D w arf SpheroidalSatellite G alaxies W ithout D ark M atter: R esults From T w o D i erent N um ericalTechniques R alfS.K lessen1 and PavelK roupa2

arXiv:astro-ph/9711350v1 28 Nov 1997

1M

ax-Planck-Institut fur A stronom ie,K onigstuhl17,D -69117 H eidelberg,G erm any

e-m ail: klessen@ m pia-hd.m pg.de 2Institut fur T heoretische A strophysik,U niversitat H eidelberg,T iergartenstr. 15,

D -69121 H eidelberg,G erm any

e-m ail: pavel@ ita.uni-heidelberg.de A B ST R A C T Self-consistent sim ulations ofthe dynam icalevolution ofa low -m ass satellite galaxy w ithout dark m atter are reported. T he orbits have eccentricities 0:41 e 0:96 in a G alactic dark halo w ith a m ass of 2:85 1012 M and 4:5 1011 M . For the sim ulations, a particle-m esh code w ith nested sub-grids and a direct-sum m ation N -body code running w ith the special purpose hardware device G rape are used. Initially, the satellite is spherical w ith an isotropic velocity distribution,and has a m ass of107 M . Sim ulations w ith 1:3 105 up to 2 106 satellite particles are perform ed. T he calculations proceed for m any orbitalperiods untilwellafter the satellite disrupts. In all cases the dynam ical evolution converges to a rem nant that contains roughly 1 per cent ofthe initialsatellite m ass. T he stable rem nant results from severe tidal shaping of the initial satellite. To an observer from Earth these rem nants look strikingly sim ilar to the G alactic dwarfspheroidalsatellite galaxies. T heir apparent m ass-to-light ratios are very large despite the fact that they contain no dark m atter. T hese com putationsshow thata rem nantw ithoutdark m atterdisplayslarger line-of-sight velocity dispersions, ,for m ore eccentric orbits,w hich is a result ofprojection onto the observationalplane. A ssum ing they are not dark m atter dom inated,it follow s that the G alactic dSph satellites w ith > 6 km /s should haveorbitaleccentricitiesofe > 0:5.Som erem nantshavesub-structurealong the line-of-sight that m ay be apparent in the m orphology ofthe horizontalbranch.

Subject headings: G alaxy: halo | galaxies: form ation | galaxies: interactions | galaxies: structure | galaxies: kinem atics and dynam ics | LocalG roup

{2{ 1. Introduction A t least about ten dwarfspheroidal(dSph) galaxies are know n to orbit the M ilky W ay at distances ranging from a few tens to a few hundred kpc. O n the sky they are barely discernible stellar density enhancem ents. Som e have internalsubstructure and appear attened. T heir velocity dispersions are sim ilar to those seen in globular clusters and they have approxim ately the sam e stellar m ass. H owever,they are about two orders ofm agnitude m ore extended. For sphericalsystem s in virialequilibrium w ith an isotropic velocity dispersion, the overall m ass of the system can be determ ined from the observed velocity dispersion. C om paring this ‘gravitational’m ass to the lum inosity ofthe system determ ines the m ass-to-light ratio,M =L (in the follow ing always given in solar units M =L ). In the solar neighbourhood this ratio is 3 < M =L < 5 (e.g. T sujim oto et al. 1997). Values for M =L ofabout10 orlargerare usually taken to im ply the presence ofdark m atterin a stellar system . Forthe dSph satellites,M =L valuesaslarge asa few hundred are inferred,im plying thatthese system sm ay be com pletely dark m atterdom inated (fora review see M ateo 1997). A carefulcom pilation oftheobserved structuralparam etersand kinem aticaldata forthe G alactic dSph galaxiescan be found in Irw in & H atzidim itriou (1995).M ore generalreview s are given by Ferguson & Binggeli (1994), G allagher & W yse (1994), M eylan & Prugniel (1994),G rebel(1997) and D a C osta (1997). T here are in principle two possibilities for achieving apparent high m ass-to-light ratios w ithout dark m atter: U nresolved binary stars m ay in ate the m easured velocity dispersion thusincreasing M =L. H owever,thise ectisnotlarge enough fora reasonable population of binary system s(H argreavesetal.1996;O lszew skietal.1996).A ssum ing N ew tonian gravity is valid in dSph galaxies,an alternative m ay be that the assum ption ofvirialequilibrium is violated: the satellite galaxies m ight be signi cantly perturbed by G alactic tides. T he structural,kinem aticaland photom etric data ofthe dSph satellites show correlations that m ay be interpreted to be the result of signi cant tidal shaping (Bellazzini, Fusi Pecci & Ferraro 1996). Extra-tidal stars indicate that m ost dSph satellites m ay be losing m ass (Irw in & H atzidim itriou 1995,K uhn,Sm ith & H aw ley 1996,Sm ith,K uhn & H aw ley 1997), and Burkert(1997)pointsoutthat,ifthetidalradiiderived from theIrw in & H atzidim itriou (1995)pro lesassum ing K ing m odelsare correct,then these radiiare sm allerthan expected from the observed large M =L values. T he \tidal scenario" has been studied in detail by a variety of authors: O h et al. (1995) m odelled the evolution ofdSph galaxies on di erent orbits in a set ofrigid spherical G alactic potentials. T he satellites are represented by 103 particles and are evolved using a softened direct N -body program over m any orbitalperiods untildisruption. T heir work allow s im portant insights into the tidalstability of such system s. Piatek & Pryor (1995)

{3{ concentrate on oneperi-galacticpassage ofa dSph galaxy in di erentrigid sphericalG alactic potentials. T heir satellite consists of 104 particles and is m odelled using a T reecode schem e. T hey nd thata single peri-galactic passage cannotperturb a satellite signi cantly enough for an observer to m easure a high M =L ratio, reaching sim ilar conclusions as O h et al. (1995). Johnston, Spergel& H ernquist (1995),w ho apply their sim ulations to the dynam icalevolution ofthe Sagittarius satellite galaxy,also arrive at sim ilar conclusions. Self-consistent sim ulations ofthe long-term evolution ofa low -m ass satellite galaxy on two di erent orbits and interacting w ith an extended G alactic dark halo are presented by K roupa (1997;hereafter referred to as K 97). T he satellite consists of3 105 particles and the w hole system isevolved applying a particle-m esh schem e w ith nested sub-grids,the aim being to study the system wellafter the satellite has m ostly dissolved and to ‘observe’its properties as it would be seen from Earth. T he satellite is projected onto the sky and its brightness pro le,line-of-sight velocity dispersion and apparentM =L ratio are determ ined. T hese quantitiescan be directly com pared to the observed valuesforG alactic dSph galaxies. H is m ain nding is that a rem nant containing about 1 per cent ofthe initialsatellite m assrem ainsasa long-lived and distinguishable entity afterthe m ajordisruption event. To an observerfrom Earth,thisrem nantlooksstrikingly sim ilarto a dSph galaxy.T he rem nant consists ofparticles that have phase-space characteristics that reduce spreading along the orbit. T hat this is a possibility to be considered had been pointed out by K uhn (1993). H owever,projection e ects are also im portant. A n observer w ho’s line-of-sight subtends a sm allangle w ith the orbitalpath ofthe rem nant sees an apparently brighter satellite w ith internalsub-clum psand an in ated velocity dispersion. T he attened structure thatm ay be apparent to the observer need nothave a m ajoraxis thatisoriented along the orbitalpath. T he observer derives values for (M =L)obs that are m uch larger than the true m ass-to-light ratio (M =L)true ofthe particles,because the object is far from virialequilibrium and has a velocity dispersion tensor that is signi cantly anisotropic. T he aim ofthe present study is to investigate ifthe conclusions ofK 97 can be arrived at w hen higher resolution sim ulations w ith m ore particles are used, and w hen a di erent num erical schem e altogether is em ployed. W e thus hope to con rm that the high M =L values ofthe G alactic dSph satellites can be explained w ithout the need for dark m atter. W e furtherm ore suggest possible observationaldiscrim inants, and continue the analysis of the two snapshots studied in K 97. In addition to theparticle-m esh m ethod applied in K 97 and here,weusea directN -body integratorin connection w ith the specialpurpose hardware device G rape,w hich allow sthe integration ofsystem s w ith 105 or m ore particles. It is therefore a usefultoolfor studying the evolution ofdSph galaxies in a G alactic dark halo.

{4{ U sing two di erent num ericalschem es enablesusto determ ine w here the m odelsagree, i.e. w hich conclusions are rm , and also w here they show deviations. T his allow s us to quantify uncertainties inherent to the num ericalm ethod,but we do not aim at an in-depth discussion ofthe detailed di erences between the two num ericalschem es. W e w illshow that the generalconclusionsagree forboth m ethods,and thatthey can both be used equivalently to explore further regions in param eter space. T he sim ulations described here are part of an extensive study ofparam eterspace to investigate w hich orbitsand w hich assum ption for the G alactic dark halo m ay lead to dSph-like objectson the sky. D etailed reportsaboutthis survey w illbe presented elsew here. In the next section we give a short introduction to the two num erical schem es applied here,followed by a description ofthe data analysis used to evaluate the sim ulations. Section 3 treats the initialconditions,and in Section 4 we discuss our results. Possible discrim inants between dark-m atter and tidalm odels are presented in Section 5. W e conclude w ith Section 6.

2. T w o N um ericalSchem es and D ata A nalysis A short description ofboth num ericalschem es used for the sim ulations is provided in Sections 2.1 and 2.2,and the data reduction m ethod is described in Section 2.3.

2.1. D irect Integration Schem e w ith G R A P E G rape is a special purpose hardware device. Its nam e is an acronym for ‘G R A vity PipE’.T he device solvesPoisson’sand the force equationsfora gravitationalN -body system by direct sum m ation on a specially designed chip,thus leading to a considerable speed-up (Sugim oto et al. 1990,Ebisuzaki et al. 1993). W e use the currently distributed version, G rape-3A F,w hich contains 8 chips on one board and therefore can com pute the forces on 8 particles in parallel. T he board is connected via a standard V M E interface to the host com puter, in our case a SU N Sparcstation. C and FO RT R A N libraries provide the software interface between the user’s program and the board. T he com putationalspeed of G rape-3A F is approxim ately 5G ops. T he force law is hardw ired to be a Plum m er law ,

Fi =

G

XN

m im j(ri rj) : rjj2 + 2i)3=2 j= 1 (jri

(1)

{5{ H ere i is the index of the particle for w hich the force is calculated and j enum erates the particles w hich exert the force; i is the gravitationalsm oothing length ofparticle i;G ,m i and m j are N ew ton’s constant and the particle m asses,respectively. W e chose allparticles to have the sam e m asses and sm oothing lengths. To increase speed,concessions in the accuracy ofthe force calculationshad to be m ade: G rape internally worksw ith a 20 bit xed pointnum berform atforparticlepositions,w ith a 56 bit xed pointnum berform atfortheforcesand a 14 bitlogarithm icnum berform atforthe m asses (O kum ura et al. 1993).C onversion to and from this internalnum ber representation is handled by the interface software. T he num ber form at lim its the spatialresolution in a sim ulation and constrainsthe force accuracy. H owever,forcollisionless N -body system s,the forces on a single particle need not be know n to better than about one per cent. In that respect,G rape is com parable to the w idely used T reecode schem es (e.g. Barnes & H ut 1986). W e utilise G rape by im plem enting the direct sum m ation approach. T his essentially involves two nested loops: an outer one over allparticles for w hich forces are calculated, and an inner loop for the interaction ofeach ofthose w ith allother particles in the system . T herefore,the num berofoperationsscalesasO (N 2)w ith the particle num berN . Typically, this scaling law lim its the particle num ber to a few thousand. H owever,G rape substitutes the inner loop and thus a considerable speed-up is gained. W e furtherm ore im plem ent variable tim e-stepsand interpolate the particle accelerationsw hen no new force calculations are needed w ithin the required accuracy. O nce the accelerations are obtained in each tim estep,the particlesare advanced using the leap-frog schem e. T he satellite galaxy isdescribed w ith 131072 particles. To give an estim ate forthe com putationaltim e needed: a sim ulation w ith 5500 tim e-steps (e.g. the run Sat-M 2 in Table 1) typically takes three days on a Sun Sparcstation w ith the G rape board.

2.2. SU P E R B O X :A P article-M esh C ode w ith N ested Sub-grids Superbox is a conventional particle-m esh code (see e.g. Sellwood 1987) but allow s high spatialresolution ofdensity m axim a by em ploying three levels ofnested grids. Each active grid has N grid = (2K )3 cells,w here K is a positive integer. T he outerm ost,coarsest grid contains the local universe. T he sub-grids of the two lower levels are positioned at the density m axim um ofa galaxy and follow its m otion through the coarse outer grid. In principle any num berofinteracting galaxiescan betreated.T heforceacting on each particle is obtained by rst solving Poisson’s equation using the Fast Fourier Transform technique, and then by num erically di erentiating the potentialat the position of the particle. T he

{6{ leap-frog integration schem e is used to advance the particles along their orbits (for a brief description ofthe code see K 97,a detailed account w illbe provided elsew here). Forthe present purpose,Superbox is used to sim ulate the interaction oftwo galaxies, nam ely oftheG alacticdark halo and thesatellitegalaxy.Typically,N grid = 323 cellspergrid and in total1:3 106 particlesareused.In addition,two sim ulationsw ith N grid = 643 cellson each leveland in total4 106 particlesarerun to testthenum ericalresolution ofSuperbox. A n 8000-tim e-step long sim ulation takes5 C PU dayson an IBM R ISC /6000 350 workstation in the rstcase,and in the lattercase ittakes55 C PU days on a SU N Sparcstation 10/514.

2.3. D ata E valuation T he m odelsatellite isanalysed by reproducing terrestrialobservationsofa dSph galaxy, as in K 97. A tevery tim e-step in the sim ulation,the position ofthe density m axim um ofthe satellite and ofits rem nant is determ ined using the fullset ofN sat particles. Every pre-chosen num ber n of integration steps, a subset of N st satellite particles is stored on disk for the detailed analysis by the hypothetical‘observer on Earth’. In the adopted C artesian coordinate system ,thisobserverislocated atR = (0;8:5;0)kpc,w here the origin isthe G alactic centre. For further analysis, only those stored particles are used that have have a distance m odulus M satisfying

M cod

M 2

M

M cod +

M ; 2

(2)

w here M cod = 5log10D cod 5 is the distance m odulus of the satellite’s density m axim um w hich lies ata distance D cod from the Sun,and M is the m agnitude range covered by the observations. T his reduced sam ple is the ‘m odelobservationalsam ple’. T hroughout this paper M = 0:8 m ag isused,exceptin Section 5.2,w here M isvaried fora detailed study oftwo snapshots. U nless stated otherw ise (in Section 5.2), the observational plane is subdivided into k = 20 circular annuliw ithin a projected radialdistance rbin = 1:5 kpc from the density m axim um ofthe satellite. T hese are used to evaluate the line-of-sight velocity dispersions and the surface brightness pro le. T he velocity dispersions are calculated using the iterated bi-weightscaleestim atorw hich istheestim ated dispersion aboutthebi-weightm ean velocity ofthe sam ple. T he bi-weight location (i.e. m ean) and scale (i.e. dispersion) estim ators are

{7{ described by Beers,Flynn & G ebhardt (1990),and are robust to outlying velocity data. T heapparentm ass-to-lightratio an observerdeducesisestim ated from theK ing-form ula (see Piatek & Pryor 1995): M L

= obs

9 2 G

2 0 0 r1=2

;

(3)

w here G is the gravitational constant, and r1=2 is the half-light radius, i.e. that radius at w hich the projected surface brightness density decreases by 0.75 m ag/pc2. T he central line-of-sight velocity dispersion, 0,is calculated w ithin the centralbin. T he centralsurface brightness, 0,isestim ated by tting an exponentialsurface density pro le to the ‘observed’ radialm odelpro le, w hich is obtained by counting the num ber of particles in the m odel observationalsam ple in the above m entioned projected radialbins,each particle having an intrinsic(M =L)true = 3,com parableto thevaluesderived forthesolarneighbourhood.O ther values m ay be used to change the lum inosity ofthe satellite.

3. T he M odels and InitialC onditions T heM ilky W ay isa highly com plex stellarand gaseoussystem .Itcan besubdivided into four m ajor com ponents: bulge,disk,the stellar halo,and a non-lum inous dark com ponent required to tthe rotation curve. T he latterdom inates the totalm ass ofthe system by far. W e thus sim plify the problem by exam ining the dynam icalinteraction ofa satellite galaxy w ith this dark halo alone. T he next sub-section details the m odels adopted for these two com ponents,and Section 3.2 discusses the initialconditions for the num ericalexperim ents.

3.1. G alaxy m odels T he dark halo of the G alaxy is taken to be an isotherm al sphere w ith a total m ass M halo = 2:85 1012 M w ithin 250 kpc. T his follow s for a halo thatis truncated at250 kpc and hasa circularvelocity of220km /s. T he crossing tim e ofthe diam etercontaining 33 per cent ofthe halo m ass,d33 = 137:2 kpc,is t33 = 588 M yr. W e also adopt a core radius of 5 kpc. In the sim ulations w ith Superbox,the dark halo is treated as a live com ponent consisting of N halo particles, w ith N halo = 1 106 or 2 106. T he sim ulations m ade for a com parison w ith G rape have a halo that is cut-o at Rc = 40 kpc w ith a total m ass

{8{ M halo = 4:5 1011 M . In this case,the inner and m iddle grids have dim ensions of303 kpc3 and 1223 kpc3,respectively. For an inter-com parison ofSuperbox sim ulations w ith a different num ber ofgrid cells and particles,R c = 250 kpc is used,in w hich case the inner and m iddle grids have dim ensions of503 kpc3 and 1883 kpc3,respectively. T he initialvelocity dispersion is always isotropic. T he isolated halo w ith R c = 40 kpc is allowed to relax to dynam icalequilibrium by integrating it for 9 t33 w ith a tim e-step of1.7 M yr. T he halo w ith R c = 250 kpc is integrated in isolation for 25 t33 w ith a tim e-step of7 M yr. Further detailsand a briefdiscussion ofthe nalslightly prolate shape ofthe halo w ith Rc = 250 kpc is provided in K 97. T he halo w ith R c = 40 kpc rem ains sphericalafter attaining dynam ical equilibrium . Both contract slightly during relaxation into equilibrium . In the sim ulations w ith G rape,the dark halo ofthe G alaxy is a rigid sphere w ith a core radius of4 kpc,a cut-o radius of40 kpc,and a totalm ass M halo = 4:5 1011 M . In allcases the satellite is initially assum ed to be a Plum m er sphere w ith a Plum m er radius R pl = 0:3 kpc,a cuto radius Rc = 1:5 kpc,and a m ass M sat = 107 M . T he initial velocity dispersion isisotropic,and the crossing-tim e ofthe diam etercontaining 33 percent ofthe m ass,d33 = 0:56 kpc,is 84 M yr. T he satellite m odelis allowed to relax to dynam ical equilibrium fortypically 8 such crossing tim esw ith a tim e-step of1.1 M yr(Superbox)and 1.5 M yr (G rape). T he nal,dynam ically relaxed satellite is spherical. In the Superbox sim ulation, the inner and m iddle grids have dim ensions 1:63 kpc3 and 83 kpc 3,respectively. T he spatialresolution is thus 50 (25) pc per celllength w ithin a distance of0.8 kpc from the satellite’s density m axim um ,and 250 (125) pc per celllength between 0.8 kpc and 4 kpc from the satellite’s density m axim um , in the 323 (643) cell sim ulation. T here are two sets ofcalculations: w ith N sat = 3 105 w ith N grid = 323,and N sat = 2 106 w ith N grid = 643. T he calculations w ith G rape use N sat = 131072 and allparticles.

= 50 pc (equation 1) equalfor

3.2. Initialconditions A s reasoned in the Introduction,the aim ofthe present paper is to investigate,using di erent num ericalrealisations, the robustness of the results from K 97. Sim ulations w ith Superbox are com pared w ith equivalent sim ulations running on G rape.A lso,Superbox sim ulations w ith in total1:3 106 particles and N grid = 323 cells are com pared w ith sim ulations w ith in total4 106 particles and N grid = 643 cells. In each case,the integration tim e-step is 1.1 M yr for sim ulations w ith Superbox,and 1.5 M yr in the lowest tim e-step

{9{ bin for the sim ulations using direct sum m ation on G rape. In allsim ulations presented here, the satellite is initially positioned on the x-axis at apo-galactic distance R apo from the G alactic centre,w ith an initialvelocity vector v0 along the y-direction.T he eccentricity ofthe orbitise = (R apo R peri)=(R apo + R peri),w here R peri is the peri-galactic distance. A n overview ofthe initialconditionsforthe eightsim ulations described here isgiven in Table 1. T he rst colum n contains the nam e ofthe sim ulation,the second colum n (Ngrid ) liststhe num ber ofgrid cellsused w ith Superbox (a G indicatesrunsw ith G rape).N halo, N sat,and N st are the num ber ofhalo,satellite and stored satellite particles used in the data evaluation,respectively;ntot isthetotalnum beroftim e-steps,and every n stepsN st particles are w ritten to com puter disk. C olum n 8 ( t) lists the totaltim e intervalsim ulated. T he next two colum ns give the initialcentre-of-m ass position,r0,and velocity,v0,vectors ofthe satellite in a C artesian coordinate system centred on the G alaxy. T he last two colum ns list the orbitaleccentricity,e,and the m ass ofthe G alactic dark halo (see Section 3.1).

4. R esults Equivalent sim ulations w ith Superbox and G rape are com pared in Section 4.1. T he dependence ofthe results obtained w ith Superbox on the num berofgrid cells and particle num ber is discussed in Section 4.2.

4.1. SU P E R B O X versus G R A P E T he initialconditions for the two pairs ofSuperbox { G rape sim ulations are listed in the top four lines of Table 1. T he evolution of the satellite galaxy on an orbit w ith eccentricity e = 0:71 (sim ulationsR S1-109 and Sat-M 1)and on an orbitw ith e = 0:46 (R S1113 and Sat-M 2) is com pared using the two di erent num ericalschem es. In both cases the apo-galactic distance is R apo = 60 kpc. T hree snapshots of the satellite in sim ulation Sat-M 1 are show n in Fig. 1. A t each particulartim e,the satellite isplotted asseen from outside the G alaxy (the solid line traces its density m axim um ). Enclosed in the circle on the left is a m agni cation ofthe central region of the satellite and its rem nant, respectively. T he upper panel show s the satellite shortly after the start ofthe calculation. T he m iddle panelshow s the dwarfgalaxy shortly after its rst apo-galacticon. C onsiderable tidaltails have developed, and there is a well bound core. T he bottom panelshow sthe galaxy shortly afteritsthird apo-galactic passage.

{ 10 { T he satellite has disrupted. H owever,there stillexists a m easurable density enhancem ent, the rem nant,w hich m ight be identi ed as a dSph satellite galaxy. T his behaviour is found in allsim ulations studied here and in K 97. In Fig.2 we show the path ofthe satellite in sim ulations R S1-109 and Sat-M 1 looking perpendicular on to the orbitalplane. T he satellite disrupts after the second peri-galactic passage. Sim ilarly,Fig.3 depicts the orbit in sim ulations R S1-113 and Sat-M 2 for the rst four peri-galactic passages. D uring passage through peri-galacticon the satellite is heated and particles escape. A n insightfuldiscussion ofthe processes involved is presented in Section 4 in Piatek & Pryor (1995).Plotting theLagrangeradiiasa function oftim e conveniently sum m arisestheoverall evolution ofthe structure ofthe satellite. T he e ects ofthe periodic passages through perigalacticon on the m assbudgetofthe satellite are show n in Fig.4 forthe eccentric orbitand in Fig.5 for the orbit w ith e = 0:46. T idalshocks expelthe outer regions ofthe satellites in both cases and excite dam ped oscillations in those m ass shells that rem ain bound. H igh valuesof(M =L)obs resultonly afterthe satellite iscom pletely disrupted and hasreached the ‘rem nant’phase. T his is sim ilar to the sim ulation discussed by K uhn & M iller (1989,see their Fig.2). In their sim ulation,w hich is a sim pli ed treatm ent ofa satellite on a circular orbitin a constanttidal eld,theobserved m ass-to-lightratio exceedsthetruevalueby m ore than a factor of ve only during the disruption phase at the end. T he essentialdi erence is that we nd long-lived rem nants w ith signi cantly in ated (M =L) obs after the disruption event. C om paring both num erical schem es, the evolution of the satellites are very sim ilar: For the eccentric orbit,the induced oscillations ofthe m ass shells are evident in both the Superbox and the G rape sim ulations,and both satellites loose m ore than 90 per cent of the initialm ass at ‘disruption’tim e t 1:3 G yr. T he tidalforces are m ilder for the less eccentric orbit (e = 0:46),and in the Superbox sim ulation R S1-113,the satellite disrupts at t = 3:6 G yr, as is evident in Fig. 5. D isruption occurs one orbital tim e (i.e. about 1.2 G yr)laterin the G rape sim ulation. H owever,both sim ulationslead to the sam e overall evolution of the satellite. A di erence in disruption tim e between the two sim ulations is to be expected because the G alactic dark halo is taken into account in very di erent ways (live and self-consistent in sim ulation R S1-113,and rigid in sim ulation Sat-M 2) leading to di erences in the tidalforces that accum ulate. A pplying the data reduction described in Section 2.3,the m easured m ass-to-light ratio (M =L)obs for the satellite rem nant is very large,despite the fact that the true m ass-to-light ratio waschosen to agree w ith the value obtained in the solarneighbourhood.Forboth sets of sim ulations w ith Superbox and G rape,(M =L)obs is plotted in Figs.6 and 7. T hese

{ 11 { gures also show the evolution ofthe centralsurface brightness, 0,and ofthe line-of-sight velocity dispersion, r1=2 ,evaluated w ithin r1=2,w hich is largely indistinguishable from 0 (see Section 5.1). T he com parison ofthe Superbox sim ulation R S1-109 w ith the G rape sim ulation SatM 1 show s excellent agreem ent. T he apparent m ass-to-light ratio is not in ated prior to disruption despite the forced oscillations of the Lagrange radiiapparent in Figs.4 and 5. A fter disruption,the rem nant stabilises w ith 0 104:5 L =kpc2. T he velocity dispersion w hich the observer m easures for the largest fraction of the tim e after satellite disruption has a value in the range r1=2 10 30 km /s,and (M =L)obs ranges from a few hundred to a few thousand. Sim ilarly, the satellite on the initially less eccentric orbit (e = 0:46) behaves alike in the Superbox (R S1-113) and G rape (Sat-M 2) sim ulations. W ith both num ericaltechniques,the rem nant stabilises w ith 0 104:5 L =kpc2, r1=2 2:5 10 km /s and (M =L)app 100 1000. T his rem nant phase is arrived at about 1.2 G yr later in the G rape sim ulation,ow ing to the earlier disruption tim e in sim ulation R S1-113,asdiscussed above. O necautionary rem ark isnecessary atthispoint:T he\observed" centralsurfacebrightnessofthe rem nantsin oursim ulationsis,w ith 0 104:5 L =kpc2,relatively low w hen com pared to the G alactic dSph satellites. T hese have centralsurface lum inosities ranging from 7 105 L =kpc2 forSextansto 0 3 107 L =kpc2 forLeo I(Irw in & H atzidim itriou 0 1995),and are thus at least about one order ofm agnitude larger than our values. H owever,so farwe have only scanned a very sm allrange ofinitialparam eters: we have lim ited the present study to an initialsatellite m ass of107 M . O ne possible way to arrive at the observed centralsurface brightness is to use a satellite galaxy w ith an initialm ass thatisapproxim ately one orderofm agnitude larger. A sim ulation ofa satellite galaxy w ith M sat = 108 M on an orbit w ith e = 0:85 but w ith properties otherw ise identicalto those ofthe satellite m odelled here (Section 3.1),show s that its behavior is sim ilar to the lower m ass satellites. D ue to its higher binding energy,it needs m any m ore orbitalperiods until it is disrupted, and thus the com putational cost is severe. In the rem nant phase it has 105:5 L =kpc2,w hich is m uch closer to the observed values. Its appearence on the sky 0 closely resem bles a dSph galaxy. T hus,ifthe size ofthe satellite is scaled up to have the sam e (relative) binding energy,the evolution is expected to be alm ost identicalto the lower m ass cases presented here. A llvaluesdiscussed here are derived adopting (M =L)true = 3.A notherway to reconcile the centralsurface brighness ofthe m odels presented here w ith the observed values,is to decrease (M =L)true. Ifwe assum e (M =L)true = 0:3,then again 0 105:5 L =kpc2. H owever, such a (M =L)true im plies rather unusualrelative num bers ofbright,evolved stars and less-

{ 12 { lum inous,unevolved stars (D irsch & R ichtler 1995). Finally,it is ofinterest to evaluate the num ber ofparticles contributing to the central part of the rem nant. To this end the num ber N ofparticles is counted in a volum e w ith a radius of 0.8 kpc and centred on the density m axim um of the rem nant. W hile not an exact quanti cation,it is a reasonable assessm ent ofthe num ber ofparticles that are either bound energetically, or have phase-space variables that inhibit spreading away from the rem nant’s density m axim um . A detailed investigation of the relative contribution ofeach type ofparticle to N awaitssim ulationsw ith initially 107 to 108 particlespersatellite galaxy. T he evolution ofN w ith tim e is show n in Fig.8 for allfour runs discussed here. A s can be seen from the gure,both the Superbox and the G rape sim ulationslead to rem nantsthat stabilise w ith 0.3 to 3 per cent ofthe initialnum ber ofparticles. T he later disruption tim e ofthe satellite in the G rape sim ulation Sat-M 2 is evident,but the outcom e is the sam e as in the Superbox run.

4.2. D i erent num ber of cells and particles T he com parison between Superbox and G rape sim ulations in the last section show s thatboth yield thesam econclusionsconcerning theevolution and fateofa low -m asssatellite galaxy w ithout dark m atter. Sm alldi erences occur,but only in as m uch as the tim e of disruption di ersby an orbitalperiod. T he satellite on the lesseccentric orbitarrivesatthe rem nant phase one orbitalperiod later in the G rape sim ulation. T he presence ofa centre to the m esh m oving w ith the satellite therefore does not arti cially prom ote the survivalof a denser core in the rem nant. In this section,Superbox sim ulations w ith N grid = 323 cells per grid,N halo = 1 106 particlesin the G alactic dark halo,and N sat = 3 105 particlesin the satellite are com pared w ith Superbox sim ulations w ith N grid = 643,N halo = 2 106 and N sat = 2 106. T he structure and m assofthe G alactic dark halo and ofthe satellite are described in Section 3.1. T wo pairs ofsim ulations are com pared,and the initialconditions are listed in the bottom fourlines ofTable 1.R uns R S1-1 and R S1-1L sim ulate the satellite galaxy on an extrem ely eccentric orbit (e = 0:96),the path ofw hich is show n in Fig.9. R uns R S1-24 and R S1-24L are sim ulations w ith the satellite galaxy on an orbit w ith e = 0:41. Its trajectory is show n in Fig.10. A s is evident from Figs.11 and 12,the overallevolution ofthe satellite is very sim ilar in allfour sim ulations. A s in the Superbox and G rape sim ulations discussed in the last section,the m ass shells are induced to oscillate w ith about the sam e period by the periodic

{ 13 { tidal eld. Shedding ofm ass proceeds on about the sam e tim e scale in the 323 and 643 cell sim ulations,although the naldisruption tim e isuncertain by aboutone orbitalperiod. O n the highly eccentric orbit,the satellite losesm ore than 90 percentofitsm assatt= 1:4 G yr in the323 cellsim ulation (R S1-1).D isruption occursatt= 2:4 G yrin the 643 cellsim ulation (R S1-1L).T he satellite on the less eccentric orbit is,however,disrupted at about the sam e tim e in both sim ulations,R S1-24 and R S1-24L. T he evolution ofthe centralsurface brightness,ofthe line-of-sight velocity dispersion w ithin the half-light radius,and ofthe apparent m ass-to-light ratio for the four sim ulations are show n in Figs.13 and 14.T he sam e resultsare obtained,nam ely thatsatellite evolution leadsto a stable rem nantthathassim ilar 0,an in ated r1=2 ,and (M =L)obs 100 orm ore. It is evident that the highly eccentric orbit leads to a brighter rem nant w ith a larger r1=2 than the rem nant on the less eccentric orbit. T his trend is also observed in Section 4.1 and results from projection e ects,as described in the Introduction. In Fig.15 the num ber ofparticles w ithin the sphericalvolum e w ith a radius of0.8 kpc is plotted as a function oftim e for allfour sim ulations. A gain this agrees w ith Section 4.1: the num ber ofparticles in the rem nant stabilises at0.5 to 3.5 percent ofthe initialnum ber ofsatellite particles.

5. P ossible discrim inants T he evidence presented so farshow s thatdark m atterm ay notbe necessary to account forthe structuraland kinem aticalpropertiesofatleastsom e ofthe G alactic dSph satellites. H owever,we do nothave unam biguousproofthatthisisso.O bservationaldata thatsupport the tidalm odelinclude extra tidalstars (Irw in & H atzidim itriou 1995,K uhn et al.1996, Sm ith etal.1997)and substructure found in som e ofthe dSph system s. T hese cluesindicate that they m ay not be com pletely dark m atter dom inated and bound system s, and that they m ay be perturbed through tidalforces. H owever,such evidence is not fully conclusive, because identifying extra-tidalstarsdepends to som e extend on the adoption ofequilibrium density pro les such as given by K ing m odels. T here is also a linear correlation between the central surface brightness and the integrated absolute m agnitude for the G alactic dSph system s: satellites w ith higher central surface brightnesstend to be m ore lum inous. Bellazzinietal.(1996)interpretthisto be evidence fortidalm odi cation.T hiscorrelation isinverse to theonefound forellipticalgalaxies and bulgesofspiralgalaxies:system sw ith largercentralsurface brightnessare lesslum inous (see e.g.Ferguson & Binggeli1994).A dem onstration thattidalm odi cation ofbound stel-

{ 14 { lar system s leads to the observed linear correlation between centralsurface brightness and integrated absolute m agnitude is to be found in Section 4 in K 97.H owever,dwarfelliptical galaxies in a num ber ofgalaxy clusters show the sam e correlation (e.g.Bothun,C aldwell& Schom bert 1989). Such a correlation cannot therefore be viewed as convincing evidence for tidalm odi cation,unless dwarfellipticalgalaxies are also tidally m odi ed. Finally,there is also a pronounced correlation between the m ean m etallicity and lum inosity ofthe G alactic dSph satellites,and it is not yet clear how the present scenario can account for this. T he form ation ofthe progenitors ofdSph satellites in di eent locations oftidalarm s pulled out ofa parent galaxy that has a radialm etallicity gradient m ay be part ofa possible solution. In the follow ing, we discuss further possible diagnostics to discrim inate between the dark-m atter-dom inated and tidalm odels.

5.1. T he preference for eccentric orbits Perusal of Figs. 6, 7, 13, and 14 show s that during the rem nant phase the average line-of-sight velocity dispersion increases w ith orbital eccentricity. To quantify this, the tim e-average centralline-of-sight velocity dispersion,h 0i,is com puted over a 2.5 G yr tim e interval, ta tb , w here (M =L)obs(t > Dta) >E 50. T he tim e-averaged line-of-sight velocity dispersion w ithin the half-lightradius, r1=2 ,isalso com puted overthe sam e tim e-interval. T hesim ulationsdiscussed hereareaugm ented by theSuperbox sim ulationsR S1-4 and R S15 w ith N grid = 323 cells and N sat = 3 105 particles analysed in K 97. T he orbits discussed there have e = 0:74 (R S1-4) and 0:60 (R S1-5),w ith R apo = 100 kpc,and the G alactic dark halo consists ofN halo = 106 particles,w ith R c = 250 kpc and M halo = 2:85 1012 M . T he result is plotted in Fig.16. T he negligible di erence between the centralvelocity dispersion and the dispersion evaluated w ithin the half-lightradiusisevident. In this gure, the largestdi erences resultforsim ulations ofnearly radialorbits,w here the m odelled tidal forces at the G alactic centre are m ost sensitively dependent on the resolution used. T he G rape and high-resolution (N grid = 643) Superbox sim ulation give essentially the sam e result,again nicely con rm ing independence ofthe num ericaltechnique used. Fig.16 show s that there is a well-de ned cor E ion between line-of-sight velocity disD relat persion and orbital eccentricity. In addition, r1=2 is consistently larger for the series of m odels w ith sm all apo-galactic distances R apo = 60 kpc, com pared to the ones w ith R apo = 100kpc. D

E

T he nding that r1=2 correlates w ith e has possibly im portant im plications. T he velocity dispersionsm easured in G alacticdSph satellitesrangefrom about6 km /sto 11 km /s

{ 15 { (Irw in & H atzidim itriou 1995,M ateo 1997). T he orbitaleccentricities are not constrained well enough yet to allow any conclusions regarding dark m atter to be m ade. O h et al. (1995) and Irw in & H atzidim itriou (1995) estim ate orbital eccentricities of the G alactic dSph satellites,but theses values cannot be used here because they assum e satellite m asses derived from the line-of-sight velocity dispersions. T he work reported here im plies that, on average,rem nants w ithout dark m atter and w ith larger velocity dispersions ought to be on m ore eccentric orbits. Fig. 16 suggests that the G alactic dSph satellites have orbital eccentricities e > 0:5. C onversely,a dSph satellite w ith e < 0:3 and r1=2 > 6 km /s would have to be dark m atter dom inated,unless such satellites have an extrem e internalvelocity anisotropy w ith a large velocity dispersion perpendicularto the direction ofthe orbitalpath (K uhn 1993). T his special case, however, would appear to be di cult to produce under presently understood galaxy form ation m echanism s.

5.2. Im plications of M A nalysis ofthe rem nantshasbeen based on particlesthatlie w ithin a m agnitude range M = 0:8 (equation 2) centred on the distance m odulus ofthe density m axim um . O bservationalsam ples used to derive kinem aticalquantities for dSph galaxies typically rely on a set ofgiant stars w ithin som e lim ited m agnitude range. A ssum ing the sam e intrinsic stellar brightness,this translates into a distance selection. Ifthe satellite is extended and has sub-structure along the line ofsight,as m ay be the case for tidally m odi ed rem nants, its colour-m agnitude diagram would exhibit a scatter that m ight m im ic populations w ith di erent agesorm etallicities,and the kinem aticalproperties m ight depend on the m agnitude range considered. T his is dem onstrated in Fig.12 ofK 97, w here a signi cant increase ofthe observed velocity dispersion is seen for increasing M = 0:1;:::;3 m ag in one of the m odels. It is im portant to notice that even for the sm allest m agnitude bin,the derived m ass-to-light ratio is extrem ely high and exceeds the true value by far. Structuraland photom etric properties ofG alactic dSph satellites rely on m ore inclusive stellar sam ples, because the stringent constraint on the nature of the stars necessary for kinem aticalstudies (lum inous stars w ith wellde ned spectralfeatures) can be relaxed. It is therefore im portant to quantify the change ofthe structuralparam eter,r1=2,and ofthe photom etric quantity, 0,w ith M . Ifthe tidal-m odi cation theory isto rem ain valid,then these quantitiesm ustnotchange m uch w ith M ,lestthe observerwould see such variations for di erent sub-populations in the H R diagram ofa dSph satellite.

{ 16 { G iven that the two snapshots of satellites R S1-4 and R S1-5 at tim es t = 6:27 G yr and 8:74 G yr, respectively, are studied in m uch detailin sections 4.2 and 4.3 of K 97, we extent that analysis here to quantify the variation ofr1=2 and 0 w ith M (Section 5.2.1), and to investigate if the m orphology of the H R diagram m ay betray tidal m odi cation (Section 5.2.2). A t t = 6:27 G yr, rem nant R S1-4 has M cod = 19:33 m ag, corresponding to a G alactocentric distance of R G C = 70:7 kpc. A t t = 8:74 G yr, rem nant R S1-5 has M cod = 19:16 m ag,corresponding to a G alactocentric distance ofR G C = 65:7 kpc.

5.2.1. D ependence ofstructuraland photom etric quantities on

M

In Fig.17,the half-lightradius r1=2 ( lled circles) and the centralsurface brightness 0 (open circles) are plotted asa function of M . T he two uppercurves describe the snapshot ofrem nantR S1-4,w hen itstrajectory isalm ostaligned w ith theobserver’sline-of-sight.T he observer sees the rem nant and parts ofthe leading and trailing tidaldebris projected onto thesam e sm allregion in thesky,producing a considerable extend along theline-of-sight,and thus a large variation ofthe observed velocity dispersion w ith M . H owever,from Fig.17 it is apparent that the inferred r1=2 and 0 values are not a ected m uch by the sam pling procedure: r1=2 170 240 pc and 0 104:7 105 L =kpc2. T he snapshot ofrem nant R S1-5 is observed at a larger angle to its orbitaltrajectory,leading to no signi cant extend along the line-of-sight. C onsequently,r1=2 and 0 do not vary m uch w ith M (lower set of curves in Fig.17).

5.2.2. T he width ofthe horizontalbranch A spread of distances leads to a broadening of the giant and horizontal branches in the H R diagram . Sub-clum ping along the line-of sight w ill lead to distinct populations thatare separated vertically in the H R diagram .T hese are im portantpossible observational discrim inants,and thehorizontalbranch isespecially wellsuited forthistypeofinvestigation because itis horizontaland blue enough to be less a ected by contam ination by foreground G alactic eld stars,assuggested by C .Pryor(private com m unication). T he signi cant lineof-sight extension ofrem nant R S1-4 provides us w ith a m odelfor exam ining the e ects of this on the horizontalbranch. To thisend,allparticlesareassum ed to have thesam elum inosity,and histogram softhe num berofparticlesin binsofdistance m oduluscentred on M cod are constructed in di erent regionsoftherem nant’sface.T hisisdoneforrem nantsR S1-4 and R S1-5 afterstoring m odel

{ 17 { observationalsam plesusing M = 3 m ag (equation 2).T he appearance ofthe rem nantson the sky and the distribution ofm ean velocities and velocity dispersions are show n in Fig.9 ofK 97,w hich also de nesthe rectilinearcoordinate system ,(xobs;yobs)on the observational plane used here. Itdem onstratesthatnotone ofthe two rem nantsiscentred precisely on its density m axim um (at position xobs = yobs = 0kpc),and that neither the velocity gradient nor the isophotalshape ofthe rem nant need be aligned w ith the orbitaltrajectory. Figures 18 and 19 show the above-m entioned histogram s for m odels R S1-4 and R S1-5, respectively,at three di erent positions along the velocity gradient across the face ofboth rem nants (upper panels). T he solid line indicates the m agnitude distribution of particles w ithin a radius of0:2kpc ofthe position ofthe density m axim um ofthe rem nants (xobs = yobs = 0). T he long-dashed line denotes the sam ple w ithin a radialdistance of0.4 kpc of the position xobs = yobs = + 0:8 kpc,and the dash-dotted line is the sam ple w ithin a radial distance of0.4 kpc ofthe position xobs = yobs = 0:8 kpc. T hese three regions are aligned along the line-of-sightvelocity gradientobserved in both rem nants. In each gure,the lower panel sam ples all particles w ithin a radius of 1:2kpc of xobs = yobs = 0, thus including the above three sm aller regions. Particles that are closer to the observer than the density m axim um have M < 0. T he large projected depth ofrem nant R S1-4,together w ith its clum piness,produces a range ofobserved m agnitudes,especially in the lower panelin Fig.18. T he three peaks are separated by 0.25 m ag and 0:85 m ag,corresponding to distancesof4.2 kpc and 12:6 kpc, respectively,from the position ofthe density m axim um (70.7 kpc). In a colour-m agnitude diagram (C M D ),the apparent scatter m ight be interpreted as com ing from three distinct stellarcom ponents ofdi erent m etallicities. H owever,the m ajorpeak at M = 0 isnarrow and wellde ned,and would be prom inent in a C M D diagram . T he C M D ofrem nant R S1-5 would appear featureless and very narrow (Fig.19).

5.2.3. W ords ofcaution A s argued in Section 5.1,the tidalm odelfavours eccentric orbits. For an observer on Earth,the angle between the line-of-sight and the trajectory ofthe dSph satellite is likely to be sm all,and the above projection phenom ena becom e im portant. T herefore,we expect the C M D s ofat least som e ofthese galaxies to exhibit som e scatter and possibly com plex substructure. H owever this prediction is stillprelim inary and has to be taken w ith caution. W e have presented a detailed analysis of only two snapshots of the sam e initiallow -m ass satellite on two slightly di erent orbits. M ore general conclusions w ill be possible once the param eter study now in progress is nished. T he present results do not exclude the

{ 18 { possibility that allknow n G alactic dSph satellites are m ore like rem nant R S1-5 than R S1-4 w ith colour-m agnitude diagram s that are not a ected by a line-of-sight extension. Spreads in stellar age or m etallicity introduce scatter to the C M D . Exam ples of the variation ofthe horizontalbranch m orphology in dependence ofm etallicity and age can be found in Lee,D em arque & Zinn (1994). T he w idth ofthe theoreticalhorizontalbranches is typically 0.2 to 0.35 m ag, as is true for globular clusters. In this case the horizontal branch m orphology could constrain thedepth iftheparticulardSph satellitehasa signi cant extension along the observer’s line-of-sight. A m ore com plex horizontalbranch m orphology resultsifa dSph satellitehad a com plex starform ation history.In thiscasedepth inform ation w illbe di cult to extract. T he C M D s of som e G alactic dSph satellites exhibit considerable scatter, and this is usually interpreted,am ong other evidence in the H R diagram ,to be a sign for a com plex star form ation history (see G rebel1997 and D a C osta 1997 for excellent review s). A longterm aim of the param eter study under way now , in w hich the orbits and initialsatellite m asses are varied,is to identify those param eters that lead to rem nants that m ost closely resem ble the know n dSph satellites in term s ofr1=2, 0, 0 and (M =L)obs. A detailed study ofindividualrem nants w illthen include the construction ofsynthetic C M D s.

6. C onclusions D i erent num ericalschem es are used to com pute the evolution ofa low -m ass satellite galaxy w ithout dark m atter on di erent orbits in di erent spheroidalG alactic dark halos. T he sim ulationsareperform ed w ith a particle-m esh code w ith nested sub-grids(Superbox) running on conventionalworkstations,and w ith a direct-sum m ation N -body code using the special-purpose hardware device G rape. For the form er num ericalapproach,323 cells per hierarchy w ith 3 105 satellite particles, as well as 643 cells per hierarchy w ith 2 106 satellite particles are used. In the latter num erical schem e, 131072 satellite particles are integrated. T he evolution is very sim ilar and thus independent of the num erical schem e em ployed. A lso,the di erent num ber ofgrid cells and particle num ber in the Superbox sim ulationsleadsto the sam e results,apartfrom sm alldi erences relating to the exacttim e ofsatellite disruption. T he com parison show s that in allcases the satellite evolves to a stable rem nant that contains on the order of1 per cent ofthe originalm ass. T his rem nant phase is arrived at afterthe naldisruption eventnearperi-galacticon,during w hich the rem aining 10 to 20 per centoftheinitialsatellite m assisthrow n o .T he structuralparam etersand theline-of-sight

{ 19 { kinem aticalproperties are sim ilar to values observed for the G alactic dSph satellites. T he present m odelrem nants have,w ith (M =L)true = 3,lower centralsurface lum inosities than the G alactic dSph satellites. H owever,larger initialsatellite m asses,or reduced (M =L)true per particle,can reconcile this di erence. Satellitesinitially on eccentric orbitslead to apparently brighterrem nantsw ith in ated line-of-sight velocity dispersions, ow ing to the observer’s line-of-sight being approxim ately aligned w ith the orbitalpath. In this case, particles ahead of and follow ing the rem nant add to w hat the observer m ay m ake out to be a dSph galaxy. A n observer looking along a very eccentric orbit nds a rem nant w ith a larger than ifits orbit were less eccentric. T he apparent M =L an observer deduces from the K ing form ula (equation 3) is very large, although the individualparticles have (M =L)true 3. T he line-of-sight velocity dispersion, ,can thus be quite unrelated to the true m ass ofthe system . T he projection onto the observationalplane has im portant im plications for deducing the dark m atter content ofG alactic dSph satellites,iftheir orbitaleccentricity were know n. T he m odeldata discussed here show that the G alactic dSph satellites,w hich have r1=2 6 10 km /s,should beon orbitsw ith eccentricitiese > 0:5.C onversely,iffutureobservations con rm e < 0:3 forsom e ofthese system sthen these m ustbe dark m atterdom inated,unless they have a very pronounced internal velocity anisotropy (K uhn 1993). Such anisotropy would appear to be di cult to produce,though,on a nearly-circular orbit,under currently know n galaxy form ation m echanism s. T hetidalm odelfurtherm orepredictsscatterin theC M D sofatleastsom edSph galaxies. For preferentially eccentric orbits, projection e ects enhance the extent of the observed rem nantalong the line-of-sight. T histranslatesinto a distribution ofstellarm agnitudesand subsequently a scatter in the C M D ,and contributes to the scatter usually interpreted to be a sign ofcom plex star form ation histories or m etallicity variations. T he m orphology of the horizontalbranch is wellsuited for the study of possible projection e ects. T he best candidate G alactic dSph satellitesto show such an e ectare those w ith internalsub-clum ps, w hich cannot be expected to be at exactly the sam e distance. T hus,each sub-clum p m ay contribute a horizontalbranch displaced vertically by a few tenths ofa m agnitude relative to the horizontalbranches ofthe other sub-clum ps. T his work strengthens the evidence that at least som e of the dSph satellite galaxies m ay not be dark-m atter dom inated,con rm ing the conclusions ofK 97. It follow s that the interpretation by K uhn (1993), that at least som e of the G alactic dSph satellite galaxies m ay be system s w ith specialphase-space characteristics that perm it long-tim e survival,is to be taken seriously. Self-consistent sim ulations of the type analysed here show how a tidally-shaped rem nant m ay be obtained through periodic m odi cations ofthe phase-space

{ 20 { properties ofthe satellite particles. T he conclusions arrived at here are in agreem ent w ith the suggestion that som e ofthe progenitors of dwarf spheroidal galaxies surrounding the M ilky W ay m ay have form ed as condensations in tidal arm s of past m erging events (see e.g. Lynden-Bell & Lynden-Bell 1995,and G rebel1997 fora review ). T he form ation ofdwarfgalaxiesin tidaltailsisstudied in detailby Elm egreen,K aufm an & T hom asson (1993) and Barnes & H ernquist (1992). W e thank A .Burkert,D .Pfenniger,and R .Spurzem for m any stim ulating discussions. W e are especially grateful to C . Pryor for his insightful com m ents and suggestions that signi cantly im proved this article,and to G .Bothun for his help. T he velocity dispersions were calculated using the R ostat software package kindly supplied by T .C . Beers. PK acknow ledges support through the Sonder-Forschungs-Bereich 328.

R E FE R E N C E S Barnes,J.E.,H ut,P.,1986,N ature,324,446 Barnes,J.E.,H ernquist,L.,1992,N ature,360,715 Beers,T .C .,Flynn,K .,G ebhardt,K .,1990,A J,100,32 Bellazzini,M .,FusiPecci,F.,Ferraro,F.R .,1996,M N R A S,278,947 Burkert,A .,1997,A pJ,474,L99 D a C osta, G .S., 1997, in "Stellar A strophysics for the LocalG roup: A First Step to the U niverse",Proceedings ofthe V IIIth C anary Islands W inter School,eds A .A paricio and A .H errero,(C am bridge: C am bridge U niversity Press),in press. D irsch,B.,R ichtler,T .,1995,A & A ,303,742 Ebisuzaki, T ., M akino, J., Fukushige, T ., Taiji, M ., Sugim oto, D ., Ito, T ., O kum ura, S., 1993,PA SJ,45,269 Elm egreen B.G .,K aufm an M .,T hom asson M .,1993,A pJ,412,90 Ferguson,H .C .,Binggeli,B.,1994,A & A R ,6,67 G allagher,J.S.,W yse,R .F.G .,1994,PA SP,106,1225 G rebelE.K .,1997,R eview s in M odern A stronom y,10,29 H argreaves,J.C .,G ilm ore,G .,A nnan,J.D .,1996,M N R A S,279,108 Irw in,M .,H atzidim itriou,D .,1995,M N R A S,277,1354

{ 21 { Johnston K .V .,SpergelD .N .,H ernquist L.,1995,A pJ,451,598 K roupa,P.,1997,N ew A stronom y,2,139 (K 97) K uhn,J.R .,1993,A pJ,409,L13 K uhn,J.R .,M iller,R .H .,1989,A pJ,341,L41 K uhn J.R .,Sm ith H .A .,H aw ley S.L.,1996,A pJ,469,L93 Lee Y .-W .,D em arque P.,Zinn R .,1994,A pJ,423,248 Lynden-Bell,D .,Lynden-Bell,R .M .,1995,M N R A S,275,429 M ateo,M .,1997,in: T he N ature ofEllipticalG alaxies,eds. M .A rnaboldi,G .S.D a C osta, & P.Saha,PA SP,Vol. 116 (also: astro-ph/9701158) M eylan,G .,Prugniel,P.(eds.),1994,D warfG alaxies,ESO C onference and W orkshop Proceedings N o. 49,ESO O h,K .S.,Lin,D .N .C .,A arseth,S.J.,1995,A pJ,442,142 O kum ura,S.K .,M akino,J.,Ebisuzaki,T .,Fukushige,T .,Ito,T .,Sugim oto,D .,1993,PA SJ, 45,329 O lszew ski,E.W .,Pryor,C .,A rm andro ,T .E.,1996,A J,111,750 Piatek,S.,Pryor,C .,1995,A J,109,1071 Sellwood,J.A .,1987,A R A & A ,25,151 Sm ith,H .A .,K uhn,J.R .,H aw ley,S.L.,1997,in: Proper M otions and G alactic A stronom y, ed. R .M .H um phreys,PA SP,in press Sugim oto,D .,C hikada,Y .,M akino,J.,Ito,T .,Ebisuzaki,T .,U m em ura,M .,1990,N ature, 345,33 T sujim oto,T .,Yoshii,Y .,N om oto,K .,M atteucci,F.,T hielem ann,F.-K .& H ashim oto,M ., 1997,A pJ,483,228

T his preprint was prepared w ith the A A S LATEX m acros v4.0.

{ 22 {

Sim ulation

N grid

N halo

R S1-109 Sat-M 1 R S1-113 Sat-M 2

323 G 323 G

1

R S1-1L R S1-1 R S1-24L R S1-24

643

2 1 2 1

323 643 323

106 | 106

1 |

106 106 106 106

N sat

N st

ntot

n

t

r0

v0

(G yr)

(kpc)

(km /s)

e

M

halo

( 1012 M

3 105 131072 3 105 131072

5 104 65536 5 104 65536

8000 3000 6000 5500

30 15 30 15

8.8 4.5 6.6 8.25

(60 = 0 = 0 ) (60 = 0 = 0 ) (60 = 0 = 0 ) (60 = 0 = 0 )

(0 = 60 = 0 ) (0 = 60 = 0 ) (0 = 120 = 0 ) (0 = 120 = 0 )

0.71 0.71 0.46 0.46

0.45 0.45 0.45 0.45

106 105 106 105

105 5 104 105 5 104

4500 5000 7500 10000

15 15 40 60

5.0 5.5 8.3 11

(100 = 0 = 0 ) (0 = 25 = 0 ) (100 = 0 = 0 ) (0 = 25 = 0 ) (60 = 0 = 0 ) (0 = 175 = 0 ) (60 = 0 = 0 ) (0 = 175 = 0 )

0.96 0.96 0.41 0.41

2.85 2.85 2.85 2.85

2 3 2 3

Table 1: Table ofinitialconditions for the sim ulations presented here.

)

{ 23 { FIG U R E C A PT IO N S

F IG .1 | Snapshotofthe evolution ofthe satellite in G rape sim ulation Sat-M 1 atthree di erent tim es. T he right side ofeach panelplots the distribution ofsatellite particles at the given tim e in the G alactic coordinate system . Each of the axes is 140 kpc long. T he solid line depicts the trajectory ofthe density m axim um ofthe satellite untilthe tim e ofthe snapshot. O n the left,the centralpartofthe satellite is show n enlarged (the totallength of each axis is 5 kpc). F IG .2 | T he orbitalpath ofthe satellite in sim ulations R S1-109 and Sat-M 1. F IG .3 | T he orbitalpath ofthe satellite in sim ulations R S1-113 and Sat-M 2. F IG .4 | T he upperpanelshow sthe evolution ofthe radiicontaining 10,20,...,90 per centofthe totalm assofthe satellite,and the lowerpanelshow sthe G alactocentric distance asa function oftim e. In both panelsthe solid curve isforsim ulation R S1-109 (Superbox), and the dashed curve is for Sat-M 1 (G rape). F IG .5 | A s Fig.4,but for Superbox sim ulation R S1-113 (solid curve) and G rape sim ulation Sat-M 2 (dashed line). F IG .6 | U pperpanel:evolution ofthe centralsurface brightness. C entralpanel:evolution ofthe line-of-sight velocity dispersion w ithin the half-light radius,r1=2. Bottom panel: evolution ofthe apparent m ass-to-light ratio evaluated using equation 3. In allpanels,the solid line is Superbox sim ulation R S1-109,and the dashed line is G rape sim ulation SatM 1. F IG .7 | A s Fig. 6, but for Superbox sim ulation R S1-113 (solid line), and G rape sim ulation Sat-M 1 (dashed line). F IG .8 | T he num ber ofparticles N (t) in the volum e w ith radius 0.8 kpc centred on the density m axim um ofthe satellite. U pper panel: the solid line is Superbox sim ulation R S1-109,and the dashed line is G rape sim ulation Sat-M 1. Lower panel: the solid line is Superbox sim ulation R S1-113,and the dashed line is G rape sim ulation Sat-M 2. In both panels,the num ber ofG rape particles is scaled up by the factor (3 105)=131072,w here the nom inator and denom inator are the initialnum ber ofparticles in the Superbox and G rape sim ulations,respectively. F IG .9 | T he orbital path of the satellite in the 323 cell sim ulation R S1-1 and the 643 cellsim ulation R S1-1L.O w ing to the slight prolate form ofthe live G alactic dark halo, the orbitalplane ips near apo-galacticon. T his renders the projection ofthe orbitalplane onto the x-y plane som ew hat irregular.

{ 24 { F IG .10 | T he orbitalpath ofthe satellite in 323 cellsim ulation R S1-24 and 643 cell sim ulation R S1-24L. F IG .11 | A sFig.4,butfor323 cellsim ulation R S1-1 (solid curve)and 643 cellsim ulation R S1-1L (dashed lines). F IG .12 | A s Fig.4,but for 323 cellsim ulation R S1-24 (solid curve) and 643 cellsim ulation R S1-24L (dashed lines). F IG .13 | A sFig.6,butfor323 cellsim ulation R S1-1 (solid line)and 643 cellsim ulation R S1-1L (dashed line). F IG .14 | A sFig.6,butfor323 cellsim ulation R S1-24 (solid line)and 643 cellsim ulation R S1-24L (dashed line). F IG .15 | T he num ber ofparticles N (t) in the volum e w ith radius 0.8 kpc centred on the density m axim um ofthe satellite. U pperpanel: the solid line is323 cellsim ulation R S11,and the dashed line is 643 cellsim ulation R S1-1L.Lower panel: the solid line is 323 cell sim ulation R S1-24,and the dashed line is 643 cellsim ulation R S1-24L.In both panels,the num berofparticlesin R S1-1L and R S1-24L isscaled dow n by the factor(3 105)=(2 106), w here the nom inator and denom inator are the initial num ber of satellite particles in the sim ulations w ith 323 and 643 cells,respectively. F IG .16 | T he tim e-averaged velocity dispersion,com puted over the rst 2.5 G yr after (M =L)obs 50 is achieved, as a function of orbitaleccentricity, e. Solid circles are h 0i for allSuperbox sim ulations presented here and in K 97 w ith N grid = 323 cells and N sat = 3 105 particles.Solid triangles:h 0iforG rape sim ulations(e = 0:46;0:71),and Superbox E D sim ulations w ith N grid = 643 and N sat = 2 106 (e = 0:41;0:96). O pen circles show r1=2 for all Superbox sim ulations w ith N grid = 323 cells and N sat = 3 105 particles, and open triangles are the corresponding G rape and Superbox sim ulations w ith N grid = 643 and N sat = 2 106. T he horizontaldashed line is the initialcentralline-of-sight velocity dispersion. N um bers next to the lled circles are Rapo in kpc. F IG .17 | T he half-light radius,r1=2 ( lled circles),and centralsurface brightness, 0 (open circles),as a function of M . T he set oftwo upper curves are for rem nant R S1-4 at t = 6:27 G yr,and the lower set ofcurves are for rem nant R S1-5 at t = 8:74 G yr. T hese data are produced using k = 100 bins w ithin rbin = 4 kpc and (M =L)true = 3. T his gure com plem ents Fig.12 in K 97. F IG .18 | T hedistance-m odulusdistribution ofparticlesacrossthefaceofrem nantR S14. Both panels show the distribution of distance m odulirelative to the distance m odulus of the rem nant’s density m axim um . In the upper panel, the solid histogram show s the

{ 25 { distribution at the rem nant’s centre, and the long-dashed and dot-dashed histogram s are the distributions in regions o set from the centre by 1.13 kpc along the velocity gradient. T he bottom panelshow sthe distribution forallparticlesappearing projected w ithin a radial distance of1.2 kpc from the position on the sky ofthe rem nant’s density m axim um . For details see Section 5.2.2. T his gure com plem ents gs.9{12 in K 97. F IG .19 | T he sam e as Fig.18,but for the snapshot ofrem nant R S1-5.

{ 26 {

Fig. 1.|

{ 27 {

Fig. 2.|

Fig. 3.|

{ 28 {

Fig. 4.|

Fig. 5.|

{ 29 {

Fig. 6.|

Fig. 7.|

{ 30 {

Fig. 8.|

Fig. 9.|

{ 31 {

Fig. 10.|

Fig. 11.|

{ 32 {

Fig. 12.|

Fig. 13.|

{ 33 {

Fig. 14.|

Fig. 15.|

{ 34 {

Fig. 16.|

{ 35 {

Fig. 17.|

{ 36 {

Fig. 18.|

Fig. 19.|