THE ASTRONOMICAL JOURNAL VOLUME 113, NUMBER 2 FEBRUARY 1997
THE KINEMATICS, ORBIT, AND SURVIVAL OF THE SAGITTARIUS DWARF SPHEROIDAL GALAXY
Rodrigo A. Ibata Department of Physics and Astronomy, University of British Columbia, 2219 Main Mall, Vancouver, Canada Electronic mail: ibata®astro.ubc.ca
Rosemary F. G. Wyse1,2 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218 Electronic mail: [email protected]
Gerard Gilmore3 Institute of Astronomy, Madingley Road Cambridge, CB3 OHA, United Kingdom Electronic mail: [email protected]
Michael J. Irwin Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 OEZ, United Kingdom Electronic mail: [email protected]
Nicholas B. Suntzeff Cerro Tololo Inter-American Observatory,4 Casilla 603, La Serena, Chile Electronic mail: [email protected] Received 1996 June 17; Revised 1996 November 7
ABSTRACT
The Sagittarius dwarf spheroidal galaxy, the closest satellite galaxy of the Milky Way, has survived for many orbits about the Galaxy. Extant numerical calculations modeled this galaxy as a system with a centrally-concentrated mass profile, following the light, and found that it should lose more than one-half of its mass every 2-4 orbits and be completely disrupted long before now. Apparently the Sagittarius dwarf spheroidal, and by implication other dSph galaxies, do not have a centrally-concentrated profile for their dark matter. We develop a model in which the stars of the Sgr dwarf are embedded in a constant-density dark matter halo, representing the core of a tidally-limited system, and show that this is consistent with its survival. We present new photometric and kinematic observations of the Sagittarius dwarf spheroidal and show these data are consistent with this explanation for the continued existence of this galaxy. The Sagittarius dwarf is being tidally distorted and is tidally limited, but is not disrupted as yet. The 9 corresponding minimum total mass is 10 Mo, while the central mass to visual fight ratio is —50 in Solar units. Our new photographic photometry allows the detection of main-sequence stars of the Sagittarius dwarf over an area of 22° X 8°. The Sagittarius dwarf is prolate, with axis ratios —3:1:1. For an adopted distance of 16± 2 kpc from the Galactic center on the opposite side of the Galaxy to the Sun, the major axis is ^9 kpc long and is aligned approximately normal to the plane of the Milky Way Galaxy, roughly following the coordinate fine /= 5 °. The central velocity dispersion of giant stars which are members of the Sagittarius dwarf is 11.4±0.7 km s-1 and is consistent with being constant over the face of the galaxy. The gradient in mean line-of-sight velocity with position along the major axis, dv/db, is — 0 kms_1/degree in the central regions and increases in amplitude to dv/db= — 3 km s-1/degree over the outermost three degrees for which we have data. A first measurement of the proper motion of the Sagittarius dwarf determines the component of its space velocity parallel to its major axis to be 250±90 kms-1, directed towards the Galactic Plane. We model these kinematic data to determine the orbit of the Sagittarius dwarf. Our best fit model has an orbital period of ^ 1 Gyr and has the Sagittarius dwarf spheroidal close to perigalacticon. This period is shorter, by about a factor of ^ 10, than the age of the bulk of its stellar population. © 1997 American Astronomical Society. [S0004-6256(97)02802-l] institute of Astronomy, Madingley Road Cambridge, CB3 OHA, UK. 2Center for Particle Astrophysics, University of California, Berkeley, CA 95124. 3Institut d’Astrophysique, 98bis Boulevard Arago, 75014 Paris, France. 4Cerro Tololo Inter-American Observatory is operated by AURA, Inc. under contract to the National Science Foundation.
634 Astron. J. 113 (2), February 1997 0004-6256/97/113(2)/634/22/$10.00 © 1997 Am. Astron. Soc. 634
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1997AJ 113. .6341 -2 roidal (Ibataetal1994,1995;hereafterIGI-IandIGI-II, Hatzidimitriou 1995),whichhasimportantimplicationsfor Milky Wayhavelongbeensuspectedtocontainlargequan- the interactionbetweenanormallargediskgalaxyandone provides anunprecedentedopportunityfordetailedstudyof the bulgeofMilkyWay(Ibata&Gilmore1995a,1995b) respectively) duringthecourseofaspectroscopicstudy tip startat«1arcminandincrementby^5arcmin“. Fig. 1.IsoplethcontoursofthetipSgrmainsequence,derivedfromAPMmeasurementsUKSTsurveyplatestoEastcenterSgr,are perigalactic passages. Thismodelalsohassignificant impli- tarius dwarfspheroidal.Wepropose anewmodelforthe nificant accretionandmergingofsmallersystemsoccurs popular hierarchicalclusteringpictureofstructureformation, lution wastruncatedbysupemovae-drivenwinds(e.g., nation fortheirlowmeanmetallicities—thatchemicalevo- dwarf spheroidalscontainstarswithabroadrangeofages the natureofconstituentsdarkhalos.Most tities ofdensedarkmatter(e.g.,Faber&Lin1983;Irwin dwarf spheroidalgalaxy. the Sgrdwarf:M54,Ter7,8,andArp2arealsoindicated,asispositionofforegroundglobularclusterM55.Contoursformainsequence tent withthesedataand thepreviouslyapparently- that areusedtoconstrainmodels oftheorbitSagit- is thison-going? during theevolutionofanormalgalaxylikeMilkyWay; of itssatellitegalaxies,andtheinternalstructurea shown superimposedonthediscoveryisoplethmapofredclumpstarsfromIGI1994.Thepositionsglobularclusterspotentiallyassociatedwith 635 IBATAETAE:THESURVIVALOFSAGITTARIUS internal structureoftheSagittarius dwarfspheroidal,consis- such asCold-Dark-Matterdominatedcosmologies,verysig- Sandage 1965;Dekel&Silk1986).Further,inthecurrently- and metallicities,whichisunexpectedinthesimplestexpla- surprising abilityoftheSgrdwarf tohavesurvivedmany This paperpresentsnewphotometricandkinematicdata At leastsomeofthedwarfspheroidalcompanions The serendipitousdiscoveryoftheSagittariusdwarfsphe- © American Astronomical Society • Provided by theNASA Astrophysics Data System 1. INTRODUCTION probable membersofthisgalaxy,eventhoughthreethem limited thisdetectiontorelativelyhighdensityregionsand, Fig. 1)revealedthattheSagittariusdwarfisvisibleoveran towards theglobularcluster M55 (Fahlmanetal.1996; IGI-I additionallynotedthatfourglobularclusterswere M54 (NGC6715).However,theintrinsicrarityofRHBstars /=5.6°,b =-14.0°,atthepositionofglobularcluster isopleth mapderivedbyIGI-IandIGI-II(superimposedon lactic starswithsimilarapparentmagnitudesandcolors.The which aremembers.The‘backgroundnoise’inthiscaseisa lized theredhorizontalbranch(RHB)andclumpgiantstars Mateo etal.1996)show,inaddition totheexpectedM55 well outsidetheoriginaldetection limits. tified membersoftheSagittarius dwarfalonglines-of-sight expectation thatthoseisoplethswerealowerlimitonthe lay welloutsidetheavailableisopleths,furthersupportingan only ^50%ofthestellarpopulationSagittariusdwarf. combination ofsmallnumberstatisticsandforegroundGa- dwarf galaxies. cations forthenatureofdarkmatterandevolution and Galacticstars, awell-definedmainsequence andthree detectable sizeoftheSgrdwarf.Thisexpectationhasbeen as IGI-IIemphasized,theiroutermostisoplethhaddetected approximately 10°X4°regioncenterednear supported intheinterimbyseveralstudieswhichhaveiden- The firstsurveytomaptheSagittariusdwarf(IGI-I)uti- Deep CCDcolormagnitudediagrams ofthelinesight 2. PHYSICALPROPERTIESOFTHESAGITTARIUSDWARF 2.1 PhotometricDeterminationoftheSpatialExtent SPHEROIDAL GALAXY 635 1997AJ 113. .6341 0o parison sequencefromSgrmembers butitisarelatively range 19.5-21.0;andamainsequencereferenceregionfrom limits ontheBjmagnitudeonly.Twosamplesweregener- bution. Unfortunately,theRsurveyplatesavailablefor region usedtoconstructtheoverallisoplethmapisonly the 5degreeseparationofsurveyfields—givinga1de- isopleth mapsarethenconstructed bysubtractingasuitably to removethevarying,mainlyforeground,Galacticcontri- regions ofcolor-magnitudespacehadtobedefinedinorder marginally affectedbyvignetting. gree overlapbetweenplates—ensuresthatthemajorityof metrical vignettingispresentbeyondaradiusof2.7degrees, UKST IllaJskysurveyplates.Althoughsignificantgeo- tude of^21.5,roughlyamagnitudeabovetheplatelimit from IGI-I. from theexcessofmain-sequencestarswhicharemembers the originalsurveyofredclumpstars.Isoplethmapsderived tarius dwarfusingmainsequencestarsthanwasobtainedin than redclumpstarsandtheyalsooccupyaregionoftheHR full depthofthesurveyplates.Starsnearmainsequence members inthereferenceregion, ^25%.TheindividualSgr fields ofinterestdonotgodeepenoughtorealistically center oftheSagittariusdwarf(IllaJskysurveyplates)are of theSagittariusdwarfonfourUKSTfieldsEast obtain amoresensitivedetectionoftheextentSagit- ground Galacticstars;thesetwoeffectsmakeitpossibleto diagram whichissignificantlylesscontaminatedbyfore- turnoff intheSagittariusdwarfaremuchmorenumerous However, furtheroutalongitsmajoraxisinthedirection down tothemainsequenceturnoffofSagittariusdwarf. Bulge, thenumberdensityandstronggradientofGalactic center definedbyIGI-I. the directiontowardsGalacticplane,theseRRLyrae the continuationofisodensitycontoursfoundbyIGI-I.In Mateo etal.1996fromtheirM55observations),butsimply what isseennotclumpytidaldebris(aswassuggestedby them tobemembersoftheSgrdwarf.Thesurfacedensity scaled, andsmoothed versionofthecomparison mapfrom small fraction,<5%,compared totheproportionofSgr 21.5-22.5. Clearly,thereissome contaminationinthecom- ated foreachplate:acomparisonregioninthemagnitude selection ofcomparisonregionsincolor-magnitudespaceto sample theSgrmainsequenceturnoffregion.Thisrestricted shown inFig.1,superimposedontheoriginalisoplethmap away fromtheGalacticplane,itispossibletoprobe and isabroad,flatdistribution:compellingevidencethat these RRLyraestarsincreasesawayfromtheGalacticplane RR Lyraestarswhichhavedistancemodulishow from theoutermostcontourreportedinIGI-I.Furthermore,a located at(/=8.8°,-23.0°),approximately3°away of memberstheSagittariusdwarfspheroidal.Thisfieldis RR Lyraestars,whichhavemagnitudesandcolorsexpected stars precludeusingphotographicsurveyplatestoprobe 5X° fieldcenteredat(/=3°,b=—I)hasdetected313 study bytheDUOmicrolensingteam(Alard1996)ina 636 IBATAETAL:THESURVIVALOFSAGITTARIUS stars arefounduptoZ?=—4°,morethan10°awayfromthe As fortheredclumpisoplethmaps,suitablecomparison The mainsequenceturnoffofSgroccursataBjmagni- At lowGalacticlatitudesinthedirectionof © American Astronomical Society • Provided by theNASA Astrophysics Data System -2 -2 -2 We cautionhowever,thattoobtain thesemeasurementsof roidal hasbeendetectedoveranareaof22X8degreesin rived frommainsequencestarsprovideadetection plane isconsistentwiththissymmetry,althoughstronger lack ofdeepphotometriccalibratorsandcomplexsystematic maps inastraightforwardquantitativefashion,including present thatprecludecombiningallfouroftheindividual tion inrandomnoise.Therearevarioussystematiceffects clump isoplethmaps,enableustoprobemuchfurtherout intervals are—1arcmin)comparedtotheearlierred bers ofSgrmainsequencemembersperunitarea(contour noise atamanageablelevel.However,themuchlargernum- tailed structureofSgrandtheneedtokeepquantization min, isacompromisebetweenthedesiretoprobede- mostly belongingtothemainsequence. limits, typicallyrevealsanimagedensityrangingfrom10 the desiredsignal. map (only),atascaleof^30arcmin,isusedtosuppress the referencemap.Theextrasmoothingofcomparison tween regionsof color-magnitudespacewhich isolatesthe ent selectioncriteria,andmost involvingadifferencebe- patch togetherdifferentdatasets, eachobtainedwithdiffer- the extentandshapeofSagittarius dwarf,wehavehadto includes thepositionsoffour globularclustermembers. extent, encompassingallpreviousmeasurements.Thisarea ent fieldsstudied.Alongtheminoraxis,ourisoplethsde- densities ofRRLyraeandmainsequencestarsinthediffer- density nearM54.Isoplethsconstructedfromeithermain- isopleth inFig.1correspondstoasurfacedensityof^1 the Esideofitsmajoraxis.Fieldsfurtheroutdonotshow dwarf spheroidalhasbeendetecteddowntob=-26°along bined isoplethmap. has tobegreaterthantheapparentboundariesincom- extent ofSgr,althoughwecautionthatthefullSgr maps areaccurateenoughtorevealthegeneralshapeand Fig. 1donotjoinupsmoothly.Inspiteofthesecaveatsthe help toexplainwhytheindividualisoplethmapsoutlinedin variation inplatelimitingmagnitude,makeselectingacon- edge. Oftheseonaveragearound10%willbeSgrmembers, arcmin atoneplateedgeto20theopposite example, countingallstellarimagesfromB=18totheplate 1 =3°to/=11°degrees.Thus,theSagittariusdwarfsphe- statements awaitbettercalibrationoftherelativesurface (/=5.6°,Z>= -14.0°).ThedetectionoftheDUORRLyrae sequence orredhorizontalbranchstarsareconsistentwitha arcmin, whichisafactor—10belowthepeaksurface any significantdetectionofstarswhicharelikelytobemain at the±0.1magnitudelevel.Thiscoupledwithnatural field effects(inadditiontovignetting)presentontheplates along bothmajorandminoraxisdirections,duetothereduc- additional quantizationnoisebeingaddedinquadratureto stars intheSagittariusdwarfwithin—5°ofGalactic stant latitudedrawnthroughthelocationofM54 system whichistofirstordersymmetricaboutalineofcon- sequence membersoftheSagittariusdwarf.Thelastdetected sistent sampleofSgrmainsequencestarsverydifficultand } The effectiveresolutionoftheisoplethmapsat^10arc- The Galacticgradientinthesefieldsisverylarge.For Combining allthisstarcountinformation,theSagittarius 636 1997AJ 113. .6341 7 7 jected ellipticities^0.3(withtheexceptionofUrsaMinor, Rjim, againalongtheminoraxis.Thecontoursshownhere has beenestimatedbyIGI-IIfromtherelativefrequencyof Way (Irwin&Hatzidimitriou1995,theirTable4—Carina, brightness radiusisalowerlimittothecharacteristic red clumpintheCMDrepresent thecore-He-bumingphase be obtainedfromthemagnitude spreadofthemorenumer- to theline-of-sightextentof^8kpc(foranassumedmean tance modulusoftheSagittariusdwarfyieldsanupperlimit tinction andapossiblerangeinRRLyraemetallicitiesthe with (j=0.3mag.Thisdispersioncontainscontributions Alard (1996)findsthattheextinction-correctedmagnitude may beusedtogiveanupperlimittheline-of-sightdepth. tudes oftheRRLyraestarsdetectedinSagittariusdwarf who deriveL—2X10. the RHB,givingLy—10L,andbyMateoetal(1996), Draco, LeoH,andUrsaMinorallhavec—0.5). typical valuederivedforthedSphcompanionstoMilky of aKingmodelfit,r(see,e.g.,Binney&Tremaine1987) fallen tohalfofitscentralvalue,andthelimitingradius, be determinedfromtheisophotalcontoursareradius Indeed, theSagittariusdwarfisasflatmostflattened high frequencystructureintheisophotalcontoursismar- have derivedarethereforenecessarilyuncertain.Addition- distribution asanabsoluteupperlimittothedifferentialdis- from photometricerrors,uncorrectedlocalvariationsinex- and thelimitingradiuswillapproximatetidalradius, give valuesofRhb—1.25°,andThehalf- along theminoraxisatwhichsurfacebrightnesshas elliptical galaxies. which hasellipticity—0.5,Irwin&Hatzidimitriou1995). other dSphcompanionstotheMilkyWay,whichhavepro- dwarf hassignificantlyhigherprojectedellipticitythanthe corresponding toanellipticityof—0.7.ThustheSagittarius ginal. Realsubstructureisnotphysicallyplausible,giventhe of intermediate-massand/orintermediate-metallicity stars ous redclumpstarsoverasmall field.Starsbelongingtothe distance of25kpc). dispersion. However,interpretingtheFWHM(2.34o-)ofthis distribution ofRRLyraevariablesmaybefitbyaGaussian King modelconcentrationparametergivesc—0.5.Thisisa r. UsingtheratioRub/R]^toestimatecorresponding short internalcrossingtimes. ally, itshouldbenotedthatthestatisticalsignificanceof and particularlythenormalizationofcontoursthatwe 637 IBATAETAL.\THESURVIVALOFSAGITTARIUS Sagittarius dwarfonlyinastatisticalway.Theshape,extent, sample. Formally,theseeffectsexplainthewholeobserved (e.g., Chiosiet al. 1992).Someintrinsicspread inthered yo o 0 t 2.2 SurfaceBrightnessProfileoftheSagittariusDwarf Field Stars.Thedispersionamongtheapparentmagni- The totalluminosityoftheSagittariusdwarfspheroidal The mostrobust,model-independentquantitiesthatcan The apparentflatteningoftheSagittariusdwarfis—3:1, Another estimateoftheline-of-sight depthatapointcan © American Astronomical Society • Provided by theNASA Astrophysics Data System 2.3 DistanceandThree-dimensionalShape 2.3.1 Line-of-sightdepth lation intheSagittariusdwarfspheroidal.Thefittedmodeldescribed The narrowlocalmaximumatV=18.25correspondstotheredclumppopu- roidal (thedataarefromthecomparisonfieldofSarajedini&Layden1995). bined modelissuperimposedonFig.2;theGaussianhasa have extractedstarsoftheappropriatecolorandmagnitude Nevertheless, eveninamixedage,abundancepopu- clump isexpected,eveninauniformage,abun- text issuperimposed. Fig. 2.Themagnitudedistributionofstarsinthecolorstrip commonahty ofdistance,radial velocityandcelestialcoor- of 0.1mag,thecorrespondingdispersioninlinesight ing aconservativedispersionof0.04mag,givingFWHM limit totheintrinsicdispersionindistancemodulus.Adopt- photometric error(<7=0.1nearU=18),soitisanupper dispersion of0.04mag. maximum-likelihood fitofthe(unbinned)datatothiscom- modeled intherange18.0
Galactic latitude (degrees) Galactic latitude (degrees)
Fig. 8. The run of mean Galactocentric radial velocity along the major axis of the Sagittarius dwarf spheroidal (assumed to be parallel to a line of constant Galactic longitude) is shown in the left-hand panel. The right-hand panel presents the velocity dispersion profile along the major axis. model would allow iteration towards a best-fitting solution. same orbit. Numerical simulations (see Sec. 4 below) show However, this approach is beyond the scope of the present that Galactic tides force a satellite galaxy into a prolate paper. shape, and at pericenter the satellite’s longest axis is aligned The present velocity and distance profiles across the ma- along its orbit. However, this alignment with the orbit is only jor axis of the Sagittarius dwarf probe the projected velocity approximate, since the internal self-gravity in the dwarf will and distance to the center of mass of the dwarf as it proceeds act to decelerate stars that lead the center of mass to lower along its orbit. We explore an assumption which greatly sim- energy orbits, and to accelerate stars that trail the center of plifies the orbit detemination: that stars in the Sagittarius mass to higher energy orbits. The prolate dwarf is thereby dwarf can be regarded as test particles which all move on the rotated about its center of mass such that the leading edge drops towards the Galactic center (see, e.g., Oh et al 1995). However, this is a small effect. The self-gravity also affects the major axis velocity profile. In the limiting case, the Sag- ittarius dwarf could be considered as a rigid, tumbling bar, in which the apparent velocity at each point along the major axis is the sum the projected velocity of the center of mass and the local streaming velocity. We consider the validity of our approach in Sec. 5.3 below, where we show the likely amplitude of the failures of our simplifying assumption is small. The mean velocity profiles of our two extreme models turn out to be similar, which suggests that the velocity gra- dient across the dwarf is primarily determined by the poten- tial of the Milky Way rather than by the rigidity of the dwarf. Following Johnston et al (1995), we model the Milky Way galaxy by the sum of three rigid potentials, with the disk component described by a Miyamoto-Nagai model (Miyamoto & Nagai 1975)
^^disk ^öisk= (2) (R2 + (a+ ^z2 + b2)2)112’
4 5 6 7 the combined halo and bulge by a spherical Hemquist poten- Galactic longitude (degrees) tial (Hemquist 1990), ^ _ _ GM sphere Fig. 9. Limits on minor axis rotation in the Sagittarius dwarf. The three (3) panels show different Galactic latitude slices through the velocity data pre- ^ sphere (r+c) , sented in Tables 2(a), 2(b) and Ibata & Gilmore (1995a), their Table Bl. Straight-fine fits provide no evidence for a gradient in velocity as a function and the dark halo by a logarithmic potential of Galactic longitude in these three fields. The adopted major axis is aligned with the Galactic coordinate fine /= 5 °. 'l'halo^halo l0g(/-2 + ri2). (4)
© American Astronomical Society • Provided by the NASA Astrophysics Data System ^r1 KOoo 647 IB ATA ET AL.: THE SURVIVAL OF SAGITTARIUS 647
^0 r-
Galactic latitude (degrees) Galactic latitude (degrees) ^ I ^ ! ! I I I , I I Fig. 10. The orbit of a test particle which best fits the kinematic data of the -50 0 50 Sagittarius dwarf in the Milky Way potential described in the text is shown x (kpc) projected on the available data. The left hand panel shows the projected velocity of the orbit along the lines of sight to the fields which determine the fit. Formally, this orbit is an acceptable description of the data, despite the Fig. 11. The orbit of the Sagittarius dwarf, integrated over 1 Gyr, is shown systematic difference between the model and the data. The right hand panel in the x-z plane. The “star” symbol represents the present position of the compares the heliocentric distance of the guiding center of the orbit fit to the Sun, while the open ellipse (drawn to scale) gives the present position of the velocity data to the heliocentric distance data presented in Fig. 4. The mod- Sagittarius dwarf spheroidal. The radial period of this orbit is 0.76 Gyr. eled orbit is an acceptable fit to these distance constraints (reduced X2= 1.8). orbit in the above potential can be made to pass through the data points better than this. The right-hand panel compares In these expressions, R and z are in cylindrical coordinates, the orbit to the heliocentric distance measurements presented while r is the radial distance in spherical coordinates; 2= n lo in Fig. 4. This fit is acceptable (reduced x 1-8). Note that Mdisk=1.0X10 Mo, Msphere=3.4XlO M0, i;halo=128 the distances to the RR Lyrae stars at low latitude (Alard km/s, a = 6.5, b - 0.26, c = 0.7, and d = 12.0, all in kpc. 1996) discussed above in Sec. 2.3.2 are not included; we We will first obtain constraints on the orbit of the Sagit- return to this point in Sec. 5.3 below. Figure 11 shows the tarius dwarf assuming that there are no systematic internal x-z shape of the best-fit center-of-mass orbit, integrated for streaming motions (essentially neglecting rotation and ex- 109 years. The radial period of this orbit—defined as the time pansion) or figure rotation (tumbling). With this, the appar- taken from apogalacticon to perigalacticon and back—is ent mean velocity at each point along the major axis of Sgr 0.76 Gyr. may be modeled by the true space velocity of a test particle We note that this orbit requires that the end of Sgr farthest at the corresponding point on the orbit. from the Galactic plane is closest to the observer, by an The test particle is started off at the present position of amount of —2.5 kpc per ten degrees along the major axis. M54, using 25 kpc as the value for the distance of the Sag- This prediction is testable by direct determination of the dis- ittarius dwarf at this point. We explicitly assume zero trans- tance to each end of the Sagittarius dwarf. Perhaps the most verse velocity perpendicular to the direction of elongation of precise such determination, given the large numbers of iden- Sgr, as in Sec. 2.7 above. The other two components of space tifiable member stars, will be comparison of the mean appar- motion are specified, by initial guesses. The equations of ent magnitude of the RR Lyraes in Sgr at either end of the motion are integrated along the orbit using a Runge-Kutta major axis. A large sample is already available from the scheme. The “amoeba” routine of Press et al (1986) is used DUO RR Lyraes, which are near the Galactic Plane and, to refine successive guesses of the initial velocities, subject thus, the more difficult to detect. Note further that this best- to the constraint that the component of the velocity of the test fit orbit gives rise to a proper motion of 2.0 mas/yr, towards particle is consistent with the mean radial velocity in the the Galactic plane, in excellent agreement with the measured observed fields presented above. 1 value of 2.1 ±0.7 mas/yr in this direction. Since the data are binned, it is natural to use the x sta- The mean velocity at each point along the major axis of tistic for this comparison: the Sagittarius dwarf is made up from the sum of the velocity of a test particle at that place moving in the Galactic poten- X ~ (f/ F j i) /Svi, (5) fields mo( e tial on the same orbit as the center of mass of the Sagittarius dwarf, together with any contribution from systematic mo- where u, is the mean velocity in field /, Svi is the standard tions associated with the internal dynamics of the Sagittarius error on and i>mo(iel is the projected velocity of the test dwarf. There are two possible such systematic motions asso- particle at the point in its orbit that corresponds to the posi- ciated with the Sagittarius dwarf itself—internal streaming tion on the sky of field i. and bulk figure tumbling. 2 The orbit of the test particle which minimizes x as de- The numerical calculations summarized in Sec. 4 below fined above, is shown in Figs. 10 and 11. The left-hand panel show that the effect of Galactic tides on a dwarf spheroidal is in Fig. 10 shows the radial velocity data in the b-v plane to generate flattening, and to orient the dwarf spheroidal such together with the test-particle orbit. This fit is formally ac- that its major axis is aligned along its orbit, at least near 2= ceptable (reduced x 1-7), in spite of the systematic differ- perigalacticon for elliptical orbits. That is, a tidally-distorted ences between the model and the mean velocity data. The dSph galaxy is effectively a bar which is spin-orbit locked curvature of the data near b=-15° is so extreme that no near perigalacticon. In this case, no bulk figure tumbling
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1997AJ 113. .6341 the apparentsystematicdeviations. Noorbit,subjecttothe formally anacceptablefittothe extant kinematics,inspiteof velocity curvenearb=—15°evident inFig.10. km/s/degree
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