Draft version May 29, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
REDEFINING THE MISSING SATELLITES PROBLEM
Louis E. Strigari
, James S. Bullock
, Manoj Kaplinghat
, Juerg Diemand
, Michael Kuhlen
, Piero
Madau
Draft version May 29, 2018
ABSTRACT
Numerical simulations of Milky-Way size Cold Dark Matter (CDM) halos predict a steeply rising
mass function of small dark matter subhalos and a substructure count that greatly outnumbers the
observed satellites of the Milky Way. Several proposed explanations exist, but detailed comparison
between theory and observation in terms of the maximum circular velocity (Vmax) of the subhalos is
hampered by the fact that Vmax for satellite halos is poorly constrained. We present comprehensive
mass models for the well-known Milky Way dwarf satellites, and derive likelihood functions to show
that their masses within 0.6 kpc (M0.6) are strongly constrained by the present data. We show that
the M0.6 mass function of luminous satellite halos is flat between ∼ 107 and 108M⊙. We use the
“Via Lactea” N-body simulation to show that the M0.6 mass function of CDM subhalos is steeply
rising over this range. We rule out the hypothesis that the 11 well-known satellites of the Milky Way
are hosted by the 11 most massive subhalos. We show that models where the brightest satellites
correspond to the earliest forming subhalos or the most massive accreted objects both reproduce the
observed mass function. A similar analysis with the newly-discovered dwarf satellites will further test
these scenarios and provide powerful constraints on the CDM small-scale power spectrum and warm
dark matter models.
Subject headings: Cosmology: dark matter, theory–galaxies
1. INTRODUCTION
It is now well-established that numerical simulations
of Cold Dark Matter (CDM) halos predict many orders
of magnitude more dark matter subhalos around Milky
Way-sized galaxies than observed dwarf satellite galax-
ies (Klypin et al. 1999; Moore et al. 1999; Diemand et al.
2007). Within the context of the CDM paradigm, there
are well-motivated astrophysical solutions to this ‘Miss-
ing Satellites Problem’ (MSP) that rely on the idea
that galaxy formation is inefficient in the smallest dark
matter halos (Bullock et al. 2000; Benson et al. 2002;
Somerville 2002; Stoehr et al. 2002; Kravtsov et al. 2004;
Moore et al. 2006; Gnedin & Kravtsov 2006). However,
from an observational perspective, it has not been possi-
ble to distinguish between these solutions.
A detailed understanding of the MSP is limited by our
lack of a precise ability to characterize the dark matter
halos of satellite galaxies. From an observational per-
spective, the primary constraints come from the veloc-
ity dispersion of ∼ 200 stars in the population of dark
matter dominated dwarf spheroidal galaxies (dSphs)
(Wilkinson et al. 2004; Lokas et al. 2005; Munoz et al.
2005, 2006; Westfall et al. 2006; Walker et al. 2005, 2006;
Sohn et al. 2006; Gilmore et al. 2007). However, the ob-
served extent of the stellar populations in dSphs are ∼
kpc, so these velocity dispersion measurements are only
able to probe properties of the halos in this limited radial
1 Center for Cosmology, Department of Physics & Astronomy,
University of California, Irvine, CA 92697
2 Department of Astronomy & Astrophysics, University of Cali-
fornia, Santa Cruz, CA 95064
3 School of Natural Sciences, Institute for Advanced Study, Ein-
stein Drive, Princeton, NJ 08540
4 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str.
1, 85740 Garching, Germany.
5 Hubble Fellow
regime.
From the theoretical perspective, dissipationless nu-
merical simulations typically characterize subhalo counts
as a function of the total bound mass or maximum cir-
cular velocity, Vmax. While robustly determined in sim-
ulations, global quantities like Vmax are difficult to con-
strain observationally because dark halos can extend well
beyond the stellar radius of a satellite. Indeed stellar
kinematics alone provide only a lower limit on the halo
Vmax value (see below). This is a fundamental limita-
tion of stellar kinematics that cannot be remedied by
increasing the number of stars used in the velocity dis-
persion analysis (Strigari et al. 2007). Thus determining
Vmax values for satellite halos requires a theoretical ex-
trapolation. Any extrapolation of this kind is sensitive
to the predicted density structure of subhalos, which de-
pends on cosmology, power-spectrum normalization, and
the nature of dark matter (Zentner & Bullock 2003).
Our inability to determinate Vmax is the primary lim-
itation to test solutions to the MSP. One particular so-
lution states that the masses of the dSphs have been
systematically underestimated, so that the ∼ 10 bright-
est satellites reside systematically in the ∼ 10 most mas-
sive subhalos (Stoehr et al. 2002; Hayashi et al. 2003). A
byproduct of this solution is that there must be a sharp
mass cutoff at some current subhalo mass, below which
galaxy formation is suppressed. Other solutions, based
on reionization suppression, or a characteristic halo mass
scale prior to subhalo accretion predict that the sup-
pression comes in gradually with current subhalo mass
(Bullock et al. 2000; Kravtsov et al. 2004; Moore et al.
2006).
In this paper, we provide a systematic investigation of
the masses of the Milky Way satellites. We highlight
that in all cases the total halo masses and maximum
http://arxiv.org/abs/0704.1817v2
2 Strigari et al.
circular velocities are not well-determined by the data.
We instead use the fact that the total cumulative mass
within a characteristic radius ∼ 0.6 kpc is much bet-
ter determined by the present data (Strigari et al. 2007).
We propose using this mass, which we define as M0.6,
as the favored means to compare to the dark halo pop-
ulation in numerical simulations. Unlike Vmax, M0.6 is
measured directly and requires no cosmology-dependent
or dark-matter-model-dependent theoretical prior.
In the following sections, we determine the M0.6 mass
function for the Milky Way satellites, and compare it
to the corresponding mass function measured directly in
the high-resolution “Via Lactea” substructure simulation
of Diemand et al. (2007). We rule out the possibility
that there is a one-to- one correspondence between the 11
most luminous satellites and the most massive subhalos
in Via Lactea. We find that MSP solutions based on
reionization and/or characteristic halo mass scales prior
to accretion are still viable.
2. MILKY WAY SATELLITES
Approximately 20 satellite galaxies can be classified
as residing in MW subhalos. Of these, ∼ 9 were
discovered within the last two years and have very
low luminosities and surface brightnesses (Willman et al.
2005; Willman et al. 2005; Belokurov et al. 2006, 2007;
Zucker et al. 2006a,b). The lack of precision in these
numbers reflects the ambiguity in the classification of the
newly-discovered objects. The nine ‘older’ galaxies clas-
sified as dSphs are supported by their velocity dispersion,
and exhibit no rotational motion (Mateo 1998). Two
satellite galaxies, the Small Megallanic Cloud (SMC)
and Large Megallanic Cloud (LMC), are most likely sup-
ported by some combination of rotation and dispersion.
Stellar kinematics suggest that the LMC and SMC are
likely the most massive satellite systems of the Milky
We focus on determining the masses of the nine most
well-studied dSphs. The dark matter masses of the dSphs
are determined from the line-of-sight velocities of the
stars, which trace the total gravitational potential. We
assume a negligible contribution of the stars to the grav-
itational potential, which we find to be an excellent ap-
proximation. The dSph with the smallest mass-to-light
ratio is Fornax, though even for this case we find that the
stars generally do not affect the dynamics of the system
(see Lokas 2002; Wu 2007, and below).
Under the assumptions of equilibrium and spherical
symmetry, the radial component of the stellar velocity
dispersion, σr, is linked to the total gravitational poten-
tial of the system via the Jeans equation,
d(ν⋆σ
= −ν⋆V 2c − 2βν⋆σ
r . (1)
Here, ν⋆ is the stellar density profile, V
c = GM(r)/r
includes the total gravitating mass of the system, and
the parameter β(r) = 1 − σ2θ/σ
r characterizes the dif-
ference between the radial (σr) and tangential (σθ) ve-
locity dispersions. The observable velocity dispersion is
constructed by integrating the three- dimensional stellar
radial velocity dispersion profile along the line-of-sight,
σ2los(R) =
I⋆(R)
rrdr√
r2 −R2
, (2)
where R is the projected radius on the sky. The surface
density of stars in all dSphs are reasonably well-fit by a
spherically-symmetric King profile (King 1962),
I⋆(R) ∝
r2king
)−1/2
r2king
)−1/2
where rt and rking are fitting parameters denoted as the
tidal and core radii. 6 The spherically symmetric stellar
density can be obtained with an integral transformation
of the surface density. From the form of Eq. (2), the
normalization in Eq. (3) is irrelevant.
Some dSphs show evidence for multiple, dynamically
distinct stellar populations, with each population de-
scribed by its own surface density and velocity dispersion
(Harbeck et al. 2001; Tolstoy et al. 2004; Westfall et al.
2006; McConnachie et al. 2006; Ibata et al. 2006). In a
dSph with i = 1...Np populations of stars, standard ob-
servational methods will sample a density-weighted av-
erage of the populations:
νi (4)
σ2r =
r,ı , (5)
where νi and σi are the density profile and radial stellar
velocity dispersion of stellar component i. In principle,
each component has its own stellar velocity anisotropy
profile, βi(r). In this case, Equation (1) is valid for ν⋆ and
σr as long as an effective velocity anisotropy is defined
β(r)=
ν⋆σ2r
βiνiσ
i . (6)
From these definitions, we also have
ı I⋆,ıσ
los,ı =
The conclusion we draw from this argument is that
the presence of multiple populations will not affect the
inferred mass structure of the system, provided that the
velocity anisotropy is modeled as a free function of ra-
dius. Since the functional form of the stellar velocity
anisotropy is unknown, we allow for a general, three pa-
rameter model of the velocity anisotropy profile,
β(r) = β∞
r2β + r
+ β0. (7)
A profile of this form allows for the possibility for β(r) to
change from radial to tangential orbits within the halo,
and a constant velocity anisotropy is recovered in the
limit β∞ → 0, and β0 → const.
In Equations (1) and (2), the radial stellar velocity dis-
persion, σr , depends on the total mass distribution, and
thus the parameters describing the dark matter density
profile. Dissipation-less N-body simulations show that
the density profiles of CDM halos can be characterized
ρ(r̃) =
r̃γ(1 + r̃)δ
; r̃ = r/rs, (8)
6 Our results are insensitive to this particular parameterization
of the light profile.
Redefining the Missing Satellites Problem 3
where rs and ρs set a radial scale and density normaliza-
tion and γ and δ parameterize the inner and outer slopes
of the distribution. For dark matter halos unaffected
by tidal interactions, the most recent high-resolution
simulations find δ + γ ≈ 3 works well for the outer
slope, while 0.7 . γ . 1.2 works well down to ∼ 0.1%
of halo virial radii (Navarro et al. 2004; Diemand et al.
2005). This interior slope is not altered as a subhalo
loses mass from tidal interactions with the MW potential
(Kazantzidis et al. 2004). The outer slope, δ, depends
more sensitively on the tidal interactions in the halo. The
majority of the stripped material will be from the outer
parts of halos, and thus δ of subhalo density profiles will
become steeper than those of field halos. Subhalos are
characterized by outer slopes in the range 2 . δ . 3.
Given the uncertainty in the β(r) and ρ(r) profiles,
we are left with nine shape parameters that must be
constrained via line-of-sight velocity dispersion measure-
ments: ρs, rs, β0, β∞, rβ , γ, δ, rking , and rt. While the
problem as posed may seem impossible, there are a num-
ber of physical parameters, which are degenerate between
different profile shapes, that are well constrained. Specif-
ically, the stellar populations constrain Vc(r) within a
radius comparable to the stellar radius rt ∼ kpc. As
a result, quantities like the local density and integrated
mass within the stellar radius are determined with high
precision (Strigari et al. 2006), while quantities that de-
pend on the mass distribution at larger radii are virtually
unconstrained by the data.
It is useful to determine the value of the radius where
the constraints are maximized. The location of this char-
acteristic radius is determined by the form of the integral
in Eq. (2). We can gain some insight using the exam-
ple of a power-law stellar distribution ν⋆(r), power-law
dark matter density profile ρ ∝ r−γ⋆ , and constant ve-
locity anisotropy. The line-of-sight velocity dispersion
depends on the three-dimensional stellar velocity disper-
sion, which can be written as
σ2r (r) = ν
Gν⋆(r)M(r)r
2β−2dr ∝ r2−γ⋆ (9)
From the shape of the King profile, the majority of
stars reside at projected radii rking . R . rt, where
the stellar distribution is falling rapidly ν⋆ ∼ r−3.5.
In this case, for β = 0, the LOS component scales as
σ2los(R) ∝
r−0.5−γ⋆(r2 −R2)−1/2dr and is dominated
by the mass and density profile at the smallest relevant
radii, r ∼ rking. For R . rking, ν⋆ ∝ r−1 and σ2los is simi-
larly dominated by r ∼ rking contributions. We note that
although the scaling arguments above hold for constant
velocity anisotropies, they can be extended by consider-
ing anisotropies that vary significantly in radius. They
are also independent of the precise form of I⋆, provided
there is a scale similar to rking .
In Strigari et al. (2007), it was shown that typical ve-
locity dispersion profiles best constrain the mass (and
density) within a characteristic radius ≃ 2 rking . For
example, the total mass within 2rking is constrained to
within ∼ 20% for dSphs with ∼ 200 line-of-sight veloc-
ities. Note that when deriving constraints using only
the innermost stellar velocity dispersion and fixing the
anisotropy to β = 0, the characteristic radius for best
constraints decreases to∼ 0.5rking (e.g. Penarrubia et al.
2007).
As listed in Table 1, the Milky Way dSphs are ob-
served to have variety of rking values, but rking ∼ 0.3
kpc is typical. The values of rking and rt are taken from
Mateo (1998). In order to facilitate comparison with
simulated subhalos, we chose a single characteristic ra-
dius of 0.6 kpc for all the dwarfs, and we represent the
mass within this radius as M0.6 = M(< 0.6 kpc). The
relative errors on the derived masses are unaffected for
small variations in the characteristic radius in the range
∼ 1.5− 2.5 rking. Deviations from a true King profile at
large radius (near rt) do not affect these arguments, as
long as there is a characteristic scale similar to rking in
the surface density. The only dSph significantly affected
by the choice of 0.6 kpc as the characteristic radius is
Leo II, which has rt = 0.5 kpc. Since the characteristic
radius is greater than twice rking , the constraints on its
mass will be weakest of all galaxies (with the exception
of Sagittarius, as discussed below).
3. DARK MATTER HALO MASSES AT THE
CHARACTERISTIC RADIUS
We use the following data sets: Wilkinson et al.
(2004); Munoz et al. (2005, 2006); Westfall et al. (2006);
Walker et al. (2005, 2006); Sohn et al. (2006); Siegal et
al. in preparation. These velocity dispersions are deter-
mined from the line-of-sight velocities of ∼ 200 stars in
each galaxy, although observations in the coming years
will likely increase this number by a factor ∼ 5 − 10.
From the data, we calculate the χ2, defined in our case
(σobs,ı − σth,ı)2
. (10)
Here σ2obs is the observed velocity dispersion in each bin,
σ2th is the theoretical value, obtained from Eq. (2), and
ǫ2ı are errors as determined from the observations.
It is easy to see that, when fitting to a single data set
of ∼ 200 stars, parameter degeneracies will be signifi-
cant. However, from the discussion in section 2, M0.6
is well-determined by the LOS data. To determine how
well M0.6 is constrained, we construct likelihood func-
tions for each galaxy. When thought of as a function
of the theoretical parameters, the likelihood function, L,
is defined as the probability that a data set is acquired
given a set of theoretical parameters. In our case L is a
function of the parameters γ, δ, rs, ρs, and β0, β∞, rβ ,
and is defined as L = e−χ
2/2. In writing this likelihood
function, we assume that the errors on the measured ve-
locity dispersions are Gaussian, which we find to be an
excellent approximation to the errors for a dSph system
(Strigari et al. 2007). We marginalize over the parame-
ters γ, δ, rs, ρs, and β0, β∞, rβ at fixed M0.6, and the
optimal values for M0.6 are determined by the maximum
of L .
We determine L for all nine dSphs with velocity dis-
persion measurements. For all galaxies we use the full
published velocity dispersion profiles. The only galaxy
that does not have a published velocity dispersion profile
is Sagittarius, and for this galaxy we use the central ve-
locity dispersion from Mateo (1998). The mass modeling
of Sagittarius is further complicated by the fact that it is
experiencing tidal interactions with the MW (Ibata et al.
4 Strigari et al.
Fig. 1.— The likelihood functions for the mass within 0.6 kpc
for the nine dSphs, normalized to unity at the peak.
1997; Majewski et al. 2003), so a mass estimate from the
Jeans equation is not necessarily reliable. We caution
that in this case the mass we determine is likely only an
approximation to the total mass of the system.
We determine the likelihoods by marginalizing over
the following ranges of the velocity anisotropy, inner
and outer slopes: −10 < β0 < 1, −10 < β∞ < 1,
0.1 < rβ < 10 kpc, 0.7 < γ < 1.2, and 2 < δ < 3.
As discussed above, these ranges for the asymptotic in-
ner and outer slopes are appropriate because we are con-
sidering CDM halos. It is important to emphasize that
these ranges are theoretically motivated and that obser-
vations alone do not demand such restrictive choices.
It is possible to fit all of the dSphs at present with
a constant density cores with scale-lengths ∼ 100 pc
(Strigari et al. 2006; Gilmore et al. 2007), although the
data by no means demand such a situation. Though
we consider inner and outer slopes in the ranges quoted
above, our results are not strongly affected if we widen
these intervals. For example, we find that if we allow
the inner slope to vary down to γ = 0, the widths of the
likelihoods are only changed by ∼ 10%. This reflects the
fact that there is a negligible degeneracy between M0.6
and the inner and outer slopes.
We are left to determine the regions of ρs and rs pa-
rameter space to marginalize over. In all dSphs, there
is a degeneracy in this parameter space, telling us that
it is not possible to derive an upper limit on this pair
of parameters from the data alone (Strigari et al. 2006).
While this degeneracy is not important when determin-
ing constraints on M0.6, it is the primary obstacle in
determining Vmax. From the fits we present below, we
find that the lowest rs value that provides an acceptable
fit is ∼ 0.1 kpc, and we use this as the lower limit in all
cases. In our fiducial mass models, we conservatively re-
strict the maximum value of rs using the known distance
to each dSph. In this case, we use 0.1 kpc < rs < D/2,
where D is the distance to the dSph.
In Figure 1 we show the M0.6 likelihood functions for
all of the dSphs. As is shown, we obtain strong con-
Fig. 2.— The velocity dispersion for Ursa Minor as a function of
radial distance, along with the model that maximizes the likelihood
function.
straints on M0.6 in all cases except Sagittarius, for which
we use only a central velocity dispersion. Table 1 sum-
marizes the best fitting M0.6 values for each dwarf. The
quoted errors correspond to the points where the likeli-
hood falls to 10% of its peak value. The upper panel of
Figure 3 shows M0.6 values for each dwarf as a function
of luminosity. In Figure 2 we show an example of the
velocity dispersion data as a function of radial distance
for Ursa Minor, along with the model that maximizes
the likelihood function. For all galaxies, we find χ2 per
degree of freedom values . 1.
The maximum likelihood method also allows us to con-
strain the mass at other radii spanned by the stellar dis-
tribution. The sixth column of Table 1 provides the inte-
grated mass within each dwarf’s King tidal radius. This
radius roughly corresponds to the largest radius where
a reasonable mass constraint is possible. As expected,
the mass within rt is not as well determined as the mass
within 2 rking. From these masses we are able to deter-
mine the mass-to-light ratios within rt, which we present
in the seventh column of Table 1. In the bottom panel
of Figure 3, we show mass-to-light ratios within rt as a
function of dwarf luminosity. We see the standard result
that the observable mass-to-light ratio increases with de-
creasing luminosity (Mateo 1998). Note, however, that
our results are inconsistent with the idea that all of the
dwarfs have the same integrated mass within their stellar
extent. We note that for Sagittarius, we can only obtain
a lower limit on the total mass-to-light ratio.
The last two columns in Table 1 list constraints on
Vmax for the dSphs. Column 8 shows results for an anal-
ysis with limits on rs as described above. In this case, the
integrated mass within the stellar radius is constrained
by the velocity dispersion data, but the halo rotation ve-
locity curve, Vc(r), can continue to rise as r increases
beyond the stellar radius in an unconstrained manner.
The result is that the velocity dispersion data alone pro-
vide only a lower limit on Vmax.
Stronger constraints on Vmax can be obtained if we
limit the range of rs by imposing a cosmology-dependent
prior on the dark matter mass profile. CDM simula-
Redefining the Missing Satellites Problem 5
TABLE 1
Parameters Describing Milky Way Satellites.
Galaxy rking rt LV Mass < 0.6 kpc Mass < rt M(< rt)/L Vmax [km s
−1] Vmax [km s
[kpc] [kpc] [106 L⊙] [10
7 M⊙] [10
7 M⊙] [M⊙/L⊙] (w/o prior) (with theory prior)
Draco 0.18 0.93 0.26 4.9
530 > 22 28
Ursa Minor 0.30 1.50 0.29 5.3
790 > 21 26
Leo I 0.20 0.80 4.79 4.3
106 > 14 19
Fornax 0.39 2.70 15.5 4.3
28 > 20 25
Leo II 0.19 0.52 0.58 2.1
128 > 17 9
Carina 0.26 0.85 0.43 3.4
82 > 13 15
Sculptor 0.28 1.63 2.15 2.7
68 > 20 14
Sextans 0.40 4.01 0.50 0.9
260 > 8 9
Sagittarius 0.3 4.0 18.1 20
> 20 > 11 > 19 —
Note. — Determination of the mass within 0.6 kpc and the maximum circular velocity for the dark matter halos of the dSphs.
The errors are determined as the location where the likelihood function falls off by 90% from its peak value. For Sagittarius, no
reliable estimate of Vmax with the CDM prior could be determined. The CDM prior is determined using the concordance cosmology
with σ8 = 0.74, n = 0.95 (see text for details).
Fig. 3.— The mass within 0.6 kpc (upper) and the mass-to-light
ratios within the King tidal radius (lower) for the Milky Way dSphs
as a function of dwarf luminosity. The error-bars here are defined as
the locations where the likelihoods fall to 40% of the peak values
(corresponding to ∼ 1σ errors). The lines denote, from top to
bottom, constant values of mass of 107, 108, 109 M⊙.
tions have shown that there is a correlation between
Vmax and rmax for halos, where rmax is the radius where
the circular velocity peaks. Because subhalo densities
will depend on the collapse time and orbital evolution
of each system, the precise Vmax-rmax relation is sen-
sitive to cosmology (e.g. σ8) and the formation his-
tory of the host halo itself (e.g. Zentner & Bullock 2003;
Power 2003; Kazantzidis et al. 2004; Bullock & Johnston
2005; Bullock & Johnston 2006). When converted to the
relevant halo parameters, the imposed Vmax-rmax rela-
tion can be seen as a theoretical prior on CDM ha-
los, restricting the parameter space we need to inte-
grate over. In order to illustrate the technique, we adopt
log10(rmax) = 1.35(log10(Vmax/kms
−1) − 1)− 0.196 kpc
with a scatter of 0.2 in log10, as measured from simulated
subhalos within the Via Lactea host halo (Diemand et al.
2007). This simulation is for a LCDM cosmology with
σ8 = 0.74 and n = 0.95. The scatter in the subhalo mass
function increases at the very high mass end, which re-
flects the fact that these most massive subhalos are those
that are accreted most recently (Zentner & Bullock 2003;
van den Bosch et al. 2005). However, as we show below
our results are not strongly dependent on the large scat-
ter at the high mass end.
Column 9 in Table 1 shows the allowed subhalo Vmax
values for the assumed prior. Note that in most cases,
this prior degrades the quality of the fit, and the likeli-
hood functions peak at a lower overall value. The mag-
nitude of this effect is not large except for the cases of
Leo II and Sagittarius. For Leo II, the peak likelihood
with the prior occurs at a value that is below the 10%
likelihood for the case without a prior on rs (i.e. the
data seem to prefer a puffier subhalo than would be ex-
pected in CDM). For Sagittarius, we are unable to obtain
a reasonable fit within a subhalo that is typical of those
expected. This is not too surprising. Sagittarius is be-
ing tidally disrupted and its dark matter halo is likely
atypical.
We emphasize that the Vmax determinations listed in
Column 9 are driven by theoretical assumptions, and can
only be fairly compared to predictions for this specific
cosmology (LCDM, σ8 = 0.74). The M0.6 values in Col-
umn 5 are applicable for any theoretical model, including
non-CDM models, or CDM models with any normaliza-
tion or power spectrum shape.
4. COMPARISON TO NUMERICAL SIMULATIONS
The recently-completed Via Lactea run is the highest-
resolution simulation of galactic substructure to date,
containing an order of magnitude more particles than
its predecessors (Diemand et al. 2007). As mentioned
above, Via Lactea assumes a LCDM cosmology with
σ8 = 0.74 and n = 0.95. For a detailed description of
the simulation, see Diemand et al. (2007). For our pur-
poses, the most important aspect of Via Lactea is its
ability to resolve the mass of subhalos on length scales
of the characteristic radius 0.6 kpc. In Via Lactea, the
force resolution is 90 pc and the smallest well-resolved
length scale is 300 pc, so that the mass within 0.6 kpc
is well-resolved in nearly all subhalos. Due to the choice
of time steps we expect the simulation to underestimate
local densities in the densest regions (by about 10% at
6 Strigari et al.
Fig. 4.— The mass within 0.6 kpc versus the maximum circular
velocity for the mass ranges of Via Lactea subhalos corresponding
to the population of satellites we study.
densities of 9 × 107M⊙/kpc3). There is only one sub-
halo with a higher local density than this at 0.6 kpc. For
this subhalo, ρ(r = 0.6 kpc) = 1.4 × 108M⊙/kpc3, so
its local density might be underestimated by up to 10%,
and the errors in the enclosed mass might be ∼ 20%
(Diemand et al. 2005). For all other subhalos the densi-
ties at 0.6 kpc are well below the affected densities, and
the enclosed mass should not be affected by more than
10% by the finite numerical resolution.
We define subhalos in Via Lactea to be the self-bound
halos that lie within the radius R200 = 389 kpc, where
R200 is defined to enclose an average density 200 times
the mean matter density. We note that in comparing
to the observed MW dwarf population, we could have
conservatively chosen subhalos that are restricted to lie
within the same radius as the most distant MW dSph
(250 kpc). We find that this choice has a negligible effect
on our conclusions – it reduces the count of small halos
by ∼ 10%.
In Figure 4, we show how M0.6 relates to the more
familiar quantity Vmax in Via Lactea subhalos. We note
that the relationship between subhaloM0.6 and Vmax will
be sensitive to the power spectrum shape and normaliza-
tion, as well as the nature of dark matter (Bullock et al.
2001; Zentner & Bullock 2003). The relationship shown
is only valid for the Via Lactea cosmology, but serves as
a useful reference for this comparison.
Given likelihood functions for the dSph M0.6 values,
we are now in position to determine the M0.6 mass func-
tion for Milky Way (MW) satellites and compare this to
the corresponding mass function in Via Lactea. For both
the observations and the simulation, we count the num-
ber of systems in four mass bins from 4 × 106 < M0.6 <
4× 108M⊙. This mass range is chosen to span the M0.6
values allowed by the likelihood functions for the MW
satellites. We assume that the two non-dSph satellites,
the LMC and SMC, belong in the highest mass bin, cor-
responding to M0.6 > 10
8 M⊙ (Harris & Zaritsky 2006;
van der Marel et al. 2002).
In Figure 5 we show resulting mass functions for MW
satellites (solid) and for Via Lactea subhalos (dashed,
Fig. 5.— The M0.6 mass function of Milky Way satellites and
dark subhalos in the Via Lactea simulation. The red (short-dashed)
curve is the total subhalo mass function from the simulation. The
black (solid) curve is the median of the observed satellite mass
function. The error-bars on the observed mass function represent
the upper and lower limits on the number of configurations that
occur with a probability of > 10−3.
with Poisson error-bars). For the MW satellites, the cen-
tral values correspond to the median number of galaxies
per bin, which are obtained from the maximum values
of the respective likelihood functions. The error-bars
on the satellite points are set by the upper and lower
configurations that occur with a probability of > 10−3
after drawing 1000 realizations from the respective like-
lihood functions. As seen in Figure 5, the predicted dark
subhalo mass function rises as ∼ M−20.6 while the visi-
ble MW satellite mass function is relatively flat. The
lowest mass bin (M0.6 ∼ 9 × 106M⊙) always contains 1
visible galaxy (Sextans). The second-to-lowest mass bin
(M0.6 ∼ 2.5×107M⊙) contains between 2 and 4 satellites
(Carina, Sculptor, and Leo II). The fact that these two
lowest bins are not consistent with zero galaxies has im-
portant implications for the Stoehr et al. (2002) solution
to the MSP: specifically, it implies that the 11 well-known
MW satellites do not reside in subhalos that resemble the
11 most massive subhalos in Via Lactea.
To further emphasize this point, we see from Figure 5
that the mass of the 11th most massive subhalo in Via
Lactea is 4 × 107M⊙. From the likelihood functions in
Figure 1, Sextans, Carina, Leo II, and Sculptor must
have values of M0.6 less than 4 × 107M⊙ at 99% c.l.,
implying a negligible probability that all of these dSphs
reside in halos with M0.6 > 4× 107M⊙.
Using the M0.6 mass function of MW satellites, we
can test other CDM-based solutions to the MSP. Two
models of interest are based on reionization suppres-
sion (Bullock et al. 2000; Moore et al. 2006) and on there
being a characteristic halo mass scale prior to subhalo
accretion (Diemand et al. 2007). To roughly represent
these models, we focus on two subsamples of Via Lactea
subhalos: the earliest forming (EF) halos, and the largest
mass halos before they were accreted (LBA) into the
host halo. As described in Diemand et al. (2007), the
LBA sample is defined to be the 10 subhalos that had
Redefining the Missing Satellites Problem 7
Fig. 6.— The solid and dashed curves show the MW satellites
and dark subhalos in Via Lactea, respectively. These lines are
reproduced from Figure 5, with error-bars suppressed for clarity.
The blue (dotted) curve represents the ten earliest forming halos
in Via Lactea, and the green (long-dashed) curve represents the 10
most massive halos before accretion into the Milky Way halo.
the highest Vmax value throughout their entire history.
These systems all had Vmax > 37.3 kms
−1 at some point
in their history. The EF sample consists of the 10 sub-
halos with Vmax > 16.2 kms
−1 (the limit of atomic cool-
ing) at z = 9.6. The Kravtsov et al. (2004) model would
correspond to a selection intermediate between EF and
LBA. In Figure 6 we show the observed mass function of
MW satellites (solid, squares) along with the EF (dotted,
triangles) and LBA (long-dashed, circles) samples. We
conclude that both of these models are in agreement with
the MW satellite mass function. Future observations and
quantification of the masses of the newly-discovered MW
satellites will enable us to do precision tests of the viable
MSP solutions. Additionally, once the capability to do
numerical simulations of substructure in warm dark mat-
ter models becomes a reality, the M0.6 mass function will
provide an invaluable tool to place constraints on WDM
models.
5. SUMMARY AND DISCUSSION
We have provided comprehensive dark matter mass
constraints for the 9 well-studied dSph satellite galax-
ies of the Milky Way and investigated CDM-based solu-
tions for the missing satellite problem in light of these
constraints. While subhalo Vmax values are the tradi-
tional means by which theoretical predictions quantify
substructure counts, this is not the most direct way
to confront the observational constraints. Specifically,
Vmax is poorly constrained by stellar velocity dispersion
measurements, and can only be estimated by adopting
cosmology-dependent, theoretical extrapolations. We ar-
gue the comparison between theory and observation is
best made using the integrated mass within a fixed phys-
ical radius comparable to the stellar extent of the known
satellites, ∼ 0.6 kpc. This approach is motivated by
Strigari et al. (2007) who showed that the mass within
two stellar King radii is best constrained by typical ve-
locity dispersion data.
Using M0.6 to represent the dark matter mass within a
radius of 0.6 kpc, we computed M0.6 likelihood functions
for the MW dSphs based on published velocity dispersion
data. Our models allow for a wide range of underlying
dark matter halo profile shapes and stellar velocity dis-
persion profiles. With this broad allowance, we showed
that the M0.6 for most dwarf satellites is constrained to
within ∼ 30%.
We derived the M0.6 mass function of MW satellites
(with error bars) and compared it to the same mass func-
tion computed directly from the Via Lactea substruc-
ture simulation. While the observed M0.6 mass function
of luminous satellites is relatively flat, the comparable
CDM subhalo mass function rises as ∼ M−20.6 . We rule
out the hypothesis that all of the well-known Milky Way
satellites strictly inhabit the most massive CDM subha-
los. If luminosity does track current subhalo mass, this
would only be possible if the subhalo population of the
Milky Way were drastically different than that predicted
in CDM. However, we show that other plausible CDM so-
lutions are consistent with the observed mass function.
Specifically, the earliest forming subhalos have a flatM0.6
mass function that is consistent with the satellite subhalo
mass function. This would be expected if the population
of bright dwarf spheroidals corresponds to the residual
halo population that accreted a significant mount of gas
before the epoch of reionization (Bullock et al. 2000).
We also tested the hypothesis that the present dwarf
spheroidal population corresponds to the subhalos that
were the most massive before they fell into the MW halo
(Kravtsov et al. 2004). This hypothesis is also consistent
with the current data.
In deriving the M0.6 mass function for this paper we
have set aside the issue of the most- recently discovered
population of MW dwarfs. We aim to return to this
issue in later work, but it is worth speculating on the
expected impact that these systems would have on our
conclusions. If we had included the new systems, mak-
ing ∼ 20 satellites in all, would it be possible to place
these systems in the ∼ 20 most massive subhalos in Via
Lactea? Given the probable mass ranges for the new
dwarfs, we find that this is unlikely. We can get a rough
estimate of their masses from their observed luminosi-
ties. We start by considering the mass-to-light ratios
of the known dSph population from figure 3 and from
Mateo (1998). If we assume that the other dwarfs have
similar M/L range, we can assign a mass range for each
of them. In all cases, the new MW dwarfs are approxi-
mately 1 to 2 orders of magnitude smaller in luminosity
than the well-known dSph population. Using the central
points for the known dSphs, we obtain M0.6/L spanning
the range from 3 − 230. Considering the width of the
likelihoods, we can allow a slightly larger range, 2− 350.
If we place the new dwarfs in this latter range, the uncer-
tainty in their masses is (2 − 350)LM⊙/L⊙. Even with
this generous range we expect most of the new dwarfs
have M0.6 . 10
The discovery of more members of the MW group,
7 These estimates are in rough agreement with recent de-
terminations from stellar velocity dispersion measurements in
the new dwarfs, as presented by N. Martin and J. Simon
at the 3rd Irvine Cosmology Workshop, March 22-24, 2007,
http://www.physics.uci.edu/Astrophysical-Probes/
http://www.physics.uci.edu/Astrophysical-Probes/
8 Strigari et al.
and the precise determination of the M0.6 mass func-
tion, could bring the status of the remaining viable MSP
solutions into sharper focus. These measurements would
also provide important constraints on warm dark matter
models or on the small scale power spectrum in CDM.
6. ACKNOWLEDGMENTS
We thank Jason Harris, Tobias Kaufmann, Savvas
Koushiappas, Andrey Kravtsov, Steve Majewski, Nicolas
Martin, Josh Simon, and Andrew Zentner for discussions
on this topic. We thank Mike Siegal for sharing his Leo
II data. LES is supported in part by a Gary McCue post-
doctoral fellowship through the Center for Cosmology at
the University of California, Irvine. L.E.S., J.S.B., and
M.K. are supported in part by NSF grant AST-0607746.
M.K. acknowledges support from PHY-0555689. J. D.
acknowledges support from NASA through Hubble Fel-
lowship grant HST-HF-01194.01 awarded by the Space
Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy,
Inc., for NASA, under contract NAS 5-26555. P.M. ac-
knowledges support from NASA grants NAG5-11513 and
NNG04GK85G, and from the Alexander von Humboldt
Foundation. The Via Lactea simulation was performed
on NASA’s Project Columbia supercomputer system.
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  Numerical simulations of Milky-Way size Cold Dark Matter (CDM) halos predict
a steeply rising mass function of small dark matter subhalos and a substructure
count that greatly outnumbers the observed satellites of the Milky Way. Several
proposed explanations exist, but detailed comparison between theory and
observation in terms of the maximum circular velocity (Vmax) of the subhalos is
hampered by the fact that Vmax for satellite halos is poorly constrained. We
present comprehensive mass models for the well-known Milky Way dwarf
satellites, and derive likelihood functions to show that their masses within
0.6 kpc (M_0.6) are strongly constrained by the present data. We show that the
M_0.6 mass function of luminous satellite halos is flat between ~ 10^7 and 10^8
M_\odot. We use the ``Via Lactea'' N-body simulation to show that the M_0.6
mass function of CDM subhalos is steeply rising over this range. We rule out
the hypothesis that the 11 well-known satellites of the Milky Way are hosted by
the 11 most massive subhalos. We show that models where the brightest
satellites correspond to the earliest forming subhalos or the most massive
accreted objects both reproduce the observed mass function. A similar analysis
with the newly-discovered dwarf satellites will further test these scenarios
and provide powerful constraints on the CDM small-scale power spectrum and warm
dark matter models.

Draft version May 29, 2018
Preprint typeset using LATEX style emulateapj v. 08/22/09
REDEFINING THE MISSING SATELLITES PROBLEM
Louis E. Strigari
, James S. Bullock
, Manoj Kaplinghat
, Juerg Diemand
, Michael Kuhlen
, Piero
Madau
Draft version May 29, 2018
ABSTRACT
Numerical simulations of Milky-Way size Cold Dark Matter (CDM) halos predict a steeply rising
mass function of small dark matter subhalos and a substructure count that greatly outnumbers the
observed satellites of the Milky Way. Several proposed explanations exist, but detailed comparison
between theory and observation in terms of the maximum circular velocity (Vmax) of the subhalos is
hampered by the fact that Vmax for satellite halos is poorly constrained. We present comprehensive
mass models for the well-known Milky Way dwarf satellites, and derive likelihood functions to show
that their masses within 0.6 kpc (M0.6) are strongly constrained by the present data. We show that
the M0.6 mass function of luminous satellite halos is flat between ∼ 107 and 108M⊙. We use the
“Via Lactea” N-body simulation to show that the M0.6 mass function of CDM subhalos is steeply
rising over this range. We rule out the hypothesis that the 11 well-known satellites of the Milky Way
are hosted by the 11 most massive subhalos. We show that models where the brightest satellites
correspond to the earliest forming subhalos or the most massive accreted objects both reproduce the
observed mass function. A similar analysis with the newly-discovered dwarf satellites will further test
these scenarios and provide powerful constraints on the CDM small-scale power spectrum and warm
dark matter models.
Subject headings: Cosmology: dark matter, theory–galaxies
1. INTRODUCTION
It is now well-established that numerical simulations
of Cold Dark Matter (CDM) halos predict many orders
of magnitude more dark matter subhalos around Milky
Way-sized galaxies than observed dwarf satellite galax-
ies (Klypin et al. 1999; Moore et al. 1999; Diemand et al.
2007). Within the context of the CDM paradigm, there
are well-motivated astrophysical solutions to this ‘Miss-
ing Satellites Problem’ (MSP) that rely on the idea
that galaxy formation is inefficient in the smallest dark
matter halos (Bullock et al. 2000; Benson et al. 2002;
Somerville 2002; Stoehr et al. 2002; Kravtsov et al. 2004;
Moore et al. 2006; Gnedin & Kravtsov 2006). However,
from an observational perspective, it has not been possi-
ble to distinguish between these solutions.
A detailed understanding of the MSP is limited by our
lack of a precise ability to characterize the dark matter
halos of satellite galaxies. From an observational per-
spective, the primary constraints come from the veloc-
ity dispersion of ∼ 200 stars in the population of dark
matter dominated dwarf spheroidal galaxies (dSphs)
(Wilkinson et al. 2004; Lokas et al. 2005; Munoz et al.
2005, 2006; Westfall et al. 2006; Walker et al. 2005, 2006;
Sohn et al. 2006; Gilmore et al. 2007). However, the ob-
served extent of the stellar populations in dSphs are ∼
kpc, so these velocity dispersion measurements are only
able to probe properties of the halos in this limited radial
1 Center for Cosmology, Department of Physics & Astronomy,
University of California, Irvine, CA 92697
2 Department of Astronomy & Astrophysics, University of Cali-
fornia, Santa Cruz, CA 95064
3 School of Natural Sciences, Institute for Advanced Study, Ein-
stein Drive, Princeton, NJ 08540
4 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str.
1, 85740 Garching, Germany.
5 Hubble Fellow
regime.
From the theoretical perspective, dissipationless nu-
merical simulations typically characterize subhalo counts
as a function of the total bound mass or maximum cir-
cular velocity, Vmax. While robustly determined in sim-
ulations, global quantities like Vmax are difficult to con-
strain observationally because dark halos can extend well
beyond the stellar radius of a satellite. Indeed stellar
kinematics alone provide only a lower limit on the halo
Vmax value (see below). This is a fundamental limita-
tion of stellar kinematics that cannot be remedied by
increasing the number of stars used in the velocity dis-
persion analysis (Strigari et al. 2007). Thus determining
Vmax values for satellite halos requires a theoretical ex-
trapolation. Any extrapolation of this kind is sensitive
to the predicted density structure of subhalos, which de-
pends on cosmology, power-spectrum normalization, and
the nature of dark matter (Zentner & Bullock 2003).
Our inability to determinate Vmax is the primary lim-
itation to test solutions to the MSP. One particular so-
lution states that the masses of the dSphs have been
systematically underestimated, so that the ∼ 10 bright-
est satellites reside systematically in the ∼ 10 most mas-
sive subhalos (Stoehr et al. 2002; Hayashi et al. 2003). A
byproduct of this solution is that there must be a sharp
mass cutoff at some current subhalo mass, below which
galaxy formation is suppressed. Other solutions, based
on reionization suppression, or a characteristic halo mass
scale prior to subhalo accretion predict that the sup-
pression comes in gradually with current subhalo mass
(Bullock et al. 2000; Kravtsov et al. 2004; Moore et al.
2006).
In this paper, we provide a systematic investigation of
the masses of the Milky Way satellites. We highlight
that in all cases the total halo masses and maximum
http://arxiv.org/abs/0704.1817v2
2 Strigari et al.
circular velocities are not well-determined by the data.
We instead use the fact that the total cumulative mass
within a characteristic radius ∼ 0.6 kpc is much bet-
ter determined by the present data (Strigari et al. 2007).
We propose using this mass, which we define as M0.6,
as the favored means to compare to the dark halo pop-
ulation in numerical simulations. Unlike Vmax, M0.6 is
measured directly and requires no cosmology-dependent
or dark-matter-model-dependent theoretical prior.
In the following sections, we determine the M0.6 mass
function for the Milky Way satellites, and compare it
to the corresponding mass function measured directly in
the high-resolution “Via Lactea” substructure simulation
of Diemand et al. (2007). We rule out the possibility
that there is a one-to- one correspondence between the 11
most luminous satellites and the most massive subhalos
in Via Lactea. We find that MSP solutions based on
reionization and/or characteristic halo mass scales prior
to accretion are still viable.
2. MILKY WAY SATELLITES
Approximately 20 satellite galaxies can be classified
as residing in MW subhalos. Of these, ∼ 9 were
discovered within the last two years and have very
low luminosities and surface brightnesses (Willman et al.
2005; Willman et al. 2005; Belokurov et al. 2006, 2007;
Zucker et al. 2006a,b). The lack of precision in these
numbers reflects the ambiguity in the classification of the
newly-discovered objects. The nine ‘older’ galaxies clas-
sified as dSphs are supported by their velocity dispersion,
and exhibit no rotational motion (Mateo 1998). Two
satellite galaxies, the Small Megallanic Cloud (SMC)
and Large Megallanic Cloud (LMC), are most likely sup-
ported by some combination of rotation and dispersion.
Stellar kinematics suggest that the LMC and SMC are
likely the most massive satellite systems of the Milky
We focus on determining the masses of the nine most
well-studied dSphs. The dark matter masses of the dSphs
are determined from the line-of-sight velocities of the
stars, which trace the total gravitational potential. We
assume a negligible contribution of the stars to the grav-
itational potential, which we find to be an excellent ap-
proximation. The dSph with the smallest mass-to-light
ratio is Fornax, though even for this case we find that the
stars generally do not affect the dynamics of the system
(see Lokas 2002; Wu 2007, and below).
Under the assumptions of equilibrium and spherical
symmetry, the radial component of the stellar velocity
dispersion, σr, is linked to the total gravitational poten-
tial of the system via the Jeans equation,
d(ν⋆σ
= −ν⋆V 2c − 2βν⋆σ
r . (1)
Here, ν⋆ is the stellar density profile, V
c = GM(r)/r
includes the total gravitating mass of the system, and
the parameter β(r) = 1 − σ2θ/σ
r characterizes the dif-
ference between the radial (σr) and tangential (σθ) ve-
locity dispersions. The observable velocity dispersion is
constructed by integrating the three- dimensional stellar
radial velocity dispersion profile along the line-of-sight,
σ2los(R) =
I⋆(R)
rrdr√
r2 −R2
, (2)
where R is the projected radius on the sky. The surface
density of stars in all dSphs are reasonably well-fit by a
spherically-symmetric King profile (King 1962),
I⋆(R) ∝
r2king
)−1/2
r2king
)−1/2
where rt and rking are fitting parameters denoted as the
tidal and core radii. 6 The spherically symmetric stellar
density can be obtained with an integral transformation
of the surface density. From the form of Eq. (2), the
normalization in Eq. (3) is irrelevant.
Some dSphs show evidence for multiple, dynamically
distinct stellar populations, with each population de-
scribed by its own surface density and velocity dispersion
(Harbeck et al. 2001; Tolstoy et al. 2004; Westfall et al.
2006; McConnachie et al. 2006; Ibata et al. 2006). In a
dSph with i = 1...Np populations of stars, standard ob-
servational methods will sample a density-weighted av-
erage of the populations:
νi (4)
σ2r =
r,ı , (5)
where νi and σi are the density profile and radial stellar
velocity dispersion of stellar component i. In principle,
each component has its own stellar velocity anisotropy
profile, βi(r). In this case, Equation (1) is valid for ν⋆ and
σr as long as an effective velocity anisotropy is defined
β(r)=
ν⋆σ2r
βiνiσ
i . (6)
From these definitions, we also have
ı I⋆,ıσ
los,ı =
The conclusion we draw from this argument is that
the presence of multiple populations will not affect the
inferred mass structure of the system, provided that the
velocity anisotropy is modeled as a free function of ra-
dius. Since the functional form of the stellar velocity
anisotropy is unknown, we allow for a general, three pa-
rameter model of the velocity anisotropy profile,
β(r) = β∞
r2β + r
+ β0. (7)
A profile of this form allows for the possibility for β(r) to
change from radial to tangential orbits within the halo,
and a constant velocity anisotropy is recovered in the
limit β∞ → 0, and β0 → const.
In Equations (1) and (2), the radial stellar velocity dis-
persion, σr , depends on the total mass distribution, and
thus the parameters describing the dark matter density
profile. Dissipation-less N-body simulations show that
the density profiles of CDM halos can be characterized
ρ(r̃) =
r̃γ(1 + r̃)δ
; r̃ = r/rs, (8)
6 Our results are insensitive to this particular parameterization
of the light profile.
Redefining the Missing Satellites Problem 3
where rs and ρs set a radial scale and density normaliza-
tion and γ and δ parameterize the inner and outer slopes
of the distribution. For dark matter halos unaffected
by tidal interactions, the most recent high-resolution
simulations find δ + γ ≈ 3 works well for the outer
slope, while 0.7 . γ . 1.2 works well down to ∼ 0.1%
of halo virial radii (Navarro et al. 2004; Diemand et al.
2005). This interior slope is not altered as a subhalo
loses mass from tidal interactions with the MW potential
(Kazantzidis et al. 2004). The outer slope, δ, depends
more sensitively on the tidal interactions in the halo. The
majority of the stripped material will be from the outer
parts of halos, and thus δ of subhalo density profiles will
become steeper than those of field halos. Subhalos are
characterized by outer slopes in the range 2 . δ . 3.
Given the uncertainty in the β(r) and ρ(r) profiles,
we are left with nine shape parameters that must be
constrained via line-of-sight velocity dispersion measure-
ments: ρs, rs, β0, β∞, rβ , γ, δ, rking , and rt. While the
problem as posed may seem impossible, there are a num-
ber of physical parameters, which are degenerate between
different profile shapes, that are well constrained. Specif-
ically, the stellar populations constrain Vc(r) within a
radius comparable to the stellar radius rt ∼ kpc. As
a result, quantities like the local density and integrated
mass within the stellar radius are determined with high
precision (Strigari et al. 2006), while quantities that de-
pend on the mass distribution at larger radii are virtually
unconstrained by the data.
It is useful to determine the value of the radius where
the constraints are maximized. The location of this char-
acteristic radius is determined by the form of the integral
in Eq. (2). We can gain some insight using the exam-
ple of a power-law stellar distribution ν⋆(r), power-law
dark matter density profile ρ ∝ r−γ⋆ , and constant ve-
locity anisotropy. The line-of-sight velocity dispersion
depends on the three-dimensional stellar velocity disper-
sion, which can be written as
σ2r (r) = ν
Gν⋆(r)M(r)r
2β−2dr ∝ r2−γ⋆ (9)
From the shape of the King profile, the majority of
stars reside at projected radii rking . R . rt, where
the stellar distribution is falling rapidly ν⋆ ∼ r−3.5.
In this case, for β = 0, the LOS component scales as
σ2los(R) ∝
r−0.5−γ⋆(r2 −R2)−1/2dr and is dominated
by the mass and density profile at the smallest relevant
radii, r ∼ rking. For R . rking, ν⋆ ∝ r−1 and σ2los is simi-
larly dominated by r ∼ rking contributions. We note that
although the scaling arguments above hold for constant
velocity anisotropies, they can be extended by consider-
ing anisotropies that vary significantly in radius. They
are also independent of the precise form of I⋆, provided
there is a scale similar to rking .
In Strigari et al. (2007), it was shown that typical ve-
locity dispersion profiles best constrain the mass (and
density) within a characteristic radius ≃ 2 rking . For
example, the total mass within 2rking is constrained to
within ∼ 20% for dSphs with ∼ 200 line-of-sight veloc-
ities. Note that when deriving constraints using only
the innermost stellar velocity dispersion and fixing the
anisotropy to β = 0, the characteristic radius for best
constraints decreases to∼ 0.5rking (e.g. Penarrubia et al.
2007).
As listed in Table 1, the Milky Way dSphs are ob-
served to have variety of rking values, but rking ∼ 0.3
kpc is typical. The values of rking and rt are taken from
Mateo (1998). In order to facilitate comparison with
simulated subhalos, we chose a single characteristic ra-
dius of 0.6 kpc for all the dwarfs, and we represent the
mass within this radius as M0.6 = M(< 0.6 kpc). The
relative errors on the derived masses are unaffected for
small variations in the characteristic radius in the range
∼ 1.5− 2.5 rking. Deviations from a true King profile at
large radius (near rt) do not affect these arguments, as
long as there is a characteristic scale similar to rking in
the surface density. The only dSph significantly affected
by the choice of 0.6 kpc as the characteristic radius is
Leo II, which has rt = 0.5 kpc. Since the characteristic
radius is greater than twice rking , the constraints on its
mass will be weakest of all galaxies (with the exception
of Sagittarius, as discussed below).
3. DARK MATTER HALO MASSES AT THE
CHARACTERISTIC RADIUS
We use the following data sets: Wilkinson et al.
(2004); Munoz et al. (2005, 2006); Westfall et al. (2006);
Walker et al. (2005, 2006); Sohn et al. (2006); Siegal et
al. in preparation. These velocity dispersions are deter-
mined from the line-of-sight velocities of ∼ 200 stars in
each galaxy, although observations in the coming years
will likely increase this number by a factor ∼ 5 − 10.
From the data, we calculate the χ2, defined in our case
(σobs,ı − σth,ı)2
. (10)
Here σ2obs is the observed velocity dispersion in each bin,
σ2th is the theoretical value, obtained from Eq. (2), and
ǫ2ı are errors as determined from the observations.
It is easy to see that, when fitting to a single data set
of ∼ 200 stars, parameter degeneracies will be signifi-
cant. However, from the discussion in section 2, M0.6
is well-determined by the LOS data. To determine how
well M0.6 is constrained, we construct likelihood func-
tions for each galaxy. When thought of as a function
of the theoretical parameters, the likelihood function, L,
is defined as the probability that a data set is acquired
given a set of theoretical parameters. In our case L is a
function of the parameters γ, δ, rs, ρs, and β0, β∞, rβ ,
and is defined as L = e−χ
2/2. In writing this likelihood
function, we assume that the errors on the measured ve-
locity dispersions are Gaussian, which we find to be an
excellent approximation to the errors for a dSph system
(Strigari et al. 2007). We marginalize over the parame-
ters γ, δ, rs, ρs, and β0, β∞, rβ at fixed M0.6, and the
optimal values for M0.6 are determined by the maximum
of L .
We determine L for all nine dSphs with velocity dis-
persion measurements. For all galaxies we use the full
published velocity dispersion profiles. The only galaxy
that does not have a published velocity dispersion profile
is Sagittarius, and for this galaxy we use the central ve-
locity dispersion from Mateo (1998). The mass modeling
of Sagittarius is further complicated by the fact that it is
experiencing tidal interactions with the MW (Ibata et al.
4 Strigari et al.
Fig. 1.— The likelihood functions for the mass within 0.6 kpc
for the nine dSphs, normalized to unity at the peak.
1997; Majewski et al. 2003), so a mass estimate from the
Jeans equation is not necessarily reliable. We caution
that in this case the mass we determine is likely only an
approximation to the total mass of the system.
We determine the likelihoods by marginalizing over
the following ranges of the velocity anisotropy, inner
and outer slopes: −10 < β0 < 1, −10 < β∞ < 1,
0.1 < rβ < 10 kpc, 0.7 < γ < 1.2, and 2 < δ < 3.
As discussed above, these ranges for the asymptotic in-
ner and outer slopes are appropriate because we are con-
sidering CDM halos. It is important to emphasize that
these ranges are theoretically motivated and that obser-
vations alone do not demand such restrictive choices.
It is possible to fit all of the dSphs at present with
a constant density cores with scale-lengths ∼ 100 pc
(Strigari et al. 2006; Gilmore et al. 2007), although the
data by no means demand such a situation. Though
we consider inner and outer slopes in the ranges quoted
above, our results are not strongly affected if we widen
these intervals. For example, we find that if we allow
the inner slope to vary down to γ = 0, the widths of the
likelihoods are only changed by ∼ 10%. This reflects the
fact that there is a negligible degeneracy between M0.6
and the inner and outer slopes.
We are left to determine the regions of ρs and rs pa-
rameter space to marginalize over. In all dSphs, there
is a degeneracy in this parameter space, telling us that
it is not possible to derive an upper limit on this pair
of parameters from the data alone (Strigari et al. 2006).
While this degeneracy is not important when determin-
ing constraints on M0.6, it is the primary obstacle in
determining Vmax. From the fits we present below, we
find that the lowest rs value that provides an acceptable
fit is ∼ 0.1 kpc, and we use this as the lower limit in all
cases. In our fiducial mass models, we conservatively re-
strict the maximum value of rs using the known distance
to each dSph. In this case, we use 0.1 kpc < rs < D/2,
where D is the distance to the dSph.
In Figure 1 we show the M0.6 likelihood functions for
all of the dSphs. As is shown, we obtain strong con-
Fig. 2.— The velocity dispersion for Ursa Minor as a function of
radial distance, along with the model that maximizes the likelihood
function.
straints on M0.6 in all cases except Sagittarius, for which
we use only a central velocity dispersion. Table 1 sum-
marizes the best fitting M0.6 values for each dwarf. The
quoted errors correspond to the points where the likeli-
hood falls to 10% of its peak value. The upper panel of
Figure 3 shows M0.6 values for each dwarf as a function
of luminosity. In Figure 2 we show an example of the
velocity dispersion data as a function of radial distance
for Ursa Minor, along with the model that maximizes
the likelihood function. For all galaxies, we find χ2 per
degree of freedom values . 1.
The maximum likelihood method also allows us to con-
strain the mass at other radii spanned by the stellar dis-
tribution. The sixth column of Table 1 provides the inte-
grated mass within each dwarf’s King tidal radius. This
radius roughly corresponds to the largest radius where
a reasonable mass constraint is possible. As expected,
the mass within rt is not as well determined as the mass
within 2 rking. From these masses we are able to deter-
mine the mass-to-light ratios within rt, which we present
in the seventh column of Table 1. In the bottom panel
of Figure 3, we show mass-to-light ratios within rt as a
function of dwarf luminosity. We see the standard result
that the observable mass-to-light ratio increases with de-
creasing luminosity (Mateo 1998). Note, however, that
our results are inconsistent with the idea that all of the
dwarfs have the same integrated mass within their stellar
extent. We note that for Sagittarius, we can only obtain
a lower limit on the total mass-to-light ratio.
The last two columns in Table 1 list constraints on
Vmax for the dSphs. Column 8 shows results for an anal-
ysis with limits on rs as described above. In this case, the
integrated mass within the stellar radius is constrained
by the velocity dispersion data, but the halo rotation ve-
locity curve, Vc(r), can continue to rise as r increases
beyond the stellar radius in an unconstrained manner.
The result is that the velocity dispersion data alone pro-
vide only a lower limit on Vmax.
Stronger constraints on Vmax can be obtained if we
limit the range of rs by imposing a cosmology-dependent
prior on the dark matter mass profile. CDM simula-
Redefining the Missing Satellites Problem 5
TABLE 1
Parameters Describing Milky Way Satellites.
Galaxy rking rt LV Mass < 0.6 kpc Mass < rt M(< rt)/L Vmax [km s
−1] Vmax [km s
[kpc] [kpc] [106 L⊙] [10
7 M⊙] [10
7 M⊙] [M⊙/L⊙] (w/o prior) (with theory prior)
Draco 0.18 0.93 0.26 4.9
530 > 22 28
Ursa Minor 0.30 1.50 0.29 5.3
790 > 21 26
Leo I 0.20 0.80 4.79 4.3
106 > 14 19
Fornax 0.39 2.70 15.5 4.3
28 > 20 25
Leo II 0.19 0.52 0.58 2.1
128 > 17 9
Carina 0.26 0.85 0.43 3.4
82 > 13 15
Sculptor 0.28 1.63 2.15 2.7
68 > 20 14
Sextans 0.40 4.01 0.50 0.9
260 > 8 9
Sagittarius 0.3 4.0 18.1 20
> 20 > 11 > 19 —
Note. — Determination of the mass within 0.6 kpc and the maximum circular velocity for the dark matter halos of the dSphs.
The errors are determined as the location where the likelihood function falls off by 90% from its peak value. For Sagittarius, no
reliable estimate of Vmax with the CDM prior could be determined. The CDM prior is determined using the concordance cosmology
with σ8 = 0.74, n = 0.95 (see text for details).
Fig. 3.— The mass within 0.6 kpc (upper) and the mass-to-light
ratios within the King tidal radius (lower) for the Milky Way dSphs
as a function of dwarf luminosity. The error-bars here are defined as
the locations where the likelihoods fall to 40% of the peak values
(corresponding to ∼ 1σ errors). The lines denote, from top to
bottom, constant values of mass of 107, 108, 109 M⊙.
tions have shown that there is a correlation between
Vmax and rmax for halos, where rmax is the radius where
the circular velocity peaks. Because subhalo densities
will depend on the collapse time and orbital evolution
of each system, the precise Vmax-rmax relation is sen-
sitive to cosmology (e.g. σ8) and the formation his-
tory of the host halo itself (e.g. Zentner & Bullock 2003;
Power 2003; Kazantzidis et al. 2004; Bullock & Johnston
2005; Bullock & Johnston 2006). When converted to the
relevant halo parameters, the imposed Vmax-rmax rela-
tion can be seen as a theoretical prior on CDM ha-
los, restricting the parameter space we need to inte-
grate over. In order to illustrate the technique, we adopt
log10(rmax) = 1.35(log10(Vmax/kms
−1) − 1)− 0.196 kpc
with a scatter of 0.2 in log10, as measured from simulated
subhalos within the Via Lactea host halo (Diemand et al.
2007). This simulation is for a LCDM cosmology with
σ8 = 0.74 and n = 0.95. The scatter in the subhalo mass
function increases at the very high mass end, which re-
flects the fact that these most massive subhalos are those
that are accreted most recently (Zentner & Bullock 2003;
van den Bosch et al. 2005). However, as we show below
our results are not strongly dependent on the large scat-
ter at the high mass end.
Column 9 in Table 1 shows the allowed subhalo Vmax
values for the assumed prior. Note that in most cases,
this prior degrades the quality of the fit, and the likeli-
hood functions peak at a lower overall value. The mag-
nitude of this effect is not large except for the cases of
Leo II and Sagittarius. For Leo II, the peak likelihood
with the prior occurs at a value that is below the 10%
likelihood for the case without a prior on rs (i.e. the
data seem to prefer a puffier subhalo than would be ex-
pected in CDM). For Sagittarius, we are unable to obtain
a reasonable fit within a subhalo that is typical of those
expected. This is not too surprising. Sagittarius is be-
ing tidally disrupted and its dark matter halo is likely
atypical.
We emphasize that the Vmax determinations listed in
Column 9 are driven by theoretical assumptions, and can
only be fairly compared to predictions for this specific
cosmology (LCDM, σ8 = 0.74). The M0.6 values in Col-
umn 5 are applicable for any theoretical model, including
non-CDM models, or CDM models with any normaliza-
tion or power spectrum shape.
4. COMPARISON TO NUMERICAL SIMULATIONS
The recently-completed Via Lactea run is the highest-
resolution simulation of galactic substructure to date,
containing an order of magnitude more particles than
its predecessors (Diemand et al. 2007). As mentioned
above, Via Lactea assumes a LCDM cosmology with
σ8 = 0.74 and n = 0.95. For a detailed description of
the simulation, see Diemand et al. (2007). For our pur-
poses, the most important aspect of Via Lactea is its
ability to resolve the mass of subhalos on length scales
of the characteristic radius 0.6 kpc. In Via Lactea, the
force resolution is 90 pc and the smallest well-resolved
length scale is 300 pc, so that the mass within 0.6 kpc
is well-resolved in nearly all subhalos. Due to the choice
of time steps we expect the simulation to underestimate
local densities in the densest regions (by about 10% at
6 Strigari et al.
Fig. 4.— The mass within 0.6 kpc versus the maximum circular
velocity for the mass ranges of Via Lactea subhalos corresponding
to the population of satellites we study.
densities of 9 × 107M⊙/kpc3). There is only one sub-
halo with a higher local density than this at 0.6 kpc. For
this subhalo, ρ(r = 0.6 kpc) = 1.4 × 108M⊙/kpc3, so
its local density might be underestimated by up to 10%,
and the errors in the enclosed mass might be ∼ 20%
(Diemand et al. 2005). For all other subhalos the densi-
ties at 0.6 kpc are well below the affected densities, and
the enclosed mass should not be affected by more than
10% by the finite numerical resolution.
We define subhalos in Via Lactea to be the self-bound
halos that lie within the radius R200 = 389 kpc, where
R200 is defined to enclose an average density 200 times
the mean matter density. We note that in comparing
to the observed MW dwarf population, we could have
conservatively chosen subhalos that are restricted to lie
within the same radius as the most distant MW dSph
(250 kpc). We find that this choice has a negligible effect
on our conclusions – it reduces the count of small halos
by ∼ 10%.
In Figure 4, we show how M0.6 relates to the more
familiar quantity Vmax in Via Lactea subhalos. We note
that the relationship between subhaloM0.6 and Vmax will
be sensitive to the power spectrum shape and normaliza-
tion, as well as the nature of dark matter (Bullock et al.
2001; Zentner & Bullock 2003). The relationship shown
is only valid for the Via Lactea cosmology, but serves as
a useful reference for this comparison.
Given likelihood functions for the dSph M0.6 values,
we are now in position to determine the M0.6 mass func-
tion for Milky Way (MW) satellites and compare this to
the corresponding mass function in Via Lactea. For both
the observations and the simulation, we count the num-
ber of systems in four mass bins from 4 × 106 < M0.6 <
4× 108M⊙. This mass range is chosen to span the M0.6
values allowed by the likelihood functions for the MW
satellites. We assume that the two non-dSph satellites,
the LMC and SMC, belong in the highest mass bin, cor-
responding to M0.6 > 10
8 M⊙ (Harris & Zaritsky 2006;
van der Marel et al. 2002).
In Figure 5 we show resulting mass functions for MW
satellites (solid) and for Via Lactea subhalos (dashed,
Fig. 5.— The M0.6 mass function of Milky Way satellites and
dark subhalos in the Via Lactea simulation. The red (short-dashed)
curve is the total subhalo mass function from the simulation. The
black (solid) curve is the median of the observed satellite mass
function. The error-bars on the observed mass function represent
the upper and lower limits on the number of configurations that
occur with a probability of > 10−3.
with Poisson error-bars). For the MW satellites, the cen-
tral values correspond to the median number of galaxies
per bin, which are obtained from the maximum values
of the respective likelihood functions. The error-bars
on the satellite points are set by the upper and lower
configurations that occur with a probability of > 10−3
after drawing 1000 realizations from the respective like-
lihood functions. As seen in Figure 5, the predicted dark
subhalo mass function rises as ∼ M−20.6 while the visi-
ble MW satellite mass function is relatively flat. The
lowest mass bin (M0.6 ∼ 9 × 106M⊙) always contains 1
visible galaxy (Sextans). The second-to-lowest mass bin
(M0.6 ∼ 2.5×107M⊙) contains between 2 and 4 satellites
(Carina, Sculptor, and Leo II). The fact that these two
lowest bins are not consistent with zero galaxies has im-
portant implications for the Stoehr et al. (2002) solution
to the MSP: specifically, it implies that the 11 well-known
MW satellites do not reside in subhalos that resemble the
11 most massive subhalos in Via Lactea.
To further emphasize this point, we see from Figure 5
that the mass of the 11th most massive subhalo in Via
Lactea is 4 × 107M⊙. From the likelihood functions in
Figure 1, Sextans, Carina, Leo II, and Sculptor must
have values of M0.6 less than 4 × 107M⊙ at 99% c.l.,
implying a negligible probability that all of these dSphs
reside in halos with M0.6 > 4× 107M⊙.
Using the M0.6 mass function of MW satellites, we
can test other CDM-based solutions to the MSP. Two
models of interest are based on reionization suppres-
sion (Bullock et al. 2000; Moore et al. 2006) and on there
being a characteristic halo mass scale prior to subhalo
accretion (Diemand et al. 2007). To roughly represent
these models, we focus on two subsamples of Via Lactea
subhalos: the earliest forming (EF) halos, and the largest
mass halos before they were accreted (LBA) into the
host halo. As described in Diemand et al. (2007), the
LBA sample is defined to be the 10 subhalos that had
Redefining the Missing Satellites Problem 7
Fig. 6.— The solid and dashed curves show the MW satellites
and dark subhalos in Via Lactea, respectively. These lines are
reproduced from Figure 5, with error-bars suppressed for clarity.
The blue (dotted) curve represents the ten earliest forming halos
in Via Lactea, and the green (long-dashed) curve represents the 10
most massive halos before accretion into the Milky Way halo.
the highest Vmax value throughout their entire history.
These systems all had Vmax > 37.3 kms
−1 at some point
in their history. The EF sample consists of the 10 sub-
halos with Vmax > 16.2 kms
−1 (the limit of atomic cool-
ing) at z = 9.6. The Kravtsov et al. (2004) model would
correspond to a selection intermediate between EF and
LBA. In Figure 6 we show the observed mass function of
MW satellites (solid, squares) along with the EF (dotted,
triangles) and LBA (long-dashed, circles) samples. We
conclude that both of these models are in agreement with
the MW satellite mass function. Future observations and
quantification of the masses of the newly-discovered MW
satellites will enable us to do precision tests of the viable
MSP solutions. Additionally, once the capability to do
numerical simulations of substructure in warm dark mat-
ter models becomes a reality, the M0.6 mass function will
provide an invaluable tool to place constraints on WDM
models.
5. SUMMARY AND DISCUSSION
We have provided comprehensive dark matter mass
constraints for the 9 well-studied dSph satellite galax-
ies of the Milky Way and investigated CDM-based solu-
tions for the missing satellite problem in light of these
constraints. While subhalo Vmax values are the tradi-
tional means by which theoretical predictions quantify
substructure counts, this is not the most direct way
to confront the observational constraints. Specifically,
Vmax is poorly constrained by stellar velocity dispersion
measurements, and can only be estimated by adopting
cosmology-dependent, theoretical extrapolations. We ar-
gue the comparison between theory and observation is
best made using the integrated mass within a fixed phys-
ical radius comparable to the stellar extent of the known
satellites, ∼ 0.6 kpc. This approach is motivated by
Strigari et al. (2007) who showed that the mass within
two stellar King radii is best constrained by typical ve-
locity dispersion data.
Using M0.6 to represent the dark matter mass within a
radius of 0.6 kpc, we computed M0.6 likelihood functions
for the MW dSphs based on published velocity dispersion
data. Our models allow for a wide range of underlying
dark matter halo profile shapes and stellar velocity dis-
persion profiles. With this broad allowance, we showed
that the M0.6 for most dwarf satellites is constrained to
within ∼ 30%.
We derived the M0.6 mass function of MW satellites
(with error bars) and compared it to the same mass func-
tion computed directly from the Via Lactea substruc-
ture simulation. While the observed M0.6 mass function
of luminous satellites is relatively flat, the comparable
CDM subhalo mass function rises as ∼ M−20.6 . We rule
out the hypothesis that all of the well-known Milky Way
satellites strictly inhabit the most massive CDM subha-
los. If luminosity does track current subhalo mass, this
would only be possible if the subhalo population of the
Milky Way were drastically different than that predicted
in CDM. However, we show that other plausible CDM so-
lutions are consistent with the observed mass function.
Specifically, the earliest forming subhalos have a flatM0.6
mass function that is consistent with the satellite subhalo
mass function. This would be expected if the population
of bright dwarf spheroidals corresponds to the residual
halo population that accreted a significant mount of gas
before the epoch of reionization (Bullock et al. 2000).
We also tested the hypothesis that the present dwarf
spheroidal population corresponds to the subhalos that
were the most massive before they fell into the MW halo
(Kravtsov et al. 2004). This hypothesis is also consistent
with the current data.
In deriving the M0.6 mass function for this paper we
have set aside the issue of the most- recently discovered
population of MW dwarfs. We aim to return to this
issue in later work, but it is worth speculating on the
expected impact that these systems would have on our
conclusions. If we had included the new systems, mak-
ing ∼ 20 satellites in all, would it be possible to place
these systems in the ∼ 20 most massive subhalos in Via
Lactea? Given the probable mass ranges for the new
dwarfs, we find that this is unlikely. We can get a rough
estimate of their masses from their observed luminosi-
ties. We start by considering the mass-to-light ratios
of the known dSph population from figure 3 and from
Mateo (1998). If we assume that the other dwarfs have
similar M/L range, we can assign a mass range for each
of them. In all cases, the new MW dwarfs are approxi-
mately 1 to 2 orders of magnitude smaller in luminosity
than the well-known dSph population. Using the central
points for the known dSphs, we obtain M0.6/L spanning
the range from 3 − 230. Considering the width of the
likelihoods, we can allow a slightly larger range, 2− 350.
If we place the new dwarfs in this latter range, the uncer-
tainty in their masses is (2 − 350)LM⊙/L⊙. Even with
this generous range we expect most of the new dwarfs
have M0.6 . 10
The discovery of more members of the MW group,
7 These estimates are in rough agreement with recent de-
terminations from stellar velocity dispersion measurements in
the new dwarfs, as presented by N. Martin and J. Simon
at the 3rd Irvine Cosmology Workshop, March 22-24, 2007,
http://www.physics.uci.edu/Astrophysical-Probes/
http://www.physics.uci.edu/Astrophysical-Probes/
8 Strigari et al.
and the precise determination of the M0.6 mass func-
tion, could bring the status of the remaining viable MSP
solutions into sharper focus. These measurements would
also provide important constraints on warm dark matter
models or on the small scale power spectrum in CDM.
6. ACKNOWLEDGMENTS
We thank Jason Harris, Tobias Kaufmann, Savvas
Koushiappas, Andrey Kravtsov, Steve Majewski, Nicolas
Martin, Josh Simon, and Andrew Zentner for discussions
on this topic. We thank Mike Siegal for sharing his Leo
II data. LES is supported in part by a Gary McCue post-
doctoral fellowship through the Center for Cosmology at
the University of California, Irvine. L.E.S., J.S.B., and
M.K. are supported in part by NSF grant AST-0607746.
M.K. acknowledges support from PHY-0555689. J. D.
acknowledges support from NASA through Hubble Fel-
lowship grant HST-HF-01194.01 awarded by the Space
Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy,
Inc., for NASA, under contract NAS 5-26555. P.M. ac-
knowledges support from NASA grants NAG5-11513 and
NNG04GK85G, and from the Alexander von Humboldt
Foundation. The Via Lactea simulation was performed
on NASA’s Project Columbia supercomputer system.
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