MEASURING FUNDAMENTAL GALACTIC PARAMETERS WITH STELLAR TIDAL STREAMS AND SIM PLANETQUEST

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1 Draft version December 5, 25 PreprinttypesetusingL A TEXstyleemulateapjv.6/22/4 MEASURING FUNDAMENTAL GALACTIC PARAMETERS WITH STELLAR TIDAL STREAMS AND SIM PLANETQUEST StevenR.Majewski 1,DavidR.Law 2,AllysonA.Polak 1 &RichardJ.Patterson 1 Draft version December 5, 25 ABSTRACT Extended halo tidal streams from disrupting Milky Way satellites offer new opportunities for gauging fundamental Galactic parameters without challenging observations of the Galactic center. In the roughly spherical Galactic potential tidal debris from a satellite system is largely confined to a single plane containing the Galactic center, so accurate distances to stars in the tidal stream can be used to gaugethegalacticcenterdistance,r,givenreasonableprojectionofthestreamorbitalpoleonthe X GC axis.alternatively,atidalstreamwithorbitalpolenearthey GC axis,likethesagittariusstream, canbeusedtoderivethespeedofthelocalstandardofrest(θ LSR ).Modestimprovementsincurrent astrometric catalogues might allow this measurement to be made, but NASA s Space Interferometry Mission(SIMPlanetQuest)candefinitivelyobtainbothR andθ LSR usingtidalstreams. Subject headings: Milky Way: structure Milky Way: dynamics Sagittarius dwarf galaxy 1. DISTANCE TO THE GALACTIC CENTER With the assumption that globular clusters trace the generalshapeandextentofthemilkyway(mw),shapley(1918)firstshowedhowtheycanbeusedtoestimate thedistance(r )tothegalacticcenter(gc),expected to lie at the center of the cluster distribution. Though Shapley s first execution of this experiment exaggerated R duetoclusterdistancescaleproblems, theoverall scheme of mapping an extended distribution of Galactic tracer objects to determine the location of its center remains a valid, if traditionally underutilized, strategy. The globular cluster sample is relatively small and concentrated to the GC, where dust effects introduce large distance uncertainties and a likely still incomplete and lop-sided cluster census. Population II tracers like RR Lyrae, blue horizontal branch(bhb) or giant stars are much more plentiful outside of the MW bulge. Unfortunately, the current census for these tracers is even more incomplete than for globulars. Though this situation may be remedied by currently planned wide angle surveys, several inherent problems remain with exploitation of these tracers as GC benchmarks. As with the clusters, themwzoneofavoidance(za)willalwaysresultin biasedsampledistributionsandpotentialr underestimates exacerbatedifsurveysdonotreachthefarside ofthemw.evenmorechallengingisthattheglobaldistributions of halo stars are far from dynamically mixed: Recent surveys of the above tracers reveal a halo streaked with substructure, likely originating as satellite disruptiondebris(e.g.,vivasetal.21,newbergetal.22, Majewski 24) and eroding simple halo axisymmetry. The very existence of numerous tidal streams motivates the present contribution. Individual tidal streams actually possess a relatively simple spatial configuration. Within spherical potentials, tidal debris arms from a dis- 1 Dept. of Astronomy, Univ. Virginia, Charlottesville, VA Caltech, Dept. of Astronomy, MS 15-24, Pasadena, CA Electronic address: srm4n, aap5u,rjpi@virginia.edu, drlaw@astro.caltech.edu rupting satellite will lie along the satellite orbital plane, which contains the GC. A sufficiently extended tidal debrisarcdefinesthatplane,whichintersectsthemwx GC axis at the GC. This simpler, almost two-dimensional geometry of tidal stream arcs removes the need for samplecompleteness: Inprinciple,R shouldbederivable from the(l, b,distance) distribution of only a large enough sampleoftidalstreamstarstodefinetheirorbitalplane. 3 In reality, non-spherical potentials precess tidal streams. Fortunately, this is a relatively small effect in themw,atleastforr GC softensofkiloparsecs.johnston et al. (25; J5 ) showed the Sagittarius(Sgr) tidal stream precession is sufficiently small to conclude the MW potential is only slightly oblate within the Sgr orbit(peri:apo-galacticon of 13:57 kpc). Moreover, as pointedoutalsobyhelmi(24),thatpartofthesgr trailing arm arcing across the southern MW hemisphere (see Majewski et al. 23, MSWO ) is so dynamically youngthatithasn thadtimetoprecess(seefig.5of J5). Unfortunately, as noted by MSWO, Sgr is in almost the worst possible orientation to undertake the proposed R -gauging: Withavirtuallynegligibleanglebetween thesgrorbitalplaneandx GC axis,smallerrorsinthe definition of the orbital plane(due to small residual precession,andthefinitewidthofthedebrisplane)leadto substantialuncertaintiesinderivationofr. TheidealtidaldebrisconfigurationforestimatingR hasapoleclosertothex GC axis. Giventhepaceof discovery,suchastreammaysoonbefound. Basedon the nearly polar orientation of the HI Magellanic Stream and the typically measured proper motions(µ s) for the Magellanic Clouds(Gardiner& Noguchi 1996, van der Mareletal.22andreferencestherein),itisclearthata stellar counterpart to the Magellanic Stream would have almosttheperfectorientationforgaugingr. Systematic errors in a tracer distance scale translate to estimatesofr. However,becausestreamscontaindifferent stellar types(e.g., giant stars, RR Lyrae, BHB), uncertainties from photometric/spectroscopic parallaxes 3 Samplesshouldbeunbiasedwithrespecttospreadperpendiculartothatplane,butthisshouldbetrivialtoachieve.

2 2 Majewski, et al. can be cross-checked. In most cases, reddening and crowding effects can be of negligible concern. Alternatively, with NASA s Space Interferometry Mission(SIM), direct trigonometric parallaxes will be well within reach: For a putative Magellanic stellar stream orbiting at 5 kpcradius,the 1-2µasparallaxesarewellabove the SIM wide-angle astrometric accuracy goal of 4µas, assumingkgiantstartracers(v 18). Asatestofwhatmightbeachieved,weranN-body simulations of different mass satellites disrupting for 5 or1gyr(whateverwasneededtoproduce>27 -long tails) in the Galactic potential that best fitted the Sgr debris stream in Law et al.(25; L5 hereafter). The orbit was constrained to match the current position, radialvelocity(rv)andµ(gardiner&noguchi1996)of the Small Magellanic Cloud(SMC), with orbital pole (l,b)=(196, 5). 4 Allothermodelparameterswere similar to those in L5. Each simulation was observed outsidea b >15 ZA,with 1 3,1 4 and1 5 tracer stars(apportioned with 9% of these in the satellite coreand 1%inthetidaltails),andwith,1and 2% random Gaussian distance errors imposed. The simplest(though not best!) analysis of these data is simply tofitaplaneandmeasureitsintersectionwiththex GC axis. With this th-order method, even for large samples of stars in dynamically cold streams(i.e., not that froma1 1 M progenitor)andnodistanceerrors,relativelylarge(<7%)systematicerrorsinr canremain (Fig. 1) because plane-fitting does not account for the residual precessional twisting of the debris arms. The direction of precession(determined by the direction of the stream angular momentum vector) drives the sense oftheimposedsystemicr error(i.e.closerorfarther), and random distance errors add additional uncertainties depending on details of the stream orientation relative to theza.abetter,nowprovenmethod(e.g.l5)istouse N-body modeling to reconstruct a given stellar stream; such modeling can precisely account not only for precession but also for stream dispersion and other higher order uncertainties, which would permit a more accurate identificationofthecenterofthemwpotentialforan appropriately oriented stream. RecentmeasurementsofstellarmotionsaroundSgrA haveledtodynamicalparallaxesgoodto5%(7.94±.42 kpc; Eisenhauer et al. 23), a measurement sure to improvewithlongersgra fieldmonitoringcampaigns. Few percent quality trigonometric parallaxes of stars nearthegcwillbemeasuredaspartofasimkey Project. In either method, the target stars are reasonablyexpectedtolieattheassumedcenterofthemw potential. The proposed use of tidal streams to measure R will provideaninterestingtest ofthis hypothesis, since tidal streams orbit the true dynamical center of the integrated potential over tens of kiloparsec scales. A comparisonofthiscentertothesgra distancecouldreveal whetherthemwmaybealop-sidedspiral(e.g.,baldwinetal.198,richter&sancisi1994,rix&zaritsky 1995). Such lop-sidedness can, in fact, be induced by mergers of large satellites(walker, Mihos& Hernquist 1997). In principle, three well-measured tidal streams 4 Wedonotmodel thepossiblycomplexinteraction between the Small and Large Magellanic Clouds since we are interested in testing a hypothetical stream with desirable properties. canverifywhetherthegcliesalong(l,b)=(,),since thetruegcshouldlieatamutualintersectionofthe three corresponding stream orbital planes. 2. VELOCITYOFTHELOCALSTANDARDOFREST Despite decades of effort, the local MW rotation rate remains poorly known, with measurements varying by 25%. Hipparcos µ s(feast& Whitelock 1997) suggest thatthelocalstandardofrest(lsr)velocityisθ LSR = (217.5±7.)(R /8)kms 1 i.e.neartheiauadopted valueof22kms 1.Butamorerecentmeasurementof µforsgra (Reid&Brunthaler24)yieldsahigher (235.6±1.2)(R /8)kms 1,whereasdirectHSTmeasurements of the µ s of bulge stars against background galaxiesinthesamefieldyield(22.4±2.8)(r /8)km s 1 (Kaliraietal. 24)and(22.8±13.6)(R /8)km s 1 (Bedinetal. 23). Ofcourse,thesemeasures(as wellasanyofthosedependingontheoortconstants) relyonanaccuratemeasureofr ( 1). Thesolarpeculiarmotionmustalsobeknown,butisasmallercorrection(e.g.,5.3±.6kms 1 ;Dehnen&Binney1998). On the other hand, considerations of non-axisymmetry of the disk yield corrections to the measurements that suggestθ LSR maybeaslowas184±8kms 1 (Olling& Merrifield 1998) or lower(kuijken& Tremaine 1994). IndependentmethodstoascertainΘ LSR areofgreatvalue because it is fundamental to establishing the MW mass scale. Eventually, aspartofakeyprojectofsim,θ LSR willbemeasureddirectlybytheabsoluteµofstarsnear the GC. Here we describe an independent method for ascertainingθ LSR usinghalotidal streamsthat overcomes several difficulties with working in the highly dustobscured,crowdedgc,andonealsoinsensitivetor (for all reasonable values of the latter). The ideal tidal streamforthismethodisonewithanorbitalpolelyingnearthey GC axis. TheSgrtidalstellarstreamnot only fulfills this requirement, but its stars, particularly its trailing arm M giants, are ideally placed for uncrowded field astrometry at high MW latitudes, and at relatively bright magnitudes for, and requiring only the most modest precisions from, SIM. Indeed, as we show, this method is even within the grasp of future high quality, groundbased astrometric studies. ItisremarkablethattheSunpresentlylieswithina kiloparsecofthesgrdebrisplane(mswo).thepoleof theplane,(l p,b p )=(272, 12),meansthatthelineof nodesofitsintersectionwiththemwplaneisalmost coincidentwiththex GC axis.thus(fig.2)themotions of Sgr stars within this plane are almost entirely contained in their Galactic U and W velocity components, whereasthev motionsofstarsinthesgrtidaltailsalmost entirely reflect solar motion. To the degree that its V distributionisnotcompletelyflatinfigure2isdueto the slight amount of streaming motion projected onto the V motionsfromthe2 SgrorbitalplanetiltfromX GC, compounded by(1) Keplerian variations in the space velocityofstarsasafunctionoforbitalphase,aswellas(2) precessionaleffectsthatleadtoλ -variabledepartures ofsgrdebrisfromthenominalbestfitplanetoallof the debris. The latter is negligible for trailing debris but ismuchlargerfortheleadingdebris,whichisonaverageclosertothegcanddynamicallyoldercomparedto the trailing debris when viewed near the Galactic poles

3 Fundamental Galactic Parameters 3 (J5). In addition, because the leading debris gets arbitrarily close to the Sun(L5), projection effects make it more complicated to use for the present purposes. Additional problems with the leading arm debris, which suggest that more complicated effects have perturbed it are discussed in L5 and J5. In contrast, the Sgr trailing tail is beautifully positioned fairly equidistantly from us for a substantial fraction of its stretch across the Southern MW hemisphere(mswo). This band of stars arcingalmostdirectly beneath uswithinthex GC -Z GC plane provides a remarkable zero-point reference against which to make direct measurement of the solar motion almost completely independent of the Sun s distance from the GC. The extensive mapping(mswo) of the Sgr tails with Two Micron All-Sky Survey(2MASS) M giants provides an ideal source list for individual stellar targetsfromthis>36 -wrapped,mwpolarring. L5usedMgiantspatial(MSWO)andRVdata(Majewskietal. 24)toconstrainmodelsofSgrdisruption,bestfittingwhena M Sgrof328kms 1 space velocity orbits with period.85 Gyr and apo:peri- Galactica of 57:13 kpc. These models fit what appears tobe 2.5orbits(2.Gyr)ofSgrmasslossinMgiants. The adopted MW potential is smooth, static and given bythesumofadisk,spheroidandhalodescribedbythe axisymmetricfunctionφ halo =v 2 halo ln(r2 +[z/q] 2 +d 2 ) whereqisthehaloflattening,randzarecylindrical coordinates and d is a softening parameter. Additional model details are given in L5. ModelfitstotheSgrspatialandvelocitydataallow predictions of the 6-D phase space configuration of Sgr debris. Figure2showspredictedU,V,W velocitycomponentsofdebrisasafunctionoflongitude,λ,inthe Sgrorbitalplane(seeMSWO).Thedebrisisshownassumingq=.9,R =7kpc,andacharacterizationof thetotalpotentialwherebythelsrspeedisθ LSR =22 kms 1.L5exploreshowvariationsinqaffectprimarily theu andw (throughprojectionalongrv).figure3 (green, yellow, and magenta points) shows how variations in the scale of the potential, expressed through variations inadoptedθ LSR,affectV. Clearly,Θ LSR rangingfrom 18to26kms 1 translatestoobviousvariationsin observed V for trailing arm stars. This effect is easily separable from any residual uncertainty in the shape of thepotentialorr : Figure3(redandbluepointsrespectively) illustrates negligible V changes produced by holdingθ LSR fixedat22kms 1 butvaryingqfrom.9 to1.25(i.e.oblatetoprolate)andr from7to9kpc. Figure3isthebasisfortheproposeduseofSgrtomeasureΘ LSR. Ideally,toexecutetheexperimentrequires obtainingv fromtheobservedµandrvsofsgrarm stars. However, because of the particular configuration ofsgrtrailingarmdebris,almostallofv isreflectedin theµofthesestars,and,morespecifically,thereflexsolar motioniscontainedalmostentirelyintheµ l cos(b)componentofµforsgrtrailingarmstarsawayfromthemw pole. Working in the observational, µ regime means that vagaries in the derivation of individual star distances can beremovedfromtheproblem,aslongasthesystemis modeledwithapropermeandistanceforthesgrstream asafunctionofλ.figure4showsthreegeneralregimes ofthetrailingarmµ l cos(b)trend:(1)λ 1 where µ l cos(b)ispositiveandroughlyconstant,(2)theregion from1 Λ 6 whereµ l cos(b)flipssignasthe debris passes through the South Galactic Pole to shift the Galacticlongitudesofthetrailingarmby 18,and (3)Λ 6,whereµ l cos(b)isnegativeandbecomes smallerwithdecreasingλ (becausethesgrstreambecomesincreasinglyfarther). Thesignflipinµ l cos(b)is ausefulhappenstanceinthecasewhereonehasµdata nottiedtoanabsolutereferenceframebutwhichisat least robust to systematic zonal errors: In this case the peaktopeakamplitudeofµ l cos(b)forthetrailingarm starsyields(twotimes)thereflexmotionofthesun 5. The intrinsic RV dispersion of the Sgr trailing arm has beenmeasuredtobe 1kms 1 (Majewskietal.24); assuming symmetry in the two transverse dimensions of thestreamgivesanintrinsicµdispersionofthesgrtrailingarmof.1masyr 1 (seefig.4a).thus,untilsimquality proper motions exist, the measurement of the reflexsolarmotionbythismethodwillbedominatedby theerrorinµ.toquantifytheaccuracyoftheproposed method, we introduce artificial random errors into the propermotionsofthefivemodelsshowninfigure3and calculate the accuracy with which we expect to recover the solar reflex motion. Simply applying the formalism described above, we recoverθ LSR valuesof 6 212,255,and279kms 1 inmodelsforwhichinputθ LSR =18,22,and26kms 1 respectively. This indicates that the method systematicallyoverpredictsθ LSR byabout3kms 1 ;thisisbecause(seefig.3)thetrendofv withλ isnotperfectly flatbutchangesby 3kms 1 betweenthepeaksat Λ =6 65 and correctingforthissystematic bias, we perform 1 tests where we randomly draw particles from the model debris streams in these ranges with artificially added random scatter in the µ, and find thatrecoveringthesolarvelocitytowithin1kms 1 requiresasampleofapproximately2starswithµmeasuredtoabout1masyr 1 precisionwithnozonalsystematics. UsingthemodelswithΘ LSR =18/22/26 kms 1,thesetestsrecovermeanvaluesof182/225/249 kms 1 respectivelywithadispersionofresultsbetween thetestsof1kms 1.AsexpectedfromFigure3,varyingqandR hasnegligibleeffect: Testsonmodelsin bothofthesemwpotentials(whereθ LSR =22km s 1 )recovermeanvaluesof225and228kms 1. Present astrometric catalogs are just short of being able to do this experiment: Hipparcos is not deep enough, the Southern Proper Motion Survey(Girard et al. 24) has not yet covered enough appropriate sky area, and UCAC2 (Zacharias et al. 24) has several times larger random errors than useful as well as comparablysized zonal systematic errors at relevant magnitudes(n. Zacharias, private communication). However, to demonstrate how only modest advances in all-sky µ precisions are needed to make a definitive measurement, Figure 4 includesadirectcomparisonoftheµ l cos(b)trendforsgr MgiantsusingUCAC2µ sfor2massmgiants.impressively,theoverallexpectedµ l cos(b)trendscanbeseen, but the large scatter and systematic shifts in the trailing arm motions belie the limits of UCAC2 accuracies at V 15. EvenafactoroftwoimprovementinUCAC2 5 WefindthatthesepeakslieatΛ =6 65 and ; notethattechnicallyv (µ/d) (µ/d) Correctingfortheassumed12kms 1 speedofthesunwith respect to the LSR.

4 4 Majewski, et al. random errors and elimination of zonal errors might lead toausefulmeasurementofθ LSR.Itisnotunreasonable to expect advances in all-sky µ catalogues at this level soon(e.g., from the Origins Billion Star Survey or Gaia), butinanycasesimplanetquestwilleasilyobtainthe necessary µ(and parallaxes) of selected Sgr trailing arm giants. We appreciate funding by NASA/JPL through the TakingMeasureoftheMWKeyProjectforSIMPlanetQuest, NSF grant AST-37851, the Packard Foundation, and the F.H. Levinson Fund of the Peninsular Community Foundation. SRM appreciates the hospitality of the Carnegie Observatories during the writing of this paper. Baldwin, J. E., Lynden-Bell, D.,& Sancisi, R. 198, MNRAS, 193, 313 Bedin,L.R.,Piotto,G.,King,I.R.,&Anderson,J.23,AJ,126, 247 Dehnen,W.,&Binney,J.J.1998,MNRAS,298,387 Eisenhauer, F., Schödel, R., Genzel, R., Ott, T., Tecza, M., Abuter, R.,Eckart,A.,&Alexander,T.23,ApJ,597,L121 Feast,M.W.&Whitelock,P.1997,MNRAS,291,683 Gardiner,L.T.,&Noguchi,M.1996,MNRAS,278,191 Girard,T.M.,Dinescu,D.I.,vanAltena,W.F.,Platais,I.,Monet, D.G.,&López,C.E.24,AJ,127,36 Johnston,K.V.,Law,D.R.,&Majewski,S.R.25,ApJ,619, 8(J5) Kalirai,J.S.,etal.24,ApJ,61,277 Kuijken,K.,&Tremaine,S.1994,ApJ,421,178 Law,D.R.,Johnston,K.V.,&Majewski,S.R.25,ApJ,619, 87(L5) Majewski, S. R. 24, Pub.Astr.Soc. Australia, 21, 197 REFERENCES Majewski,S.R.,etal.24,AJ,128,245 Majewski, S. R., Skrutskie, M. F., Weinberg, M. D.,& Ostheimer, J.C.23,ApJ,599,182(MSWO) vandermarel,r.p.,alves,d.r.,hardy,e.,&suntzeff,n.b. 22, AJ, 124, 2639 Newberg,H.J.,etal.22,ApJ,569,245 Olling,R.P.&Merrifield,M.R.1998,MNRAS,297,943 Reid,M.J.,&Brunthaler,A.24,ApJ,616,872 Richter,O.-G.,&Sancisi,R.1994,A&A,29,L9 Rix,H.,&Zaritsky,D.1995,ApJ,447,82 Shapley,H.1918,ApJ,48,154 Vivas,A.K.,etal.21,ApJ,554,L33 Walker,I.R.,Mihos,C.,&Hernquist,L.1996,ApJ,46,121 Zacharias,N.,Rafferty,T.J.,&Zacharias,M.I.2,ASPConf. Ser., 216, 427

5 Fundamental Galactic Parameters 5 Fig. 1. Results of plane-fitting to simulations of disrupting satellites of different masses orbiting similarly to the Magellanic Clouds. Filled, half-filled and open symbols are for simulations with no, 1% and 2% distance errors imposed on the member stars.

6 6 Majewski, et al Fig. 2. Predicted U, V, W velocities (right-handed system) as a function of longitude in the Sgr orbital plane (Λ = at present Sgr position). Debris lost on last half (yellow) and previous full (magenta) orbits are shown (see Law et al. 25). Trailing debris stretches from Λ = to 18 ; leading arm debris goes from Λ = to 27. The Galactic model has a q =.9 halo with ΘLSR = 22 km s 1 and R = 7 kpc.

7 Fundamental Galactic Parameters Fig. 3. Variation of the Sgr trailing arm V velocities for a range of ΘLSR, R, and q. Green, yellow, and magenta points represent debris from satellites disrupting in potentials where q =.9 and R = 7 kpc, but ΘLSR is 18, 22, and 26 km s 1, respectively. Red and blue points represent satellite debris in potentials where ΘLSR = 22 km s 1 but q = 1.25 (R fixed at 7 kpc) and R = 9 kpc (q fixed at.9), respectively. Note how changes in the shape of the potential have little to no effect on V, while changes in the scale of the potential produce large, approximately linear shifts.

8 8 Majewski, et al. Fig. 4. (a)sameasfig.2(i.e.,samemwmodel),butforpredictedµ l cos(b)andfor2.5orbitsofmassloss. (b)observeducac2 µ l cos(b)for2massmgiantsingalacticregionsdominatedbysgrstreamstars(allmgiantswithin7kpcofthenominalsgrplaneand having heliocentric distances of 15-3 kpc). Note the differing vertical scales of the two panels.