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15Mon. Not. R. Astron. Soc. 451, 2524–2535 (2015) Printed 23 June 2015 (MN LATEX style file v2.2)

A Robust Determination of Milky Way Satellite Properties

using hierarchical mass modelling

Gregory D. Martinez1⋆1 The Oskar Klein Center, Department of Physics, Stockholm University, Albanova, SE-10691 Stockholm, Sweden

23 June 2015

ABSTRACT

We introduce a new methodology to robustly determine the mass profile, as well as theoverall distribution, of Local Group satellite galaxies. Specifically we employ a statisti-cal multilevel modelling technique, Bayesian hierarchical modelling, to simultaneouslyconstrain the properties of individual Local Group Milky Way satellite galaxies andthe characteristics of the Milky Way satellite population. We show that this method-ology reduces the uncertainty in individual dwarf galaxy mass measurements up to afactor of a few for the faintest galaxies. We find that the distribution of Milky Waysatellites inferred by this analysis, with the exception of the apparent lack of high-masshaloes, is consistent with the Λ cold dark matter (ΛCDM) paradigm. In particular wefind that both the measured relationship between the maximum circular velocity andthe radius at this velocity, as well as the inferred relationship between the mass within300pc and luminosity, match the values predicted by ΛCDM simulations for haloswith maximum circular velocities below 20 km s−1. Perhaps more striking is that thisanalysis seems to suggest a more cusped “average” halo shape that is shared by thesegalaxies. While this study reconciles many of the observed properties of the MilkyWay satellite distribution with that of ΛCDM simulations, we find that there is stilla deficit of satellites with maximum circular velocities of 20-40 km s−1.

Key words: methods: statistical — galaxies: dwarf — galaxies: kinematics anddynamics — Local Group — galaxies: statistics — galaxies: structure

1 INTRODUCTION

The Λ cold dark matter (ΛCDM) paradigm makes far-reaching predictions about galaxy formation and cosmol-ogy. Observations of the CMB and large-scale structurehave confirmed many of these predictions (Efstathiou et al.1992; Riess et al. 1998; Komatsu et al. 2011) making thisparadigm the favoured cosmological model. AlthoughΛCDM has enjoyed much success at large scales, there aresome indications of discrepancies at smaller scales. Thesediscrepancies can be categorized in two ways: individual halodensity profiles and overall distributions of sub-structure.Particularly, it has been suggested that the Milky Way Lo-cal Group dwarf spheroidal (dSph) galaxies have a muchflatter mass–luminosity relation and possess much shal-lower inner density profiles than those predicted by ΛCDM(Goerdt et al. 2006; Gilmore et al. 2007; Evans et al. 2009;Walker & Penarrubia 2011; Agnello & Evans 2012). Thishas motivated development of alternative hypotheses suchas warm and self-interacting dark matter (Bond et al.

⋆ E-mail: gregory.martinez@fysik.su.se

1980; Spergel & Steinhardt 2000; Dalcanton & Hogan 2001;Strigari et al. 2006; Boyarsky et al. 2009; Carlson et al.1992; Burkert 2000; Colın et al. 2002; Ahn & Shapiro 2005;Loeb & Weiner 2011; Maccio et al. 2012; Rocha et al. 2013).Many of these models give flatter inner density pro-files at dSph scales while retaining CDM’s successfullarge-scale predictions. However, recent studies have sug-gested that shallow inner density profiles may be a con-sequence of the flattening of inner cusps from baryoniceffects (Mashchenko et al. 2006, 2008; Arraki et al. 2014;Brooks et al. 2013; Governato et al. 2012; Parry et al. 2012;Pontzen & Governato 2012; Zolotov et al. 2012).

Perhaps just as troubling is the apparent disagree-ment between the sub-structure distributions observed lo-cally and those predicted by ΛCDM simulations. Oneof the best studied discrepancies is the apparent lackof numerous low-mass sub-structures predicted by theΛCDM paradigm (Kauffmann et al. 1993; Klypin et al.1999; Moore et al. 1999; Bullock 2010). This so-called “Miss-ing Satellites Problem” persisted even after the discovery ofseveral additional ultra-faint satellite galaxies (Grebel 2000;Willman et al. 2005; Irwin et al. 2007; Simon & Geha 2007;

c© 2015 RAS

2 Martinez

Belokurov et al. 2008; Liu et al. 2008; Martin et al. 2008;Watkins et al. 2009; Belokurov et al. 2010; Martinez et al.2011; Simon et al. 2011). This problem is further compli-cated by the fact that these galaxies may share a com-mon mass scale over several orders of magnitude in lu-minosity (Strigari et al. 2007b, 2008). Many of these dis-crepancies have been addressed by including astrophysi-cal and observational effects, such as suppression of starformation due to reionization and feedback (Quinn et al.1996; Thoul & Weinberg 1996; Navarro & Steinmetz 1997;Barkana & Loeb 1999; Klypin et al. 1999; Bullock et al.2000; Gnedin 2000; Benson et al. 2002; Hoeft et al. 2006;Madau et al. 2008; Alvarez et al. 2009), and selection biases(Willman et al. 2004; Simon & Geha 2007; Tollerud et al.2008; Koposov et al. 2009; Walsh et al. 2009; Rashkov et al.2012). However, a slight correlation between the mass andluminosity is still common in these improved dark mat-ter analyses (Ricotti & Gnedin 2005; Koposov et al. 2009;Maccio et al. 2009; Okamoto & Frenk 2009; Busha et al.2010; Font et al. 2011; Rashkov et al. 2012). To make mat-ters worse, there is also a deficit of observed satellites at thehigh-mass end of the dwarf satellite mass spectrum. Dubbedthe “Too Big to Fail” problem, Boylan-Kolchin et al. (2011,also see Vera-Ciro et al. 2013) noted that the Aquariussimulations predict at least 10 sub-haloes with a maxi-mum circular velocity greater than 25 km/sec. Attemptsto place the most luminous known satellites into haloesof this size result in halo densities inconsistent with CDMsimulations (Boylan-Kolchin et al. 2012). One possible so-lution to this discrepancy may originate in the same bary-onic processes used to explain the cusp/core problem – inthat core-like central regions created by supernova feedbackand tidal stripping make these galaxies more suspectibleto disruption by the Milky Way disc (Penarrubia et al.2010; di Cintio et al. 2011; Brooks et al. 2013; Zolotov et al.2012; Penarrubia et al. 2012). On the other hand, if themass of the Milky Way halo has been overestimated, thisapparent lack of high-mass sub-haloes may be due to astatistical anomaly (Wang et al. 2012). However, the ef-ficacy of these solutions has been disputed by variousauthors (Boylan-Kolchin et al. 2012; Strigari & Wechsler2012; Garrison-Kimmel et al. 2013). For a general review ofthese issues see Strigari (2013).

While much effort has been focused on reconcil-ing CDM predictions with various dwarf galaxy obser-vations, little attention has been paid to the statisti-cal consistency of the measurements themselves. Cur-rent Jeans modelling methods used to constrain dSphshalo properties are dominated by assumed prior probabil-ities (Martinez et al. 2009; Walker et al. 2009a; Wolf et al.2010; Walker 2013). This can potentially be over-come by using more advanced methods such as phasespace, Schwarzschild, or higher-order Jeans modelling( Lokas et al. 2005; Wu & ApJ 2007; Amorisco & Evans2012; Jardel & Gebhardt 2012; Richardson & Fairbairn2012; Breddels et al. 2013; Jardel et al. 2013). However,these methodologies introduce systematics such as binarycontributions and membership effects that, to date, havenot been included in such analyses (Walker et al. 2009b;Minor et al. 2010; Martinez et al. 2011). Therefore, priordominance is crucial to this discussion as it not only af-fects the characterization of individual dSphs, but has an

unknown effect on the inferred parameters of the population.In this paper we aim to address this issue through the pow-erful statistical technique of multilevel modelling (MLM)(Mandel et al. 2009; Loredo & Hendry 2010; Mandel et al.2011; Soiaporn et al. 2012). This broad class of modellingtechniques base prior probabilities on the actual modelparameter distribution implied between data sets. MLMconstrains the actual prior distribution by requiring thatthe distribution derived from the individual measurementsmatch the prior distribution assumed.

In the next section we will introduce the MLM method-ology and outline the specific technique used here: BayesianHierarchical Modelling. In Section 3 we specify our model as-sumptions. Finally, we present our results and discuss theirimplications for characterizing Local Group dSphs.

2 MULTILEVEL MODELLING

Constraining dark matter halo properties from individualstellar line-of-sight velocity measurements in Local GroupdSph spheroidal galaxies is a difficult problem. Mass con-straints from dispersion measurements are riddled with un-constrained degeneracies that affect mass measurementsfar from the stellar half-light radius (Walker et al. 2009a;Wolf et al. 2010). This causes inferred mass probabilities tobe dominated by prior probabilities. In Bayesian analysis,this is problematic because of the “degree of belief” prob-abilistic interpretation that is usually assigned to the priorand posterior probabilities (Cox 1946). In other words, themass posterior beyond the half-light radius is dominatedby the observer’s (sometimes arbitrary) prior belief ratherthan being dominated by data. One solution is to apply thestrict frequentist interpretation of probability to the prior –e.g., restrict the interpretation of the prior probability den-sity function (PDF) to represent the frequency of observingan halo property given a sufficiently large galaxy sample.Within this interpretation, the choice of prior is constrainedto match that of the overall galaxy sample. This causes re-sultant mass posteriors to be much more stringent. How-ever, the accuracy of these posteriors is highly dependenton the agreement between the assumed prior probabilityand the actual galaxy sample distribution. For the LocalGroup dSphs, the properties of the source galaxy sample isusually inferred from numerical simulations. Unfortunately,even if these simulations are an adequate description of theunderlying distribution, the actual observable distributionwill only be a subset of this sample. Because the subset isdetermined by astrophysical interactions that currently arenot well understood, it is very likely that strict application ofnumerical simulations, in this regard, will lead to erroneousresults.

In this paper we address the issue of prior dominanceby applying a multilevel statistical modelling technique todirectly constrain the prior probabilities. MLM divides theparameters of the dSph galaxies into various ‘levels’, eachwith its own set of observables. Starting with the most basic‘lowest’ level, the posteriors on the observables at each levelare used as input into subsequent levels.

Starting at the lowest level, the distribution being con-strained consists solely of the set of observables, D

(0) = {di},where the probability of observing a single data point, di,

c© 2015 RAS, MNRAS 451, 2524–2535

Hierarchical mass modelling 3

given a parametrization, M(0)0 , is

P(di|M (0)0 ). (1)

Here, M(n)x represents a parametrization that was intro-

duced at the xth level and is the nth superset consistingof the sets {M (n−1)

x,i } (for n > 0). Because there is an actual

distribution of individual data points ({di}), a likelihoodfunction can be defined that can be used to gauge the qual-ity of fit of the assumed set of model parameters, M

(0)0 :

L(M(0)0 |D(0)) ≡ P(D(0)|M (0)

0 ) =∏

i

P(di|M (0)0 ). (2)

But in order to infer a posterior probability of

P(M(0)0 |D(0)) = L(M

(0)0 |D(0))

P(M(0)0 )

P(D(0))(3a)

=P(M

(0)0 )

P(D(0))

∏

i

P(di|M (0)0 ). (3b)

a prior PDF, P (M(0)0 ), must be supplied. Since only one

set of model parameters, M(0)0 , can be inferred from a sin-

gle data set, D(0), we possess little knowledge about the set

of {M (0)0 } needed to constrain the prior probability distri-

bution. But this can be alleviated if a superset of data isavailable – e.g., a set of D

(0) (D(1) ≡ {D(0)i }) from which a

set of individual M(0)0 ’s (M

(1)0 ≡ {M (0)

0,i }) can be estimated

(note that D(n) represents the nth superset of data consist-

ing of the sets {D(n−1)i }). Given this superset of data, D

(1),a new level of modelling can be introduced that simultane-ously constrains the full set of M

(1)0 . The only truly free

parameters at this level are the ones used to parametrizethe prior, M

(0)1 . In this context, this level’s likelihood now

becomes a function of M(0)1 :

L(M(0)1 |M (1)

0 ,D(1)) ≡ P(M(1)0 ,D(1)|M (0)

1 ) (4a)

=∏

j

P(M(0)0,j ,D

(0)j |M (0)

1 ) (4b)

=∏

j

P(M(0)0,j |M (0)

1 )∏

i

P(dj,i|M (0)0,j ) (4c)

where dj,i is i-th data point of the j-th data set. By addingthis additional level of modelling, the prior PDF from theprevious level is reinterpreted as a contributing term in thehigher level likelihood. Unfortunately, the probability distri-bution, P(M

(0)1 ), is not specified for the same reason that

P(M(0)0 ) could not be constrained in the previous level: at

this level, only one M(0)1 can be inferred. This necessitates

the assumption of an unrestrained prior PDF to determinethe joint posterior probability distribution,

P(M(0)1 ,M

(1)0 |D(1)) = L(M

(0)1 |M (1)

0 ,D(1))P(M

(0)1 )

P(D(1)). (5)

Thus, in this modelling methodology, information containedwithin the internal distribution of each data set, as wellas the distribution of data between each data set, is con-currently used to constrain the combined probability dis-tributions P(di|M (0)

0 ) as well as the prior distribution,

P(M(0)i |M (0)

1 ).

Of course, given a set of D(1) (D(2) = {D(1)

i }), P (M(0)1 )

can be constrained by the application of an additional levelof modelling. In this manner, MLM can be used recursively

to constrain newly introduced prior probabilities from pre-vious iterations – provided the appropriate superset of data(D(n)) is available. Used recursively, the n-th implementa-tion (n-th level) has a likelihood function of

L(M (0)n |M (n)

0 ,M(n−1)1 , . . . ,M

(1)n−1,D

(n)) (6a)

≡ P(M(n)0 ,M

(n−1)1 , . . . ,M

(1)n−1,D

(n)|M (0)n ) (6b)

=∏

i

P(M(0)n−1,i|M (0)

n )L(M(0)n−1,i| . . . ,D(n−1)

i ) (6c)

where M(n)x ≡ {M (n−1)

x,i } and D(n) ≡ {D(n−1)

i }.Regardless of the number of levels applied, the prior in-

troduced at the top-level is still completely unconstrained.Thus, to utilize the complete posterior distribution in a sta-tistically consistent manner, either the top-level prior mustbe inferred from external information or the interpretationof probability must be expanded to include the Bayesian‘degree of belief’ probabilistic interpretation. Although log-ically consistent, the ‘degree of belief’ interpretation intro-duces subjectivity that is unsettling to some scientists. And,the applicability or accessibility of prior information canmake the former methodology difficult to implement. How-ever, the subjectivity of the top-level prior has only an in-direct, and thus mitigated, effect on lower-level posteriors.This is because this prior affects lower-level posteriors onlyindirectly through lower-level priors. The strict frequentistinterpretation applied to the likelihoods at various levels andtheir associated lower-level priors ensure that these lower-level priors are constrained by the intrinsic distribution ofthe data. This, thereby, mitigates the effect of the top-levelprior assumptions.

In this paper, we utilize the full top-level posterior

P(M (0)n , . . . |D(n)) = L(M (0)

n | . . . ,D(n))P(M

(0)n )

P(D(n)). (7)

to obtain model parameter uncertainties. Although therather arbitrary Bayesian ‘degree of belief’ probabilistic in-terpretation is assigned to the top-level prior, these so-called‘Bayesian hierarchical models’ (Loredo & Hendry 2010) pro-vide a straight forward and statistically consistent method-ology to apply MLM. Another option would be to use‘Bayesian empirical modeling’ which (ironically) avoids theintroduction of the Bayesian probabilistic interpretation bysolely utilizing the top-level maximum likelihood solution(Berger 1985; Petrone et al. 2012). But, these methods areusually not as conservative (as if the full posterior were used)because they do not explore the full top-level parameterspace.

3 MODEL ASSUMPTIONS

In this paper we assume a two level model. The bottom-level describes the astrophysical properties of each individ-ual dSph and its underlining dark matter potential, whereasthe top-level details the overall distribution of halo proper-ties. For the bottom-level, the total set of observables are theline-of-sight velocities, metallicites, and positions of individ-ual stars in the galaxy, as well as the total galaxy luminos-ity. The total model parameter set is composed of the stellarprofile, dark matter profile, and stellar velocity anisotropyparameters. Normally, a likelihood would be created that

c© 2015 RAS, MNRAS 451, 2524–2535

4 Martinez

models the intrinsic dispersion of each galaxy via the Jeansequation. To save computational time, we utilize the approx-imation that the Jeans equation only constrains the enclosedmass at the (3D) half-light radius. Specifically, we utilizethe relationship between the mass at the half-light radius(M(r1/2)) and total measured velocity dispersion (σ2

tot) de-rived in Wolf et al. (2010):

M(r1/2) = 3σ2tot/G. (8)

Here, it is the mass profile (M(r)) that is parametrized by

the lower-level parameters (M(0)0 ). This approximation has

been found to be very accurate because the degeneracy be-tween the enclosed mass and the stellar velocity anisotropyis drastically minimized at the stellar half-light radius caus-ing the mass posterior at this point to be data dominated(Walker et al. 2009a; Wolf et al. 2010). This approximationhas the additional benefit of removing any potential biasescaused by the assumed form of the velocity anisotropy pro-file. Using this approximation, the bottom-level data set isnow composed of the mass enclosed within the half-lightradius (M1/2, or equivalently, the measured velocity disper-sion), the measured half-light radius (r1/2), the total lumi-nosity (L), and their associated errors (ǫM1/2

, ǫr1/2 , and ǫL)– thus,

D(0) = {M (obs)

1/2 , ǫM1/2, r

(obs)

1/2 , ǫr1/2 , L(obs), ǫL} (9)

(Wolf et al. 2010; McConnachie 2012, and referenceswithin). To better match the actual distributions, we modelM1/2 and L as lognormal distributions, and r1/2 as a normaldistribution:

L(M(0)0 |D(0)) = N (log(M(r1/2)), ǫM1/2

|log(M(obs)

1/2 ))

×N (log(L), ǫL|log(L(obs))) ×N (r1/2, ǫr1/2 |r(obs)1/2 ) (10)

Here, N (µ, σ|x) denotes a normal distribution in x definedby:

N (µ, σ|x) ≡ Pnorm(x|µ, σ) =1√2πσ

exp

[

− (x− µ)2

2σ2

]

.

(11)As r1/2 has little implication for the underlying dark mattertheory, we assume a non-informative uniform Jeffery’s priorand marginalize over all possible r1/2 values. The remainingbottom-level prior then consists of L and the parametersthat model the enclosed mass (M(r)). Since the enclosedmass is dominated by the dark matter contribution, itsproperties represent the dark matter distribution within thegalaxy. Various dark matter halo properties have been foundto have tight correlations in the ΛCDM paradigm. Particu-larly, relationships between the halo concentration and mass(Diemand et al. 2007; Neto et al. 2007; Strigari et al. 2007a;Springel et al. 2008) as well as between the mass and lumi-nosity (Koposov et al. 2009; Busha et al. 2010; Font et al.2011; Rashkov et al. 2012). We use these correlations as mo-tivation for the selection of the functional form of the prior.Using the profile-independent quantities of the maximumcircular velocity (vmax) and radius corresponding to this ve-locity (rmax) as proxies for halo concentration and mass re-spectively, the bottom-level prior becomes

P(log(vmax), log(rmax), log(L))

= P(log(rmax)| log(vmax))P(log(vmax)| log(L))

×P(log(L)). (12)

The first term in the RHS of Equation 12 representsthe rmax-vmax relation. Simulations show that this relationclosely resembles a linear relationship between log(rmax) andlog(vmax) with an intrinsic dispersion (Diemand et al. 2007;Strigari et al. 2007a; Springel et al. 2008):

P(log(rmax)| log(vmax)) ≈ N (αrv log(vmax)+βrv, σrv) (13)

Here, we keep this functional form, but do not in-fer the model parameters from simulations. Rather, weconstrain these parameters using the next level’s likeli-hood. For the mass–luminosity relation encapsulated inP(log(vmax)| log(L)), we again assume a log–log relationshipwith some intrinsic dispersion:

P(log(vmax)| log(L)) ≈ N (αvl log(L) + βvl, σvl) (14)

Our choice is motivated by observations implying thatthis relationship is flat (Strigari et al. 2007b, 2008) andby the functional form of mass distributions in simula-tions that account for completeness and re-ionization ef-fects (Koposov et al. 2009; Busha et al. 2010; Font et al.2011; Rashkov et al. 2012). Since these probability distri-butions depend only conditionally on luminosity, the func-tional forms of these probability distributions are not ex-pected to be influenced by observational bias in luminosityand therefore describe the underlying dark matter distribu-tion properties. This is not true for the luminosity function,P(log(L)), which is affected significantly by observationalbias. Thus, the functional form of P(log(L)) is expected tobe a complicated convolution of this bias and the actual un-derlying luminosity function. However, this effect tends toflatten the P(log(L)) at faint luminosities (Koposov et al.2008). Here, we assume the simplified functional form:

P(log(L)) ∝ Lαl . (15)

For the underlying dark matter density profiles (ρ(r)) weconsider the four models listed in Table 1: NFW, CoredNFW, Burkert, and Einasto defined as

ρ(r) =

ρSr3S

r(rS+r)2NFW

ρSr3S

rcr2S+r(rS+r)2

Cored NFW

ρSr3S

(rS+r)(r2S+r2)

Burkert

ρS exp

[

−2n

(

(

rrS

)1/n

− 1

)]

Einasto

(16)where rS and ρS are derived from the profile-independentquantities vmax and rmax.

Therefore, the complete set that parametrize thebottom-level likelihood is

M(0)0 = {vmax, rmax, L}, (17)

whereas the complete sets that parameterizes the bottom-level prior for each density profile model are

M(0)1 =

{

αrv, βrv, σrv,αvl, βvl, σvl, αl

}

NFW/Burkert

{

αrv, βrv, σrv, rc/rS,αvl, βvl, σvl, αl

}

Cored NFW{

αrv, βrv, σrv, n,αvl, βvl, σvl, αl

}

Einasto.

(18)These sets are in turn constrained by the top-level likelihood

c© 2015 RAS, MNRAS 451, 2524–2535

Hierarchical mass modelling 5

Table 1. Summary of the Bayes factors.

Model loge(Bayes factor⋆)

NFW 0

Burkert 1.03 ± 0.36

Cored NFW 2.19 ± 0.34

Einasto −16.42 ± 0.35

⋆Here the Bayes factor is the ratio of the evidences (E ) relative to the

evidence of the NFW model (Emodel/ENFW).

(Equation 6). The top-level superset of data (D(1)) is com-posed of individual data sets (Equation 9) of the 20 dSphslisted in Table 3. Standard non-informative priors were as-sumed for the top-level priors: uniform in αx, βx, and log(σx)whose prior ranges are given in Table 2. We also assume uni-form priors for the cored NFW scaled core radius (rc/rS) andEinasto index (n) limited to the range 0 < rc/rS < 1 and0.5 < n < 10 respectively. The allowed range for the lumi-nosities is also marginalized over with the exception of theEinasto model whose luminosity range, because of compu-tational complications, was set to be 2.0 < log10(L) < 8.0.For the other models, the lower limit to log10(L) was al-lowed to vary between 0 and 5 whereas the upper limit wasallowed to vary between 5 and 10. However, in next sectionwe show that the form of the luminosity function, such asthe slope or allowed luminosity range, affects neither thefinal lower-level posteriors nor the higher-level parametersof interest. In total, this model contains 67–68 parameters(20×M

(0)0 +M

(0)1 ), though each galaxies’ Li parameter is an-

alytically integrated to reduce the number of parameters by20. To explore this parameter space, we employ a metropo-lis nested sampling technique (Skilling 2004; Brewer et al.2010) obtaining approximately 500 000 sample points perrun. In the next section we show that these results not onlydiminish systematic uncertainties due to the degeneracy be-tween the enclosed mass and the velocity anisotropy, butalso directly constrain the properties of the Milky Way dSphsatellite distribution.

4 RESULTS

The previously mentioned anisotropy-mass degeneracy thatis intrinsic in Jeans modelling makes mass modelling in-herently dominated by prior assumptions. Because MLMconstrains the overall distribution (and thus the lower-level priors), this approach can significantly reduce the ef-fect of this degeneracy. Fig. 1 (left-hand panel) illustratesthe advantage of this methodology in limiting prior domi-nance on lower-level posteriors. This figure shows the im-pact of varying prior assumptions on the astrophysical con-tribution of the dark matter annihilation flux, the J fac-tor (Strigari et al. 2007b; Martinez et al. 2009; Abdo et al.2010; Llena Garde et al. 2011; Charbonnier et al. 2011;Ackermann et al. 2011). Shown is one of the most prior dom-inated galaxies: Segue 1. Compared are the solutions of as-suming two priors, one non-informative and the other bias-ing the solution to low masses, with and without the use ofhierarchical modelling. Varying prior assumptions withoutthe use of hierarchical priors can alter the J factor posteri-ors by more than an order of magnitude. However, this effectis minimized when hierarchical priors are used.

The constraint that MLM provides to prior distribu-tions is not only useful in limiting the effect of prior assump-tions on lower-level posteriors, but is also useful in inferringthe distribution of galaxy properties. For example, the dis-tribution of the mass within 300pc as well as the relationbetween vmax and rmax have specific simulated predictions.In the first column in Fig. 3, we show the joint αrv–βrv

posteriors for the four models considered here. These pos-teriors are in excellent agreement with CDM simulations(Diemand et al. 2007; Springel et al. 2008; Strigari et al.2007a, shown by the blue mark). Unfortunately, while theeffect of the degeneracy between the enclosed mass and ve-locity anisotropy is minimized at the lower-level posteriors,this degeneracy manifests itself through an ‘αrv–βrv degen-eracy’. This is a consequence of the Milky Way dwarfs ap-proximately sharing the same scale. But even so, the spreadin scale radii is enough to partly constrain this degeneracy.This effect also manifests itself in the αvl–βvl joint posteriors(second column), but to a much lesser extent. It is actuallythe constraint of the P(log(vmax)| log(L)) prior that givesthis method this drastic improvement. This is to be expectedsince we are including information (luminosity) that was notincluded in previous studies. In short, we are constrainingour selection of galaxies to be part of a physical galaxydistribution that is dependent on luminosity – thus, con-straints on this underlying distribution will naturally leadto more robust individual galaxy property determination.The last column in Fig. 3 shows the posterior of the slopeof the luminosity function. Out of the lower-level priors con-strained, the luminosity function is the only prior directlyaffected by observation bias. This could very well affect theform of the luminosity function, even though this effect maystill produce a luminosity function close to a power law (seeTollerud et al. 2008). Fortunately, even though oversimplifi-cation of the observation bias may lead to systematic biaseson αvl and βvl, there is only a minimal, if any, influence onthe individual lower-level posteriors or on the top-level pos-teriors αrv and βrv. The reason for this is that the parame-ters of interest do not explicitly depend on the luminosity inthe lower-level likelihoods. Thus, we may replace the jointvmax–L prior with

Peff(log(vmax)| log(L(obs))) ∝∫

d(log(L))P(log(L))

×P(log(L(obs)))| log(L))P(log(vmax)| log(L))

where it is Peff(log(vmax)| log(Lmeas)) that is effectively be-ing constrained. Consequently, if the lower-level parametersare the only parameters of interest, then one may com-pletely forgo specifying the luminosity function and modelPeff(log(vmax)| log(Lmeas)) directly. This effect can be seenin Fig. 1 (right-hand panel) where changes in the luminosi-ties function’s shape does not affect the lower-level poste-riors. However in this study, we use the full joint prior inorder to produce vmax–L posteriors that can be consistentlycompared to future simulations.

In Fig. 4 we plot the relevant parameter constraintsfor each galaxy: log(vmax) versus log(L), log(rmax) versuslog(vmax), and log(M(300)) versus log(L). Overlaid is themedian fit prior distribution. These plots show that indi-vidual posterior constraints for each galaxy agree well withthe inferred overall galaxy distribution. The log(rmax) ver-sus log(vmax) plots show the net effect of the ‘αlv – vβlv

c© 2015 RAS, MNRAS 451, 2524–2535

6 Martinez

16 18 20 22 24

00.

51

Prob

. Den

sity

log10(J [GeV2 cm−5])

flat prior

cdm prior

mlm prior

mlm (bias)

Figure 1. Left: this figure illustrates the ability of this methodology to limit prior dominance on lower-level posteriors. Compared arethe J factors posteriors using MLM to that using assumed lower-level priors. Plotted are the J factor assuming a prior that resemblesthe distribution predicted by CDM simulations (CDM prior, dotted red line), priors uniform in log(rmax) and log(vmax) (dash-dottedred line), and two posteriors that employ the MLM methodology presented in this paper. Because simulations predict a large amountof low mass sub-haloes, the CDM prior assumptions bias the J posterior artificially to lower values. The result is a posterior that issignificantly lower than if uniform priors where assumed. For comparison, the posteriors of two MLM runs are also plotted: one resultassumes the usual non-informative priors (black solid line) and the other result drastically biases our posterior results to artificially lowconcentrations. Not only are these two posteriors more robust to top-level assumptions, but the resulting posteriors are better constrained.Right: shown is the joint posterior of the maximum circular velocity (vmax) and the slope of the luminosity function. Unsurprisinglythere is no apparent correlation between the shape of the luminosity function (e.g. the slope) and lower-level posteriors. This is because

of, to the lower-level posteriors, the effective prior that influences the result is the prior integrated over all possible luminosities (seeSection 4, especially Equation 19).

degeneracy’ in the extreme values of log(vmax). This effectis most prominent at low vmax values where the posteriorswidths are the largest. This is due to the scale radii be-ing far from the stellar half-light radius – an unfortunatebyproduct of the approximate common scale shared by theMilky Way dSph galaxies. The effect of this degeneracy alsomanifests at the low-luminosity end of the log(vmax)–log(L)relation. But this effect is minimal compared to the over-all effect on the log(rmax)–log(vmax) relation. Most notable,though, is the implied log(M(300))–log(L) relation. Whilethis relation is fairly constant, there is a definite impliedsmall positive slope consistent with the value of 0.088±0.024from simulations (Rashkov et al. 2012, compare to Table 2of this paper). Whereas, the implied intrinsic dispersion of∼ −0.7 (Springel et al. 2008, in log) is consistent with thevalue derived here (see Table 2). The median and 68% cred-ible levels for the individual galaxies parameters log(rmax)and log(vmax), as well as the overall distribution parame-ters, are summarized in Tables 2 and 3. It is important tonote that these bottom-level posteriors contain informationof both the individual galaxy fit as well as the fit of the fulldata set to the lower-level prior. Thus, the width of the pos-teriors reflect both the uncertainty of the individual galaxyparameters as well as the quality of fit of the lower-levelprior. Models that produce distributions that fit the lower-level prior well allow for a larger range in the lower-levelparameters since these models naturally produce more so-lutions that are a good overall fit to the data. Conversely,models that produce distributions that poorly fit the lower-

level prior allow a shorter range in the lower-level posteriorsfor the same reason. Since the posteriors contain informationabout the full parameter space, the posterior width is the re-sult of both the individual galaxy distribution as well as theallowed range due to the fit of the prior distribution. Thus,a narrower posterior width is not necessarily indicating abetter overall fit. This is indeed the case with the Einastoprofile that even with narrower posteriors, the Bayes factor(see Table 1) indicates that this profile is disfavoured rel-ative to the other, more cuspy, profiles. Also, from Fig. 2,we see that the constraints on the common halo shape pa-rameters for the cored NFW and Einasto models indicated amore cuspy common halo shape as well. However, this studydoes not allow for different density profile shapes among thegalaxies, but rather imposes a common shape among thefull sample. Nevertheless, this result may have interestingimplications on galaxy formation as discussed in the nextsection.

5 DISCUSSION

Our result of a more cuspy halo, as compared tovery core-like (such as the Einasto or profiles witha very shallow slope), has interesting implicationsgiven recent literature (Evans et al. 2009; Strigari et al.2010; Walker & Penarrubia 2011; Agnello & Evans 2012;Breddels & Helmi 2013). First, it is important to note thatthese results should be viewed as the aggregate solution to

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Figure 2. These figures plot the posteriors for the scaled core radius of the cored NFW model (rc/rS) and the Einasto index of theEinasto model (n). Interestingly, the constraints on the common halo shape parameters for the cored NFW and Einasto models indicatea more cuspy common halo shape. However, this study does not allow for different halo profile shapes among the galaxies, but ratherimposes a common shape among the full sample. Thus, these results should be viewed as the aggregate solution to the full sample ratherthan a statement about any individual halo shape. Even so, this is an interesting result given recent literature (see Section 5).

the full sample rather than a statement about any individ-ual halo shape. It is entirely possible that, while taken asa whole, these galaxies prefer a cuspy dark matter profile,an individual galaxy’s profile may indeed exhibit more core-like behavior. This is especially true for the more extended(and luminous) galaxies whose half-light radii do not probethe very inner portions of their dark matter distributions. Ifthis hypothesis is correct, it may suggest that these haloes’once cuspy inner regions were softened due to subsequentastrophysical interactions. Another possibility is that theserecent slope measurements suffer from a constant systematicbias due to asphericity in the shape of the stellar and darkmatter density profiles. A stellar profile of high aspheric-ity can shift the measured mass at the half-light radius andimplied slope by a factor of a few (Kowalczyk et al. 2013).However, it is expected that reliable lower-limits to the in-ner slope can be achieved regardless of dark matter halo tri-axiality (Laporte et al. 2013). Because these haloes wouldbe randomly originated within the sample, the bias due totriaxiality is not expected to sensitively affect central priormeasurements. Thus, the expected net effect on hierarchicalmodelling is an increase in the dispersion of the priors.

This work utilizes only the line-of-sight second ordervelocity moments (dispersion) of a single population. Withonly line-of-sight dispersions of a single population, it isnot possible to uniquely determine the full mass profile(Walker et al. 2009a; Wolf et al. 2010). So in this sense,when analysing a single galaxy, this data cannot distin-guish between different mass profile models. This work at-tempts to alleviate this by including information of thefull galaxy sample. But, if more information beyond theline-of-sight velocity dispersions were included, such ashigher order velocity moments or multiple populations, itmay be possible to individually constrain the inner slopesolely by analysing a single galaxy (Walker & Penarrubia2011). Specifically, incorporating multiple population intothis analysis would likely increase the ‘average’ profileconstraint and is subject of future work. Other authorshave obtained mass profile constraints from the use ofhigher order moments [for example, using methods such as

phase space, Schwarzschild, or higher-order Jeans modelling( Lokas et al. 2005; Wu & ApJ 2007; Amorisco & Evans2012; Jardel & Gebhardt 2012; Richardson & Fairbairn2012; Breddels et al. 2013; Jardel et al. 2013)]. But thesemethods are more sensitive to membership issues due toforeground contamination and any physical process thatwould affect higher order moments (e.g. unresolved binaries;Minor et al. 2010). Thus, in ordered to avoid these system-atics, we avoid such methodologies in this study.

Finally, we should mention that, while our results areconsistent with a shared cuspy halo profile, there is a smallpreference for shallow core-like properties towards the cen-tre. With the exception of the Einasto profile, the Bayes fac-tor (Table 1) between the various models indicate that, whilenot definitively preferring one model, models with core-likebehaviour (especially the cored NFW) have a small prefer-ence over the NFW model. However, even if these galaxies’inner regions truly are shallow, their outer regions are veryNFW-like.

The primary benefit of hierarchical modelling to dy-namical mass modelling is the inclusion of relevant galaxypopulation distribution information. This is particularlytrue in prior dominated systems in which exclusion of suchinformation will lead to posteriors dominated by prior as-sumptions rather than by data. This is the origin in theapparent difference between the results presented here andthose presented in Strigari et al. (2007b). Because some ofthe galaxies are prior-dominated, use of an arbitrary (non-hierarchical) prior can drastically shift the results. In someinstances, especially with the prior-dominated ultrafaintgalaxies, the differences in these posteriors differ by morethan two standard deviations. This is a typical consequenceof applying a arbitrary prior instead of constraining thesepriors from the complete data set. Specifically this can beseen at the high and low ends of the luminosity function.With the use of the data-driven priors used here, we gen-erally find lower rmax and vmax values at low luminositieswhereas these values were a bit higher at high luminosities(when compared to Strigari et al. (2007b)).

While hierarchical modelling has done quite a lot rec-

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Figure 3. This figure shows the unrestrained top-level parameter posteriors (P). Shown in each column, from left to right, are the jointposteriors αrv–βrv and αlv–βlv, as well as the posterior for αl. From top to bottom, each row contains the posteriors assuming the NFW,cored NFW, Burkert, and Einasto models. The blue dot represents the αrv and βrv values predicted by simulations (Diemand et al.2007; Strigari et al. 2007a; Springel et al. 2008). The solid lines represent the 68% and 95% credible levels (all logs are base 10).

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Figure 4. This figure shows the various bottom-level posteriors for rmax, vmax, and M(300). Shown in each column, from left to right,are the individual galaxies posteriors of log(rmax) versus log(vmax), log(vmax) versus log(L), and log(M(300)) versus log(L). From top tobottom, each row contains the posteriors assuming the NFW, cored NFW, Burkert and Einasto models. Overlaid is the median fit priordistribution showing the distribution peak (solid line) and intrinsic one sigma variance (dashed line). These plots show that individualposterior constraints for each galaxy agrees well with the inferred overall galaxy distribution. The log(rmax) versus log(vmax) plots showthe net effect of the ‘αlv – vβlv

degeneracy’ in the extreme values of log(vmax). This effect is most prominent at low vmax values wherethe posteriors widths increase the at more extreme vmax values. This is due to the scale radii being far from the stellar half-light radius– an unfortunate byproduct of the approximate common scale shared by the Milky Way dSph galaxies. The effect of this degeneracyalso manifests at the low-luminosity end of the log(vmax) versus log(L) relation. But this effect is minimal compared to the overalleffect on the log(rmax) versus log(vmax) relation. Most notably though is the implied log(M(300)) versus log(L) relation. While thisrelation is fairly constant, there is a definite implied small positive slope consistent with simulated value of 0.088± 0.024 (Rashkov et al.2012). Note that these bottom-level posteriors contain information of both the individual galaxy fit as well as the fit of the full dataset to the lower-level prior. Thus, the width of the posteriors reflect both the uncertainty of the individual galaxy parameters as well asthe quality of fit of the lower-level prior. Models that produce distributions that fit the lower-prior well allow for a larger range in thelower-level parameters since these models naturally produce more solutions that are a good overall fit to the data. Conversely, modelsthat produce distributions that poorly fit the lower-level prior allow a shorter range in the lower-level posteriors for the same reason.Since the posteriors contain information about the full parameter space, the posterior width is the result of both the individual galaxydistribution as well as the allowed range due to the fit of the prior distribution. Thus, a narrower posterior width is not necessarilyindicate a better overall fit.

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10 Martinez

onciling the Local Group dwarf spheriodal distribution withthat predicted by CDM at the lower end of the mass func-tion, it has only exacerbated the Too-Big-To-Fail problem.Here we have shown that, not only would the local dwarfshave concentrations inconsistent with CDM if they werehosted by haloes with a vmax of 20−40 km/sec, but that it isstatistically inconsistent for them to be hosted by haloes of avmax this size. In other words, the improved results from thisanalysis undoubtedly shows that there is a deficit of MilkyWay haloes with a vmax of 20 − 40 km/sec.

We have shown here that MLM may drastically improveresults and can be applied to any problem that involves anensemble of data sets – given that this ensemble originatefrom the same underlying distribution. Problems that meetthis criterion, such as determining the mass and period dis-tribution of planets (Howard et al. 2012) or the stellar initialmass function of star clusters (Bastian et al. 2010) standto benefit from this type of analysis. In particular, MilkyWay dSph measurements stand to benefit from this anal-ysis because they not only meet this criterion, but previ-ous analyses suggest that these galaxies truly follow an un-derlying distribution (Diemand et al. 2007; Neto et al. 2007;Strigari et al. 2007b; Springel et al. 2008). Most notable, thegalaxies at ultrafaint luminosities have benefited the mostfrom this analysis. This is entirely due to the fact that theuncertainty is simply an representation of our total ‘degreeof belief’. Here, it is important to realize that this beliefis based on all available information: information containedin the individual data set and information contained in theunderlying distribution. Since these galaxies’ constraints ontheir mass profiles are dominated by the lack of knowledgeof the underlying distribution, it is these constraints thathave the most to gain from this methodology. These galax-ies will also be affected most if our knowledge of the under-lying distribution changes. Indications that some of the ul-trafaint satellites may have dispersion lower than previouslymeasured Kirby et al. (2013, e.g.) would not only affect themass measurements of those individual galaxies, but alsoall the ultrafaint galaxies as a whole. However, such indica-tions would only serve to exclude solutions with high con-centrations (e.g. large log(rmax)–log(rmax) slopes, see Fig.3) that are inconsistent with CDM simulations. Therefore,if this were indeed the case, we suspect that this would onlystrengthen our main conclusion that the Milky Way dSph’sdistribution is consistent within the ΛCDM paradigm forhaloes with vmax < 20 km/sec.

We reiterate that these conclusions are based on dataconstraints rather than prior assumptions. Although it mayseem that the subjectivity increased with the inclusion ofseven new top-level priors, this is indeed not the case be-cause the 20 lower-level priors are being interpreted as ac-tual physical distributions. As a matter of fact, the onlyassumption MLM requires is that the total galaxy set sam-ples an overall distribution described by the prior proba-bility. But this is equivalent to assuming that each individ-ual data set samples a larger distribution of data describedby the likelihood – an assumption that is necessary to per-form any likelihood analysis. Furthermore, the issues thatplague a normal likelihood analysis also hold for the com-plete set of lower-level posteriors. One example that is oftenoverlooked is the effect of the choice of parametrization ofthe likelihood. If a likelihood is parametrized with too few

parameters, relevant detail may be lost or misinterpreted,whereas too many parameters may cause over-fitting. Like-wise, prior parametrization (e.g. Equation 12) is also an is-sue for MLM for the same reasons. An interesting directionof this work is to explore varying forms of P(vmax|L) (e.g.P(M(300)|L)). Our main motivation in the selection of alognormal form of P(vmax|L) was the apparent flat M(300)-L relation claimed by Strigari et al. (2007a). Specifically, wequestioned whether posteriors derived using this prior infor-mation would yield the same results. Of course, we foundthat they did not. But, from Fig. 4 it is conceivable that thehigh luminosity galaxies follow a different M(300)–L rela-tionship than low luminosity galaxies. If so, this may haveprofound consequences on our conclusions considering thatthe main reason for our improved constraints is the additionof luminosity information. While it is hard to surmise whateffect, if any, these issues have on the allowed density profilesand the subsequently constrained rmax–vmax distribution,study of alternative prior forms may give insights to the con-sistency of alternative dark matter theories. Because of thelarge dimensional parameter space that is usually involvedin Bayesian hierarchical modelling we have taken advantagehere of every technical simplification available. Specifically,we have integrated over the luminosities of each individualgalaxy to reduce the parameter space by a third. But if morecomplicated luminosity mass relations were considered, thenthe integral over luminosity would not be analytic. For thisreason, these technical difficulties preclude us from includ-ing these issues in this current study. Thus, we leave thisanalysis to future work.

6 CONCLUSION

In this work we introduced a new methodology to derivemass profiles for the Milky Way Local Group dwarf dSphs.This new methodology, based on MLM, exploits the factthat these individual galaxies sample an underlying distri-bution. By simultaneously constraining both the individ-ual galaxy and the overall galaxy distribution parameters,not only do individual galaxies posteriors become more ro-bust, but the overall distribution properties may also beinferred. This is done by interpreting the individual galaxyprior probabilities as the probability of observing a partic-ular galaxy from this overall distribution. In other words,we interpret the prior probability as a frequentist probabil-ity sampling from an actual physical galaxy distribution.Thus, in much the same way that single data point prob-abilities can be combined to form a ‘lower-level’ likelihoodthat can be used to constrain individual galaxy parameters,the posteriors from the full galaxy sample can be combinedto form a ‘upper-level’ likelihood that can be used to con-strain both the overall galaxy distribution and the individ-ual galaxy parameters. This interpretation of the individualprior distributions then becomes beneficial for the followingreasons. First, it removes the subjectivity normally associ-ated with the Bayesian ‘degree of belief’ interpretation ofprobability on the lower-level priors. And secondly, it allowsus to use of the combined data set to directly constrain theprior probabilities via this newly defined ‘higher-level’ likeli-hood. Because the Bayesian interpretation of probability hasnot been used in the formation of this likelihood, multilevel

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Hierarchical mass modelling 11

analysis can be done in both the frequentist and Bayesianframeworks. For this analysis, we utilized the Bayesian mul-tilevel methodology, Bayesian hierarchical modelling.

Application of this methodology resulted not only inmore robust individual galaxy mass profile constraints, butalso in fairly robust constraints on the overall distribution.The galaxies that benefited the most from this analysis werethe ultrafaint dwarf satellite galaxies. These galaxies, be-cause of their extreme prior dominance, had the most togain from the extra information gained from constrainingthe prior PDF. Although the anisotropy-mass degeneracywas greatly minimized, we found that this indirectly causeda somewhat constrained degeneracy between the slope andintercept of the overall distribution’s rmax–vmax relation.Even so, the overall inferred relationship between rmax andvmax as well as the inferred relationship between the masswithin 300pc and luminosity are in excellent agreement withCDM simulations. Also, we found that a cuspy halo is a good‘average’ fit to the Milky Way satellites density profiles. Al-though this does not exclude the possibility that individ-ual galaxies (especially extended high-luminosity satellites)from having cored dark matter profiles, it may suggest thatthese galaxies’ central regions were once cusped but mayhave ‘softened’ due to astrophysical interactions.

7 ACKNOWLEDGEMENTS

GDM is very appreciative to Alex Drlica-Wagner, Jan Con-rad, Louie Strigari, Annika Peter, Tuan Do, Manoj Kapling-hat, and Christian Farnier for useful comments and discus-sions. GDM also acknowledges support from the Wennergrenfoundation.

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Table 2. Summary of unrestrained top-level model parameters and results

Prior Prior Derived value Derived value Derived value Derived value

parameters range (NFW) (Cored NFW) (Burkert) (Einasto) Description

log10(σrv) [−10, 3] −1.00+0.36−0.37 −1.01+0.37

−0.44 −1.44+0.54−0.38 −1.03+0.40

−0.61 (log) dispersion of the log(rmax)–log(vmax) relation

αrv [−30, 30] 2.44+1.30−1.20 2.59+1.43

−1.10 2.43+0.97−0.95 1.89+1.57

−1.23 Slope of the log(rmax)–log(vmax) relation

βrv [−30, 30] −3.10+1.46−1.58 −3.41+1.27

−1.74 −3.32+1.07−1.13 −2.31+1.47

−1.84 Intercept of the log(rmax)–log(vmax) relation

log10(σvl) [−4, 3] −1.10+0.16−0.16 −1.15+0.15

−0.15 −1.20+0.12−0.13 −1.14+0.16

−0.16 (log) dispersion of the log(vmax)–log(L) relation

αvl [−10, 10] 0.05+0.03−0.04 0.05+0.03

−0.03 0.06+0.02−0.02 0.05+0.03

−0.03 Slope of the log(vmax)–log(L) relation

βvl [−10, 10] 0.93+0.24−0.18 0.89

+0.16−0.16 0.87

+0.14−0.13 0.91

+0.20−0.18 Intercept of the log(vmax)–log(L) relation

αl [−3, 3] −0.07+0.08−0.09

−0.07+0.08−0.08

−0.06+0.08−0.08

0.00+0.06−0.06

Slope of the luminosity function

rc/rS [0, 1] – 0.40+0.39−0.30 – – Scaled core radius

n [0.5, 10] – – – 6.87+2.18−2.72 Einasto index

Derived prior Derived value Derived value Derived value Derived value

parameters (NFW) (Cored NFW) (Burkert) (Einasto) Description

log10(σml) −0.88+0.14−0.13

−0.86+0.15−0.13

−0.88+0.12−0.12

−0.90+0.13−0.11

(log) dispersion of the log(M300)–log(L) relation

αml 0.07+0.07−0.06 0.08

+0.06−0.05 0.10

+0.06−0.05 0.07

+0.06−0.05 Slope of log(M300)–log(L) relation

βml 6.76+0.32−0.38

6.67+0.31−0.35

6.57+0.32−0.32

6.71+0.28−0.32

Intercept of log(M300)–log(L) relation

Note: All vmax in km/s. All rmax and r1/2 in kpc. All Masses in M⊙. All Luminosities in L⊙. All log to the base of 10.

Table 3. Summary of galaxy model parameters and results

NFW Cored NFW Burkert Einasto

Galaxy log10(vmax)⋆

log10(rmax)† log10(vmax)

⋆log10(rmax)

† log10(vmax)⋆

log10(rmax)† log10(vmax)

⋆log10(rmax)

†

Carina 1.09+0.14−0.04 −0.35

+0.72−0.33 1.09

+0.09−0.03 −0.58

+0.58−0.26 1.09

+0.04−0.03 −0.69

+0.25−0.15 1.10

+0.10−0.04 −0.18

+0.61−0.39

Draco 1.26+0.07−0.04 −0.12+0.28

−0.24 1.24+0.04−0.04 −0.28+0.19

−0.19 1.23+0.04−0.03 −0.36+0.16

−0.17 1.26+0.08−0.05 −0.01+0.42

−0.30

Fornax 1.27+0.03−0.02 0.01+0.31

−0.21 1.28+0.03−0.02 −0.11+0.32

−0.18 1.28+0.03−0.02 −0.22+0.17

−0.16 1.28+0.06−0.02 0.11+0.56

−0.30

Leo I 1.24+0.09−0.05

−0.06+0.40−0.28

1.23+0.07−0.05

−0.24+0.35−0.24

1.22+0.05−0.04

−0.37+0.27−0.19

1.24+0.09−0.05

0.02+0.52−0.33

Leo II 1.15+0.15−0.07

−0.18+0.54−0.40

1.13+0.12−0.06

−0.38+0.46−0.33

1.12+0.11−0.04

−0.57+0.48−0.21

1.15+0.11−0.06

−0.08+0.57−0.37

Sculptor 1.24+0.08−0.05 −0.07+0.35

−0.27 1.23+0.06−0.04 −0.24+0.31

−0.22 1.23+0.05−0.03 −0.36+0.24

−0.18 1.24+0.08−0.05 0.02+0.49

−0.32

Sextans 1.13+0.04−0.03 −0.36+0.28

−0.19 1.15+0.04−0.03 −0.48+0.16

−0.15 1.16+0.03−0.03 −0.51+0.10

−0.11 1.13+0.05−0.04 −0.22+0.44

−0.25

Ursa Minor 1.29+0.05−0.04 0.01+0.27

−0.24 1.28+0.04−0.04 −0.13+0.22

−0.19 1.28+0.04−0.03 −0.24+0.17

−0.17 1.29+0.06−0.04 0.08+0.43

−0.29

Bootes 1.18+0.10−0.08 −0.23+0.36

−0.30 1.16+0.07−0.06 −0.43+0.25

−0.24 1.15+0.06−0.06 −0.53+0.20

−0.18 1.17+0.10−0.07 −0.12+0.44

−0.33

Canes Venatici I 1.15+0.05−0.05 −0.32+0.32

−0.22 1.15+0.04−0.04 −0.46+0.20

−0.18 1.16+0.04−0.03 −0.52+0.14

−0.13 1.15+0.06−0.05 −0.16+0.47

−0.29

Canes Venatii II 1.09+0.13−0.10

−0.39+0.46−0.48

1.09+0.10−0.09

−0.56+0.33−0.41

1.08+0.09−0.08

−0.68+0.26−0.27

1.10+0.11−0.10

−0.23+0.49−0.43

Coma Berentices 1.09+0.13−0.09

−0.40+0.46−0.46

1.07+0.10−0.08

−0.63+0.32−0.39

1.05+0.09−0.07

−0.75+0.26−0.27

1.08+0.12−0.09

−0.26+0.48−0.44

Hercules 1.11+0.11−0.08 −0.37+0.42

−0.36 1.09+0.07−0.07 −0.61+0.30

−0.29 1.08+0.06−0.06 −0.71+0.21

−0.20 1.10+0.10−0.08 −0.24+0.49

−0.40

Leo IV 1.11+0.12−0.10 −0.37+0.44

−0.44 1.09+0.09−0.09 −0.58+0.32

−0.37 1.09+0.08−0.08 −0.68+0.24

−0.24 1.11+0.11−0.10 −0.22+0.49

−0.43

Leo T 1.19+0.10−0.08 −0.21+0.34

−0.30 1.16+0.07−0.06 −0.44+0.23

−0.25 1.16+0.06−0.06 −0.53+0.19

−0.19 1.19+0.09−0.07 −0.12+0.45

−0.33

Segue I 1.07+0.15−0.12 −0.50+0.46

−0.51 1.04+0.11−0.10 −0.74+0.32

−0.43 1.02+0.09−0.08 −0.84+0.24

−0.28 1.05+0.14−0.12 −0.35+0.49

−0.52

Ursa Major I 1.12+0.09−0.07 −0.38+0.41

−0.33 1.10+0.06−0.06 −0.56+0.26

−0.26 1.09+0.05−0.05 −0.68+0.18

−0.17 1.11+0.09−0.07 −0.22+0.47

−0.36

Ursa Major II 1.12+0.11−0.08 −0.39

+0.39−0.35 1.10

+0.08−0.07 −0.58

+0.28−0.29 1.09

+0.07−0.06 −0.68

+0.20−0.20 1.12

+0.11−0.08 −0.22

+0.45−0.38

Willman 1 1.08+0.15−0.12

−0.45+0.46−0.49

1.05+0.10−0.11

−0.71+0.30−0.39

1.04+0.09−0.09

−0.80+0.23−0.27

1.06+0.13−0.11

−0.31+0.47−0.49

Segue 2 1.07+0.15−0.12 −0.45+0.48

−0.52 1.05+0.10−0.11 −0.69+0.31

−0.41 1.03+0.09−0.09 −0.81+0.24

−0.27 1.05+0.13−0.11 −0.31+0.50

−0.50

⋆All vmax in km/s.

†All rmax in kpc.

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