stars(middlepanel)andthedensityprofileofthedarkmat- Power et al. (2003) and is shown in Table 1. For the equi- ter (right panel). For clarity on the plots, we compare the librium tests, our simulations were found to conserve total resultsofoursimulationstodatafromtheDracodSphonly, energy to better than one part in 103 over the whole simu- on theassumption that Draco is representative of the most lation timeof 5Gyrs. dark matter dominated Local Group dSphs (see section 1 Finally, it is not trivial to mass and momentum centre and Figure 1). In Figure 6 we show the effect of different thesatellitewhenperforminganalysisofthenumericaldata. viewing angles in projection on the sky, using model A as An incorrect mass centre can lead to spurious density and anexample.InFigure8weshowthemassloss,duetotides, velocity features (Pontzen et al. 2005). We use the method as a function of time for all four models A-D. ofshrinkingspherestofindthemassandmomentumcentre of the satellite (Power et al. 2003). 4.1 The no-dark matter hypothesis: model A InmodelA,wetestthehypothesisthatwecaneliminatethe 3.3 The choice of initial conditions: models A-D needfordarkmatterindSphgalaxiesbyinsteademploying WechosefourmodelsforourinitialconditionslabelledA-D tidal heating to raise the velocity dispersions. One of the asshown inTable 1and Figure3.Inall models thedensity appealingaspectsofsuchamodelisthatitproducestidally distribution forthestars anddarkmatterwere giveneither distortedisophotes,suchasthosewhichmayhavebeenseen byPlummer spheres (Binney & Tremaine 1987) or by Split in some dSphs (see e.g. Mart´ınez-Delgado et al. 2001 and Powerlawprofiles(SP)(Hernquist1990,Saha1992,Dehnen Palma et al. 2003). 1993 and Zhao 1996). The density-potential pairs for these In Figure 4, top left panel, we can see that distortions distributions are given byrespectively: occur naturally in such a tidal model and the S-shaped isophotes caused by tides can be clearly seen in this pro- 3M 1 jection. However, once the kinematic data are taken into ρ = (11) plum 4πa3(1+ r2)5/2 account, it is clear that such a model runs into serious a2 difficulties. The projected velocity shows a strong gradient GM Φplum = − (12) (10km/s/kpc)duetothenear-dominanttidalarms,whereas √r2+a2 no such velocity gradients have been observed in the Local Group dSphs so far (Wilkinson et al. 2004). Furthermore, M(3 α) 1 ρSP= 4πa−3 (r/a)α(1+r/a)4−α (13) theprojected velocity dispersion rises out towards theedge of the light, whereas the Local Group dSphs like Draco GM have flat or falling projected velocity dispersions (see e.g. ΦSP = a(2 α) (1+a/r)α−2−1 (14) Wilkinson et al.2004,Munõz et al.2005,Kleynaet al.2004 − and Figure 1). It is possible, however, that such tidal fea- (cid:2) (cid:3) whereM andaarethemassandscalelengthinbothcases, turescouldbehiddeninprojection.Figure6showstheeffect and α is the central log-slope for the SP profile (note that ofdifferentviewingprojectionsontheskyformodelA.The theSP profile always goes as ρSP r−4 for r a). top and bottom panels show two different projections: the ∝ ≫ Asdetailedinsection1,eachofthesemodelswaschosen top is a more typical projection, while the bottom, which totest a differentscenario for theevolution of dSphsin the kinematically hides one of the tidal tails, is more rare. By presence of tides. In all of themodels, thedSphwas placed rare, we mean that such obscurations only occur for 1/5 on a plunging orbit which took it through the Milky Way of all projection angles. While the orbit of any given∼dSph disc. This ensured that both tidal stripping and shocking would rule out some projection angles, when studying the were important (see section 2).The initial conditions, orbit generic effect of tides on all of the Local Group dSphs, it and output time for each of the models are given in Table seems reasonable to consider such random projections. 1. Models B and C placed the dSph on an orbit which is Notice from the middle left panels, that the primary consistent with current constraints from the proper motion cause of the rising velocity dispersion in projection is the ofDraco(Kleynaet al.2001).However,thesepropermotion cylindrical averaging which picks up both tidal tails. This measurements are notoriously difficult and the errors large is why,when a suitable projection is found which hidesone (see e.g. Piatek et al. 2002). Thus in models A and D, we also consider more extreme (i.e. more radial) orbits. 6 Note that throughout this paper, all velocity dispersions are calculatedusingthelocalmeanvelocity;thisisincontrasttoob- servationaltechniqueswhichoftenusetheglobalmeanforallthe 4 RESULTS stars.Theuseoftheglobalmeanisonlyjustifiedwherethereare no velocity gradients measured; this is the case for the observa- The results for models A-D are presented in Figures 4 and tions.Itisnotthecase,however,forsomeofthemodelspresented 5.InFigure4,weshowtheprojectedsurfacebrightness(left herewherevelocitygradients areinducedthroughtides. 8 Read et. al. Figure4.Projectedsurfacedensity(left),velocity(middle)andvelocitydispersion(right)formodelsA(top)throughD(bottom).The greycontoursshowparticlenumbersperunitarea(left)andvelocitiesinkm/s(middle).Over-plottedontheprojectedvelocitydispersion plot (right) are the data for the Draco dSph galaxy taken from Wilkinsonetal. (2004). The blue boxes overlaidon the contours mark approximatecurrentobservational limits.Theprojectionsshownaretypical;differentprojectionsformodelAareshowninFigure6. ofthetails,thevelocitydispersion becomesflatratherthan and Sextans which all show falling velocity dispersions in rising(bottompanels).Inthiscasethevelocitygradientsare projection (see Figure 1). visible (middle right panel) only at large radii. Interior to It is worth recalling at this point that we are not try- 1kpc,the velocity gradient is less than 5km/s. Such a ing to explicitly model the Draco, UMi or Sextans dSphs. ∼ ∼ small velocity gradient would bedifficult to detect observa- Instead we show that the generic effect of tides is to pro- tionally (Wilkinson et al. 2004). However, even this special duce velocity gradients and flat or rising projected velocity projection is inconsistent with the data from Draco, UMi dispersions. That thisisnot seen in Draco, SextansorUMi suggests that while tidal stripping may be important be- The importance of tides for dSphs 9 A B C D Figure 5. Projected surface brightness profile of the stars (left panel), velocity dispersion of the stars (middle panel) and the density profileof the darkmatter (rightpanel). Over-plotted arethree theoretical tidal radiicalculated as described inReadetal. (2005); the blue/green/red vertical lines arefor the prograde/radial/retrograde tidal radiirespectively. The black lines show the initial conditions, whilethe redlinesshow the evolved profiles atthe output times giveninTable1.In thevelocity dispersionplot,the solid,dotted and dashedlinesshowther,θ andφcomponents ofthevelocitydispersionrespectively.ModelsA-Dareshownfromtoptobottom. Model A did not contain any dark matter which is why no plot is shown for the dark matter in this case. The tidal radius for prograde star orbits(vertical blueline)cannot beseenformodelAsinceitliesatverysmallradii(9pc). 10 Read et. al. Figure 6. The effect of viewing projection for model A. Top panels show a typical projection; bottom panels show a rare projection which kinematically hides one of the tidal tails. From left to right panels show: projected surface density, individual projected star velocities,meanprojectedvelocityandprojectedvelocitydispersion.ThegreycontoursshowparticlenumbersperunitareaasinFigure 4. Over-plotted on the projected velocity dispersion plot (right) are the data for the Draco dSph galaxy taken from Wilkinsonetal. (2004).Theblueboxes overlaidonthecontoursmarkapproximatecurrentobservational limits.Notethatthevelocitygradients shown hereareaveragedoverthey andz coordinates andarenecessarilydifferenttothetwodimensionalgradientsshowninFigure4. yondtheedgeofthecurrentlyobservedlight ( 1kpc),it is velocity dispersion. The decline in model A was present in ∼ unlikely to havebeen important within this radius. theinitial conditionsandisnotassharpasthatseen inthe An interesting feature of model A is that the small data from Draco. pericentre of the satellite (9kpc) causes strong tidal shocks ThiscanalsobeseeninFigure5,secondandthirdpan- which over 10Gyr lower the central density of the stars by elsfromtop.Noticethat,asinmodelA,tangentialvelocity a factor of 2. This can be seen in Figure 5, top panels. anisotropyappearsattheprogradestrippingradius(vertical ∼ These tidal shocks, as discussed in Read et al. (2005) wash blueline)inbothmodelsBandC.Noticealsotheactionof outthetidalfeatureswhichwouldotherwisebeexpectedat tidalshocks,particularlyonthedarkmatterinmodelC.As theanalytic tidalradii (blue,red andgreen vertical lines in the satellite is puffed up by shocks, its velocity dispersion Figure 5). However, the effects of tides can still be seen in at all radii is lowered. This may seem counter-intuitive at the velocity dispersions. Notice that tangential anisotropy, first:thesatelliteisheatedbytidalshocksandyetitsveloc- which should be present beyond the prograde stripping ra- itydispersion falls. However,it isthesubsequentexpansion diusispresentatallradiioverthesatellite(seesection2.1). of the satellite after the injection of energy from the tidal For model A, the prograde stripping radius lies at just 9pc shock which causes a drop in its velocity dispersion at a and hence does not show up in Figure 5. given radius7. Notice furtherthat it is only in projection that the velocity dispersions appear hot beyond the tidal radius. The truevelocitydispersionsarestillgentlyfalling,asintheini- 4.2 Weak tides: models B and C tial conditions, at this point. This occurs for two reasons. In models B and C we consider the effect of weak tides on First, and this is usually the dominant effect for a given theLocalGroupdSphgalaxies.Inthiscaseweimaginethat projection, the cylindrically averaged tidal tails cause the tidesgentlyshapetheouterregionsofdSphsoveraHubble projected velocity dispersion to rise (see Figure 6). Second, time. theonsetoftangentialanisotropymakestheoutermoststars In Figure 4, second and third panels from top, we can see that tides havelittle affected models B and C. The surface brightness distributions are nearly spherical, as in the 7 Inprojection this becomes alittlemorecomplicated sincehot initialconditions,andtherearenovisiblevelocitygradients starswhichhavemovedouttolargeradiicaninflatetheprojected across the galaxy. Tidal heating of the outermost stars is velocitydispersion.Thisdoesnothappeninpractice,however,be- present in both models from about 2kpc outwards. No- causethesestarsmovebeyondthetidalstrippingradius,become ∼ tice that in neither model is there a drop in the projected unboundandmoveawayfromthesatellite.