estimating the dark matter halo mass of our milky way using dynamical tracers

8/9/2019 Estimating the Dark Matter Halo Mass of Our Milky Way Using Dynamical Tracers

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Mon. Not. R. Astron. Soc. 000 , 000–000 (0000) Printed 13 February 2015 (MN L ATEX style le v2.2)

Estimating the dark matter halo mass of our Milky Wayusing dynamical tracers

Wenting Wang 1, Jiaxin Han 1 , Andrew Cooper 1, Shaun Cole 1, Carlos Frenk 1,Yanchuan Cai 1, Ben Lowing 11 Institute for Computational Cosmology, University of Durham, South Road, Durham, DH1 3LE, UK

13 February 2015

ABSTRACTThe mass of the dark matter halo of the Milky Way can be estimated by ttinganalytical models to the phase space distribution of dynamical tracers. We test thisapproach using realistic mock stellar halos constructed from the Aquarius N-bodysimulations of dark matter halos in the ΛCDM cosmology. We extend the standardtreatment to include a Navarro-Frenk-White (NFW) potential and use a maximumlikelihood method to recover the parameters describing the simulated halos from thepositions and velocities of their mock halo stars. We nd that the estimate of halomass is degenerate with the estimate of halo concentration. The best-t halo masseswithin the virial radius, R 200 , are biased, ranging from a 40% underestimate to a5% overestimate in the best case (when the tangential velocities of the tracers areincluded). There are several sources of bias. Deviations from dynamical equilibriumcan potentially cause signicant bias; deviations from spherical symmetry are relativelyless important. Fits to stars at different galactocentric radii can give different massestimates. By contrast, the model gives good constraints on the mass inside 0 .2R 200 .

Key words: Galaxy: Milky-Way

1 INTRODUCTION

Our Milky Way (MW) galaxy provides a wealth of informa-tion on the physics of galaxy formation and the nature of the dark matter. This information can, in principle, be un-locked from studies of the positions, velocities and chemistryof stars in the Galaxy, its satellites and globular clusters,which can be observed with high precision.

Many inferences derived from the properties of theMilky Way (MW) depend on the precision and accuracy

with which the mass of its dark matter halo can be es-timated. An example is the much-publicized “too big tofail” problem, the apparent lack of MW satellite galaxieswith central densities as high as those of the most massivedark matter subhalos predicted by ΛCDM simulations of ‘Milky Way mass’ hosts ( Boylan-Kolchin et al. 2011 , 2012;Ferrero et al. 2012 ). In these simulations the number of massive subhalos depends strongly on the assumed MWhalo mass and the problem disappears if the MW halomass is sufficiently small ( < 1 ×1012 M ; Wang et al. 2012 ;Cautun et al. 2014 ).

There are many different ways of constraining

the MW dark matter halo mass 1 . These include tim-ing argument estimators ( Kahn & Woltjer 1959 ) cali-brated against N -body simulations ( Li & White 2008 );the kinematics of bright satellites ( Sales et al. 2007b ,a;Barber et al. 2014 ; Cautun et al. 2014 ), particularly LeoI (Boylan-Kolchin et al. 2013 ) and the Magellanic Clouds(Busha et al. 2011 ; González et al. 2013); the kinematics of stellar streams ( Newberg et al. 2010 ), especially the Sagit-tarius stream ( Law et al. 2005 ; Gibbons et al. 2014 ); mea-surements of the escape velocity using nearby high veloc-ity stars, such as those from the RAVE survey ( Smith et al.2007; Piffl et al. 2014); and combinations of photometric and

kinematic data such as Maser observations and terminal ve-locity curves ( McMillan 2011 ).Some authors have used large composite samples of ob-

jects assumed to be dynamical tracers in the halo, such asstars, globular clusters and planetary nebulae. For example,the halo circular velocity, V circ , may be inferred from theradial velocity dispersion of tracers, σr (r ), using the spher-ical Jeans equation. Such methods require the tracer veloc-ity anisotropy and density proles to be known or assumed.Battaglia et al. (2005) made use of a few hundred stars and

1 We use M 200 to denote the mass of a spherical region with meandensity equal to 200 times the critical density of the Universe.

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globular clusters from 20 to 120 kpc; Xue et al. (2008) used2401 BHB stars from SDSS/DR6 ranging from 20 to 60 kpc;Gnedin et al. (2010) used BHB and RR Lyrae stars rangingfrom 25 to 80 kpc; and Watkins et al. (2010) used 26 satel-lites within 300 kpc. Most recently Kae et al. (2012, 2014)used a few thousand BHB stars extending to 60 kpc andK-giants beyond 100 kpc.

Most measurements based on dynamical tracers involveassumptions about the tracer density proles and veloc-ity anisotropies. However, Wilkinson & Evans (1999) intro-duced a Bayesian likelihood analysis, based on tting amodel phase space distribution function to the observed dis-tances and velocities of tracers. In their analysis the tracerdensity prole and velocity anisotropy can be consideredas free parameters of the distribution function, to be con-strained together with parameters of the host halo such as itsmass and characteristic scalelength. The sample of stars usedby Wilkinson & Evans (1999) was small and their best-thost halo mass for a truncated at rotation curve model was1.9+3 . 6− 1.7 ×1012 M (see also Sakamoto et al. 2003 ). Most re-cently, Deason et al. (2012 ) used a few thousand BHB starsfrom SDSS up to r 50 kpc.Fig. 1 summarizes the results of these studies. Thex-axis is the measured MW halo mass. We have con-verted results to M 200 by assuming an NFW density pro-le (Navarro et al. 1996a , 1997a) and using the mean haloconcentration relation of Duffy et al. (2008) in cases wherea value for concentration is not given in the original study.The measurements are grouped by methodology, indicatedby colours and labeled along the y-axis. We group thosemethods that use large samples of dynamical tracers intotwo sets: 1) those based on the radial velocity dispersion of the tracers and spherical Jeans equation to infer the circularvelocity and underlying potential; 2) those based on ttingto model distribution functions, which attempt to constrain

both halo mass and the velocity anisotropy of the tracerssimultaneously. Errorbars correspond to those quoted bythe original authors; we have converted 90% or 95% con-dence intervals to 1 σ errors assuming a Gaussian distri-bution. Wilkinson & Evans (1999), Sakamoto et al. (2003)and Watkins et al. (2009) included systematic errors in theirmeasured masses, which makes their errors relatively large.Fig. 1 shows that existing measurements of the most likelyMW halo mass differ by more than a factor of 2.5, evenwhen similar methods are used, although apart from a fewoutliers, the estimates are statistically consistent.

Here we are particularly interested in methods such asthat of Wilkinson & Evans (1999), which treat the spatialand dynamical properties of tracers as free parameters tobe constrained under the assumption of theoretical phasespace distribution functions. The primary aim of this pa-per is to test the model distribution functions used in thisapproach. We extend the distribution function proposed byWilkinson & Evans (1999) to one based on the NFW poten-tial, and model the radial proles of tracers with a more gen-eral double power-law functional form. The model functionis then t to the phase space distribution of stars in real-istic mock stellar halo catalogues constructed from the cos-mological galactic halo simulations of the Aquarius project(Springel et al. 2008 ), to understand its reliability and pos-sible violations to the underlying assumptions. Our resultshave implications that are not limited to the specic form

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Figure 1. Measured MW halo masses in the literature ( x-axis),converted to M 200 and categorized by methodology ( y-axis). Mea-surements using similar methods and/or tracer populations areplotted with the same colour. Categories include the timing argu-ment estimator (black); constraints from luminous MW satellitessuch as the Magellanic Clouds (magenta), Leo I(cyan), the orbitsor radial velocity dispersion of other bright satellites (yellow) andtheir V max distribution (light green); modeling of the Sagittariustidal stream (grey); high velocity stars from the RAVE survey(blue); combinations of maser observations and and the terminalvelocity curve (pink); and (of most relevance to this work) dy-namical modeling using large samples of dynamical tracers (redand green). Methods involving large samples of dynamical tracersare split into two categories, 1) those based on the radial veloc-ity dispersion of tracers (green) and 2) those using model dis-tribution function to constrain both halo properties and velocityanisotropies of tracers simultaneously (red). We have convertedresults to M 200 by assuming an NFW density prole and a com-mon mass-concentration relation. 95% or 90% condence intervalshave been converted to a 1 σ error by assuming a Gaussian errordistribution.

of the distribution function that we test, but are applicableto the method itself.

This paper is structured as follows. The mock stellarhalo catalogues are introduced in Section 2. Detailed de-scriptions of the model distribution function and the max-imum likelihood approach are provided in Section 3. Ourresults are presented in Section 4, with detailed discus-sions of reliability and systematics in Section 5 and Sec-tion 6. We conclude in Section 7. Throughout this paperwe adopt the cosmology of the Aquarius simulation series(H 0 = 73 km s − 1 Mpc − 1 , Ωm = 0 .25, ΩΛ = 0 .75 and n = 1).

2 MOCK STELLAR HALO CATALOGUE

We use mock stellar halo catalogues constructed from theAquarius N-body simulation suite (Springel et al. 2008 )with the particle tagging method described by Cooper et al.

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(2010), to which we refer the reader for further details. Inthis section we summarise the most important features of these catalogues.

2.1 The Aquarius simulations

The Aquarius halos come from dark matter N-body simu-lations in a standard ΛCDM cosmology. Cosmological pa-rameters are those from the rst year data of WMAP(Spergel et al. 2003 ). Our work uses the second highest res-olution level of the Aquarius suite, which corresponds to aparticle mass of 104 h− 1 M .The simulation suite includes six dark matter halos withvirial masses spanning the factor-of-two range of Milky Wayobservations discussed in the previous section. We have onlyused ve out of the six halos for our analysis (labeled haloA to halo E according to the Aquarius convention). Thehalo we have not used (halo F) undergoes two major mergerevents at z < 0.6, and is thus an unlikely host for a MW-like disc galaxy. We list in Table 1 the host halo mass, M 200 ,and other properties of the ve halos, which are taken fromNavarro et al. (2010).

2.2 The galaxy formation and evolution model

The Durham semi-analytical galaxy formation model, GAL-FORM, has been used to post-process the Aquarius simula-tions, predicting the evolution of galaxies embedded in darkmatter halos. To construct the mock stellar halo cataloguesused in this paper, the version described by Font et al.(2011) was adopted. This model has several minor differ-ences from the model of Bower et al. (2006 ), such that theFont et al. (2011) model matches better the observed lu-minosity function, luminosity-metallicity relation and radial

distribution of MW satellites. The main changes are a moreself-consistent calculation of the effects of the photoioniza-tion background and a higher chemical yield in supernovaefeedback.

2.3 Particle tagging

The GALFORM model predicts the amount of stellar masspresent in each dark matter halo in the simulation at eachoutput time, as well as properties of stellar populationssuch as their total metallicity. However, GALFORM doesnot provide detailed information about how these starsare distributed in galaxies. The particle tagging methodof Cooper et al. (2010) is a way to determine the six-dimensional spatial and velocity distribution of stars fromdark matter only simulations, by associating newly-formedstars with tightly bound dark matter particles.

At each simulation snapshot, each newly formed stellarpopulation predicted by GALFORM is assigned to the 1%most bound dark matter particles in its host dark matterhalo. Each “tagged” dark matter particle then represents afraction of a single stellar population, the age and metallicityof which are also known from GALFORM. Traced forwardto the present day, these tagged particles give predictions forthe observed luminosity functions and structural propertiesof MW and M31 satellites that match well to observations.Recently, Cooper et al. (2013) have applied this technique to

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Figure 2. The radial density proles of stellar mass (red pointsand lines with errors) in the mock stellar halo catalogue of theve Aquarius halos. The errors are in most cases of comparablesize to the symbols and are almost invisible. Dashed black curvesare double power-law ts to the red data points obtained froma χ 2 minimization. Green dashed curves are the best-t densityproles from the maximum likelihood method.

large-scale cosmological simulations and have shown that itproduces galactic surface brightness proles that agree wellwith the outer regions of stacked galaxy proles from SDSS.

Our study is based on tagged dark matter particlesfrom accreted satellite galaxies. We ignore particles asso-ciated with in situ star formation in the central galaxy.Strictly, our results thus only apply in the case where mostMW halo stars originate from accretion. This is supportedby the data o fBell et al. (2008, 2010) although other worksuggests that a certain fraction of the halo stars are con-tributed by in-situ star formation, especially close to thecentral galaxy ( r < 30 kpc) (see, e.g. Carollo et al. 2007 ,2010; Zolotov et al. 2010 ; Helmi et al. 2011 ). Ignoring thepossible in-situ component is thus a weakness of our mockstellar halo catalogue. Nevertheless, our mock halo stars en-able us to test and constrain the theoretical distributionfunction and, in practice, most of our conclusions (see Sec. 4and Sec. 5) do not depend on whether the MW halo starsformed in-situ or were brought in by accretion.

3 METHODOLOGY

In this section we discuss the theoretical context of ourmethod for constraining dark matter halo properties us-ing dynamical tracers and a maximum likelihood approachbased on theoretical distribution functions. In Sec. 3.1, wedescribe how the phase-space distribution of the tracer pop-ulation is modeled. Sec. 3.2 gives details about the explicitform of the distribution function. The likelihood function isintroduced in Sec. 3.3. Finally, we describe how we weight

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tagged particles and how errors are estimated in Sec. 3.4.Our method follows that of Wilkinson & Evans (1999 ) butintroduces signicant modications to the form of the darkmatter halo potential and the assumed tracer density prole.

3.1 Phase-space distribution of Milky Way halostars

The phase-space distribution function of tracers (e.g. stars)bound to a dark matter halo potential (binding energy E >0) can be described by the Eddington formula ( Eddington1916). The simplest isotropic and spherically symmetric caseis

F (E ) = 1√ 8π 2d

dE E

Φ( r max , t )

dρ(r )dΦ( r )

dΦ(r )

E −Φ(r ), (1)

where the distribution function only depends on the bindingenergy per unit mass, E = Φ( r ) −v

2

2 . Φ(r ) and v2 / 2 are the

underlying dark matter halo potential and kinetic energy perunit mass of tracers. The integral goes from the potential atthe tracer boundary 2 to the binding energy of interest. Usu-ally both the zero point of potential and tracer boundary,r max , t , are chosen at innity, and thus Φ( r max , t ) = 0.

In reality the velocity distribution of tracers may beanisotropic and depend both on energy and angular mo-mentum, L. In the simplest case, the distribution functionis assumed to be separable:

F (E, L ) = L− 2β f (E ), (2)

where the energy part, f (E ), is expressed as (Cuddeford1991)

f (E ) = 2β − 3/ 2

π 3/ 2 Γ(m −1/ 2 + β )Γ(1 −β ) ×d

dE E

Φ( r max , t )(E −Φ)

β − 3/ 2+ m dm [r 2β ρ(r )]dΦm

dΦ

= 2β − 3/ 2

π 3/ 2 Γ(m −1/ 2 + β )Γ(1 −β ) ×

E

Φ( r max , t )(E −Φ)β

− 3/ 2+ m dm +1 [r 2β ρ(r )]dΦm +1

dΦ.

(3)

Here β is the velocity anisotropy parameter dened as

β = 1 − vθ 2 − vθ 2 + vφ 2 − vφ 2

2( vr 2

−vr 2 )

, (4)

with vr , vθ and vφ being the radial and two tangential com-ponents of the velocity. The integer, m, is chosen to makethe integral converge and depends on the value of β . In ouranalysis the parameter range of β is −0.5 < β < 1 andm = 1. β > 0 represents radial orbits, while tangential or-bits have β < 0. β = 0 corresponds to the isotropic velocitydistribution.

In real observations, the tangential velocities of tracersare often unavailable. We thus test two different cases, in

2 To dene the binding energy, we adopt the convention thatΦ(r ) > 0.

which i) only radial velocities are available and ii) both ra-dial and tangential velocities are available. For case (i), thephase-space distribution in terms of radius, r , and radial ve-locity, vr , is given by the integral over tangential velocity,vt = v2θ + v2φ , as

P (r, v r |C ) = L− 2β

f (E )2πv t dvt , (5)where C denotes a set of model parameters. With theLaplace transform, this can be written as

P (r, v r |C ) = 1√ 2πr 2β

E r

Φ( r max , t )

dΦ√ E r −Φ

dr 2β ρsdΦ

, (6)

where E r = Φ( r ) − v2r / 2. All factors of m cancel in theLaplace transform and hence Eqn. 6 does not depend on m.For case (ii), the distribution function is simply Eqn. 2, i.e.

P (r, v r , vt |C ) = L− 2β f (E ), (7)

where E = Φ( r ) −v2r / 2 −v2t / 2 and L = rv t .

3.2 NFW potential and double power-law densityproles of the tracer population

Wilkinson & Evans (1999) and Sakamoto et al. (2003)adopted the so-called truncated at rotation curve modelfor the underlying dark matter potential. In our analysis,we will extend Eqn. 2 to the NFW potential ( Navarro et al.1996b, 1997b)

Φ(r ) = −4πGρ s r2s

ln(1 + r/r s )r/r s

+ 11 + r max ,h /r s

, (8)

when r < r max , h , and

Φ(r ) = −4πGρ s r 2sln(1 + r max , h /r s )

r/r s+ rmax , h /r s

(r/r s )(1 + r max ,h /r s ),

(9)when r > r max , h .

There are two parameters in Eqn. 8 and Eqn. 9, thescalelength, rs , and the scaledensity, ρs , dened at r = r s .r max , h is the halo boundary. If the halo is innite, the secondterm in Eqn. 8 vanishes. In most of our analysis, we willassume the NFW halo is innite. We test different choicesof halo boundary in the Appendix C.

To derive analytical expressions for Eqn. 6 and Eqn. 7,we need an analytical form for the tracer density prole,ρ(r ). Fig. 2 shows the radial density prole of stellar mass(red points) in each of the ve Aquarius halos. Error barsare obtained from 100 realizations of bootstrap resampling.In most of the cases, these proles can be described well by adouble power law (black dashed lines are double power-lawts that minimize χ 2 ). Signicant deviations from a doublepower law are most obvious in the outskirts of the halos. Forexample, halo E has a prominent bump at r 100 kpc dueto a tidal stream.

There are indications that the real MW has a two-component prole, with density falling off more rapidlybeyond 25 kpc, whereas M31 has a smooth proleout to 100 kpc with no obvious break (e.g. Watkins et al.

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2009; Deason et al. 2011 ; Sesar et al. 2011 ). Recently,Deason et al. (2014) report evidence for a very steep outerhalo prole of the MW. If we believe that MW halo starsoriginate from the accretion of dwarf satellites, whether theprole is broken or unbroken depends on the details of accre-tion history ( Deason et al. 2013 ; Lowing et al. 2014). Thereis an as yet unresolved debate over whether the stellar halo

of the MW has an additional contribution from stars formedin situ, in which case a break in the prole may reect thetransition from in situ-dominated regions to accretion dom-inated regions.

As our mock halo stars (which are all accreted) andobserved MW halo stars can be approximated by a doublepower-law prole, we adopt the following functional form tomodel tracer density proles:

ρ(r ) rr 0

α

+ rr 0

γ − 1

. (10)

This equation has three parameters: the inner slope, α , theouter slope, γ , and the transition radius, r0 .

Previous studies have adopted a single power law todescribe the density prole of MW halo stars beyondr 20 kpc (e.g. Xue et al. 2008; Gnedin et al. 2010 ;Deason et al. 2012 ; Wilkinson & Evans 1999 ). Our doublepower-law form naturally includes this possibility as a spe-cial case. We also note that Sakamoto et al. (2003) consid-ered the case of “shadow” tracers with a radial distributionthat shares the same functional form with the underlyingdark matter. We emphasize that our mock halo stars are not“shadow” tracers; their radial distribution is signicantlydifferent from that of the dark matter.

Assuming these analytical expressions for Φ( r ) andρ(r ), Eqn. 6 and Eqn. 7 can be written more explicitly as

P (r, v r |ρs , r s ,β ,α ,γ , r 0 ) = − r2β − α − γ s√ 2πr 2β vs

R max , t

R inner

R ′ 2β − 1

(r ) −φ(R ′ )×

(2β −α)( R′

r 0 )α r − γ s + (2 β −γ )( R

′

r 0 )γ r − αs

[( R′

r 0 )α r− γ s + ( R

′

r 0 )γ r− αs ]2

dR ′ ,

(11)

and

P (r, v r , vt |ρs , r s ,β ,α ,γ , r 0 ) =

− r − α − γ s l− 2β

23/ 2− β π 3/ 2 v3s Γ(β + 1 / 2)Γ(1 −β ) ×

R max , t

R innerdR ′ ( (r ) −φ(R

′ )) β − 1/ 2×(2β + 1) R ′ 2β R

′

1+ R ′ −ln(1 + R ′ ) − 1(1+ R ′ ) 2 − 11+ R ′ R ′ 2β +1 R ′1+ R ′ −ln(1 + R ′ )

2 ×

(2β −α ) R′

r 0

αr − γ s + (2 β −γ ) R

′

r 0

γ r − αs

R ′r 0

αr − γ s + R

′

r 0

γ r − αs

2 +

R ′ 2β +1

R ′1+ R ′ −ln(1 + R ′ ) R

′

r 0

αr − γ s + R

′

r 0

γ r − αs

2 ×

(2β −α)r− α − γ s

αr 0 −

2γ r 0

R ′

r 0

α + γ − 1

+

(2β −γ )r − α − γ s γ r 0 − 2αr 0 R′

r 0

α + γ − 1

−(2β −α)r

− 2γ s

αr 0

R ′

r 0

2α − 1

−(2β −γ )r− 2αs

γ r 0

R ′

r 0

2γ − 1

.

(12)

Here, analogously to Wilkinson & Evans (1999), we haveintroduced a characteristic velocity, vs = rs √ 4πGρ s . Thebinding energy, , angular momentum, l, potential, φ, andradius, R, have all been scaled by vs and rs and are thusdimensionless, as follows,

= E v2s

, l = Lr s vs

, φ = Φv2s

, R = rr s

. (13)

As mentioned above, Rmax , t is the boundary of the tracerdistribution and, for most of our analysis, we take R max , t =

∞. Note that Eqn. 11 and Eqn. 12 are deduced by assumingthe tracer boundary, R max , t , is smaller or equal to the haloboundary, Rmax , h . In both Eqn. 11 and Eqn. 12 there aresix model parameters.

The phase-space probability of a tracer at radius, r ,whose radial and tangential velocities are vr and vt , canbe derived from Eqn. 11 or Eqn 12. The lower limit of theintegral is determined by solving

φ(R inner ) = , (14)

where equals φ(R) −v2r / (2v2s ) when only the radial ve-locity is available, and equals φ(R)−

v2r / (2v2s )

−v2t / (2v2s )

when tangential velocity is also available. The fact that theintegral goes from R inner to Rmax , t indicates that the phase-space distribution at radius r has a contribution from trac-ers currently residing at larger radii, whose radial excursionincludes r .

3.3 Likelihood and window function

The probability of each observed tracer object, labeled i,with radius, r i , radial velocity, vri , and tangential velocity,vti , is

P i (r i , vri , vti |ρs , r s ,β ,α ,γ , r 0 ). (15)c 0000 RAS, MNRAS 000 , 000–000


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Dynamical tracers, such as MW globular clusters, BHBstars and satellites, are subject to selection effects. For ex-ample, sample completeness is often a function of appar-ent magnitude (hence distance). If we assume that all selec-tion effects can be described by a window function, then theprobability of nding each tracer object, i, within the datawindow is given by the normalized phase-space density

F i = P i

window P d3 r d3 v. (16)

The integral in the denominator runs over the phase-spacewindow. The likelihood function then has the following form:

L =i

F i . (17)

It can easily be shown that this likelihood function isequivalent to the extended likelihood function marginalizedover the amplitude parameter of the phase-space density(e.g. Barlow 1990 ), which we are not interested in. For ourmock MW halo star catalogue, we deliberately exclude stars

in the innermost region of the halo. These stars have ex-tremely high phase-space density and so make a dominantcontribution to the total likelihood, strongly biasing the t.We nd that excluding all stars at r < 7 kpc removes thisbias3 . The window function in our analysis is then simplyP = 0 at r < 7 kpc. In real observations, the window func-tion can be much more complicated.

We seek parameters that maximize the value of the like-lihood function dened in Eqn. 16 and Eqn. 17. In order tosearch the high-dimensional parameter space efficiently, weuse the software IMINUIT , which is a python interface of the MINUIT function minimizer ( James & Roos 1975 ).

There are six parameters in Eqn. 11 or Eqn. 12. Tomake best use of the likelihood method, we treat all six

parameters as free. In previous work using this approach thethree parameters of the spatial part of the tracer distributionare often xed to their observed values. We have carried outtests and found that, as expected, three parameter modelsgive results consistent with those using six parameters onlywhen the choice of tracer density prole is close to the truedistribution. We recommend that all six parameters shouldbe left free if the observed sample size is large enough toavoid introducing unnecessary bias.

Another source of potential bias in the halo mass es-timates of previous studies arises from the use of universalmean mass–concentration relations for dark matter halos.In CDM simulations, the relation between halo mass andconcentration has very large scatter (e.g. Neto et al. 2007).

Taking halo A as an example, if we use the mass concen-tration relation from Duffy et al. (2008), the estimated con-centration would be around 5.7, which is almost three timessmaller than the true value (see Table 1). This would resultin an overestimate of halo mass by almost an order of magni-tude, and the corresponding scalelength, r s , would be threetimes larger. The huge scatter in the mass-concentration re-lation can cause catastrophic problems unless we are fortu-nate enough that the host halo of the MW does in fact lieon the mean mass-concentration relation.

3 A detailed discussion of the radial dependence of results fromour model is given in Sec. 6.

3.4 Weighting tagged particles

As described in Sec. 2.3, our mock catalogues are createdby assigning stars from each single age stellar population tothe 1% most bound dark matter particles in their host haloat the time of their formation. The total stellar mass of eachpopulation will obviously vary from one population to thenext (according to our galaxy formation model), as will thenumber of dark matter particles actually tagged (accordingto the number of particles in the corresponding formationhalo). The result is that stellar masses associated with in-dividual dark matter particles range over several orders of magnitude. Particles tagged with larger stellar masses cor-respond to more stars, and thus in principle should carrymore weight in the likelihood t.

To reect this we could simply reweight each particleaccording to its associated stellar mass, M ,i . However, in-dividual stars are not resolved: the phase-space coordinatesof the underlying dark matter particles comprise the max-imum amount of dynamical information available from thetagging technique. Therefore, we give each particle a weight(M ,i / Σ i M ,i )N tags . This conserves the total particle num-ber, N tags , but re-distributes this among particles in propor-tion to the fraction of the total stellar mass they represent.In this way we maintain a meaningful error estimated fromthe likelihood function.

We also randomly divide stars into subsamples and ap-ply our maximum likelihood analysis to each of these to es-timate the effects of Poisson noise. To do so, we assign eachweighted particle a new integer weight drawn from a Poissondistribution with mean equal to the weight given by the ex-pression above. We repeat this procedure 10 times, so thatwe have 10 different subsamples. The expectation values of the total weight for all tagged particles in these subsamplesare the same, so this approach can be regarded as analogousto bootstrap resampling. We nd this procedure yields con-sistent error estimates with that obtained from the Hessianmatrix of the likelihood surface.

We restrict our analysis to the 10% oldest tagged parti-cles in the main halo. This is to reect the fact that, in realobservations, old halo stars such as blue horizontal branch(BHB) and RR Lyrae stars are most often used as dynamicaltracers, because they are approximately standard candles.We also exclude stars bound to surviving subhalos.

3.5 Testing the method

Before tting the model distribution function to our realis-tic mock stellar halo catalogues, we test the method withideal samples of particles that obey Eqn. 12. We applied ourmaximum likelihood method to 750 sets of 1000 phase-spacecoordinates ( r , vr and vt ) each drawn randomly from thesame distribution function of the form given by Eqn. (12).Fig. 3 shows a comparison between the input halo parame-ters and the recovered best-t halo parameters. The x axis isthe ratio between the best-t and true-input halo mass, andthe y axis the ratio between best-t and true concentration.The red cross indicates the mean ratios averaged over all the750 realizations, which is very close to unity (horizontal andvertical dashed lines).

The best-t halo mass and concentration varies amongrealizations as a result of statistical uctuations, as shown in

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c t r u

e

Figure 3. The ratio between input and best-t halo masses ( x-axis) versus the ratio between input and best-t halo concentra-tions ( y-axis). The red cross is the mean ratio over 750 differentrealizations, which is very close to 1 on both axes (horizontal and

vertical dashed lines). Black solid contours mark the region in pa-rameter plane enclosing 68.3% (1 σ) and and 95.5% (2 σ) of bestt parameters among the 750 realizations.

Fig. 3. We note that the shape of these contours indicate acorrelation between the recovered halo mass and concentra-tion parameters. The correlation coefficient (i.e., normalizedcovariance) is -0.89, which implies a strong degeneracy in themodel parameter. We will discuss this degeneracy further inSec. 5. The above exercise reassures us that our methodworks with ideal tracers, so we can move on to apply it tothe more realistic mock halo stars in our simulations.

4 RESULTS

In this section we investigate how well the true halo masscan be recovered by tting Eqn. 11 to mock halo stars incases where: (a) only radial velocities are available (Sec. 4.1)and (b) both radial and tangential velocities are available(Sec. 4.2). In both cases we model the underlying potentialwith innite halo boundaries. We refer to parameters esti-mated with the maximum likelihood technique as best-t(or measured) parameters, to be compared with the real (ortrue) parameters taken directly from the simulation.

The total number of tagged particles we used in theve halos is shown in Table 2. These are of the order

of 105

, one or two orders of magnitude larger than thetracer samples used by Deason et al. (2012) or Kae et al.(2012). This permits a robust test of the method free fromthe effects of sampling uctuations. Future samples of ob-served tracers will grow with ongoing and upcoming surveyssuch as APOGEE ( Ahn et al. 2013 ) and Gaia ( Prusti 2012 ;Gilmore et al. 2012 ).

4.1 Radial velocity only

Fig. 4 shows, as black points, the best-t M 200 , c200 and β for our ve halos in the case where only radial velocities areknown. These best-t parameters are given in Table 1 along

A B C

D E

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Figure 4. The best-t values of M 200 , c200 and β for the vehalos (black dots with errors). Error bars are 1 σ uncertainties

obtained from the Hessian matrix and are almost invisible. The1σ errors are comparable to the scatter among the 10 subsamplesconstructed in Sec. 3.4. For direct comparison we show the truevalues of M 200 , c200 and β as red dots.

Table 2. The total number of tagged particles in each of our vesimulated halos.

A B C D Enumber 181995 225030 184197 365280 120806

with the true halo or tracer properties (shaded in grey),which are plotted as red points in Fig. 4.

Table 1 lists the true and best-t values of the host halomass ( M 200 ), halo concentration ( c200 ), scalelength ( r s ),scaledensity ( ρs ) and virial radius ( R200 ). Note only two of these parameters are independent. The best-t M 200 valuesare overestimates of the true values for halos B and D by140% and 7%, and underestimates for halos A, C and E by30%, 10% and 35% respectively. These biased estimates of halo mass and concentration are very signicant comparedwith the small tting errors.

The measured spatial parameters ( α , γ and r0 ) agreewell with the true values obtained from a double power-lawt to the stellar mass density, shown as black dashed lines in

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ters (red dashed curves) are a poor description of the mockstars in the left hand column, especially for halos A, C andD where we see a signicant over prediction of low angu-lar momentum particles. Halo E is the exceptional case inwhich we nd good agreement. The strongest disagreementin the other halos is, interestingly, mainly due to the biasedmeasurement of β . In the central column, where we x the

value of β to its best-t value (obtained using both velocitycomponents) we see that the model predictions agree muchbetter with the true phase-space distribution, although somediscrepancies remain.

For comparison, the right hand column of Fig. 7 showsthat model predictions based on the best-t parameters givean equally good match to the simulation data. Judging byeye alone, it would be hard to tell whether the middle col-umn shows better or worse agreement than the right handcolumn. However, judging according to the likelihood ratios,the best-t halo parameters are indeed a much better de-scription of the data than the true halo parameters ( 3σ).This is also reected in the small formal errors of the t.

5 SOURCES OF BIAS

Fig. 7 indicates that the model is able to recover the generalphase-space distribution of the mock halo stars, althoughthere are some subtle factors which signicantly bias ourbest-t parameter values relative to their true values. Thereare several possible sources of this bias:

• Both the underlying potential and the spatial distribu-tion of tracers may not satisfy the spherical assumption.• There are ambiguities in how to model the boundariesof halos.• The true dark matter distribution may deviate from theNFW model.• The velocity anisotropy ( β ) is not constant with radiusas assumed in the model.• Correlations among parameters make the model moresensitive to perturbations and, in some cases, a poor t to

one parameter will propagate to affect the others.

• Tracers may violate the assumption of dynamical equi-librium.We have investigated each of these possibilities and

found that violations of the spherical symmetry assumptionand ambiguity in the treatment of halo boundaries are rel-atively unimportant; hence we describe their effects in Ap-pendix B. We have investigated the density proles of theAquarius halos and found that halo A is not well t by anNFW prole; instead its inner and outer density proles arebetter described by two different NFW proles of differentmass and concentration. This might explain the systematicunderestimation of M 200 . For the other halos, the NFW formis a good approximation and thus deviations from it are notthe dominant source of bias.

In Fig. 6 we showed that the velocity anisotropy, β ,varies strongly with radius. The best-t value of β (which isassumed to be constant) turns out to show a signicant bias.However, when we investigate correlations among parame-ters (in the following subsection) we nd that, fortunately,this bias does not propagate to the other parameters of themodel. We therefore defer discussion of the origin of the bias

Table 3. Correlation matrix of model parameters for halo B

β M 200 c200 α γ r 0β 1.0 0.014 0.026 0.023 0.004 0.058M 200 0.014 1.0 -0.940 -0.300 0.427 -0.178c200 0 .026 -0.940 1.0 0.348 -0.408 0.060α 0.023 -0.300 0.348 1.0 0.068 0.563γ 0.004 0.427 -0.408 0.068 1.0 0.314r 0 0.058 -0.178 0.060 0.563 0.314 1.0

in β to Appendix A. In the following, we focus on correla-tions among model parameters and the dynamical state of tracers, which are the most important factors.

5.1 Correlations among model parameters

Fig. 3 demonstrated a strong degeneracy between M 200 andc200 . From a modeling perspective, this is dangerous: thereare multiple combinations of halo parameters that can givealmost equally good t to both the tracer density prole and

velocity distribution. In this subsection we ask what causesthis degeneracy and whether there are similar correlationsamong other parameters. In particular, we have seen thatthe model gives strongly biased estimates of the velocityanisotropy of stars, β . We want to check whether this biaspropagates to the other parameters. To this end, Table 3gives the normalized covariance matrix of the parametersrecovered for halo B, using both radial and tangential veloc-ities as constraints. The covariance matrix is qualitativelysimilar for the other halos.

The degeneracy between M 200 and c200 is very strong(covariance close to -1). To understand the origin of this de-generacy, we have explored the velocity distribution of trac-ers predicted by the model using different combinations of

M 200 and c200 . We verify that, if M 200 is increased, the pre-dicted velocity distribution of stars in the centre of the haloextends to larger velocities, with a corresponding reductionin the probability of smaller velocities. A decrease in c200 canroughly compensate for this change in the velocity distribu-tion. In terms of mass proles, the degenerate parameterspredict similar halo masses inside a certain characteristicradius. This radius is very close to the half-mass radius of the tracer population (more discussions will be presented inHan et al., in preparation). Conversely, the mass inside thecharacteristic radius changes rapidly for M 200 and c200 alonglines perpendicular to the degenerate contour. This suggeststhat the mass inside a xed radius close to the half-mass ra-dius of tracers is better constrained than the total halo mass.

In Fig. 8, we examine the halo mass proles with thebest-t parameters, normalized by proles with the true pa-rameters. Except for halo B, which gives an acceptable resultat all radii, the measurements are very close to the true valuefor r 0.2R200 but become biased at larger radii.

In contrast to the strong degeneracy between M 200 andc200 , the correlation between β and all the other parametersis very weak. This is fortunate, as it suggests the systemat-ically biased estimate of β will not introduce further bias tothe other parameters.

Correlations between halo parameters ( M 200 or c200 )and tracer spatial parameters ( α , γ and r 0 ) are at the levelof a few tens of percent. An increase in the tracer density

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Figure 8. The best-t total mass inside a xed radius comparedto the true mass inside that radius. Errors are obtained througherror propagation from the covariance matrix of ρs and r s , withthe correlated error between ρs and r s included. The black dashedline marks equality between the measured and true mass.

outer slope would cause an increase in the recovered halomass and a corresponding decrease in halo concentration;conversely, an increase in the inner slope would cause a de-crease in the halo mass and an increase in the concentra-tion. As a result, uncertainties in the t to the tracer den-sity prole may further bias the best t halo parameters.For example, the best-t (green dashed) curve in the halo Cpanel of Fig. 2 agrees well with the true prole (red points)inside 170 kpc but is somewhat shallower at larger radii.If we x the three spatial parameters in our t to halo Cto those given by a conventional reduced- χ 2 best-t to thetracer density (dashed black curve in Fig. 2) the best-t halomass is boosted by about 10%. If the tracer density prolesdeviate from the double power-law form, these correlationsbetween halo parameters and spatial parameters would in-troduce further bias to the best-t halo mass.

Lastly, we note that correlations between the three spa-tial parameters are strong as well. This quanties our earliernding that, in the case of halo A, adding tangential veloc-ities as constraints in the t makes the outer slope of thetracer density prole shallower and the break radius smaller,but results in very little perceptible change in the overallprole shape. Hence, good ts to the tracer density prolesmay not be unique. An increase in r0 can be roughly com-pensated by a corresponding increase of both α and γ .

5.2 Unrelaxed dynamical structures

The model distribution function used in our analysis as-sumes that the tracer population is in dynamical equilibriumand hence the phase-space density of tracers is conserved.Our mock halo stars are all accreted from satellite galaxies,

with a range of accretion times. Some prominent phase-spacestructures, such as stellar streams, may therefore violate theassumption of dynamical equilibrium. In this section we askhow the presence of unrelaxed dynamical structures affectsour results.

We expect the dynamical state of stars in our mock cat-alogue to depend on the infall redshift of their parent satel-

lite, at least approximately (satellites on different orbits willhave different rates of stellar stripping). We might expect toobtain an improved mass estimate if we use only stars fromsatellites that fell in earlier, because they have had moretime to relax in the host potential. To test this, we rankhalo stars according to their infall time 4 . We measure thehost halo mass and concentration with samples dened bydifferent cuts in infall time, corresponding to roughly thesame fraction of stellar mass in each halo. The top and mid-dle rows of Fig. 9 present these parameters as a function of the fraction of stars selected by each cut, for the ve differenthalos. A small fraction corresponds to an earlier mean infalltime, but also (obviously) to a smaller sample size. A frac-tion of 1 means all the mock stars have been included, hence

the corresponding parameters are those listed in Table 1.We see uctuations in the measured halo propertieswith infall time, but no obvious trends. Using samples of stars with earlier mean infall times does not seem to reducethe bias between best-t and true parameters. This maybe because the dynamical state of tracers depend on manyother factors, such as the orbit of their parent satellites 5 .Samples for which the measured halo masses increase pro-duce a corresponding decrease in the measured concentra-tions, again reecting the strong degeneracy between M 200and c200 .

To gain more intuition regarding the dynamical stateof halo tracers, Fig. 10 shows phase-space scatter diagramsfor mock halo stars (radius, r , versus radial velocity, vr ).

Points are colour coded according to the infall time of theirparent satellite, with black points corresponding to satellitesfalling in earliest and blue, magenta, red and yellow points tosuccessively later infall times. Stars with earlier infall timesare clearly more centrally concentrated. For points in Fig. 6with decreasing fraction of stars that fell in earliest, the cor-responding particles in Fig. 10 can be found by excludingyellow, red, magenta and blue points by sequence and look-ing at the remaining points.

Green curves in Fig. 10 are contours of constant angularmomentum and binding energy. There are six contours in to-tal, corresponding to three discrete values of binding energyand two discrete values of angular momentum: dashed lineshave a higher angular momentum than solid lines. Smaller

maximum radius indicates higher binding energy. It is thusstraightforward to see that particles with higher binding en-ergy have smaller velocities and are more likely to be found

4 Dened as the simulation output redshift at which the parentsatellite of each star reaches its maximum stellar mass, which isgenerally within one or two outputs of infall as dened by SUB-FIND.5 We have carried out an analogous exercise in which we rankstars by the time at which they are stripped from their parentsatellite. We found that this stripping time correlates with theinfall time of the parent satellite, and the conclusions regardingthe recovered halo parameters are similar.

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0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

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Figure 9. The host halo masses ( M 200 ) and concentrations ( c200 ) measured using a certain fraction of stars that have the earliest infalltime. The ve columns are for ve Aquarius halos, as labeled at the top. The red dashed lines show the true values of M 200 and c200 .The top and middle rows are based on the 10% oldest tagged particles. The bottom row is analogous to the top one, except that starswhose parent satellites have not been entirely disrupted yet are excluded.

in the inner regions of the halo. Comparing the solid anddashed contours, we see that increasing angular momentumat xed binding energy causes signicant differences in theinner regions of the halo, while at larger radii the two setsof contours almost overlap.

We can see that points with the same colour trace thesecontours with some scatter, implying that stars whose par-ent satellites fall in at a particular epoch share similar orbits.This can be seen more clearly in the bottom right panel,which shows a scatter plot of binding energy versus angularmomentum for stars in halo A. Points with the same colouroccupy regions covering a narrow range of binding energy.The correlation between infall time and binding energy of subhalos has been studied by Rocha et al. (2012). Here we have shown that stars from stripped subhalos show a corre-lation between infall redshift and binding energy as well.

Although mock stars trace the green contours overall,we can still see some prominent structures. For example,there are two yellow spurs in the outskirts of halo D andone yellow spur in halo E. These correspond to particlesthat have only just been stripped from their parent satellites.These stars are far from equilibrium: their exclusion causesthe rapid change in M 200 and c200 in Fig. 9 between fractionsof 1 and 0.7 in halos D and E.

To conrm that these unrelaxed phase-space structurescan affect our results, we have repeated the above exerciseexcluding all stars whose parent satellites have not been en-

tirely disrupted. Corresponding results are shown in the bot-tom row of Fig. 9, again ranking stars by their infall time.Measured halo masses are clearly affected by excluding starswhose parent satellites still survive. For halos A and C, wesee some small uctuations in the measured halo mass, butthe systematic underestimate of the true halo mass still re-mains. The most dramatic changes occur for halos B, Dand E. First, the point corresponding to a fraction of 1 forhalos D and E show a signicant increase in the recoveredmass towards the true values, reinforcing our conclusion thatunrelaxed structures are causing signicant underestimatesof M 200 in these halos. In fact, most of the yellow dots inhalo D panel of Fig. 10 are stars that have been strippedfrom satellites that still survive. After excluding these, the

two highest-fraction points in the halo D panel are basedon almost the same sample of stars. We also notice thatuctuations around the true value for the different fractionsare reduced in the bottom row (for example, the two lowestfraction points in halo D).

The effects of excluding halo stars from surviving satel-lites are more ambiguous in halos B and E. Excluding starsthat are considerably unrelaxed should better match the as-sumption of dynamical equilibrium, but these measurementsare entangled with other uncertainties that might get worse.Our conclusion is thus for halo D (and perhaps E) the un-derestimates of their host halo masses when all particles areused are mainly due to unrelaxed dynamical structures; for

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Figure 10. Phase-space scatter plot (radial velocities versus radial positions) for the 10% oldest tagged particles in the ve Aquariushalos. Data points are colour-coded according to their infall time. In sequence, points with yellow, red, magenta, blue and black coloursare stars that have earlier and earlier infall times. The bottom right panel is a scatter plot of energy and angular momentum for halo A.Only one out of every 20 points are plotted in order to avoid saturation.

the other halos, the effects of unrelaxed dynamical struc-tures are not obvious. A more detailed study quantifyingthe dynamical state of tracers will be carried out in Han etal. (in preparation).

6 MODEL UNCERTAINTIES IN THE RADIALAVERAGE AND IMPLICATIONS FOR REALSURVEYS

We have seen in the previous section that our maximumlikelihood technique recovers different halo mass from setsof tracers with different infall redshifts, or more fundamen-tally, different binding energies. The sense and magnitudeof these differences show no obvious correlations with eitherquantity, however. Stars falling in earlier typically have high

binding energy and are mostly concentrated in the centralregions of the halo; since binding energy correlates with ra-dius, we may also expect uctuations in the recovered haloparameters when using samples of stars drawn from a par-ticular radial range. In this section we investigate this radialdependence. This helps to understand the behaviour of thefull model, which averages over all radii. Variations withradius are relevant to observational applications as well, be-cause in practice tracers are often selected from relativelynarrow radial ranges, and these ranges may be different fordifferent tracers.

We assign stars to four bins of galactocentric radius:(7-20) kpc, (20-50) kpc, (50-100) kpc and > 100 kpc. Themodel distribution function is then t to stars in each binseparately. However, in each case the three spatial parame-

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ters of the tracer distribution are xed to their best-t valuesobtained from tracers over the entire radial range, otherwisewe would end up with extremely poor extrapolations basedon the local density slope. All the other parameters, M 200 ,c200 and β , are left as free parameters. The window function,Eqn. 16, is modied appropriately for each bin.

Fig. 11 shows the measured M 200 as functions of the

mean radius of each bin, normalized by the halo virial ra-dius ( R200 ). The value of M 200 varies signicantly with thetracer radius. For halos A, C and E, stars in the outermost(r > 100 kpc) and innermost ( r < 20 kpc) bins give underes-timates, while stars at 20 < r < 100 kpc give signicant over-estimates. Similarly, for halos B and D, stars at r > 100 kpcgive underestimates, whereas stars at smaller radii give over-estimates.

The velocity anisotropy of tracers, β , is a function of radius, whereas the model distribution function assumes asingle value of β . To test whether the radial average of β mayaffect our estimates in the host halo mass, we repeated theanalysis of Fig. 11 but xed the value of β in each radial binto the best-t value obtained from the whole population.

These measurements are almost identical to the measure-ments presented in Fig. 11, which conrms our result fromSec. 5.1 that the radial averaging of β does not cause furtherbias in the other parameters.

One feature in Fig. 11 is puzzling at rst glance: thebest-t halo masses obtained from the four radial bins indi-vidually are all larger than the best-t halo mass ( M 200 =1.15) obtained using stars over the whole radial range. Thisseems odd, as we might expect that the best-t M 200 wouldbe close to the average of the values estimated from the fourradial subsamples. The situation is not that straightforward,however, because the likelihood surfaces from the subsam-ples are superimposed in two dimensional ( M 200 and c200space when the full sample is used. Coupled with the strong

degeneracy between the two parameters, the peak of the -nal likelihood surface is located around a region where thedegeneracy lines from different subsamples intersect.

In real observations, there is often a maximum radius of tracers corresponding to the instrumental ux limit. In thepresent literature this limit is typically much smaller thanthe expected halo virial radius. Beyond this maximum ra-dius, extrapolations are required to t the distributions of both the dark matter and tracers. We explore the implica-tions of this directly in Fig. 12 by adopting several outerradial cuts ( r < r cut ) and reporting the estimated halo massas a function of the cut radius normalized to the virial radius(r cut /R 200 ). Unlike Fig. 11, the three spatial parameters aretreated as unknown and left free to be constrained by thet, in order to mimic real observations where the densityproles of tracers is taken directly from the available data.

The overall trends with rcut in Fig. 12 are very clear:the recovered halo mass is constant at large rcut , and turnsup once r cut becomes small (about 40 per cent of R200 ). Wechecked the best-t values of the tracer spatial parametersin each case, and found they do not vary much with the ra-dial cut as long as r cut < 0.4R 200 . This is because the breakradius of tracer density proles in our mock catalogues aresmaller than 0 .4R200 for all the ve halos, and so the extrap-olation in tracer density is not severe. However, once rcutreduces below 0 .4R200 , the outer slope becomes essentiallyunconstrained. We believe the turn-up behaviour is due to

1 0

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M

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0

0

[

1

0

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]

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1 0

- 1

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2 0 0

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M

2

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[

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H a l o B

1 . 0

2 . 0

3 . 0

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Figure 11. Best-t host halo masses using samples of stars in fourradial bins, (7-20) kpc, (20-50) kpc, (50-100) kpc and > 100 kpc.

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

r

c u t

/ R

2 0 0

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0

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r

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Figure 12. best-t host halo masses as a function of the outerradius limit of the tracer population ( r < r cut ).

the changing dynamical state of tracers and the extrapola-tions required to know the underlying potential where thereare no tracers.

Previous constraints on the MW halo mass have beenderived from tracers roughly covering the range 0.1 to 0.4R 200 (R200 250 kpc; Deason et al. 2012 ). Our results sug-gest that masses derived from the tting distribution func-tion of these tracers may be signicantly biased even with

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respect to ‘asymptotic’ results from the same method us-ing all stars in the halo. Furthermore, instead of being asharp cut, the radial selection functions of real surveys areoften complicated, with non-trivial incompleteness as func-tion of radius and angular position. These selection effectsmay cause additional bias in the measured host halo mass.

7 CONCLUSIONS

Several authors have measured parameters of the host haloof our MW, in particular its total mass, by tting spe-cic forms of the distribution function to the observeddistances and velocities of dynamical tracers such as oldBHB and RR Lyrae stars, globular clusters and satellitegalaxies ( Wilkinson & Evans 1999 ; Sakamoto et al. 2003 ;Deason et al. 2012 ). These models assume that the tracersare in dynamical equilibrium within the host potential. Withthe help of Jeans theorem, the distribution function of thetracers is further assumed to depend only on two integralsof motion, the binding energy, E , and the angular momen-tum, L. In the case of a separable function of E and L, thedistribution function can be obtained through Eddingtoninversion of the tracer density prole.

In this paper we have extended earlier analytical formsof the MW halo distribution function to the case of the NFWpotential, which is of most relevance to CDM-based mod-els. We generalized the radial distribution of tracers (halostars) to a double power law, which is suggested by recentobservational results and simulations. We used a maximumlikelihood approach to t this model distribution function toa realistic mock stellar halo catalogue of distances and ra-dial velocities, constructed from the high resolution Aquar-ius N -body simulations using the particle tagging techniqueof Cooper et al. (2010). Our aim was to test the model per-formance and assumptions. We considered cases with andwithout additional tangential velocity data. Our conclusionsare as follows:

• The best-t host halo virial masses and concentrationsare biased from the true values, with the level of bias varyingfrom halo to halo.

• Adding tangential velocity data substantially reducesthis bias, but does not eliminate it. For example, for haloB the agreement between measured and true halo mass isvery good (a 5% overestimate) if tangential velocities areused, but for halo A, a 40% underestimate persists even withthis additional constraint. The inclusion of tangential veloc-ities therefore is crucial for accurate measurements of bothhost halo and tracer properties, especially for the velocityanisotropies of the tracers.

• A strong negative correlation between the host halomass and the halo concentration is found in our analysis.The two parameters are almost completely degenerate, withcovariance very close to -1.

• The model gives a strongly biased measurement of thevelocity anisotropies of stars.• There are various sources that contribute to the biasedestimates of halo properties. The two most important factors

in our analysis are violation of the dynamical equilibriumassumption and correlations among different model param-eters, especially the strong degeneracy between M 200 andc200 .

• In contrast to the signicantly biased measurements of M 200 or c200 , the model gives good constraints on the totalmass inside a xed radius of about 0 .2R200 .

The strong degeneracy between halo mass and concen-tration arises because changes in the corresponding tracervelocity distribution due to the increase of one of these pa-

rameters can be roughly compensated by the other. Thiswarns us that it is dangerous to constrain the total mass of the MW halo with this specic form of the distribution func-tion. Small perturbations, for example from dynamically hotstructures, may cause signicant bias. Similar degeneraciesbetween model parameters and the robustness of the bestconstrained mass within a xed radius have been reportedand discussed in previous work (e.g. Deg & Widrow 2014 ;Kae et al. 2014 ; Wolf et al. 2010 ), although these modelsare quite different from ours. Further information needs tobe incorporated into the model to break this strong degen-eracy and give better constraints on halo properties.

In addition to the degeneracy between halo mass andconcentration, relatively weak but still signicant correla-

tions exist between these halo parameters and the three pa-rameters describing the spatial variation of tracer density.We found that a steeper inner slope gives a lower estimateof halo mass, while a steeper outer slope gives an higher es-timate. If the true tracer density prole deviates from thedouble power-law form, the resulting bias will be propagatedto the best-t values of the halo parameters.

The correlations between velocity anisotropy, β , and allother parameters are very weak. This is fortunate, becausewe know that the model can give highly biased estimatesof β for stars; this particular bias is not propagated to theother parameters.

The model distribution function requires tracers to be indynamical equilibrium, with time-independent phase-space

density. In reality, stars stripped from satellite galaxies canhave highly correlated orbits that violate this assumption.We were able to test how well the assumption holds forour mock halo stars. Perhaps surprisingly, we do not ndany systematic correlation of the recovered halo mass withthe infall redshift of tracer subsamples. This suggests thatthe dynamical state of halo tracers depends on other fac-tors, such as their orbits, and not only their infall time. Dy-namical relaxation is nevertheless a factor: excluding starsstripped from surviving satellites improves the agreementbetween best-t and true halo masses in two cases (halos Dand E). This cut eliminates dynamically hot structures thatcan be identied by eye in these halos.

Beyond all these assumptions and uncertainties in themodel itself, in real observations the maximum observableradius of dynamical tracers may be much smaller than thehalo virial radius. We found tracer subsamples selected overdifferent ranges of radius can give signicantly different es-timates of the host halo mass, even if the three parametersdescribing the density of tracers are xed to be those derivedfrom the whole tracer population. An outer radius limit re-sults in biased measurements if it signicantly smaller thanthe virial radius. For example, the recovered halo masses of halos A, B, C and D converge for outer radius limits largerthan r 0.4R200 but give systematically larger masses forsmaller radial limits. For one halo, E, this overestimationoccurs for limits r 0.8R200 . There are two reasons behind

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this radial dependence: stars at different radii have signi-cantly different dynamical state and extrapolations to largerand smaller radii become less accurate when only a limitedradial range is sampled.

We conclude that methods to estimate the mass of theMilky Way halo using the kind of distribution functions wehave investigated here need to be used with extreme cau-

tion. This is particularly true when estimating the totalvirial mass. Restricting the estimate to the mass interior tor 0.2R200 is considerably more reliable. In any case, mockcatalogues like those we have analyzed here (and made pub-licly available in Lowing et al. (2014)) are required to assessthe reliability of any particular mass estimation method.

ACKNOWLEDGMENTS

This work was supported by the Euopean Research Coun-cil [grant number GA 267291] COSMIWAY and Scienceand Technology Facilities Council Durham ConsolidatedGrant. WW acknowledges a Durham Junior Research Fel-

lowship. The simulations for the Aquarius project were car-ried out at the Leibniz Computing Centre, Garching, Ger-many, at the Computing Centre of the Max-Planck-Societyin Garching, at the Institute for Computational Cosmol-ogy in Durham, and on the STELLA supercomputer of theLOFAR experiment at the University of Groningen. Thiswork used the DiRAC Data Centric system at DurhamUniversity, operated by the Institute for ComputationalCosmology on behalf of the STFC DiRAC HPC Facility(www.dirac.ac.uk). This equipment was funded by BIS Na-tional E-infrastructure capital grant ST/K00042X/1, STFCcapital grant ST/H008519/1, and STFC DiRAC Operationsgrant ST/K003267/1 and Durham University. DiRAC ispart of the National E-Infrastructure. WW is grateful for

useful discussions with Yipeng Jing and Till Sawala.

REFERENCES

Ahn C. P., Alexandroff R., Allende Prieto C., Anders F.,Anderson S. F., Anderton T., Andrews B. H., AubourgÉ., Bailey S., Bastien F. A., et al. 2013, ArXiv e-prints

Barber C., Starkenburg E., Navarro J. F., McConnachieA. W., Fattahi A., 2014, MNRAS, 437, 959

Barlow R., 1990, Nuclear Instruments and Methods inPhysics Research A, 297, 496

Battaglia G., Helmi A., Morrison H., Harding P., OlszewskiE. W., Mateo M., Freeman K. C., Norris J., ShectmanS. A., 2005, MNRAS, 364, 433

Bell E. F., Xue X. X., Rix H.-W., Ruhland C., Hogg D. W.,2010, AJ, 140, 1850

Bell E. F., Zucker D. B., Belokurov V., Sharma S., JohnstonK. V., Bullock J. S., Hogg D. W., Jahnke K., de JongJ. T. A., Beers T. C., Evans N. W., Grebel E. K., IvezícŽ., Koposov S. E., Rix H.-W., Schneider D. P., SteinmetzM., Zolotov A., 2008, ApJ, 680, 295

Bower R. G., Benson A. J., Malbon R., Helly J. C., FrenkC. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS,370, 645

Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2011,MNRAS, 415, L40

Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2012,MNRAS, 422, 1203

Boylan-Kolchin M., Bullock J. S., Sohn S. T., Besla G., vander Marel R. P., 2013, ApJ, 768, 140

Busha M. T., Marshall P. J., Wechsler R. H., Klypin A.,Primack J., 2011, ApJ, 743, 40

Carollo D., Beers T. C., Chiba M., Norris J. E., Freeman

K. C., Lee Y. S., Ivezić ˇZ., Rockosi C. M., Yanny B., 2010,ApJ, 712, 692

Carollo D., Beers T. C., Lee Y. S., Chiba M., NorrisJ. E., Wilhelm R., Sivarani T., Marsteller B., Munn J. A.,Bailer-Jones C. A. L., Fiorentin P. R., York D. G., 2007,Nature, 450, 1020

Cautun M., Frenk C. S., van de Weygaert R., HellwingW. A., Jones B. J. T., 2014, MNRAS, 445, 2049

Cautun M., Hellwing W. A., van de Weygaert R., FrenkC. S., Jones B. J. T., Sawala T., 2014, MNRAS, 445, 1820

Cooper A. P., Cole S., Frenk C. S., White S. D. M., Helly J.,Benson A. J., De Lucia G., Helmi A., Jenkins A., NavarroJ. F., Springel V., Wang J., 2010, MNRAS, 406, 744

Cooper A. P., D’Souza R., Kauffmann G., Wang J., Boylan-

Kolchin M., Guo Q., Frenk C. S., White S. D. M., 2013,MNRAS, 434, 3348Cuddeford P., 1991, MNRAS, 253, 414Deason A. J., Belokurov V., Evans N. W., 2011, MNRAS,

416, 2903Deason A. J., Belokurov V., Evans N. W., An J., 2012,

MNRAS, 424, L44Deason A. J., Belokurov V., Evans N. W., Johnston K. V.,

2013, ApJ, 763, 113Deason A. J., Belokurov V., Evans N. W., Koposov S. E.,

Cooke R. J., Pe˜narrubia J., Laporte C. F. P., FellhauerM., Walker M. G., Olszewski E. W., 2012, MNRAS, 425,2840

Deason A. J., Belokurov V., Koposov S. E., Rockosi C. M.,

2014, ArXiv e-printsDeg N., Widrow L., 2014, MNRAS, 439, 2678Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., 2008,

MNRAS, 390, L64Eddington A. S., 1916, MNRAS, 76, 572Ferrero I., Abadi M. G., Navarro J. F., Sales L. V.,

Gurovich S., 2012, MNRAS, 425, 2817Font A. S., Benson A. J., Bower R. G., Frenk C. S., Cooper

A., De Lucia G., Helly J. C., Helmi A., Li Y.-S., McCarthyI. G., Navarro J. F., Springel V., Starkenburg E., WangJ., White S. D. M., 2011, MNRAS, 417, 1260

Gibbons S. L. J., Belokurov V., Evans N. W., 2014, ArXive-prints

Gilmore G., Randich S., Asplund M., Binney J., Bonifacio

P., Drew J., Feltzing S., Ferguson A., 2012, The Messen-ger, 147, 25

Gnedin O. Y., Brown W. R., Geller M. J., Kenyon S. J.,2010, ApJL, 720, L108

Gonz ález R. E., Kravtsov A. V., Gnedin N. Y., 2013, ApJ,770, 96

Helmi A., Cooper A. P., White S. D. M., Cole S., FrenkC. S., Navarro J. F., 2011, ApJL, 733, L7

James F., Roos M., 1975, Computer Physics Communica-tions, 10, 343

Jing Y. P., Suto Y., 2002, ApJ, 574, 538Kae P. R., Sharma S., Lewis G. F., Bland-Hawthorn J.,

2012, ApJ, 761, 98

c 0000 RAS, MNRAS 000 , 000–000


17/21

17

Kae P. R., Sharma S., Lewis G. F., Bland-Hawthorn J.,2014, ApJ, 794, 59

Kahn F. D., Woltjer L., 1959, ApJ, 130, 705Law D. R., Johnston K. V., Majewski S. R., 2005, ApJ,

619, 807Li Y.-S., White S. D. M., 2008, MNRAS, 384, 1459Lowing B., Wang W., Kennedy R., Cooper A., Helly J.,

Frenk C., Cole S., 2014, ArXiv e-printsMcMillan P. J., 2011, MNRAS, 414, 2446Navarro J. F., Frenk C. S., White S. D. M., 1996a, ApJ,

462, 563Navarro J. F., Frenk C. S., White S. D. M., 1996b, ApJ,

462, 563Navarro J. F., Frenk C. S., White S. D. M., 1997a, ApJ,

490, 493Navarro J. F., Frenk C. S., White S. D. M., 1997b, ApJ,

490, 493Navarro J. F., Ludlow A., Springel V., Wang J., Vogels-

berger M., White S. D. M., Jenkins A., Frenk C. S., HelmiA., 2010, MNRAS, 402, 21

Neto A. F., Gao L., Bett P., Cole S., Navarro J. F., Frenk

C. S., White S. D. M., Springel V., Jenkins A., 2007, MN-RAS, 381, 1450Newberg H. J., Willett B. A., Yanny B., Xu Y., 2010, ApJ,

711, 32Piffl T., Scannapieco C., Binney J., Steinmetz M., Scholz

R.-D., Williams M. E. K., de Jong R. S., Kordopatis G.,2014, A&A, 562, A91

Prusti T., 2012, Astronomische Nachrichten, 333, 453Rocha M., Peter A. H. G., Bullock J., 2012, MNRAS, 425,

231Sakamoto T., Chiba M., Beers T. C., 2003, A&A, 397, 899Sales L. V., Navarro J. F., Abadi M. G., Steinmetz M.,

2007a, MNRAS, 379, 1475Sales L. V., Navarro J. F., Abadi M. G., Steinmetz M.,

2007b, MNRAS, 379, 1464Sesar B., Jurić M., Ivezić Ž., 2011, ApJ, 731, 4Smith M. C., Ruchti G. R., Helmi A., Wyse R. F. G., Ful-

bright J. P., Freeman K. C., Navarro J. F., 2007, MNRAS,379, 755

Spergel D. N., Verde L., Peiris H. V., Komatsu E., NoltaM. R., Bennett C. L., Halpern M., Hinshaw G., Jarosik N.,Kogut A., Limon M., Meyer S. S., Page L., Tucker G. S.,Weiland J. L., Wollack E., Wright E. L., 2003, ApJS, 148,175

Springel V., Wang J., Vogelsberger M., Ludlow A., JenkinsA., Helmi A., Navarro J. F., Frenk C. S., White S. D. M.,2008, MNRAS, 391, 1685

Wang J., Frenk C. S., Navarro J. F., Gao L., Sawala T.,2012, MNRAS, 424, 2715

Wang J., Navarro J. F., Frenk C. S., White S. D. M.,Springel V., Jenkins A., Helmi A., Ludlow A., Vogels-berger M., 2011, MNRAS, 413, 1373

Watkins L. L., Evans N. W., An J. H., 2010, MNRAS, 406,264

Watkins L. L., Evans N. W., Belokurov V., Smith M. C.,Hewett P. C., Bramich D. M., Gilmore G. F., Irwin M. J.,Vidrih S., Wyrzykowski L., Zucker D. B., 2009, MNRAS,398, 1757

Wilkinson M. I., Evans N. W., 1999, MNRAS, 310, 645Wojtak R., Lokas E. L., Mamon G. A., Gottl¨ ober S., 2009,

MNRAS, 399, 812

Wojtak R., Lokas E. L., Mamon G. A., Gottl¨ ober S., KlypinA., Hoffman Y., 2008, MNRAS, 388, 815

Wolf J., Martinez G. D., Bullock J. S., Kaplinghat M.,Geha M., Mu˜noz R. R., Simon J. D., Avedo F. F., 2010,MNRAS, 406, 1220

Xue X. X., Rix H. W., Zhao G., Re Fiorentin P., Naab T.,Steinmetz M., van den Bosch F. C., Beers T. C., Lee Y. S.,

Bell E. F., Rockosi C., Yanny B., Newberg H., WilhelmR., Kang X., Smith M. C., Schneider D. P., 2008, ApJ,684, 1143

Zolotov A., Willman B., Brooks A. M., Governato F., HoggD. W., Shen S., Wadsley J., 2010, ApJ, 721, 738

APPENDIX A: ORIGIN OF THE BIAS IN β

In Fig. 6 we showed that there is a systematic bias betweenthe best-t and true value of β . We now show in the leftpanel of Fig. A1 the phase-space distribution of stars inhalo A (top panel, binding energy, E , versus angular mo-mentum, L) and the one dimensional angular momentumdistribution at two xed values of E (the second and thirdpanels from the top, respectively) indicated by the horizon-tal dashed lines in the top panel .6 E and L have been nor-malized so that they are dimensionless (see Eqn. 13). Weonly show results based on halo A; for the other halos ourconclusions are the same.

In the top panel we see that, for a xed value of E , thereis an upper limit to L which increases with decreasing E .This is the maximum allowed value of angular momentum,corresponding to circular orbits with zero radial velocity atxed E . In the next two panels, we see that the angular mo-mentum distributions at different values of E have similarfeatures. They are both at at small values of L and dropquickly when L approaches its upper limit. For comparison,we also plot two lines of the form F (E, L ) L− 2β , whereβ is the velocity anisotropy obtained from particles in theenergy slice or the full sample. Neither model could give asatisfactory description of the empirical distribution. Thisimplies that the physical interpretation of the power-law in-dex in our distribution function as β is inaccurate. The truedistribution function must be more complex.

We also notice that the best-t value of β for halo A is0.458 (Table 1), which predicts a power-law slope that is stilla poor match to the empirical distributions in Fig. A1. If wex the power-law slope in the model according to the trueanisotropy of the full sample, this results in better agree-ment with tangential velocity distribution but a much pooreragreement with the radial velocity distribution. Afterall, ourmaximum likelihood approach is designed to t the velocityand spatial distributions of stars, not the distributions of binding energy or angular momentum.

The β prole of dark matter particles have been stud-ied in earlier works. For example, Wojtak et al. (2008, 2009)looked at the distribution functions of dark matter parti-cles in halos of mass 1014 to 1015 M . Although the details

6 Note the quantity we are plotting here is F (L) = F (L|E ). Toobtain this distribution empirically one has to properly accountfor the density of state in ( E ,L) space (see Wojtak et al. 2008 ,for more details).

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Figure A1. phase-space distributions of mock stars in halo A (left) and of dark matter in the same halo (right). The top panels in eitherplots show 2D distributions in the plane of binding energy, E , and angular momentum, L . E and L have been scaled to be dimensionless.The middle and bottom panels show 1D angular momentum distributions of stars (likewise DM) at two xed values of E , indicated by thehorizontal black dashed curves in the top panels. Red and green dashed lines are power-law distributions with arbitrary normalizations,predicted from the velocity anisotropy of all stars (or DM) and stars (or DM) in each binding energy range.

of their modeling and the mass range of halos are differ-ent from ours, their model distribution function can recover

well the true β of dark matter particles in their simulation.We therefore examine the angular momentum distributionof dark matter particles in our simulations in the three rightpanels of Fig. A1. We nd the mean velocity anisotropy fordark matter particles is about 0.3 (see black dashed line inFig. 5). This agrees better with the shape of dark matterL distributions (green dashed lines), although there are stillobvious discrepancies. Red dashed lines are predicted fromthe velocity anisotropy of dark matter particles in the par-ticular binding energy range being probed. At E 10− 0. 4 v2s ,the agreement between the red dashed lines and the shapeof the L distributions is quite poor, whereas at a lower bind-ing energy ( E 10− 1. 1 v2s ), we see a better agreement. Wehave looked at many different choices of E in this regard,and found that for less bound dark matter particles, theirvelocity anisotropy correctly predicts the power-law slopeof their L distribution. However, for dark matter particlesthat are more tightly bound, the velocity anisotropy is notas well correlated with the power-law slope of the L distri-bution. This is the same as the stellar case, although thediscrepancy for dark matter particles is smaller.

Stars in the stellar halo are clearly a biased populationof tracers with respect to dark matter particles in the sim-ulation. Their orbits are more radial (Fig. 5) correspondingto a higher β . However, the difference in β is not becausest

estimating the dark matter halo mass of our milky way using dynamical tracers

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