Sagittarius stream 3-d kinematics from SDSS Stripe 82

3 MB·English

by  Sergey E. Koposov

Draftversion January31,2013
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
SAGITTARIUS STREAM 3-D KINEMATICS FROM SDSS STRIPE 82
Sergey E. Koposov1,2, Vasily Belokurov1, N. Wyn Evans1
Draft version January 31, 2013
ABSTRACT
Using multi-epochobservationsofthe Stripe 82 regiondone by SloanDigitalSky Survey,we measure
precise statistical proper motions of the stars in the Sagittarius stellar stream. The multi-band pho-
3
tometry and SDSS radial velocities allow us to efficiently select Sgr members and thus enhance the
1
proper motion precisionto 0.1masyr−1. We measure separately the proper motion of a photomet-
0 ∼
rically selected sample of the main sequence turn-off stars, as well as of a spectroscopically selected
2
Sgr giants. The data allow us to determine the proper motion separately for the two Sgr streams in

n the SouthfoundinKoposov et al.(2012). Togetherwiththe precisevelocitiesfromSDSS,ourproper
a motion provide exquisite constraints of the 3-D motions of the stars in the Sgr streams.
J
Keywords: Galaxy: halo, stars: kinematics, methods: statistical, surveys
9
2
1. INTRODUCTION Carlin et al. (2012) have provided the first measure-
] ments of the proper motion of the Sgr trailing tail.
A The disintegrating Sagittarius (Sgr) dwarf galaxy re-
Theytookadvantageofarchivalphotographicplatedata
mains a riddle, wrapped in a mystery, inside an enigma.
G in some of Kapteyn’s Selected Areas which provides a
Large scale photometric surveys, such as the Two Mi-
h. cronAll-Sky Survey(2MASS) andthe SloanDigitalSky 9400′yea4r0′bfiaesledlisnceo.veTrhinegylodcearitvioenpsroonpetrhemtortaiiolinnsgftoarilfobuer-
p Survey (SDSS) have now revealed the structure of the twe×en 70◦ and 130◦ from the Sgr core. However, the
- tidal tails of the Sgr over more than 2π radians on the
o Sky (Majewski et al. 2003; Belokurov et al. 2006). By numberofstarsineachfieldremainsmodest( 15 55),
∼ −
r and so the precision of the proper motion measurement
tallyingallthestellardebrisinthestreamsandremnant,
st we now know that the progenitor galaxy had a luminos- is still quite low ( 0.2 0.7 mas yr−1).
∼ −
[a ity of ∼ 108L⊙, comparable to the present day Small moHteioren,swoef twhielltprauirlsinuge satrdeiaffmereinntrtoaucgkhltyotohbetasianmperoapreear
Magellanic Cloud (Niederste-Ostholt et al. 2010, 2012).
of sky. As part of a project to detect supernovae, the
1 Theingestationofsuchalargeprogenitor,togetherwith
Sloan Digital Sky Survey scanned a 290 square de-
v its dismantling under the actions of the Galactic tides,
∼
gree region on the Celestial Equator, known as Stripe
0 can provide us with a wealth of information about both
7 the Galaxy and the Sgr, if we can only decode it. 82 (e.g., Abazajian et al. 2009). Proper motions
0 Radial velocities, and sometimes metallicities and can be derived by matching objects between the 80
∼
7 chemical abundances, are now known for many hun- epochs (Bramich et al. 2008) obtained over time period
1. dreds of stars in the Sgr tails (e.g., Majewski et al. of ∼ 7 years,whilst the co-added opticaldata is roughly
2magnitudesdeeper thanasingleepochSDSSmeasure-
0 2004;Monaco et al.2007;Chou et al.2007,2010). There
ment. Althoughthebaselineissmallsotheprecisionofa
3 are also 10 globular clusters associated with the Sgr
∼ measurementofpropermotionofasinglestarisstilllow,
1 tails (Law & Majewski 2010b). This rich mosaic of
wecantakeadvantageofthelargenumberofSgrtracers
: positions and velocities of Sgr tracers has proved sur-
v togetahighprecision( 0.1masyr−1)measurementfor
i prisingly difficult to understand. Although there is no the proper motion of th∼e ensemble.
X shortage of Sgr disruption models in the literature (see
The paper is arranged as follows. Section 2 describes
e.g., Helmi 2004; Law et al. 2005; Johnston et al. 2005;
r theextractionofpropermotionsforstarsfromtheStripe
a Fellhauer et al. 2006), they all have significant short-
82datausingbackgroundquasarstoprovideanabsolute
comings, and fail to reproduce a substantial portion of
referenceframe. Section 3 shows how to identify the Sgr
the datasets. The most successful recent attempt is
stars in Stripe 82, where they occupy a distinctive niche
by Law & Majewski (2010a), though they do not ex-
in magnitude and radial velocity space. Section 4 dis-
plain the striking two stream morphology seen in the
cusses our modelling of the proper motions using both
SDSS data (Belokurov et al. 2006; Koposov et al. 2012;
photometric and spectroscopic samples. We extract the
Slater et al.2013). Additionallytheyadvocatetheuseof
proper motion for both the bright and faint Sgr streams
a triaxial halo for the Galaxy with minor axis contained
identifiedbyKoposov et al.(2012). Insection5,wecom-
with the Galactic plane, which is unattractive on other
pare our proper motions both with the earlier work of
grounds (e.g. Kuijken & Tremaine 1994). Given the im-
Carlin et al. (2012) and with the simulation data.
passe, it is natural to look to proper motions of the Sgr
stream as so as to obtain a clearer picture of its space 2. STRIPE82PROPERMOTIONDETERMINATION
motion.
Stripe 82 has already been the subject of numerous
studies. Its multi-epoch and multi-band imaging al-
1Institute of Astronomy, Madingley Road, Cambridge CB3
lowsstudy ofthe variableskyandidentificationofmany
0HA,UK
2Moscow MV Lomonosov State University, Sternberg Astro- kindsoftransientphenomena(seee.g.,Sesar et al.2007;
nomicalInstitute, Moscow119992, Russia Becker et al. 2008; Kowalski et al. 2009; Watkins et al.
2 Koposov, Belokurov & Evans
2009). The Stripe 82 dataset has also been used to de- the positional offsets is
rivepropermotionsbyBramich et al.(2008). Thislight-
P(∆t,∆ ,µ,σ,f)=fR(∆)+
motioncataloguewassubsequentlyexploitedtobuildre- 0
|
duced proper motion diagrams (Vidrih et al. 2007) and 1 f ∆ ∆ µt 2
0
analyse kinematical properties of Galactic disk and halo − exp − − (2)
populations (Smith et al. 2009a,b, 2012). √2πσ −(cid:18) σ (cid:19) !
Even so, the proper motion measurements pioneered
where µ is the proper motion of the star, t is the date
by Bramich et al. (2008) can be improved. For the bulk
of the observation, f is the fraction of outliers, and σ is
proper motion of Sgr, we are interested in the statistical
thescatteraroundthe linearrelation,whilstR(∆)isthe
properties of a large ensemble of faint tracers, so proper
rectangular function to account for the outliers.
motions with the smallest possible systematic errors are
Theresultinglikelihoodisthenminimizedwithrespect
highlydesirable. TheoriginalcataloguebyBramichdoes
to the 4 parameters µ, ∆ , f, σ with the error-bars de-
not provide proper motions for stars fainter than r 0
∼ termined from the Hessian at the minimum. Repeating
20.5 and is known to have some noticeable systematics.
this procedure for the offsets in declination gives us the
Since the Sgr stream has a very large number of tracers
propermotionsandtheirerrors,µ ,σ , µ ,σ forall
in Stripe 82 (Watkins et al. 2009), we do not require a α µ,α δ µ,δ
the sources with a spectroscopic or a photometric QSO
proper motion measurement for every star, but rather
nearby.
need small systematic errors and well understood error-
barsforanensemble. Forthispurpose,itmakessenseto
2.2. Systematic Errors in the Proper Motions
measure the proper motions relative to quasars (QSOs).
Stripe 82 has a number of both spectroscopically After performing the computation of the proper mo-
and photometrically identified QSOs. In this work, we tion for individual stars, and before trying to measure
have used the catalogue of spectroscopic QSOs from thestatisticalpropermotionsforensemblesofstars,itis
Schneider et al. (2010) and the sample of photometri- importanttocheckforthepresenceofpossiblesystemat-
cally identified QSOs from Richards et al. (2009) to ex- ics, as well as to examine the accuracyof the error bars.
tract denizens of Stripe 82. The purity of the spectro- Fig.1presentssuchanassessment. Inthispaper,weare
scopic catalogue of QSOs is guaranteed – all the objects focusing on the particular part of Stripe 82, which in-
are QSOs and must have zero proper motion. How- tersects with the Sgr stream. As systematic effects may
ever, the photometric catalogue is known to have some dependontherightascension,Fig.1usesonlytheproper
contamination by stars. In order to minimize contam- motions in the right ascension range 20◦ <α<50◦.
inants, we use the cut good 1, as recommended by The left panel of Fig. 1 shows the histogramof proper
≥
Richards et al. (2009). This guarantees a small stellar motions of the spectroscopic QSOs measured relative to
contamination, certainly <5%. the photometric QSOs and normalized by the error bar
provided by our fitting procedure. The red overplotted
curveshowsaGaussianwiththecenteratzeroandunity
dispersion. The excellent match of the histogram with
2.1. Relative Proper Motions
the Gaussian curve shows that that the proper motions
GivenasampleofQSOseachwithzeropropermotion, donotpossessnoticeablesystematicoffsets,andthatthe
then for each star in the vicinity of the QSO, we may error bars on the proper motions are a faithful descrip-
determine proper motion relative to the quasar. As an tion of the precision. It is also quite clear from the plot
input catalogfor the stars,we took the Stripe 82 co-add that this is not true for proper motion measurements by
dataset(Annis et al.2011),fromwhichweselectprimary Bramich et al.(2008),whichhavenoticeablesystematics
objects, classified by the SDSS pipeline as stars. The and error underestimation. The middle panel of Fig. 1
individual source detections are taken from the Stripe shows the the proper motion of the spectroscopic QSOs
82 portion of the SDSS DR7 database (O’Mullane et al. relative to the photometric QSOs versus the g r color
2005) using only those fields having acceptable and difference. The points with error bars show the−median
good data quality flags. Matching co-added sources to propermotioninbinsofg r,whiletheerrorbarsare1.48
−
detections at individual epochs is done with the 0.5 arc- times the median absolute deviation of proper motions
sec radius using the Q3C module for the PostgreSQL within a corresponding g r bin. Since all the points lie
−
database (Koposov & Bartunov 2006). This procedure within1σ ofzero,we concludethatthe color-dependent
makes the catalogue incomplete for high proper motion terms in the proper motions are negligible down to the
objects (with proper motions & 100 mas yr−1), but we precision 0.1 0.2 mas yr−1. This holds true at least
−
are not interested in such objects in our current study. within the color range 0.2 . g r . 1, which is the
− −
Then, for each pair (star, QSO) observed multiple color-range applicable to 99% percent of the photomet-
times by SDSS within one field, we analyse the posi- ric QSOs. Later we will see that the proper motion of
tional offsets and errors. For the right ascension, these the Sgr streamdetermined fromthe photometricsample
are defined via with g r 0.3, g i 0.3 and the spectroscopicsample
− ∼ − ∼
withg r 0.5,g i 0.75agreeeachother,furthercon-
− ∼ − ∼
firming the smalllevelofcolor-relatedsystematic errors.
∆ =α α ,
i star,i QSO,i And last, the right panel of Fig. 1 shows the precision
−
σ =(σ2 +σ2 )1/2 (1) of our proper motion measurements as a function of r-
∆,i α,star,i α,QSO,i
bandmagnitude. Thereisaconsiderablescatter,butthe
median proper motion precision of 2mas yr−1 is sig-
∼
withsimilarequationsforthedeclination. Themodelfor nificantly better than that of Bramich et al. (2008) for
Sgr Stream kinematics 3
10
0.35 this work 1.5 this work
00..2305 α GBraaumssic(0h,018) mas/yr] 01..50 s/yr] 68 Bramich08
000...211005 [<µ>α −−010...005 [maσµ,α 4
0.05 −1.5 2
0.00 10
0.35 1.5
00000.....2231100055 δ [mas/yr]<µ>δ −−00101.....05050 [mas/yr]σµ,δ 468
2
0.05 −1.5
0.00
−10 −5 0 5 10 −0.4 −0.2 0.0 0.2 0.4 15 16 17 18 19 20 21
µ/σ (g r) (g r) r [mag]
µ − 1− − 2
Figure 1. Left panels: Normalized histogram of proper motions of spectroscopic QSOs measured relative to the photometric QSOs
normalizedbytheerrorbarprovidedbythemodeling. RedlinesareGaussianswithzeromeanandunitydispersion. Bluedashedcurves
showthehistogramsofthepropermotionsofspectroscopicQSOsfromBramichetal.(2008). Central panels: Themedianpropermotion
of spectroscopic QSOs relative to the photometric QSOs versus their color-difference. Right panels: The measured error bars on the the
propermotionsasafunctionoftherbandmagnitude. Theredlinesshowthemedianvaluesofourmeasurementerrors. Thebluedashed
linesshowthemedianofpropermotionerrorsfromBramichetal.(2008).
is shown in the middle panel of Fig. 2. The Sgr streams
Table 1 are clearly evident, with the brighter stream visible at
Radialvelocities anddispersions α 35◦ and fainter one at α 15◦. Since the distances
ofSgrstarsalongthestreamas ∼ ∼
tracedbySDSS. to the streams are not constantwith α, and since Stripe
82 crosses the stream at an angle, we observe a clear
Λ VGSR σV distance gradient with right ascension.
deg kms−1 kms−1 TherightpanelofFig. 2showsthe radialvelocityasa
function of right ascension for giant stars, extracted by
87.5 −85.6±2.1 13.7±2.1
92.5 −100.5±1.4 14.8±1.5 the cut log(g) < 4 using the SDSS spectroscopic mea-
97.5 −106.0±1.5 11.1±1.6 surements. This time notonly arestarswithinStripe 82
102.5 −121.0±1.0 13.0±1.0 included, but also those satisfying δ < 20◦ and lying
107.5 −132.5±1.2 13.8±1.3 near the stream ( 15◦ < B < 5◦)|. |As already found
112.5 −140.5±1.6 19.5±1.4
−
117.5 −146.7±1.5 16.7±1.4 by Watkins et al.(2009), the Sgr streamis visible at the
125.0 −154.7±2.0 19.9±2.8 V 150kms−1 3. The figure also shows the vari-
135.0 −156.8±2.8 27.2±3.3 GSR ∼ −
ation of radial velocities as the stream stars return from
large range of magnitudes. the apocenter of the Sgr orbit. As they will be needed
later, we extract the radial velocities along the stream,
3. THESAGITTARIUSSTREAM together with the velocity dispersion, by fitting a Gaus-
The Sgr stream in the southern Galactic hemisphere sian to the Sgr signal. Although Figure 2 shows the
is known to have a complex structure. Koposov et al. radialvelocitiesas afunction ofrightascension,it is im-
(2012) used main-sequence turn-off (MSTO) stars ex- portanttorealisethatweperformthesefitsinbinsalong
tractedfromSDSSDataRelease8(DR8)todemonstrate Λ,whichistheanglealongthestreamfromtheremnant,
the existence of two streams – a thicker,brighter stream measured positive in the trailing direction, as defined in
and a thinner, fainter stream displaced by 10◦. The Majewski et al. (2003)). The resulting measurements as
brighterstreamhasmultipleturn-offsaswell∼asapromi- a function of Λ are shown in Table 1.
nent red clump, whereas the fainter stream does not. Importantly, Fig. 2 demonstrates that the Sgr stream
Koposov et al. (2012) argued that the brighter stream starsarevisibleasadistinctbrightfeatureinmagnitude
was composed of more than one stellar population, in- and velocity space. Therefore, this information can be
cludingsomemetal-richstars,whereasthefainterstream usedtoselectSgrmemberstarsandstatisticallymeasure
is dominated by a single metal-poor population. their properties such as proper motion.
Here, we are primarily interested in the intersection
of the Sgr stream with the SDSS Stripe 82. Fig. 2 4. MODELINGPROPERMOTIONS
shows the density of MSTO stars extracted via the cuts As demonstrated above, in the Southern Galactic
0.25 < g i < 0.35 and 19.8 < r < 22.5 in the south- hemisphere, it is possible to achieve clean separation of
−
ern hemisphere, together with Stripe 82 demarcated by the Sgr trailing tail stars and the Galactic foregroundin
the red lines. Again, two distinct streams structures are
cSlteraiprley8v2isaibtlaer–igahtbraisgchetnesrioonneofcros3s5in◦ganthdeaEdqimuamtoerraonnde (B3ovyHeetreal.an2d00t9h;roRuegihdouettatlh.e2p00a9p;erD, ewaesounseetVaLl.SR2=01213)5aknmd−a1
crossing the Equator at a right a∼scension of 15◦. The solar peculiar velocity of (U,V,W)=(-8.5,13.38,6.49)kms−1 from
magnitude distribution of MSTO stars alon∼g Stripe 82 (Co¸skunoˇglu etal.2011)
4 Koposov, Belokurov & Evans
60 23 300
50 22
200
40
21
100
g] 30 g] 20 m/s]
, [deδ 2100 r [ma 19 V [kGSR 0
−100
18
0
−200
−10 17
−20 16 −300
60 50 40 30 20 10 0 −10 −20 60 50 40 30 20 10 0 −10 −20 60 50 40 30 20 10 0 −10 −20
α, [deg] α [deg] α [deg]
Figure 2. Leftpanel: DensityofMSTOstars(0.25<g−i<0.35,19.8<r<22)fromSDSSDR8inthesouthernGalactichemisphere. The
SDSSStripe82regionisdelineatedbytheredline. Fourhorizontalerror-barsshowtherangeofrightascensionsusedtoperformproper
motionmeasurementsinStripe82;rederror-barscorrespondtophotometricsamples,blueerror-barscorrespondtothespectroscopicsample.
Centerpanel: ThedensityofMSTOstarsasafunctionofr bandmagnitudeandrightascensionillustratingadistancegradientalongthe
stream. RightpanelRadialvelocitiesofstarsneartheStripe82. TheseareSDSSmeasurementsofstarswithlog(g)<4andlocatednearthe
Sgrorbit−15◦<B<5◦andnearStripe82|δ|<20◦. TheSgrstreamisobviousatradialvelocitiesof−200kms−1<VGSR<−100kms−1.
Thechangeofradialvelocityalongthestreamisalsoquiteprominent.
Figure 3. The brightSgr stream component at 25◦ <α<40◦ Left and Middle panels: Greyscale shows the 2D distributionof proper
motions andmagnitudes, whiletheredcontours show thetotal error-deconvolvedGaussianmixturemodel. Thecomponent ofthemodel
correspondingtothestreamisshowninblue. Rightpanel: 1Dprojectionontotheapparentmagnitudeaxisoftheleftpanel. Thehistogram
of the data is shown in black, red curve shows the Gaussian mixture model of the luminosity function, whilst the model of the stream
contributionisshowninblue.
both apparent magnitude and the radial velocity space. motion of an ensemble of stars belonging to the stream
However, even for high-confidence stream members, the canbeobtainedthroughsimplebackgroundsubtraction.
uncertainty of individual proper motion measurements Alternatively, the overall stellar distribution in the
( 2 3masyr−1;seeSection2)iscomparableorhigher space of observables can be modeled, yielding the di-
∼ −
than the expected tangential velocity of the stream ( rectcontributions of the Galaxy and the stream. If such
1 3mas yr−1; Law & Majewski 2010a). It is therefor∼e modelscanbecastinthe3Dspaceofpropermotionand
cr−ucialtocombinethesignalfromasmanystreammem- magnitude or radial velocity, it is feasible that a supe-
bers as possible to beat down the noise. Koposov et al. rior measurement of the proper motion can be achieved
(2010) have shown that, for the regions of apparent as the contribution of the background to both system-
magnitude and radial velocity space dominated by the atic and random noise will be reduced. Unfortunately,
stream,anaccuratemeasurementofthesystemicproper adequate analytical models of the Sgr stream and the
Sgr Stream kinematics 5
Figure 4. Left and Middle panels: Greyscale shows 2D histograms of proper motions and the GSR radial velocities corrected for the
stream’s radial velocity gradient for all stars with SDSS spectra and log(g) <4 in Stripe 82. Red contours show the error-deconvolved
Gaussian mixture model of the data. Right panel: The histogram of the radial velocities. The Gaussian mixture model of the radial
velocitiesisshownbytheredcurveandthestreamcomponent inblue.
Table 2
Propermotionsmeasurements
Field α1 α2 µαcos(δ)b µδb σµ,αcos(δ) σµ,δ µlcos(b)b µbb σµ,l σµ,b
deg deg mas/yr mas/yr mas/yr mas/yr mas/yr mas/yr mas/yr mas/yr
FP1 5.00 22.00 0.05 -2.51 0.11 0.11 0.10 -2.51 0.11 0.12
FP2 25.00 40.00 -0.50 -2.24 0.09 0.09 0.81 -2.15 0.09 0.11
FP3 42.00 52.00 0.19 -2.36 0.14 0.14 1.87 -1.45 0.11 0.15
FS4a 14.00 50.00 -0.42 -2.65 0.11 0.13 1.07 -2.46 0.11 0.11
Note. — α1, α2 columns denote the edges of theboxes inrightascensioninStripe82used toperformthe
propermotionmeasurements
b notcorrectedfortheSolarreflexmotion
a Spectroscopic sample
Table 3
PositionsandvelocitiesoftheSgrstream.
Field Λ B X Y Z Distance U σU V σV W σW
deg deg kpc kpc kpc kpc kms−1 kms−1 kms−1 kms−1 kms−1 kms−1
FP1 89.5 10.0 -15.0 9.6 -22.6 25.4 194 14 -46 14 23 8
FP2 106.1 0.6 -23.5 5.2 -24.4 29.1 290 12 33 14 -20 9
FP3 118.6 -6.7 -31.4 0.4 -25.0 33.9 270 17 -57 24 -48 16
FS4a 105.6 0.8 -23.2 5.3 -24.3 28.9 308 15 -16 18 -40 11
Note. — Λ,B,X,Y,Zcorrespondtothecenters ofthefieldswherethepropermotionsaremeasured.
aSpectroscopicSample
Galaxy are not readily available. Therefore, we choose ments (as well as the missing data) are naturally incor-
to approximate these distributions by a sum of multi- poratedinto the model. Secondly, there exists a guaran-
dimensionalGaussians. This so-calledGaussianmixture teed fast convergence procedure – the Expectation Min-
is a well known semi-parametric technique widely used imisation (EM) algorithm (Dempster et al. 1977). In
tomodelmulti-dimensionaldatasets(McLachlan & Peel thiswork,wehaveusedtheextreme-deconvolutionpack-
2000). Gaussian mixtures have several key properties age, the open-source Gaussian mixture implementation
that make the model-fitting straightforward and fast. by Bovy et al. (2011).
First, the uncertainties associated with the measure-
6 Koposov, Belokurov & Evans
Gaussian. When applying the extreme deconvolution,
we force the covariancematrix for the Sgr componentto
60 be diagonal, e.g. assuming zero covariance between the
MSTO
propermotionandthe apparentmagnitude. The proper
50 spec motions for the stream stars were not corrected for the
solar reflex motion. The errors in the proper motion
measurement, i.e. the uncertainties in determining the
40
centreoftheGaussianrepresentingtheSgrStream,were
determined either from bootstrap procedure or from the
30
Hessian matrix of the likelihood function.
g] Fig. 3 shows the data analysed as well as the best
de 20 fit Gaussian mixture model of the main Sgr stream
[δ 25◦ <α<40◦. It is reassuring to see that the Gaussian
mixturemodelwasabletodescribethedatadistribution
10
adequately. The resulting measurements of the proper
motion together with the uncertainties are given in Ta-
0
ble 2. Among 7000 stars analyzed, according to the
∼
model, 4000 stars belong to Sgr, while 3000 stars
∼ ∼
−10 belong to the background/foregroundpopulation.
−20 4.2. Spectroscopic sample
60 50 40 30 20 10 0 −10 −20
α [deg] The number of members of the Sgr stream with SDSS
spectroscopic measurements is significantly smaller than
Figure 5. Sgr streams in the South with the proper motion
the number of MSTO stars used in the previous section.
vectors overplotted. The red vectors indicate the measurements
performed using the photometrically selected MSTO stars at Despite this, knowledge of the radial velocities and sur-
three different locations along the Stripe. Blue vector is for facegravitiesallowsustohavemuchpurersamplesofthe
the spectroscopic sample. The measured error-bars of proper Sgr streammembers. In this section,we performa com-
motions are shown in pink. The proper motion vectors have
plementary measurement of the stream’s proper motion
been corrected for the solar motion, assuming the distances from
Koposovetal.(2012)andVLSR=235kms−1 andpeculiarveloc- using spectroscopic members only.
ityfromCo¸skunoˇglu etal.(2011). Tothisend,weselectallstarswithspectrainStripe82
lyingat14◦ <α<50◦ (welabelthisareaFS4andshow
4.1. Photometric sample itwiththe blue error-baronFig.2)andclassifiedby the
As evident from the dissection of the SDSS dataset SDSS spectroscopic pipeline as giants log(g) < 4. De-
shown in Fig. 2, main sequence turn-off stars (MSTO) spite very wide range of selected right ascensions, most
are the most numerous Sgr stream specimens available of the Sgr members with spectroscopy in the Stripe 82
to carry out the proper motion analysis. The typical region located at the center of the bright stream, at
density ofthe MSTO starsin the streamis 150starsper α 30◦. According to the right panel of Fig. 2, the
∼
square degree, compared to the foreground density of Sgr stream’s radial velocity changes along the Stripe.
around 50. Stripe 82 slices through both the bright and Therefore, to ease the modeling, we subtract the vari-
thefaintstreamsatanangle,resultinginaslightlytilted ation of the radial velocity centroid from the data using
bi-modaldistributioninthe plane ofrightascensionand the measurements presented in Table 1. The resulting
r band magnitude. A faint Eastern wing to the bright radialvelocityisapproximatelyconstantasafunctionof
Stream can also be discerned at higher right ascension RA,andthe measurementsofpropermotionsandradial
(see middle panel of Fig. 2). velocities µ ,µ ,V˜ =V V (Λ) can now
α δ GSR GSR model,GSR
−
Motivated by the evolution of the distance and the be represented by a mixture of Gaussians.
structure of the stream along the Stripe, we split the We run the extreme deconvolution on the sample of
MSTO sample into three parts: the faint stream at 1500 stars and find that 3 Gaussian components are
5◦ < α < 22◦, the bright stream at 25◦ < α < 40◦, and ∼sufficientto describethe dataset. We followthe strategy
the Eastern wing of the bright stream at 42◦ <α<52◦ outlined in Section 4.1, with the difference that the co-
(welabelthosefieldsasFP1,FP2,FP3respectively;they variancematrix of the Gaussianrepresentingthe stream
are shown by red error-bars on Fig. 2). In each of the component is now set free. Fig. 4 shows the density dis-
three right ascension bins, the overall 3D distribution tribution of the data, together with the best-fit Gaus-
of MSTO stars in the space of (µ ,µ ,r) was modeled sian mixture model for the Stripe 82 stars with spectra.
α δ
with a mixture of 5 Gaussians, The initial guess for the By comparing the grey-scale density with the red con-
free parameters was obtained by running the K-means tours, we can confirm that the model has captured the
algorithm (MacQueen 1967). The choice of number properties of the dataset reasonably well. The last line
of Gaussians (N ) was motivated by looking at the of Table 2 reports the values of the proper motion for
gau
cross-validated log-likelihood (Arlot & Celisse 2010) as the spectroscopic sample, and confirms that the mea-
a function of N . This initially rises as a function of surementsforthetwoindependentstreamsamplesagree
gau
N , peaks at N 5 and stays roughly constant for within 2σ. It is also particularly reassuring since Sgr
gau gau
∼
N >5, demonstrating the goodness of fit and the ab- members in the spectroscopic sample have very different
gau
sence of overfitting. We have also made sure that the colors (g i 0.75) from the Sgr members in the photo-
− ∼
objects in the Sgr stream are represented by a single metric sample (g i 0.3) of the Sgr giants. Therefore
− ∼
Sgr Stream kinematics 7
theagreementofthepropermotionsfromtwosamplesis
aproofofasmalllevelofcolor-relatedsystematiceffects. 2
The proper motion signals we have measured have a
very large contribution of the solar reflex motion. Since yr] 1
wehavedistanceestimatestotheSgrstreamintheSouth as/
m
mcoeraresuctrefdoreltsheiwshaenrde (ceh.egc.kKwohpeotshoevretthael.p2ro0p12er),mwoeticoanns [s(b) 0 LM10 δ=−20
are actually aligned with the streams. Figure 5 shows co −1 LM10 δ=0
µl LM10 δ=20
the proper motion vectors after applying the solar reflex
Koposov13
corrections. As we can clearly see, the proper motions −2 Carlin12
are indeed properly aligned with the streams and are 1
consistentwitheachother. Forthiscalculationwehave
0
still used the distance to the faint stream as given by
KinoSploastoevr eettaall..((22001132),t2h0a1t3)t,healstthroeuagmh,thtehefraeinitssetvreidaemncies s/yr] −1
3 5 kpc closer. ma −2
− [
5. COMPARISONSTOEARLIERWORK µb −3
There are many models of the Sgr stream in the lit- −4
erature, all purporting to provide the distances, veloci-
ties and proper motions as a function of position on the −5
160 140 120 100 80 60
sky (e.g. Fellhauer et al. 2006; Pen˜arrubia et al. 2010;
Λ [deg]
Law & Majewski 2010a). Although quite detailed, all
themodelsfailtoreproduceatleastsomeofthefeatures
Figure 6. ThecomparisonofthemeasuredStripe82propermo-
that we see on the sky (e.g., Niederste-Ostholt et al.
tions (red with errorbars) with both the earlier measurements of
2010;Koposov et al.2012). But,itstillinstructivetosee Carlinetal. (2012, grey with error bars) and the simulations of
where our data measurements lie relative to the existing Law&Majewski (2010a, blue,green,yellow dots). Our measure-
models. We have chosen the Law & Majewski (2010a) ment of the proper motion of the fainter of the two trailing tails
is identified by an empty red circle. Note that the datapoints
modelasacomparisonbenchmark,becauseitisarguably
fromLaw&Majewski (2010a) are selected from trailingtail only
the most comprehensive and up-to-date. and fromthe region near Stripe82 |δ|<20◦. The points are col-
There are several caveats to be borne in mind. The oredaccordingtoδsuchthatpointswithδ∼−20◦ aredarkblue,
first is related to the fact that while our central mea- pointswithδ∼20◦areorange,andpointsnearStripe82arelight-
surement in Stripe 82 at α 35◦ or (Λ,B) (100◦,0◦) green. TwoofthefieldsfromCarlinetal.(2012)liewithinStripe
82, while two other fields are ∼ 15 degrees away from the Stripe
∼ ∼
corresponds directly to the center of the trailing tail in 82.
the simulation, the fainter stream which crosses Stripe V = 235kms−1. The distances taken from the work
LSR
82 at α 15◦ or Λ,B (90◦,10◦) doesn’t have by Koposov et al. (2012), and the error on the velocities
∼ ∼
a counterpart in the simulation by Law & Majewski does not take into account any possible systematic error
(2010a). The second is related to the choice of rota- distance of 10%. Readers wishing to compare Sgr
∼
tion velocity of the Local Standard of Rest. As shown disruption models with our observations should use Ta-
in Carlin et al. (2012), the observed proper motion sig- ble3onlyforroughchecks. Apropercomparisonentails
nal is sensitive to the adopted V . The models of measuring the average proper motions within the same
LSR
Law & Majewski (2010a) have been computed and fit- area of the sky as we do and comparing these numbers
ted under the assumption of the IAU standard value of directly with our Table 2.
the V =220kms−1, while the current best estimates We also compare our measurements with the data
LSR
are slightly higher at 230 250kms−1(Bovy et al. 2009; from Carlin et al. (2012). Out of 6 fields analysed by
−
Reid et al. 2009; Deason et al. 2011). Throughout the Carlin et al. (2012), 4 have measurable Sgr signal and
paper, we adopted the value of 235kms−1. only2ofthose(SA 93andSA94)lie nearthe Stripe 82.
Fig. 6 shows our data points overplotted onto the The other 2 fields with detectable Sgr signal are 15
∼
distribution of tracers from Law & Majewski (2010a) degreesawayfromthe Stripe 82. As Figure6 shows,the
model, where we have selected tracers from the trail- error-bars of our measurements are much tighter than
ing tail only (Lmflag = 1 and Pcol <= 7), and lying Carlin et al. (2012), and the proper motions themselves
within20degreesofStripe−82(δ <20). TheΛvaluesof agree approximately within the combined errors.
| |
our measurements correspond to the centers of the rect-
angular areas used to perform the measurements, and 6. CONCLUSIONS
the error-bars indicate the extent of those areas. The WehavemeasuredthepropermotionoftheSgrstream
agreement for the central field (Λ 105◦), as well as using Stripe 82 data. By tying the astrometry to the
for the edge of the bright stream (∼Λ 120◦), is quite known QSOs and combining the measurements for large
∼
good. At the location of the fainter Sgr stream (empty samples of stars, we have been able to achieve high pre-
circle on the plot), the model doesn’t have many par- cision proper motions. Our measurements have been
ticles, as expected. In order to facilitate further com- performedfor twodistinct groups–spectroscopicallyse-
parisons, Table 3 gives the positions and velocities cor- lected red giants/subgiants with SDSS radial velocities
responding to our measurements. Here, (X,Y,Z) and and photometrically selected MSTO stars. The proper
(U,V,W) are in a right handed Galactocentric coordi- motions of those agree very well and after correcting for
nate system with the Sun located at X = 8.5kpc, and the solar reflex motion are tightly aligned with the Sgr
−
8 Koposov, Belokurov & Evans
stream. Our results are in agreement with earlier mea- Abazajian,K.N.,Adelman-McCarthy,J.K.,Agu¨eros,M.A.,et
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