Stellar radial density proles

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 of

dis-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(yellowupward

trian-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.

In document Title: Structure and substructure in the stellar halo of the Milky Way (pagina 38-47)