Title: Structure and substructure in the stellar halo of the Milky Way

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The handle http://hdl.handle.net/1887/33295 holds various files of this Leiden University dissertation.

Author: Pila Díez, Berenice

Title: Structure and substructure in the stellar halo of the Milky Way

Issue Date: 2015-06-16

A skewer survey of the

Gala ti halo from deep

CFHT and INT images

Authors

B.Pila-Díez,J.T.A.deJong,K.Kuijken,R.F.J.vanderBurgandH.Hoekstra

Abstra t

We study the density prole and shape of the Gala ti halo using deep multi-

olour images from the MENeaCS and CCCP proje ts, over 33 elds sele ted

to avoid overlap with the Gala ti plane. Using multi olour sele tionand PSF

homogenizationte hniques we obtain ataloguesof Fstars (near-main sequen e

turnostars)outtoGala to entri distan esupto60kp . Groupingnearbylines

ofsight,we onstru tthestellardensityprolesthroughthehaloineightdierent

dire tionsbymeansofphotometri parallaxes. Smoothhalo modelsarethent-

tedtotheseproles. Wend leareviden eforasteepeningofthedensityprole

power law index around

R = 20

^{kp , from}

−2.50 ± 0.04

^to

−4.85 ± 0.04

^{, and}

foraattening ofthehalo towardsthepoles withbest-taxis ratio

0 .79 ± 0.02

^.

Furthermore,we annotruleoutamildtriaxiality(

w = 0.88 ± 0.07

^{). Were over}

thesignaturesofwell-knownsubstru tureandstreamsthat interse tourlinesof

sight. These results are onsistent with those derived from wider but shallower

surveys,andaugurwellforup oming,wide-eldsurveysof omparabledepth to

ourpen ilbeamsurveys.

A eptedforpubli ationinAstronomy&Astrophysi s

PreprintinarXiv:1502.02460[astro-ph.GA℄

2.1 Introdu tion

ThestellarhalooftheMilkyWayonly ontainsatinyfra tionofitsstars,yetit

providesimportant luesabouttheformationoftheGalaxyandgalaxyformation

in general. Within the paradigm of hierar hi al stru ture formation, galaxies

evolveovertime, growingbymeansof mergersanda retionof smallersystems.

Whilein the entral parts of galaxies the signatures of su h events are rapidly

dissipated, thelong dynami al times alesallow a retion-indu edsubstru tures

tolingerfor Gigayears in theiroutermostregions. Thus, thestellarstru ture of

theouterhalosofgalaxiessu hastheMilkyWay anhelp onstrainnotonlythe

formationhistoryofindividualgalaxies,butalso osmologi almodelsofstru ture

formation.

Owing to the intrinsi faintness of stellar halos, the Milky Way is our best

bet fora detailedstudyof su h stru tures. However,even studyingtheGala ti

stellarhalo is fraught withdi ulties; very sensitive dataare requiredto probe

starsatthese largedistan es(outto 100kp ),andspread over su ientlylarge

areas to onstrain the overall stru ture as well as lo alized substru tures. In

re entde adestheadventofCCD-basedall-skysurveyssu hastheSloanDigital

Sky Survey (SDSS York et al. 2000; Ahn et al. 2014) in the opti al and the

2 Mi ron All Sky Survey (2MASS Skrutskie et al. 2006) in the infrared have

unlo ked unpre edented views of the outer regions of the Galaxy. This has led

tothedis overy ofmanypreviouslyunknownsubstru tures (e.g. Newberget al.

2002;Belokurovetal.2006b;Grillmair2006b;Belokurovetal.2007b;Juri¢etal.

2008;Belletal.2008)andtoimprovedknowledgeoftheoverallstru tureinthese

outskirts(e.g.Chenetal.2001;Juri¢etal.2008;deJongetal.2010;Sesaretal.

2010a,2011;Fa iolietal.2014). Nevertheless,mostofthesere entanalysesare

still limited to either the inner parts of the stellar halo (

R GC ≤ 30

^{kp ) or to}

parti ular,sparsestellartra ers(e.g. K-giantsorRRLyrae).

In this paper we use deep photometry obtained with the Canada-Fran e-

HawaiiTeles ope(CFHT)MegaCamandtheWide FieldCamera(WFC)atthe

Isaa Newton Teles ope(INT),s atteredovera largerangeofGala ti latitudes

and longitudes to probe main sequen e turn-o (MSTO) stars outto distan es

of60kp . Combiningourdataintoeightindependentlines ofsightthrough the

Gala ti halo,weareableto onstraintheoverallstru tureoftheouterhalo,and

toprobethesubstru tureintheseoutermostregions. Inse tion2wedes ribethe

dataset used forthis analysis andthe onstru tion ofourdeep star atalogues.

Se tion3presentsthederivedstellardensityprolesandsmoothGala ti model

ts. Wedis ussourresultsin se tion4andpresentour on lusionsinse tion5.

2.2 Observations and data pro essing

2.2.1 Survey and observations

Weuse

g

^and

r

^{images from theMENeaCSand theCCCPsurveys(Sand et al.}

2012; Hoekstra et al. 2012; Bildfell et al. 2012) together with several ar hival

−50 0

50 100

150 200

250 RA (degrees)

−40

−20 0 20 40 60 80

D EC ( de gr ee s) A

B C

D

E

G F H

Figure2.1: Equatorialmapshowingthepositionofalltheeldsusedinthiswork.

Thedierent oloursand symbols indi ate how theelds havebeengrouped to

al ulatethedierentdensity proles. The ba kgroundimage isthe SDSS-DR8

map from Koposov et al. (2012), whi h shows the footprint of the Sagittarius

stream and the lo ation of the Sagittarius dwarf galaxy. When grouping the

elds,wehavealsotakenintoa ountthepresen eofthisstream,theTriangulum-

Andromedaoverdensity,andtheanti entresubstru tures(ACS,EBS,andMono-

eros),intrying to ombine theiree t in ertainprolesandavoiditin others.

lustereldsfromtheCFHT-MegaCaminstrument. We ombinethesedatawith

U

^and

i

^{imagesfrom a follow-up ampaign withtheINT-WFC instrument(van}

derBurgetal.,inprep.). Whereasthesesurveystargetedapresele tedsampleof

galaxy lusters,thepointings onstitutea"blind"surveyoftheMilkyWaystellar

halosin etheirdistribution is ompletelyindependentofanypriorknowledgeof

thehalo's stru tureandsubstru ture.

Our pointings are distributed over the region of the sky visible to both the

CFHT and the INT (see Figure 2.1). To optimize the star-galaxy separation

(see se tion 2.2.2) we restri t our analysis to exposures with image quality of

subar se ond seeing, typi ally

<≈ 0.9 arcsec

^{in the}

r

^{band. This} limitation, ombinedwith thevarying elds of view and observing onditions between the

datasets,leadstopointingfootprintsizesthatrangebetween

0.24

^and

1.14 deg ²

^.

2.2.2 Image orre tion of the PSF distortion [and impli a-

tions for the star-galaxy separation℄

Previousresear hbyourgrouphasshownthattheperforman eofstandardstar-

galaxyseparationmethodsbasedonthesizeandellipti ityofthesour es anbe

improvedbyhomogenizingthepoint-spreadfun tion(PSF)a rossanimageprior

toitsphotometri analysis(Pila-Díezet al.2014). Inaddition,su h a orre tion

alsoprovidesthebenetofallowingustoperformxedaperturephotometryand

olourmeasurements.

InordertohomogenizethePSFofourimages,weusea ode(Pila-Díezetal.

2014)that,asarststep,takestheshapeofthebrightstarsinagivenimageand

usesittomapthevaryingPSFand, asase ondstep, onvolvesthismapwitha

spatiallyvariable kerneldesignedto transformeverywheretheoriginalPSFinto

agaussianPSF.

2.2.3 Catalogues

From the PSF-homogenized exposures we reate photometri atalogues using

Sour eExtra tor (Bertin &Arnouts 1996). For the

g

^{and the}

r

^{data, we sta k}

thedierentexposures in ea h band to reatea single alibratedimage, and we

extra t the band atalogues from them. We perform a star-galaxy separation

based on the brightness, size and ellipti ity of the sour es and we mat h the

surviving sour es in the two atalogues to produ e a

gr

^{- atalogue of stars for}

ea held ofview (seePila-Díezetal.(2014)). Thelimitingmagnitudesof these

gr

^{star atalogues rea h}

m AB ∼ 25.0

^atthe

5 .0σ

^{levelin the}

r

^band.

Forthe

U

^andthe

i

^{eldsofview,weprodu eseveral}photometri atalogues, oneforea hindividualexposure. We orre tthemagnitudesinthe

i

^ataloguesfor

thedependen yoftheilluminationonpixelposition. Forea hpointingandband,

theexposure ataloguesare alibratedto a ommonzeropointand ombinedto

produ e a single-band atalogue. Inthese single-band atalogues, the resulting

magnitudeforea h sour eis al ulatedas themedianof the ontributionsofall

theindividualexposures. Atthispointthe

U

^andthe

i

^{magnitudesare onverted}

from the INT to the CFHT photometri system using the following equations,

whi hwederiveby alibratingourmixedINT-CFHT olourstothe olourstellar

lo ioftheCFHTLega ySurvey(Erbenetal.(2009),Hildebrandt etal.(2009)):

i M egaCam = i IN T − 0.12 ∗ (r M ega − i IN T )

(2.1)

u M egaCam = u IN T − 0.15 ∗ (u IN T − g M ega ) .

^(2.2)

Finallyweposition-mat hthesour esfromthe

U

^-,the

i

^-andthe

gr

- atalogues to reateanal atalogueofstellarsour esforea heldofview. Thesenal

ugri

^-

ataloguesareshallowerthanthe

gr

- ataloguesbe auseofthelesserdepthofthe

i

^andthe

U

observations(seeTable2.1). Figure2.2showsthe olour-magnitude diagrams (CMDs) for the nal

ugri

^and

gr

^{atalogues (top and entre, respe -}

tively),andthedieren ebetweenthem(bottom). Thebottompanelhighlights

that,inthe olourregimeofthehalo(

0 .2 < g − r < 0.3

^),the ombinationofthe fourbandsremovesmainlyveryfaint,unresolvedgalaxies.

We orre tforinterstellarextin tionusingthemapsfromS hlegeletal.(1998)

andtransform themagnitudes in the

ugri

^{-stellar atalogues from theCFHT to}

theSDSS photometri system. For this we use the equations on the Canadian

Table2.1: GroupsofpointingsasshowninFigures2.1,2.5,2.6and2.8. Thetable

showsthe entral oordinates for ea h group,the number of individual elds of

view ontributingto it, itstotalarea and thestellar ompleteness limitin ther

band.

Group RA(deg) De (deg)

l

^(deg)

b

^(deg)

n fields Σ

^(deg

²

^{) mag}

_lim,r,∗

A 160.654338 43.98310 171.335811 59.15040 8 5.60 22.8

B 231.593130 29.13513 45.577138 55.93598 5 3.98 22.7

C 229.347757 6.91624 9.425402 49.92775 4 3.44 24.1

D 210.062933 51.67173 99.735627 62.24580 2 0.64 23.4

E 121.918411 41.20348 179.233500 31.26694 5 2.73 22.7

F 342.735895 17.09581 86.019738 -36.99391 3 2.17 23.2

G 157.028363 17.15674 222.142793 55.48268 3 2.02 23.1

H 220.659749 2.00187 354.337092 53.38989 3 2.04 24.2

AstronomyData CenterMegaCamwebsite 1

u M egaCam = u SDSS − 0.241 · (u SDSS − g SDSS )

(2.3)

g M egaCam = g SDSS − 0.153 · (g SDSS − r SDSS )

(2.4)

r M egaCam = r SDSS − 0.024 · (g SDSS − r SDSS )

(2.5)

i M egaCam = i SDSS − 0.003 · (r SDSS − i SDSS )

(2.6) andinvertthemtoturn ourmeasurementsintoSDSSmagnitudes. Subsequently

we alibrate ea helddire tlytoSDSSusingstellarphotometryfromDR8. The

resultingphotometrymat hesthe olour- olourstellarlo iofCoveyetal.(2007)

asshownin Figure2.3. Unlessexpli itlystatedotherwise,allmagnitudesin this

paper areexpressedin theSDSSsystem.

Inordertoredu ethenoisewhenanalysingtheradialstellardensitydistribu-

tionofthehalo,we ombinethe ataloguesfromnearbypointings,groupingthem

a ordingto theirposition in thesky. Thisstepisimportantbe auseofthena-

tureofoursurvey,whi his omposedofrelativelysmall,s atteredeldsofview.

Weusea friends-of-friends(FoF)algorithmto groupthedierentpointings. We

requesttwofriendsnotto beapartbymore than20degrees,andin a few ases

we leanorsplitaresultinggroup(redpentagonsorblueandorangetrianglesin

Figure2.1) or ombineothers (purple diamonds)to a ountforthepositions of

thegala ti diskor majorhalosubstru tures. Be ause thedierentpointingsin

oursurveys havedierent ompletenesslimits,these groupedor ombined ata-

logueswhi hwenameA,B,C,... Harenallylteredtomeetthe ompleteness

magnitudethresholdoftheirmostrestri tive ontributor 2

.

1

www2. ad - da.hia-iha.nr - nr .g . a/megapipe/do s/lters.html

2

Todeterminethe ompletenesslimitofea heldofview,wetitsmagnitudedistribution

to a gaussian representing thepopulation of faint galaxies and another variable fun tion

representingthe stellar distribution alongthe whole magnituderange. We hoose as the

ompletenesslimiteitherthetransitionpointbetween thetwodistributions(thevalley)or,if

insteadthereisaplateau,theturningpointofthewholedistribution(theknee).

Figure2.2: Hessdiagrams showingthe number ofsour esper olour-magnitude

binin the

ugri

^{atalogue (top), in the}

gr

^{atalogue ( entre) and the dieren e}

betweenboth(bottom)foreld A1033. Mostofthesour eslostwhen ombining

the atalogues orrespondtofaintmagnitudes,be ausethe

i

^andthe

U

^observa-

tionsareshallower. Theee tistheremovalofmostofthefaintgalaxies(lo ated

in the

−0.2 < g − r < 0.7

^and

r > 23

^{region in the entral panel), mostof the}

faintestdiskMdwarves(

1 .1 < g − r < 1.3

^{)andanumber offaintobje ts(inthe}

i

^orthe

U

^{bands)s atteredthroughoutthe}

( g − r, r)

^diagram.

Figure2.3: Colour- olourdiagrams(CCDs) orrespondingto theelds ingroup

A(pointingsmarkedaslightgreen ir lesinFigure2.1). Thesour esinthe

ugri

atalogues(bla k)andthesubsetofnear-MSTOstars(red)havebeen alibrated

to SDSS using DR8 stellar photometry. The main sequen e stellar lo i (green

dashedlines)aretheonesgiveninTables3and4ofCoveyetal.(2007). Quasars

andwhitedwarf-Mdwarfpairsareabundantinthe

u − g < 1

^,

−0.3 < g − r < 0.7

spa e.

0.0 0.2 0.4 0.6 0.8 1.0

g - i

−10

−5 0 5 10

M ag r

−7 −6 −5 −4 −3 −2 −1 0 1

[Fe/H]

−10

−5 0 5 10

M ag r

Figure 2.4: Estimated absolute magnitude in the

r

^{band (}

M r

^{) and estimated}

metalli ity(

[ F e/H]

^{)forgroupAforthesour estypi ally onsideredashalostars}

(blue)and those that we havesele ted as near-MSTO stars(red). The sour es

sele tedas halo members meet

0 .2 < g − r < 0.3

^and

g, r, i > 17

^{. The subset}

of near-MSTO stars, additionally meets

M r > −2

^,

−2.5 ≤ [F e/H] ≤ 0

^and

0 .1 < g − i < 0.6

^.

2.3 Stellar radial density proles

2.3.1 Star sele tion and onstru tion of the radial stellar

density proles

The oordinatesandthe ompletenesslimitsofthegroupsaregiveninTable2.1.

Weusehalomainsequen eturnostarsinoureldsastra erofthestellarhalo:

atthe ompletenesslimitsofthedatasu hstars anbeidentiedasfaroutas

60

kp fromtheGala ti entre. WetseveralGala ti stellardistributionmodelsto

thesedensityprolesandderiveanumberofstru turalparametersforthestellar

halo. Previous works have already used main sequen e turno point (MSTO)

stars, near-MSTO stars, BHB and blue stragglers of type A and RRLyrae as

stellartra ersfortheGala ti stellarhalo. We ompareanddis ussourndings

totheirsin se tion2.4.2.

In order to sele t the near main sequen e turno stars we make use of two

empiri al photometri variables. The ratio

[ F e/H]

^{is al ulated following the}

photometri metalli ityrelationbyBondetal.(2010),andtheabsolutemagnitude

M r

^{is al ulated following the} photometri parallax relation from Ivezi¢ et al.

(2008):

[ F e/H] = −13.13 + 14.09x + 28.04y − 5.51xy − 5.90x ²

− 58.68y ² + 9 .14x ² y − 20.61xy ² + 58 .20y ³ ,

^(2.7)

where

x = u − g

^and

y = g − r

^{. This relation is valid in the}

g − i < 0.6

^and

−2.5 ≤ [F e/H] ≤ 0

^{range,whi his ompatiblewiththeregimeofournear-MSTO}

starsele tion.

M r = −0.56 + 14.32z − 12.97z ² + 6 .127z ³ − 1.267z ⁴

+ 0 .0967z ⁵ − 1.11[F e/H] − 0.18[F e/H] ² ,

^(2.8)

where

z = g − i

^{. The tested validity regime of this equation} en ompasses the

0 .2 < g − i < 1.0

^{range, meaning that the absolute} brightnesses of our near- MSTO starshave been properlyestimated. We extrapolatethe relationfor the

0 .1 < g − i < 0.2

^{range, whi h isjustied owingtothesmoothandslow hange}

of

M r

^with

z

^.

Wesele tthehalo near-MSTOstarsbyrequiring

0 .2 < g − r < 0.3 ;

^(2.9)

g, r, i > 17 ;

^(2.10)

0 .1 < g − i < 0.6 ;

^(2.11)

5 .0 > M r > −2 ;

^(2.12)

−2.5 ≤ [F e/H] ≤ 0 .

^(2.13)

Thersttworestri tions(2.9and2.10)retrievestarstypi allyasso iatedwith

thehalo,inparti ulardistantmainsequen eFstars(seeTable3fromCoveyetal.

(2007)). This sele tion however, an be signi antly ontaminated by quasars

and white dwarf-M dwarf pairs, whi h are abundant in (but not restri ted to)

the

−0.2 < g − r < 0.3

^{range (seeFigure 2.3). To redu e thepresen e of these}

interlopersandsele tthebulkoftheFstarspopulation,weapplyrestri tions2.11

(basedonTable4in Coveyetal.(2007))and2.12. Constraint2.13ensuresthat

the nal sour es are at most as metal ri h as the Sun (to a ount for possible

ontributionsfrommetal-ri hsatellites)andnotmoremetal-poorthan0.003times

theSun.

The de rease in interlopers attained by applying restri tions 2.11, 2.12, and

2.13 omparedtoonlyapplyingrestri tions2.9and2.10isillustratedinFigure2.3,

wherethered dots indi ate thenalsele tion ofhalo near-MSTOstarsand the

bla k dots represent the whole atalogue of star-like sour es. It is lear that

thenalsele tionof near-MSTOstars doesnotspanthewhole rangeofsour es

en ompassedbetween

g − r = 0.2

^and

g − r = 0.3

^{. Theee t ofthe}

[ F e/H]

^and

M r

^{sele tionisfurther} illustratedin Figure2.4.

Using the estimated absolute brightness, we al ulate the distan e modulus

and the helio entri distan e for all the near-MSTO stars. We dene distan e

modulusbinsofsize

∆ µ = 0.2

^magand

∆ µ = 0.4

^{mag,and ountthenumber of}

near-MSTOstarsper binforea hgroup of elds (A,B,C,...). The hoi e ofdis-

tan ebinsismotivatedbya ompromisebetweenmaximisingtheradial distan e

resolutionandminimisingthePoissonnoiseinthestellarnumber ounts. Wetest

this ompromise byexploring two distan emodulusbinsizes, whi h orrespond

todistan ebinsizesoftheorderof

10 ²

p and

10

kp ,respe tively.

Wethen al ulatethenumberdensityper binanditsun ertaintyasfollows:

ρ l,b,D = N l,b,∆µ

0 .2 · ln(10) · D hC ³ · ∆Ω · ∆µ ,

^(2.14)

E ρ =

r ( ρ

√ N

) ² + ( ρ

√ n f ields

) ² ,

^(2.15)

where

∆Ω

isthearea overed byea h group,

D hC

^{is the}helio entri distan e,

l

and

b

^{arethegala ti} oordinatesand

N l,b,∆µ

^{isthenumberofstarsper binina}

givendire tionofthesky. Parti ularly,

∆Ω = 4 π

41253 Σ(deg ² )

(2.16)

and the area of ea h group (

Σ

) depends on the individual area of ea h eld ontributingtoit(Table2.1).

The resultsfor these number density al ulations an beseen in Figure 2.5,

whereweplotthelogarithmi numberdensityagainstthegala to entri distan e 3

,

R GC

^{, forea h group (orline of sight). For thisand thesubsequentanalysis, we}

only onsiderbinswith

R GC > 5kpc

^,

|z| > 10

^{kp (toavoidtheinner regionsof}

theGalaxy)andadistan emodulusof

µ ≤ mag lim −4.5

^{(toguaranteea omplete}

sampleofthefaintestnear-MSTOstars 4

).

Figure2.5showsthat thedensityprolesde reasequitesmoothlyfor

40 − 60

kiloparse sandformostofthelinesofsight.

2.3.2 Fitting pro edure

Wet several models of theGala ti stellar number density distribution to the

data,rangingfromabasi axisymmetri powerlawtomore omplexmodelswith

triaxiality and a break in thepower law. Themodels take the following math-

emati alforms, with

x

^,

y

^{, and}

z

^{being the artesian} gala to entri oordinates withtheSunat(8,0,0) kp (Malkin2012):

- Axisymmetri model

ρ(x, y, z) = ρ 0 ·



x ² + y ² + z ² q ²



n/2 ,

^(2.17)

where

q = c/a

^{isthepolaraxis ratioortheoblatenessofthehalo;}

- Triaxial model

ρ(x, y, z) = ρ 0 ·

 x ² + y ²

w ² + z ² q ²



n/2 ,

^(2.18)

where

w = b/a

^{istheratiobetweentheaxesintheGala ti plane;}

- Brokenpower law,withvaryingpower indexat

R break

ρ(x, y, z) =

 ρ 0 · (R ellip ) ^{n in} , R ellip < R break

ρ 0 · (R ^ellip ) ^{n out} · R ⁿ break ^{in −n out} , R ellip ≥ R ^break

^(2.19)

R ellip =



x ² + y ² + z ² q ²



1/2 ;

3

R GC = p R ² + z ²

where

R

^and

z

^{aretheradialandverti al} oordinatesonthe ylindri algala to entri referen e system.

4

This onstraintguaranteesthattherearenodistan e ompletenessissuesduetoourspe i

typeofstellartra ersandduetothedierentdepthsofourelds. Theonlysubsetae tedby

in ompletenessisthatof

mag lim − 5.0 < µ < mag lim − 4.5

^{forthestarsinthe}

4.5 < M r < 5.0

range;anditsaveragelossisof

20%

overthetotalnumberofnear-MSTOstars(

−2.0 < M r <

5.0

^{)inthesamedistan e range. Several testson dierentupperdistan ethresholds forthe}

density proles show that thedistan e modulus onstraint of

µ ≤ mag lim − 4.5

^isenough

toguaranteethatallthelinesofsight ontributerobustdensitymeasurementsatthefurthest

distan esandthatthein ompletenessin

mag lim −5.0 < µ < mag lim −4.5

^forthe

4.5 < M r < 5.0

near-MSTOstarshasnostatisti allysigni antee tonthebesttparameters.

Figure2.5: Logarithmi stellardensityprolesversusdistan e forthenearMain

Sequen e turno point stars (near-MSTO) from the elds in groups A (green

ir les),B( yansquares),C(bluedownwardtriangles),D(yellowupwardtrian-

gles),E(redpentagons),F(pinkhexagons),G(purplediamonds)andH(orange

leftwardtriangles). Theirsymbolsmat h thoseinFigure2.1.

- Brokenpower law,withvaryingpower indexandoblatenessat

R break

ρ(x, y, z) =







ρ 0,in · 

x ² + y ² + ^{z 2}

q ² _in

 n in /2 , R GC ≤ R break

ρ 0,out · 

x ² + y ² + ^{z 2}

q _out ²

 n out /2 , R GC > R break ,

(2.20)

wheretheinner power lawistto datathat meets

R GC ≤ R break

^andthe

outerpower lawisappliedtodatathat meets

R GC > R break

^.

Wetallthesemodelstothedatausingthe" urve-t"methodfromPython's

S ipy.optimize, whi h uses the Levenberg-Marquardt algorithm for non-linear

least squares tting. The obje tive fun tion takes the form of a

χ ²

^{, and we}

also al ulatearedu ed

χ ²

^{foranalysispurposes,}

χ ² =

N data

X

i=1

 ρ data,i − ρ ^model,i E ρ,i

 2

,

^(2.21)

χ ² _red = χ ² N data − N params

,

^(2.22)

where

N data

^and

N params

^{arethenumber ofdata pointsandthenumber offree}

parameters,respe tively.

The inuen e of the photometri un ertainties on the density proles and

the best t parameters is evaluated through a set of Monte Carlo simulations

that randomly modify the

g

^,

r

^,

i

^,

u

^{magnitudes of ea h star within the limits of}

thephotometri un ertainties. Through this method wend that the variation

oftheMonte Carlo bestt parameters aligns with theun ertainties of ourbest

t parameters (derived from the se ond derivative of thets by the " urve-t"

method). The entre ofthesevariationsiswithin

1 σ

^{ofourdire t ndings.}

Wet all models tofour data sets: with and without[known℄ substru tures

andbinnedin

0 .2

^and

0 .4

^{magnitude ells.Inthiswaywe an he ktherobustness}

ofourresultstodierentbinningoptionsandweareableto omparewhatwould

betheee tofsubstru tureonourunderstandingofthesmoothhalo ifwewere

to ignore it or unable to re ognize it as su h. Spe i ally, we ut the distan e

binsat

R GC < 25

^{kp in groupEto avoid} ontributionsbythestru turesinthe dire tionofthegala ti anti entre(theMono erosring,theAnti entreStru ture

andtheEasternBandStru ture),thedistan ebinswithin

15 < D hC < 40

^{kp in}

groupGto avoid ontributionsbytheSagittariusstream, and thedistan ebins

within

20 kpc < D hC < 60

^{kp in group H to avoid} ontributions againby the Sagittariusstream.

2.3.3 Results

Thebesttparametersforea hmodelresultingfromttingthesefourdatasets

aresummarizedin Tables 2.2to2.5. Table2.2 ontainstheresultsofttingthe

∆ µ = 0.2

^{magbinneddataex ludingregionswith}substru ture,whereasTable2.3 ontainstheresultsofttingto allthe

0 .2

^{mag bins. SimilarlyTable2.4 overs}

thetsto

∆ µ = 0.4

^{magdatawithout}substru turebins,andTable2.5,toall

0 .4

magbins. Theredu ed

χ ²

^{andtheinitialparametershavealsobeenre ordedin}

thesetables.

We ompare the tting results for the four dierent data sets re orded in

Tables2.2to2.5andndthatthetsforwhi hthesubstru turehasbeenmasked

signi antlyoutperformthosethathavebeenallowedtotalltheavailabledata.

Thedieren eon

χ ² _red

^{forallthesemodelsandbinsizesisinevery aseatleasta}

fa torof

2 .3

^{orlarger. Wendthatallowingthemodelstotdatathat ontains}

substru turedoesnotae tlargelymostofthestru turalparameters(polaraxis

ratiosare ompatible within the un ertainties and power law indi es have lose

values)ex eptthatitde reasesthediskaxisratio

w

^byatleast

10%

,suggestinga strongdeparturefromtheaxisymmetri modelthatisnotimpli itin theltered

data sets. Hen eforth we will restri t the remaining dis ussion to the results

derivedfromthe leanestdatasets.

Comparingtheparametersresultingfromthebesttstothemasked

0 .2

^mag

and

0 .4

^{mag data, wend that thets to}

0 .2

^{mag binned data performbetter}

forallthemodels(

χ ² _red

^ratiooftwo). Nonetheless,allthemeasurementsforthe dierentstru turalparametersinthetwodatasetsare ompatiblewithea hother

withintheun ertainties. Thebesttsforthefourmodelsandtheirresidualsfor

oureightlinesofsightareshowninFigures2.6aand2.6bforthemasked

0 .2

^mag

binneddata. Itis learthatthedieren esbetweenthettedmodelsalongthese

sightlinesaresmall.

Ourdataarein on lusiveregardingtriaxiality,butare ompatiblewitheither

amildlytriaxialhaloorwithnotriaxiality. Forthe

0 .2

^{magdataset, thetriaxial}

model tsslightly better thanthe axisymmetri model and returns

w = 0.87 ± 0 .09

^{. For the}

0 .4

^{mag data set, however, the} axisymmetri model ts slightly betterandthetriaxialmodelreturnsadiskaxisratio ompatiblewith

1

. Inboth data sets the other best-tting parameters are pra ti ally identi al for the two

models. Thisindi ates that the ostoftheextraparameter isnotsupportedby

the

0 .4

^{magdata. Thus,itishardtoderiveapre isevalueforthediskaxisratio}

andto on ludeifitistrulytriaxial,butaweightedaverageof

w

^{andthegeneral}

analysisshow ondentlythat

w > 0.8

^.

Wein reasethe omplexityoftheaxisymmetri modelbyaddingtwodegrees

offreedomand onsideringa hangeinthepower lawindex

n

^{at aspe i break}

distan e

R break

^{(a brokenpower law). For this purpose, weusea grid ofvalues}

toexplorealltheparametersex eptthedensitys alefa tor

ρ 0

^{,whi hweleftfree}

to t (see below for the grid hara terization). This model de reases the

χ ² _red

inboththe

0 .2

^andthe

0 .4

^{magbinned ases,indi atingthat ourdataisbetter}

t by a broken power law than by a simple axisymmetri model or a triaxial

model. Itturnsthesinglepowerlawindexfrom

n = −4.26 ± 0.06

^{intoalesssteep}

inner index

n in = −2.50 ± 0.04

^{and a steeper outer index}

n out = −4.85 ± 0.04

(measurementshereareforweightedaveragesbetweenthe

0 .2

^and

0 .4

^magdata).

ewersurveyoftheGala ti halofromdeepCFHTandINTima

distan emodulus ells.

Model

χ ² _red ρ 0 (pc ⁻³ ) · 10 ⁻³ R break (kpc) n n in n out q q in q out w

axisymmetri

1.90 14 ± 6

^

−4.31 ± 0.09

 

0.79 ± 0.06

^{  }

triaxial

1.86 14 ± 6

^

−4.28 ± 0.09

 

0.77 ± 0.06

^{ }

0.87 ± 0.09

brokenp.l.

n 1.52 0.071 ± 0.003 19.0 ± 0.5

^

−2.40 ± 0.05 −4.80 ± 0.05 0.77 ± 0.03

^{  }

brokenp.l.

n, q 1.99, 1.51 1 ± 3 19fixed

^

−3.3 ± 0.6 −4.9 ± 0.2



0.7 ± 0.2 0.88 ± 0.07

^

initialparameters 

0.001 40.0 −3.00 −3.00 −3.50 0.70 0.70 0.8 1.00

Table2.3: SameasinTable2.2butthistimettingalltheavailabledata(in ludingthoseregions ontainingstellar ountsfromknown

substru turesanddete tedoverdensities).

Model

χ ² _red ρ 0 (pc ⁻³ ) · 10 ⁻³ R break (kpc) n n in n out q q in q out w

axisymmetri 4.71

8 ± 3

^

−4.15 ± 0.08

 

0.83 ± 0.06

^{  }

triaxial 4.59

7 ± 2

^

−4.07 ± 0.08

 

0.82 ± 0.06

^{ }

0.77 ± 0.07

brokenp.l.

n

^4.24

0.17 ± 0.01 21.0 ± 0.5

^

−2.80 ± 0.05 −4.80 ± 0.05 0.84 ± 0.03

^{  }

brokenp.l.

n, q

^3.36,4.79

1 ± 2 21fixed

^

−3.3 ± 0.4 −5.0 ± 0.2



0.7 ± 0.2 0.89 ± 0.08

^

initialparameters 

0.001

^{40.0 -3.00 -3.00 -3.50 0.70 0.70 0.8 1.00}

Table2.4: SameasinTable2.2butthistimettingthedatabinnedin

0.4

^{magdistan emodulus ells.}

Model

χ ² _red ρ 0 (pc ⁻³ ) · 10 ⁻³ R break (kpc) n n in n out q q in q out w

axisymmetri 3.89

12 ± 4

^

−4.26 ± 0.08

 

0.77 ± 0.05

^{  }

triaxial 3.97

12 ± 5

^

−4.25 ± 0.08

 

0.77 ± 0.06

^{ }

0.9 ± 0.1

brokenp.l.

n

^2.61

0.11 ± 0.01 20.0 ± 0.5

^

−2.60 ± 0.05 −4.90 ± 0.05 0.81 ± 0.03

^{  }

brokenp.l.

n, q

^4.95,2.34

1 ± 1 20fixed

^

−3.2 ± 0.4 −5.0 ± 0.3



0.7 ± 0.2 0.82 ± 0.08

^

initialparameters 

0.001

^{40.0 -3.00 -3.00 -3.50 0.70 0.70 0.8 1.00}

Table2.5: SameasinTable2.4butthistimettingalltheavailabledata(in ludingthoseregions ontainingstellar ountsfromknown

substru turesanddete tedoverdensities).

Model

χ ² _red ρ 0 (pc ⁻³ ) · 10 ⁻³ R break (kpc) n n in n out q q in q out w

axisymmetri 9.13

7 ± 2

^

−4.10 ± 0.07

 

0.81 ± 0.05

^{  }

triaxial 9.19

7 ± 2

^

−4.07 ± 0.07

 

0.81 ± 0.06

^{ }

0.86 ± 0.09

brokenp.l.

n

^7.74

0.058 ± 0.005 20.0 ± 0.05

^

−2.40 ± 0.05 −4.8 ± 0.05 0.84 ± 0.03

^{  }

brokenp.l.

n, q

^6.05,9.2

0.6 ± 0.9 20fixed

^

−3.1 ± 0.4 −4.9 ± 0.2



0.7 ± 0.2 0.86 ± 0.07

^

(a)Fitteddensityprolesforthe

0.2

^{magbinneddata.}

Figure2.6: Densityprolesin de imallogarithmi s aleandthebest tmodels

fromTable2.2 (ttedto masked

0 .2

^{binned data). Thedierent linesrepresent}

theaxisymmetri (bla k solid line), the triaxial (greendashed line), the broken

powerlawwithvaryingpower index (reddottedline)and thebrokenpower law

withvaryingpowerindexandoblateness(bluedashed-dotted-dottedline)models.

Thegreyareasdenotedatathathavebeenmaskedfromthettingtoa ountfor

thepresen eofsubstru ture.

(b)Data-to-modelresidualsforthe

0.2

^{magbinneddata.}

Figure2.6: Residualsbetweenthedata andthebestt modelsfrom panel2.6a.

Thedierentlinesandtheshadedareasfollowthesame olourandsymbol ode.

Italsoin reasesthe entralvalueofthepolaraxisratio

q

^withintheun ertainties, fromaweighted

q = 0.77 ± 0.04

^{toa weighted}

q = 0.79 ± 0.02

^{. Globally,thedisk}

axisratioseemsto bethemoststableparameterthroughout thedierentmodel

tstoourdata, returninga moderatelyoblatehalo.

Finally we x the break distan e at the best t value found by the broken

power law model (

R break = 19

kp and

20

kp for the

0 .2

^and

0 .4

^{mag binned}

data,respe tively)andaddanotherparametertoit,allowingnotonly

n

^,butalso

q

^{to hangeatthebreakdistan e. Wendthatthebesttstothismodelreturn}

su h largeerrorbarsfortheinner halo that,inpra ti e, ityieldsun onstrained

measurements:

∆ ρ 0 ≤ ρ 0

^,

∆ n in

^{is12-18% of}

n in

^and

∆ q in

^is30%of

q in

^.

Weexplore ea h modelto investigatepossible parameter degenera ies,toler-

an e rangesand potentiallo al minima in our best ts. For this we x all the

parametersinthefourmodelsex eptthedensitys alefa tor

ρ 0

^,andwerunthe

tsa rossa gridofparameter values. Inparti ular, thegridsarebuiltfollowing

q ² , w ² ∈ [0.1, 2.0; δ = 0.05]

^,

n ∈ [−5.0 − 1.0; δ = 0.1]

^,

n in ∈ [−4.0, −1.0; δ = 0.1]

^,

n out ∈ [−7.0, −3.0; δ = 0.2]

^and

R br ∈ [15, 50; δ = 1]

^{, where}

δ

^{is the}in remental stepforea hparameter. Wendthatthereisadegenera ybetween

R br

^and

n in

forthesimplebrokenpowerlawmodelforbothbinnings(seeFigure2.7).

Finallyourmeasurementsforthedensitys alefa tor

ρ 0

⁽

ρ

^at

R GC = 1

kp ) aretheresultoflargeextrapolationsandmerelyserveas normalizationsforour

ts. Forthat reasonwedonotdis ussthese valuesin detail.

2.4 Dis ussion

2.4.1 Robustness of the best t stru tural parameters

Inordertodeterminehowthedataavailabletousinuen estheresultsfromour

best ts,weremovethedierentlinesof sightone ata timeandrepeatthets.

Inthis waywe andeterminewhi harethemost riti allinesofsightandwhat

istheiree t onourresults.

Wendthatmostofthemhavenosigni antinuen eonthebesttparam-

etersofthedierenthalomodels. However,startingwiththepolaraxisratiowe

ndthat removinggroup A in reases slightlyitsvalue (

q ≈ 0.85

^{)and removing}

groupsCor E de reasesitslightly(

q ≈ 0.70

^{)in boththe}axisymmetri andtri- axialmodelinthetwodatasets. Regardingthepowerlawindex,againgroupsA

orChaveaninuen e,butgroupBaswell. RemovinggroupsAorBin reases

n

to

≈ −4.1 ± 0.1

^{,whereasremovingCde reasesitto}

n ≈ −4.6

^{. When} onsidering a triaxialhalo, wend that groups A, Bor Cin reasethe disk axis ratio

w

^by

∼ 0.10

^{, andthat removinggroups E or Fde reasesit to}

w ≈ 0.7

^. Additionally, in onditionsoftriaxiality,thela kofgroup Eredu es

q

^furtherto

q ≈ 0.60

^.

ThusremovinggroupEturnsouttobe riti alforboth

q

^and

w

^,representing a rather dierently looking halo (signi antly oblate and quite ellipti al in the

plane). GroupF also has a similar ee t on

w

^{but not on}

q

^{. Thereason why}

groupEhassu ha stronginuen e inthedeterminationofapossibletriaxiality

is that it is by far the losest group to the Gala ti anti entre. Other groups

(a)

χ ² _red

^{mapfortheltered}

0.2

^{magbinneddataset.}

(b)

χ ² _red

^{mapfortheltered}

0.4

^{magbinneddataset.}

Figure2.7:

χ ² _red

iso ontoursmapsfor

n in

^and

R br

^{fromthesimplebrokenpower}

law model. The minimum is indi ated with a white star. The bla k solid iso-

ontours rangefrom

min(χ ² _red ) + 0.1

^{to the maximumvalue, whereasthewhite}

dashediso ontours rangefrom

min(χ ² _red ) + 0.01

^to

min(χ ² _red ) + 0.05

^{. Themaps}

illustrateadegenera ybetweenbothparametersin thebest ts.

alsoinuen e the measurementsof the dierentparameters, but havea smaller

inuen e on thegeneral pi ture we would derive. Overall wesee that the lines

of sight we use an havea drasti ee t on the

w

^{resultsand a signi ant but}

moderateee ton

q

^and

n

^{. Thismeansthataglobalviewofthehaloisessential}

owingto its omplexstru ture.

2.4.2 Comparison to previous studies

Previousinvestigationsusingnear-MSTOstarshaveexploredboththeinnerand

the outer halo out to moderate distan es (

30 − 40

^{kp ), and similar regimes}

have been probed with blue horizontal bran h stars and blue struggler stars,

MSTO stars or multiple stellar halo tra ers. Studies involving RRLyrae stars

haverea hed furtheroutto

50

^{kp . Remarkably, thedepthof ourdataallowsus}

to probe further than any previous study (out to

60

^{kp ) in several} dire tions, independentlyofthestellartra er.

In this se tionwe ompare our ndingsregarding thestru tural parameters

ofthestellarhalo tothoseofthefollowingresultsin theliterature:

- Juri¢et al. (2008) use near-MSTOstars from theSDSS-DR3 and DR4as

stellartra ers, and over the

5 kpc < R GC < 15

^{kp range. They omprise}

5450

^deg

²

^{inthenorthernGala ti hemisphereand}

1088

^deg

²

^inthesouth.

- Sesar et al. (2011) use as well near-MSTO stars from the CFHT Lega y

Survey, and explore the

5 kpc < R GC < 35

^{kp range. Two of their four}

eldsexploretheSouthGala ti Cap.

- Deason et al. (2011) use type A blue horizontal bran h (BHB) stars and

bluestragglers (BS),rea hingoutto

R GC = 40

^{kp .}

- deJongetal.(2010)useCMDttingofSEGUEstellarphotometrytoprobe

thetotalstellarmassdensityfrom

R GC = 7

^{kp to}

R GC = 30

^{kp alonga}

"pi ketfen e"of

2.5

^{degreewidestripsatxedGala ti longitudespanning}

alargerangeofGala ti latitudes.

- Chen et al. (2001) use more general MSTO stars from two high latitude

regionsofSDSStotheNorthandtheSouthoftheGala ti plane(

49 deg <

|b| < 64 deg

^{). Theyexploretheinner haloregime(}

R GC . 30

^{kp ).}

- Bellet al.(2008) usealsomoregeneralMSTOstarsfromSDSS-DR5span-

ning

5 < R GC < 40

^{kp .}

- Fa ioli et al. (2014) use RRLyrae in the

9 kpc < R GC < 49

^{kp range.}

Theirmultiepo hdata omesfromtheXuyiS hmidtTeles opePhotometri

Survey(XSTPS)in ombinationwithSDSS olours,and overs

376.75

^deg

²

at

RA ≈ 150

^{deg and}

Dec ≈ 27

^deg.

- Sesaretal.(2010a)useRRLyraestarsfromSDSS-IIinthestripe82region.

Althoughtheirdataoriginallyspans

5 kpc < R GC < 110

^{kp ,thereanalysis}

performedbyFa iolietal.(2014)toderivestru turalparameterstrun ates

thesampleat

49

kp .

- Watkinset al.(2009) useaswellRRLyraefromSDSSinstripe82,andthe

omparative derivation of stru tural parameters by Fa ioli et al. (2014)

alsotrun atesitat

49

kp . Stripe82islo atedintheSouthGala ti Cap.

Theresultof this omparisonis summarized in Table2.6. Wenote that the

oblatenessvaluesforFa iolietal.(2014),Sesaretal.(2010a)andWatkinsetal.

(2009) are not the result of absolute best ts to a set of free parameters, but

thebest ts to free

R br

^,

n in

^and

n out

^{with xed priorvalues for a quite oblate}

(

q = 0.59 ^+0.02 _−0.03

^{)anda moderatelyoblatehalo(}

q = 0.70 ± 0.01

^).

Allsurveysthatrea hbeyond

R GC = 30

kp oin ideintheneedforabreakin thepower-lawindexofthehalodensity. Regardingpossibletriaxiality,onlyafew

ofthestudiesreport onstraintson

w

^{. Thosethatdo,haveeitherreported'nding}

unreasonable values' (Sesar et al. 2011) or have obtained limits on triaxiality

similartoours(

w > 0.8

^{,Belletal.(2008)).}

On thebreak radius, there is a general onsensus towards

R break ≈ 27

^{kp .}

Theonlyex eptionisthatofBelletal.(2008),whondavaluevery losetoour

measurement(

∼ 20

^{kp ). These}dis repan ies,however, anbeexplainedbythe ee tofthe

R break

^-

n in

^{degenera ydis ussedinse tion2.3.3.}

Theinnerandouterhalopowerlawindi esmostlyfallinthe

[ −2.3, −3.0]

^and

[ −3.6, −5.1]

^{ranges. Ourinnerpower lawindex}

n in = −2.50 ± 0.04

^{is onsistent}

with these results, parti ularly with the lower end. In the ase of the outer

halo power index (

n out = −4.85 ± 0.04

^{), the omparison is less trivial. First,}

only Sesar et al. (2011) and Deason et al. (2011) have provided measurements

for

n out

^{basedonts with a free}

q

^{parameter (}

n out = −3.8 ± 0.1

^and

−4.6 ^+0.2 −0.1

^,

respe tively). Se ond,onlyone work with

n out

measurements(Sesaretal.2011) uses a stellar tra er similar to ours (the others use A-BHB and BS stars, or

RRLyraestars). Most important, a good onstraint on

n out

^{requiresdeep data,}

andnoneofthese earliersurveys rea h asdeepas ourdataset. Oursteep outer

index,although well in therangeofprevious measurements, might wellindi ate

aprogressivesteepeningofthehalodensity,thoughitwouldbegoodtotestthis

with additional sight lines of omparable depth. In any ase, it seems safe to

on ludethat

n out < −4.0

^.

Thebest t values forthe polaraxis ratioor oblateness

q

^{range from}

0 .5

^to

0 .9

^{,withmostofthe}measurements on entratedwithin

(0 .55, 0.70)

^{. Thevalues}

of

q

^{donotseemtodepend onwhethera break was dete tedor not,noronthe}

limitingdistan eofthesurveyoronthestellartra er. Thedis repan ies anthus

beattributed either tomethodologi al dieren esorto dieren esin thespatial

overage of the data samples. However, it is di ult to determine the a tual

ause. Our results(

q = 0.79 ± 0.02

^{)do nott well within themost} onstri ted rangebut rathermat htheupperpart ofthebroaderrange.

Finally it is noteworthy that the hoi e of stellar tra er a ross the dierent

worksdoesnotseemto auseanysigni antbiasonthebest tparameters.

Dis ussion

ofthe

0 .2

^and

0 .4

^{mag datasets)and thosereported byothergroups inprevious works. Thedierent workshavebeen}

labelledasfollows: J08(Juri¢et al.2008),S11(Sesaret al.2011),D11 (Deasonetal.2011),dJ10(deJonget al.2010),

Ch01(Chenetal.2001),B08(Belletal.2008),F14(Fa iolietal.2014),andS10(Sesaretal.2010a)andW09(Watkins

et al.2009) as reanalysedin F14. Thettedmodelsin F14,S10 and W09 havexed oblatenessand testtwo dierent

valuesmotivatedbythepreviousndingsinS11 andD11.

Work stellartra er dist.range(kp )

χ ² _red R br (kpc) n n in n out q w

thiswork-axisym. near-MSTO

[10, 60]

^{1.9 }

−4.28 ± 0.06

^{ }

0.78 ± 0.04

^

thiswork-triax. near-MSTO

[10, 60]

^{1.9 }

−4.26 ± 0.06

^{ }

0.77 ± 0.04 0.88 ± 0.07

thiswork-broken near-MSTO

[10, 60]

^1.5

19.5 ± 0.4

^

−2.50 ± 0.04 −4.85 ± 0.04 0.79 ± 0.02

^

J08 near-MSTO

[5, 15] [2, 3]

^{ }

−2.8 ± 0.3

^

0.65 ± 0.15

^

S11 near-MSTO

[5, 35] 3.9 27.8 ± 0.8

^

−2.62 ± 0.04 −3.8 ± 0.1 0.70 ± 0.02

^{ex luded}

D11 A-BHB,-BS

[−, 40]

^

27.1 ± 1

^

−2.3 ± 0.1 −4.6 ^+0.2 _−0.1 0.59 ^+0.02 _−0.03

^

dJ10 multiple

[7, 30] [3.9, 4.2]

^

−2.75 ± 0.07

^{ }

0.88 ± 0.03

^

Ch01 MSTO

[−, 30]

^{ }

−2.5 ± 0.3

^{ }

0.55 ± 0.06

^

B08 MSTO

[5, 40] 2.2 ∼ 20 −3 ± 1

^{ }

[0.5, 0.8] ≥ 0.8

F14 RRLyrae

[9, 49] 0.8 28.5 ± 5.6

^

−2.8 ± 0.4 −4.4 ± 0.7 q f ix = 0.70 ± 0.01

^

" RRLyrae

[9, 49] 1.04 26.5 ± 8.9

^

−2.7 ± 0.6 −3.6 ± 0.4 q f ix = 0.59 ^+0.02 _−0.03

^

S10 RRLyrae

[9, 49] 1.1 34.6 ± 2.8

^

−2.8 ± 0.2 −5.8 ± 0.9 q f ix = 0.70 ± 0.01

^

" RRLyrae

[9, 49] 1.52 26.2 ± 7.4

^

−3.0 ± 0.3 −3.8 ± 0.3 q f ix = 0.59 ^+0.02 _−0.03

^

W09 RRLyrae

[9, 49] 1.1 27.6 ± 3.3

^

−2.5 ± 0.3 −4.3 ± 0.4 q f ix = 0.70 ± 0.01

^

" RRLyrae

[9, 49] 0.69 26.9 ± 3.1

^

−2.1 ± 0.3 −4.0 ± 0.3 q f ix = 0.59 ^+0.02 _−0.03

^

2.4.3 Dete tion of overdensities and identi ation

We analyse the data-to-models residuals for the dierent lines of sight in Fig-

ure 2.6b in sear h for overdensities. We nd that, in general, all the lines of

sightpresentregionswithdata-to-modelsdeviationsofamaximumfa toroftwo.

Additionally, ertainlinesof sightC,D, G,andHpresentmoresigni antde-

viations spanning from a few kiloparse s to tens ofkiloparse s in distan e. We

dis ussthese overdensitiesin greater detailbelow, and wealso dis uss expe ted

overdensitiesthat shownosignaturein ourdata.

Themost prominentoverdensities in thedata-to-modelresiduals orrespond

to the northern wrap of the Sagittarius (Sgr) stream. This stream overlaps in

proje tion with groups G and H (see Figure 2.8). For group G, the residuals

indi ateoverdensitiesin thedistan erangewhereweexpe ttondboththeSgr

andthe Orphanstream (

20 < D hC . 40

^{kp or}

25 < D GC . 44

^{kp ,Pila-Díez}

et al. (2014)). The overdensities indeed peak between

R GC = 25

^{kp and}

45

kp , rea hing

ρ/ρ M = 7 ± 2

^{,and drop sharply}afterwards. GroupH probesthe Sgrstream losertotheGala ti entrebut alsoforlarger distan esthangroup

G. Based both on extensive data (summarized in Pila-Díez et al. (2014)) and

in models (Law & Majewski (2010b) and Peñarrubia et al. (2010)), we expe t

this stream to span the

20 < D hC < 60

^{kp or}

16 < R GC < 55

^{kp range at}

these oordinates. Thisexpe tation ismetallalong: theysteadily in reasefrom

R GC ≈ 15

^{kp ,departfrom}

ρ/ρ M = 3 ± 1

^at

R GC = 30

^{kp ,rea h}

ρ/ρ M = 6 ± 2

at

R GC = 40

^{kp and peak at}

R GC = 45

^{kp with}

max(ρ/ρ M ) = (12 , 15) ± 2

^.

However,theydonotde reasenear

R GC = 55

^{kp but seemtostaystable with}

a signi ant

ρ/ρ M > 7 ± 2

^{). This suggests a thi ker bran h than predi ted by}

themodels,butin agreementwithpreviousRRLyraemeasurements(Ibataetal.

(2001 ),Totten&Irwin(1998)andDohm-Palmeretal.(2001)assummarizedin

Figure17ofMajewskietal.(2003)).

Twomoremodestoverdensitiesthatdonotappearintheliteratureseemtobe

presentingroups CandD.Ingroup C,a weakbut onsistentoverdensityspans

adistan e rangeof

R GC ≈ 35

^{kp to}

R GC ≈ 60

^{kp . Ingroup D,asharpbump}

extendsoverafewkiloparse s around

R GC ≤ 20

^{kp .}

Wehavelookedforotherknownoverdensitiesthatposition-mat hourlinesof

sight(seeFigure2.8),butfoundnoindi ationofthemintheresiduals. Therst

one orresponds to the tidal tails of the NGC5466 globular luster (Belokurov

etal. 2006a),whi h overlap withone eld in groupA and anotherone ingroup

B (A1361 entred at

(RA, Dec) = (176.09, 46.39)

^{and A1927 at}

(RA, Dec) = (217.92, 25.67)

^{). Thisisaveryweak old}substru turelo atedat

R GC ≈= 16

^kp

andextending for

45 deg

^{with anaverage widthof}

1.4 deg

^{(Grillmair&Johnson}

2006). Assu h,itisnotsurprising tondnosignaturein thedensityproles.

These ondone istheensembleof threeknown overdensities inthedire tion

ofgroupE:theAntiCenterStream(

R GC = 18 ± 2

^{kp ,}Ro ha-Pintoetal.(2003) andLietal.(2012)),theMono erosring(

R GC ≈ 18

^{kp ,Lietal.(2012))andthe}

EasternBandStru ture(

R GC = 20 ±2

^{kp ,Lietal.(2012)). These}substru tures

Dis ussion

−50 0

50 100

150 200

250 RA (degrees)

−40

−20 0 20 40 60 80

D EC ( de gr ee s)

EBS G&D ACS

Pisces NGC5466

Orphan

Tri-And A

B

C D

E

G F H

Figure2.8: Equatorialmapshowingthepositionofalltheeldsusedinthisworkandthe losest oldstellaroverdensities

tothem. Theseoverdensitiesareusedfor omparisonanddis ussionofthestellardensityproledata-to-modelresiduals

throughout se tion 2.4.3. The labels in the gure orrespond to the Anti entre Stru ture (ACS), the Eastern Band

Stru ture (EBS),theNGC5466stream, theGrillmair &Dionatosstream (G&D),theOrphanstream, theTriangulum-

Andromedaoverdensity(Tri-And)andthePis esoverdensity. Theba kgroundimageistheSDSS-DR8mapfromKoposov

et al. (2012), whi h shows the footprint of the Sagittarius stream. The Mono eros ring also appears partially in this

ba kgroundimage,as adarkregionoverlappingthewesternpartoftheGala ti diskintheanti entreregion,eastwards

oftheACS.

aremaskedfrom ourtsandresidualswhenweimpose

|z| > 10

^{kp toavoidthe}

inuen eofthethi kdisk,andtherefore,they annotbedete ted.

TheTriangulum-Andromeda overdensity((Martin et al. 2007)) falls lose to

oneoftheeldsingroupF.Despitethisproximity,theresidualsshownoeviden e

foranoverdensityattheexpe teddistan eof

R GC ≈ 30

^{kp ,indi atingthatthe}

overdensitydoesnotextendfurtherin thisdire tion.

2.5 Con lusions

Inthispaper wehaveused wide-eldimages from theCFHTand theINT tele-

s opes in eight broad lines of sight spread a ross the sky to produ e deep pho-

tometri ataloguesofhalo nearmain sequen eturno(near-MSTO) stars. Our

imageshavebeen orre tedforPSFinhomogeneities,resultingin atalogueswith

xed-aperture olourmeasurementsandimprovedstar-galaxyseparation. Thanks

tothedepthandqualityofourdata,werea hstellar ompletenesslimitsranging

from

22 .7

^{mag to}

24 .2

^maginthe

r

^{band,whi htranslate intoa}

60

kp distan e limitfornear-MSTOstars.

We al ulate gala to entri distan es forthe starsbasedon thephotometri

parallax method by Ivezi¢ et al. (2008) and the metalli ity estimator by Bond

etal.(2010). Webinthembydistan emodulus,and al ulatethestellarnumber

densitydistribution alongtheeightdierentlinesofsight.

In sele ting the halo near-MSTO stars, we have used additional onstraints

than the standard

0 .2 < g − r < 0.3

^and

g, r, i > 17

^{uts in order to obtain}

a leanersample. Parti ularly, by applyingadditional utsbased ong-i olour,

absolutemagnitudeandmetalli ity,wegetasampleofmainlyFstarssigni antly

de ontaminated fromquasarsandwhitedwarf-Mdwarfpairs.

Wetseveralgala ti halomodelsofthestellardistributiontooureightlines

ofsight,andexplorethestru turalparametersresultingfromthebestts,aswell

astheinuen eofsubstru tureinthoseparameters. Wendthatthehaloisbest

represented by a broken power law with index

n in = −2.50 ± 0.04

^{in the inner}

halo(

R < R break = 19 .5 ± 0.04

^)and

n out = −4.85 ± 0.04

^{intheouterhalo. Our}

data annot onstrainwhethera hangein thepolaraxis ratioalsoa ompanies

the break in the halo. The best t values for the polar axes ratio indi ate a

moderatelyoblatehalo:

q = 0.79 ± 0.02

^{. Thesimpler}(non-broken)triaxialpower lawmodelsfavourapra ti ally axisymmetri halo,with

w ≥ 0.88 ± 0.07

^andthe

restofparametersequal tothoseoftheaxisymmetri one.

Wendthatttingmodelstodatathat ontainssubstantialsubstru ture an

biassigni antlytheper eptionoftriaxiality,de reasingthediskaxis ratio

w

^by

10%

. Wealso ndthatdierentdistan e modulusbinsizes andthein lusionor ex lusionofparti ularlinesofsight anmoderatelyinuen eourmeasurementsof

somestru turalparameters. This allsfor arefully raftedanalysis andtailored

tests in any future studies. When ompared to previous works, the hoi e of

stellartra erseemstohavenosigni antinuen eonthevaluesofthestru tural

parameters,atleastforthesedistan e ranges.

Comparingourdensityprolestothesmoothmodelts,were over thepres-

en eoftheSagittariusstream ingroupsGandH. TheSagittariusstream inthe

dire tion ofgroup H seemsto extendfurther out from theGala ti entre than

themodelshaveso farpredi ted,and onrmsprevious RRLyraedete tionsas-

so iated withthestreamat su h distan es(Ibata et al.(2001 ), Totten&Irwin

(1998) and Dohm-Palmer et al. (2001)). We also nd eviden e of more modest

substru turesextendingoveralongrangeofdistan esingroupC(

35 ≤ R GC ≤ 60

kp ) andquite on entratedindistan ein groupD (

R GC ≈ 20

^{kp ).}

Ourpen ilbeamsurveyhasdemonstratedthatevenarelativelysmallnumbers

ofnarroweldsofview,providedtheyaresampledsu ientlydeepandwithan

abundant tra er, an pla e ompetitivelimits on the global density prole and

shape of the Gala ti halo. The advent of similarly deep, wide-area surveys -

like KiDS,VIKING andLSST- therefore promisesto enhan e substantiallyour

understandingofthehalo.