IPMU22-0038

Cosmological prior for the J-factor estimation of dwarf
spheroidal galaxies

arXiv:2207.10378v1 [astro-ph.CO] 21 Jul 2022

Shun’ichi Horigome

1,a, Kohei Hayashi

2,3,4,b, and Shin’ichiro Ando

1

5,1,c

Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba, 277-8583, Japan
National Institute of Technology, Ichinoseki College, Takanashi, Hagisho, Ichinoseki,
Iwate, 021-8511, Japan
3
Astronomical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi
980-8578, Japan
4
Institute for Cosmic Ray Research, The University of Tokyo, Chiba, 277-8582, Japan
5
GRAPPA Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam,
The Netherlands
2

Abstract
Dark matter halos of dwarf spheroidal galaxies (dSphs) play important roles in dark
matter detection. Generally we estimate the halo profile using a kinematical equationof dSphs but the halo profile has a large uncertainty because we have only a limited
number of kinematical dataset. In this paper, we utilize cosmological models of dark
matter subhalos to obtain better constraints on halo profile of dSphs. The constraints
are realized as two cosmological priors: satellite prior, based on a semi-analytic modelof the accretion history of subhalos and their tidal stripping effect, and stellar-to-halo
mass relation prior, which estimates halo mass of a galaxy from its stellar mass using
empirical correlations. In addition, we adopt a radial dependent likelihood function
by considering velocity dispersion profile, which allows us to mitigate the parameterdegeneracy in the previous analysis using a radial independent likelihood function withaveraged dispersion. Using these priors, we estimate the squared dark matter density
integrated over the region-of-interest (so-called J-factor) of 8 classical and 27 ultra-faintdSphs. Our method significantly decreases the uncertainty of J-factors (upto about20%) compared to the previous radial independent analysis. We confirm the model
dependence of J-factor estimates by evaluating Bayes factors of different model setups
and find that the estimates are still stable even when assuming different cosmologicalmodels.
ashunichi.horigome@ipmu.jp
kohei.hayashi@g.ichinoseki.ac.jp
cs.ando@uva.nlb

Contents
1 Introduction

1

2 Method2

2.1 Models and likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.2 Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.1 Photometry prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.2 Satellite prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2.3 SHMR prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 Results

9

4 Discussion

14

4.1 Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4.2 J-factor and Bayes factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

5 Summary and conclusion

16

A Posteriors

17

1

1

Introduction

The presence of dark matter in our universe is one of the most important open questionsin the current physics. Even though cosmological observations agree with predictions of
Λ-cold dark matter (CDM) model with surprising accuracy [1], we still do not know whatdark matter is. In order to answer this question, many candidates and detection methods
have been proposed [2]. Among various detection methods, those using dwarf spheroidal
galaxies (dSphs) are interesting. DSphs are a kind of satellite galaxy of the Milky Way withlarge mass-to-light ratio, which implies that they are dark matter rich objects. Such a large
amount of dark matter enables us to explore the nature of dark matter. In the indirect
detection method, we can constrain the annihilation cross section by observing signal fluxof annihilation products. This method is very useful to detect well-motivated dark mattercandidates such as the weakly interacting massive particles (WIMPs) [3, 4], because they
have large annihilation cross section thanks to a non-relativistic quantum effect, so-called
the Sommerfeld effect [5, 6]. The sensitivity of the detection depends on the estimatedamount of signal flux. This amount is proportional to an astrophysical quantity, so-calledJ-factor:ZZJ(∆Ω) ≡dΩdl ρ 2 (r)(1)

where Ω is the solid angle, ∆Ω is the region-of-interest, l denotes the line-of-sight distance,r is the radius from the center of a target dSph, and ρ(r) denotes the dark matter density
profile. In order to obtain reliable and useful results, we need to know the accurate andprecise value of J-factor, that is, the shape of dark matter density profile ρ(r).
Although dSphs are useful tools to study the dark matter, their dark matter density
profiles have large astrophysical uncertainty compared to other uncertainties from particle
physics models. Generally, the dark matter density profile is estimated by fitting spectroscopic data set of dSph member stars using the Jeans equation [7]. However, the stellar data
set cannot completely determine the dark matter profile because we generally use empiricalmodels of the dark matter profile through the fitting and their parameters are degenerated.
Fortunately, from the viewpoint of cosmology, structure formation models predict the distribution of dSph profiles in the universe, which is useful to select theoretically favored density
profile model parameters and mitigate the problematic degeneracy. For instance, Ref. [8]use the conventional theory of spherical collapse to roughly constrain the parameter spaceof dark matter density profile.
Recently, semi-analytic models of the tidal mass striping effect on the cold dark matter
halo are developed [9, 10], which allows us to construct a multivariate distribution of dSph
mass and its tidal truncation radius. This probability distribution was applied as a prior(called satellite prior) for the J-factor estimation of dSphs by fitting their averaged velocity dispersion [11] and it was shown that the satellite prior has a potential to break thedegeneracy among parameters of the NFW profile [12].
While the satellite prior gives a statistical trend for the whole dSphs, cosmology offers
us another way to constrain the dark matter mass of individual dSph. Since the dark matter
plays an essential role to form the structure of the universe including dSphs, the stellar
components of dSphs has a strong relation to their dark matter halo, known as the stellar-tohalo mass relation (SHMR) [13]. The dark matter halo mass in each dSph can be therefore
constrained by its stellar mass by using this relation.
In this paper, we perform a more detailed analysis of the satellite prior in Ref. [11]1(hereafter SA20) by considering the radial dependence of the velocity dispersion to optimize given kinematical data sets. The radial dependency weakens the degeneracy among
dark matter halo parameters and gives more precise estimation than the radial independent
analysis. Moreover, we consider some SHMR models to obtain more reasonable estimation
results from the viewpoint of cosmology. These models help us to obtain more accurate
constraints of the halo parameters than the previous satellite prior only analysis. From theother point of view, our analysis also provides a method for evaluating the credibility ofeach SHMR model using dark matter halos of dSphs. This paper is organized as follows: In
Section 2, we discuss our analysis method. In Section 2.1, we describe our model setups
and assumptions on the dSph system and introduce our likelihood function. In Section 2.2,
we explain the construction of the satellite prior and the stellar-to-halo mass relation prior.
We set up several choice of these prior reflecting the uncertainty of cosmological models. In
Section 2.3, we show the table of dSphs including their half-light radii and discances. The
kinematical data set and preanalysis method for each dSph is also described. In Section 3,
we show results of the analysis and estimated J-factor values. Here we compare results ofdifferent priors to verify the stability of estimated J-factor by changing cosmological models.
In Section 5, we discuss and summarize our results.

2

Method2.1

Models and likelihood

We assume that dSphs are spherical and steady systems according to conventional analyses1 [8, 15]. Their velocity dispersions are determined by the spherical Jeans equation
GM (r)1 ∂ (ν(r)σ r (r))+ 2σ2r (r)β(r) = −
,
ν(r)
∂r
r2
2

(2)

where G is the gravitational constant, r denotes the distance from the dSph center, ν(r) is thestellar number density and M (r) denotes the dark matter mass enclosed within the radius r.
The anisotropy of the stellar motion β(r) is defined by the ratio of the velocity dispersions
σ2 +σ2σ2r (r), σθ2 (r) and σφ2 (r) as β(r) ≡ 1 − θ2σ2 φ . By definition, β(r) satisfies −∞ < β(r) ≤ 1.
r
In this paper, we assume that β(r) = βani (const.) for simplicity.
We describe the stellar profile ν(r) as the Plummer profile, a widely-used fitting functionof the stellar number density of dSphs [16].2 . Its stellar number density ν(r) and surface
density Σ(R) are given by 2 −5/23r
1+,
ν(r) =
3
4πR e
Re

−2
1
R2
Σ(R) =
1+ 2
,
πR2e
Re

(3)
(4)

The sphericity of dSphs is also assumed when deriving the satellite prior [11]. However, Ref. [14] suggeststhat axisymmetric models show better fitting performance than spherical models in terms of Bayes factor
comparison. Hence, further study on the axisymmetric formalization of the satellite prior would be useful for
future J-factor analyses, which is however beyond the scope of this paper.
2
Note that the stellar profile fitting by using generalized function suggests that some dSphs have multiple
components. [17, 18]12

where R denotes the radius projected on the celestial sphere and R e is the half-light radiusof the surface density profile.
For the dark matter density profile, there are many discussions and no consensus exists
yet. This is known as the core-cusp problem [19]; N-body simulation shows the cuspy dark
matter density profile (ρ(r) ∝ r −1 around the center), while observations suggest the coredprofile (ρ(r) ∝ const.). In this paper, we assume the cold dark matter model, then the dark
matter density profile can be well described
R by the truncated NFW profile [12], whose massdensity ρ(r) and enclosed mass M (r) ≡ dr 4πr 2 ρ(r) are respectively written as
ρ(r) =(  −1 −2
1 + rrs
ρs rrs(0 ≤ r ≤ r t ),0(r t < r)¨

4πρs rs3 fNFW (r/rs )(0 ≤ r ≤ r t )4πρs rs3 fNFW (r t /rs )(r t < r),

M (r) =

(5)

(6)

where ρs and rs is scale density and radius of the profile, respectively, r t denotes the truncation radius, and an auxiliary function fNFW (x) is defined as follows:
fNFW (x) = ln (1 + x) −1.
1 + x −1

Using these quantities, we define our likelihood function as follows:Y2
L (Θ) =
N [vi ; vdSph , σlos
(R i ) + δσ2i ],

(7)

(8)

i


where vdSph is the systemic velocity of each dSph, Θ ≡ R e , rs , ρs , r t , βani , vdSph represents
the parameter set in our model, N [x; µ, σ2 ] denotes the normal (Gaussian) distribution
with the mean µ and the variance σ2 , vi is the observed velocity of i-th star and δσi is2
its observational error. The function σlos(R) is the projected velocity dispersion along theline-of-sight at projected radius R, given by the following formula:Z∞ 
2R2 ν(r)σ r (r)22
dr 1 − β(r) 2 p.
(9)σlos (R) =Σ(R) Rr
1 − R2 /r 2
When β(r) = βani (const.), Eq. (9) is simplified to [20]Z∞1
GM (s)
K(s/R),ds ν(s)Σ(R) R
s
 




p
3
3
1
3
2
2
K(u) = 1 − u−2 u− βani 2 F1 1, − βani ; ; 1 − u − .
2
2
2
2
2
σlos
(R) =

(10)
(11)

where 2 F1 (a, b; c; z) is the Gaussian hypergeometric function.
We note that our likelihood function has R-dependence in contrast with that SA20 usedR∞2
the averaged (R-independent) velocity dispersion σlos
= 4πG
dr rν(r)M (r). The advan30tage of the our R-dependent analysis is that it weakens the degeneracy between parameters22
by probing the shape of σlos(R) even when σlos
is not changed.

3

2.2

Priors

2.2.1

Photometry prior

The half-light radius R e is constrained by the result of photometric observations, which isrealized as a photometric prior. We adopt log-normal distribution for the half-light radius
to construct the prior as follows:
πphoto. (log10 R e /[pc]) = N (log10 R e /[pc]| log10 re,circ /[pc], δ log10 re,circ /[pc]),(12)

where we calculate the mean log10 re,circ and standard deviation δ log10 re,circ based on theerror propagation law by using θ̂ listed in Table. 1 on the supplement material of SA20. 3
2.2.2

Satellite prior

Structure formation models of subhalos in Milky Way predict structural parameters of subhalo profile ρ(r): rs , ρs and r t . In this paper we use the satellite prior proposed in SA20,
briefly reviewed in the following: The formation of subhalos are well described by the extended Press-Schechter formalism [21], which gives the differential number of accreted
d2 N
subhalos dza dmsha . Here Nsh denotes the number of subhalo, za and ma are the redshift andmass of a subhalo when the subhalo accreted onto its host. Here ma can be reinterpreted as
halo parameters ρs,a , rs,a and r200 by considering two conditions: i.) The subhalo is virial3
ized ma = 4πρcrit (za )200r200/3, where the virial radius r200 is calculated from rs,a by using
the concentration parameter ca = rs,a /r200 , whose probability density distribution P(ca ) is
given by the log-normal distribution with mean c200 (ma , za ) [22] and standard deviation
σln c = 0.13 [23]. ii.) The dark matter distribution of the subhalo is given by the NFW
profile m200 = 4πρs,a rs3 fNFW (r200 /rs,a ), where fNFW is the same as defined in Eq. (7).
After the accretion, the tidal force of Milky Way starts striping subhalo mass. In a semianalytic strategy, the mass-loss rate through this process is modeled asm(z) m(z) ζ
dm
= −A
dt
τdyn (z) M (z)(13)

where τdyn (z) denote the dynamical timescale [24], m(z) and M (z) are subhalo and hosthalo mass at redshift z, respectively. The two parameters A and ζ are calibrated by the resultsof N-body simulations. The solution of this equation with the initial condition m(za ) = magives current (z = 0) subhalo mass m0 = m(0). As the subhalo mass evolve, structuralparameters ρs,a and rs,a also evolve to ρs,0 and rs,0 (or simply ρs and rs ) according empiricalfitting formula [23]. Finally, current truncation radius r t,0 (or simply r t ) are determined by3
the NFW condition m0 = 4πρs,0 rs,0fNFW (r t,0 /rs,0 ).
d2 N

We combine the two distributions of parameters at accretion dza dmsha and P(ca ) with theparameter evolution model to obtain the distribution of parameters at present.4 Insteadof calculating the distribution of current parameters directly by using the Jacobian of the3

In SA20 they assume the normal distribution for the half-light radius, but we found no significant difference between the normal and log-normal distributions because the standard deviation is so small that the
log-normal distribution can be approximated by the normal distribution. In this paper, we adopt log-normal
distribution, reflecting the fact that the radius must be positive.
4
Here we assume that there are no correlation between (ma , za ) and c t,a .

4evolution formula, we obtain finite samples of the parameters. We subdivide (ln ma , za , ca )linearly and calculate the weight of i-th grid according to
wi = Nd2 Nsh(∆ca )i(∆za )i (∆ma )i P(ca )dza dma z =z ,m =m
c =c
a

a,i

a

a

a,i

(14)

a,i

RR
P
d2 N
where N is a normalization factor to satisfy the condition i w i = Nsh,tot ≡dza dma dza dmsha .
Each point (ln ma,i , za,i , ca,i ) is interpreted to (ρs,i , rs,i , c t,i ) according to the stripping model,
then we obtain finite samples of (ρs , rs , c t ) with its weight.5
Some subhalos do not host any stars because baryons in too small a halo cannot lose
their energy due to its ionizing background, known as reionization suppression [25, 26]. In
order to consider the effect, we multiply w i by the formation probability of a satellite forthe given subhalo Pform , as follows:
Vpeak − V50
1
1 + erf,
Pform (Vpeak ) =
p
2
2σ

(15)

where Vpeak denotes the maximum circular velocity of the satellite at accretion time, given
by Vpeak = (4πGρs,a /4.625)1/2 rs,a for a NFW subhalo and it is calculated for each parameter
grid (ρs,i , rs,i , c t,i ). For the lower bound parameter V50 , we have two choices: V50 = 18 km/s,
motivated by conventional theory of reionization [27–30], and V50 = 10.5 km/s, based on
the result of more resent analysis [26]. For σ, we adopt σ = 2.5 km s−1 , following Ref. [26].
For classical dSphs, we adopt V50 = 25 km/s according to [31]. Here we assume σ =0 km s−1 for simplicity. In this case Pform (Vpeak ) is equivalent to a step function Θ(Vpeak − V50 ).
Using these quantities, the probability density distribution of the three profile parametersis then given byd3 NshPform (Vpeak ),(16)
πsat. (rs , ρs , r t ) ∝drs dρs dr t
where πsat. should be properly normalized to be a probability density distribution function.
For the discrete sample points generated above, it is realized as
w i Pform (Vpeak,i )
πsat.,i = P.
i w i Pform (Vpeak,i )(17)Finally we smooth these samples πsat.,i to reconstruct a prior function πsat. (rs , ρs , r t ) byusing weighted kernel density estimation implemented in scipy [32].
2.2.3

SHMR prior

In addition to the satellite prior, we use another prior motivated by the stellar-to-halo mass
relation (SHMR) (see [33] for a review). This relation is obtained by fitting the structureformation model by using observed cosmological data sets with a simple assumption; thelarger halo becomes, the more stars it hosts. Stellar mass m∗ is then written as a monotonic
function of halo mass mz for given redshift z. We identify m(z) with the subhalo mass at theaccretion time and relate it to the current subhalo mass using the semi-analytic approach
5

Available in https://github.com/shinichiroando/dwarf_params.
named SASHIMI is in https://github.com/shinichiroando/sashimi-c.

5More integrated version

m(z)semi-analytic
model

m(0)

SHMR
m∗

const.

m∗

Figure 1: A schematic picture to illustrate how to construct our SHMR prior. Horizontal one-sidearrows indicate time evolution. Shapes with bold edges are values at present (z = 0), appeared in
the definition of the SHMR prior in Eq. (18).

Figure 2: SHMR function m∗ (Mh , z) normalized by the halo mass Mh . Here we fix z = 0.1 forillustration purpose.

mentioned in Section 2.2.2. Here, for the simplicity, we assume that the stellar mass does
not change after accretion and identify m∗ as the current stellar mass.6 The schematic figureof this procedure is shown in Fig. 1.
In order to check model dependence, we adopt four SHMR models [34–37] (hereafter
PB13, BM13, PB19 and BM18, respectively). These model have the following features:• PB13: calibrated by the Bolshoi simulation, using a multi-power law fitting function
for the SHMR and fitting the SHMR intrinsic parameters with other systematic parameters.
• BM13: calibrated by the Millennium simulation, using a double-power law fitting
function for the SHMR and simply fitting SHMR parameters.
• PB19: updated data sets from PB13, cosmological models calibrated by the BolshoiPlanck simulation, using a double-power law fitting function for the SHMR and simply
fitting SHMR parameters.
• BM18: updated data sets from BM13, cosmological models assuming double-powerlaw for the evolution of baryon conversion efficiency calibrated by an independent
6

Basically, a galaxy is hosted at the center of a dark matter halo which is more massive than stellar mass,and the tidal force strips mainly the dominant component of the halo mass. Therefore stellar mass is not
significantly changed until the most of halo mass is striped and halo mass and galaxy mass become comparable.
However, once the halo mass becomes comparable, it cannot bound the galaxy within the halo and the galaxy
gets collapsed, thus we cannot find such galaxies anymore.

6

simulation, using double-power law for the SHMR and simply fitting SHMR parameters.
The probability density distribution of stellar mass m∗ is then written as follows:P(m∗ |m(z)) = N [m∗ ; m∗ (m(z), z), δm∗ ](18)

where m∗ (Mh , z) is expected stellar mass for given halo mass Mh at accretion redshift z. δm∗denotes the model uncertainty of each SHMR model. We show the function m∗ (Mh , z) for
each model in Fig. 2.
In terms of the Bayesian statistics, we can compare credibility of a model (model 1)
to a reference model (model 0) by using Bayes factor BF, defined as the ratio of Bayesianevidences E :R
dΘ1 L1 (Θ1 )π1 (Θ1 )
E1
BF =
.
(19)
=R
E0
dΘ0 L0 (Θ0 )π0 (Θ0 )Here the minus logarithm of the Bayesian evidence − ln E is approximated by the widely
applicable Bayesian information criterion (WBIC) [38]:RdΘ (ln L (Θ))L (Θ)β π(Θ)R,
(20)
− ln E ' WBIC = −dΘ L (Θ)β π(Θ)β = 1/ ln(#data).

(21)

In this work, we calculate WBIC of each model using the MCMC algorithm and evaluate theBayes factor according toln BF = ln E1 − ln E0 ' −WBIC1 + WBIC0 .

(22)

Here BF > 1 or ln BF > 0 means that model 1 is more credible than model 0. According
to Ref. [39], there is a scale for interpreting ln BF into the strength of evidence as follows:
Decisive for ln BF ¦ 4, very strong for 3 ® ln BF ® 4, strong for 2 ® ln BF ® 3, substantial for1 ® ln BF ® 2, and barely worth mentioning for 0 ® ln BF ® 1.

2.3

Data

We analyze the dSphs listed in Tables 1 and 2 according to SA20, where we show the halflight radius, distance, and stellar mass of each dSph. The half-light radius and distances
are from SA20 and also we use the values in Ref. [40] for the stellar masses. For dSphs
without stellar mass values in Ref. [40], we calculate their stellar masses from apparent
magnitudes and distances, assuming M /L = 1 according to Ref. [40]. The last column
in Tables 1 and 2 indicates references of the kinematical data set. In general, kinematical
data set includes member stars and foreground stars. For data sets having membership flag,
we extract stars identified as members. For those containing membership probability PM ,we choose member-like stars (PM > 0.95). For the other data sets having no membership
information, we adopt the selection criteria illustrated and described in the reference. In
addition, we remove member stars identified as binary stars in order to avoid accidental
increase of the velocity dispersion.

7

Table 1: The half-light radius, distance from Earth, stellar mass and reference list for the 27 ultrafaint dSphs analyzed in this paper. We derive the half-light radius and its error based on the value
in SA20. The values of distance and stellar mass are from SA20 and Ref. [40], respectively (see text
for more details). The last column shows the source of kinematical data set.
log10 R e /[pc]NameAquarius 2
Boötes 1
Boötes 2
CanesVenatici 1
CanesVenatici 2
Carina 2
ComaBerenices
Draco 2Eridanus 2
Grus 1
HerculesHorologium 1
Hydrus 1
Leo 4
Leo T
Leo 5
Pegasus 3
Pisces 2
Reticulum 2
Segue 1
Segue 2
Triangulum 2
Tucana 2
Tucana 3
UrsaMajor 1
UrsaMajor 2
Willman 1

2.09 ± 0.08
2.20 ± 0.02
1.52 ± 0.07
2.53 ± 0.02
1.73 ± 0.09
1.87 ± 0.05
1.76 ± 0.03
1.12 ± 0.18
2.20 ± 0.05
1.27 ± 0.46
2.08 ± 0.04
1.49 ± 0.10
1.73 ± 0.03
2.01 ± 0.05
2.13 ± 0.05
1.57 ± 0.18
1.62 ± 0.16
1.68 ± 0.07
1.49 ± 0.02
1.30 ± 0.06
1.53 ± 0.04
1.10 ± 0.13
2.21 ± 0.07
1.64 ± 0.06
2.18 ± 0.02
1.93 ± 0.02
1.30 ± 0.05

D∗ [kpc]

M∗ [M ]

Refs.

108
66
42
218
160
36
44
20
380
120
132
79
28
154
417
178
215
182
30
23
35
30
57
25
97
32
38

3

[41]
[42]
[43, 44]
[45]
[45]
[46]
[45]
[47]
[48]
[49]
[45]
[50]
[51]
[45, 52]
[45]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[49]
[59]
[45]
[45]
[60]

4.7 × 10
2.9 × 104
1.0 × 103
2.3 × 105
7.9 × 103
5.4 × 103
3.7 × 103
1.0 × 103
6.5 × 104
2.1 × 103
3.7 × 104
2.2 × 103
6.5 × 103
1.9 × 104
1.4 × 105
1.1 × 104
3.6 × 103
8.6 × 103
3.0 × 103
3.4 × 102
8.6 × 102
4.5 × 102
2.8 × 103
7.9 × 102
1.4 × 104
4.1 × 103
1.0 × 103

Table 2: Same as Table 1, but for classical dSphs.
Name

log10 R e /[pc]D∗ [kpc]

M∗ [M ]

Refs.

CarinaDraco
Fornax
Leo 1
Leo 2
Sculptor
Sextans 1
UrsaMinor

2.392 ± 0.005
2.256 ± 0.005
2.849 ± 0.003
2.353 ± 0.004
2.217 ± 0.005
2.359 ± 0.004
2.538 ± 0.004
2.434 ± 0.006

105
76
147
254
233
86
86
76

3.8 × 105
2.9 × 105
2.0 × 107
5.5 × 106
7.4 × 105
2.3 × 106
4.4 × 105
2.9 × 105

[61]
[62]
[61]
[63]
[64]
[61]
[61]
[65]

8

Table 3: Scanning region of each parameter.

2.4

parameter

min.

max.

log10 R e /[pc]log10 rs /[pc]log10 ρs /[M pc−3 ]log10 r t /[pc]− log10 (1 − βani )vdSph /[km s−1 ]

1.0
0.0
−4.0
0.0
−1.0
−1000

3.5
5.0
4.0
5.0
1.0
1000

AnalysisBased on the likelihood and priors defined above, we can calculate the posterior probability
density distribution (or simply posterior) P(Θ|D) by using the Bayes’ theorem:
P(Θ|D) = R
where
π=


πphoto.

πphoto. πsat.
πphoto. πsat.+SHMR

L (Θ)π(Θ)
dΘ L (Θ)π(Θ)

,

(23)

(without any cosmological priors)(satellite prior only)(24)(satellite & SHMR prior)

Here, as mentioned in Section 2.2, the satellite prior πsat. is selected from two candidates
sat.10.5 and sat.18 , and the SHMR model for πsat.+SHMR is chosen from PB13, BM13, PB19
and BM18.
Instead of calculating Eq. (23) straightforwardly, we obtain samples from the posterior
by using the Markov Chain Monte-Carlo methods. In this paper, we use the Affine invariant
ensemble sampler implemented in emcee [66]. We scan the parameter region as shown in
Table 3. For R e , rs , ρs and r t we adopt the logarithmic scale, reflecting that they are positive.
The range for the anisotropy βani is set to include both of radial and tangential cases. Since
vdSph is strongly constrained by the likelihood function, we choose its limits large enough toinclude the estimated value.

3

Results

Figure 3 shows the posterior projected onto rs -ρs plane. The value of likelihood function isshown by blue dots and contours. Colored contours denote posteriors assuming the satellite
and/or SHMR priors. The satellite prior itself is shown by gray contours and gray shaded
area shows the cosmological constraint adopted in Ref. [8]. For illustration purpose, weonly show the result of Segue 1. For other dSphs, see Figs. 7 and 8.
In order to clarify the advantage of R-dependent analysis, we compare results of Rdependent and R-independent likelihood analyses in Fig. 4, where R-independent likelihood2
is defined similarly to Eq. (8) but the velocity dispersion σlos(R) is replaced by averaged2
dispersion σlos
. In Fig. 4 the color of the heat map corresponds to the value of likelihood
functions.

9Figure 3: Posterior probability density function projected onto rs -ρs plain for the case of Segue
1. The left and right panels show result assuming the sat.10.5 and sat.18 model, respectively. Blue
dots are distributed according to likelihood only analysis (without any cosmological priors), whilecolored contours show posteriors with cosmological priors. Gray shaded area shows the cosmologicalconstraint adopted in Ref. [8]. For the other dSphs, see Figs. 7 and 8.

Figure 4: Results of likelihood only (without any cosmological priors) analyses. The left panel isfor the R-dependent likelihood function, (Eq. (8)), while the right panel is for the R-independent
likelihood (see text), respectively. Color of heat map corresponds to the value of profiled likelihood
function Lprof (rs , ρs ) ≡ max r t L (rs , ρs , r t ). Gray contours denote the 1σ and 2σ regions of thesatellite prior and gray shaded region shows the rough cosmological prior adopted in Ref. [8]. For
illustration purpose, we show only the Coma Berenices case.

10

11

Willman1

UrsaMajor2

UrsaMajor1

Tucana3

Tucana2

Triangulum2

Segue2

Segue1

Reticulum2

Pisces2

Pegasus3

LeoT

Leo5

Leo4

Hydrus1

Horologium1

Hercules

Grus1

Eridanus2

Draco2ComaBerenices

Carina2

CanesVenatici2

CanesVenatici1

Bootes2

Bootes1

Aquarius2

19.50+0.44−0.44

19.54+0.38−0.38

18.66+0.30−0.29

15.71+1.79−4.35

18.14+0.58−0.52

14.36+2.91−3.86

18.02+0.70−2.06

19.74+0.39−0.41

18.98+0.37−0.36

17.24+0.91−1.17

17.82+1.01−2.10

17.61+0.43−0.44

17.18+0.78−0.82

15.63+1.85−5.06

18.52+0.36−0.32

19.08+0.68−0.63

17.89+0.37−0.37

17.42+0.88−0.89

17.29+0.38−0.36

16.80+2.54−4.77

18.95+0.34−0.36

18.38+0.56−0.55

17.94+0.50−0.45

17.59+0.32−0.19

16.64+2.80−4.90

18.85+0.34−0.38

19.01+0.29−0.28

18.30+0.22−0.22

17.52+0.45−0.56

17.87+0.38−0.36

17.75+0.67−0.86

18.01+0.46−0.50

19.20+0.33−0.39

18.65+0.30−0.30

16.74+0.55−0.56

16.72+0.60−0.72

16.95+0.31−0.31

16.92+0.45−0.48

16.72+0.53−0.61

18.33+0.29−0.27

18.11+0.41−0.40

17.51+0.30−0.27

17.09+0.46−0.48

16.91+0.25−0.24

18.30+0.62−0.78

18.58+0.28−0.29

18.06+0.41−0.37

17.40+0.37−0.36

17.42+0.13−0.12

17.69+0.65−0.81

18.03+0.22−0.21

18.95+0.29−0.34

19.10+0.25−0.24

18.34+0.20−0.20

18.06+0.34−0.33

18.16+0.36−0.35

18.54+0.43−0.49

18.42+0.37−0.41

19.32+0.28−0.33

18.80+0.27−0.28

17.15+0.46−0.48

17.23+0.44−0.51

17.01+0.27−0.29

17.28+0.36−0.43

17.17+0.47−0.46

18.50+0.31−0.28

18.26+0.33−0.37

17.69+0.28−0.27

17.44+0.44−0.44

16.98+0.24−0.23

18.93+0.43−0.49

18.70+0.26−0.26

18.35+0.37−0.36

17.58+0.30−0.31

17.45+0.12−0.12

18.41+0.46−0.53

18.11+0.21−0.21

17.84+0.35−0.36

17.62+0.37−0.36

18.25+0.61−0.55

18.24+0.28−0.26

sat.18sat.10.5

flat

w/o SHMR

18.04+0.27−0.28

18.56+0.23−0.24

18.03+0.21−0.19

17.73+0.25−0.27

17.57+0.26−0.25

17.56+0.29−0.29

17.66+0.26−0.26

18.22+0.29−0.31

18.31+0.21−0.20

16.70+0.27−0.27

16.41+0.29−0.28

16.82+0.22−0.22

16.73+0.27−0.26

16.96+0.27−0.27

18.25+0.21−0.19

17.53+0.24−0.23

17.43+0.21−0.21

16.77+0.26−0.25

16.76+0.20−0.19

18.12+0.29−0.29

18.13+0.23−0.23

18.00+0.25−0.26

16.98+0.23−0.25

17.41+0.13−0.12

17.48+0.28−0.29

17.95+0.19−0.18

17.21+0.26−0.24

sat.10.5

PB13

18.14+0.27−0.30

18.63+0.24−0.24

18.09+0.19−0.20

17.80+0.26−0.25

17.68+0.25−0.23

17.63+0.27−0.28

17.72+0.27−0.28

18.36+0.31−0.33

18.35+0.21−0.20

16.72+0.30−0.30

16.45+0.31−0.31

16.87+0.23−0.23

16.87+0.26−0.26

17.01+0.27−0.29

18.30+0.21−0.21

17.59+0.25−0.24

17.48+0.22−0.23

16.85+0.28−0.26

16.81+0.19−0.19

18.15+0.29−0.27

18.19+0.24−0.23

18.06+0.26−0.25

17.08+0.23−0.25

17.45+0.14−0.12

17.53+0.32−0.29

17.99+0.18−0.17

17.31+0.26−0.24

sat.1818.59+0.30−0.32

18.91+0.26−0.25

18.26+0.20−0.20

17.63+0.35−0.29

17.84+0.33−0.30

17.68+0.55−0.49

17.93+0.40−0.44

18.95+0.30−0.37

18.57+0.27−0.28

16.87+0.42−0.38

16.78+0.46−0.51

17.07+0.26−0.29

17.07+0.34−0.34

17.09+0.39−0.37

18.37+0.26−0.26

17.95+0.31−0.34

17.62+0.26−0.27

17.05+0.42−0.41

16.97+0.22−0.23

18.28+0.49−0.48

18.49+0.25−0.27

18.15+0.33−0.32

17.38+0.27−0.31

17.45+0.11−0.11

17.64+0.53−0.49

18.09+0.21−0.21

17.57+0.31−0.32

sat.10.5

BM13

18.86+0.29−0.32

19.05+0.25−0.26

18.27+0.20−0.19

17.65+0.35−0.30

18.02+0.36−0.35

17.96+0.65−0.68

18.11+0.44−0.60

19.23+0.27−0.30

18.70+0.29−0.32

16.94+0.47−0.45

17.00+0.50−0.66

17.09+0.24−0.28

17.16+0.37−0.39

17.15+0.37−0.41

18.38+0.31−0.26

18.18+0.30−0.36

17.61+0.26−0.25

17.26+0.46−0.50

16.96+0.22−0.22

18.40+0.69−0.61

18.60+0.24−0.26

18.18+0.39−0.36

17.48+0.27−0.32

17.44+0.12−0.12

17.84+0.69−0.67

18.09+0.21−0.21

17.73+0.34−0.37

sat.1818.70+0.29−0.33

18.98+0.25−0.25

18.30+0.20−0.20

17.53+0.46−0.38

17.96+0.33−0.35

17.77+0.57−0.70

18.11+0.37−0.51

18.98+0.30−0.34

18.68+0.25−0.28

16.93+0.48−0.60

16.88+0.47−0.74

17.12+0.26−0.30

17.17+0.36−0.43

16.95+0.53−0.50

18.42+0.29−0.31

18.05+0.31−0.34

17.69+0.26−0.29

17.15+0.41−0.48

17.03+0.21−0.21

18.35+0.54−0.70

18.55+0.25−0.27

18.17+0.39−0.42

17.47+0.27−0.28

17.47+0.12−0.12

17.74+0.58−0.73

18.12+0.22−0.20

17.71+0.31−0.35

sat.10.5

PB19

18.84+0.26−0.28

19.04+0.25−0.24

18.30+0.20−0.19

18.05+0.43−0.50

18.10+0.32−0.32

18.44+0.36−0.42

18.37+0.30−0.37

19.21+0.25−0.29

18.75+0.25−0.26

17.15+0.42−0.46

17.17+0.35−0.39

17.12+0.27−0.32

17.29+0.32−0.37

17.18+0.43−0.47

18.51+0.26−0.27

18.18+0.28−0.30

17.69+0.27−0.28

17.43+0.36−0.39

17.03+0.23−0.23

18.89+0.37−0.43

18.63+0.23−0.24

18.39+0.33−0.35

17.51+0.27−0.28

17.46+0.13−0.12

18.33+0.39−0.42

18.12+0.21−0.20

17.80+0.31−0.35

sat.1818.86+0.32−0.33

19.12+0.26−0.27

18.39+0.19−0.19

18.29+0.27−0.24

18.20+0.32−0.30

18.43+0.37−0.38

18.38+0.32−0.30

19.20+0.31−0.35

18.81+0.27−0.27

17.47+0.33−0.33

17.35+0.39−0.42

17.06+0.10−0.12

17.56+0.30−0.30

17.70+0.28−0.27

18.69+0.25−0.22

18.25+0.35−0.36

17.93+0.18−0.20

17.53+0.37−0.36

17.15+0.14−0.16

18.90+0.37−0.35

18.71+0.25−0.26

18.53+0.30−0.27

17.68+0.30−0.31

17.46+0.10−0.07

18.33+0.41−0.38

18.31+0.19−0.18

17.92+0.34−0.33

sat.10.5

BM18

Table 4: Median and 1σ interval of the estimated J-factors. sat.10.5 and sat.18 denote the satellite prior assuming V50 = 10.5 and 18 km/s, respectively.

18.92+0.31−0.35

19.13+0.27−0.27

18.38+0.20−0.19

18.25+0.27−0.23

18.19+0.32−0.29

18.39+0.43−0.36

18.39+0.35−0.34

19.23+0.30−0.36

18.82+0.26−0.27

17.50+0.34−0.34

17.35+0.39−0.39

17.05+0.10−0.14

17.51+0.33−0.31

17.69+0.27−0.28

18.68+0.25−0.22

18.29+0.34−0.36

17.95+0.18−0.21

17.51+0.39−0.35

17.14+0.14−0.15

18.93+0.38−0.35

18.71+0.26−0.27

18.56+0.30−0.27

17.68+0.30−0.30

17.46+0.10−0.07

18.35+0.44−0.41

18.30+0.18−0.17

17.91+0.33−0.34

sat.18

Table 5: Same as Table 4, but for classical dSphs.
w/o SHMR

CarinaDraco
Fornax
Leo1
Leo2
Sculptor
Sextans1
UrsaMinor

PB13

BM13

PB19

BM18

flat

sat.

sat.

sat.

sat.

sat.

17.86+0.09−0.07

17.86+0.07−0.06

17.85+0.06−0.06

17.85+0.06−0.06

17.86+0.07−0.06

17.87+0.06−0.06

17.93+0.20−0.08

18.03+0.11−0.10

18.02+0.10−0.08

18.00+0.09−0.07

18.02+0.09−0.08

17.96+0.07−0.06

18.92+0.06−0.06
17.80+0.22−0.14
17.82+0.25−0.20
18.56+0.07−0.05
18.09+0.40−0.16
18.47+0.13−0.09

18.89+0.06−0.06
17.71+0.10−0.09
17.70+0.16−0.14
18.55+0.04−0.04
18.12+0.15−0.13
18.46+0.09−0.08

18.85+0.06−0.06
17.73+0.08−0.08
17.64+0.13−0.11
18.55+0.04−0.04
18.09+0.14−0.11
18.50+0.09−0.08

18.85+0.06−0.06
17.73+0.11−0.09
17.69+0.15−0.13
18.55+0.04−0.04
18.09+0.14−0.12
18.46+0.09−0.08

18.85+0.06−0.06
17.74+0.12−0.09
17.71+0.15−0.14
18.55+0.05−0.04
18.12+0.15−0.12
18.46+0.08−0.07

18.88+0.06−0.06
17.78+0.14−0.11
17.73+0.17−0.14
18.56+0.04−0.04
18.19+0.15−0.13
18.47+0.09−0.08

Table 6: The natural logarithm of Bayes factors of each model calculated according to Eq. (22). Column 1 shows the Bayes factor of sat18 to a reference model sat10.5 for each dSph. Columns 2-5 shows
the Bayes factors of the satellite prior and SHMR analyses to the satellite prior only analysis sat10.5
as a reference, so as Columns 6-9 not for sat10.5 but sat18 cases. By definition, positive (negative)
values mean that the corresponding model is more (less) credible than the reference model.
sat.18 /sat.10.5
w/o SHMR
Aquarius2
Bootes1
Bootes2
CanesVenatici1
CanesVenatici2
Carina2
ComaBerenices
Draco2Eridanus2
Grus1
HerculesHorologium1
Hydrus1
Leo4
Leo5
LeoT
Pegasus3
Pisces2
Reticulum2
Segue1
Segue2
Triangulum2
Tucana2
Tucana3
UrsaMajor1
UrsaMajor2
Willman1

0.77
-0.01
-0.09
0.49
1.29
-0.14
1.06
0.16
0.79
-0.30
0.88
1.12
-0.17
-0.16
-0.03
1.39
1.28
0.27
0.96
1.89
0.08
-0.65
-0.13
-2.75
1.06
1.25
2.07

(sat.10.5 + SHMR)/sat.10.5
PB13 BM13 PB19 BM18

(sat.18 + SHMR)/sat.18
PB13 BM13 PB19 BM18

-1.16
0.34
0.05
0.36
-0.70
0.65
-1.71
0.10
-0.22
0.34
0.58
-3.87
0.30
0.34
0.34
0.65
-0.19
0.49
-1.08
-2.63
0.11
-0.01
-0.23
0.56
-5.26
-4.98
-3.05

-2.16
0.12
0.07
0.33
-2.61
0.76
-3.07
-0.07
-0.62
0.27
-0.04
-4.13
0.38
0.44
-0.47
-0.01
-1.65
0.29
-1.85
-4.29
-0.17
0.57
-0.54
3.39
-5.30
-5.93
-4.90

0.17
0.20
0.06
0.49
0.64
0.44
-0.09
0.10
0.76
0.14
0.96
-0.78
0.23
0.02
0.51
1.85
0.37
0.26
0.03
-1.00
0.12
0.14
0.16
0.18
-0.16
-1.11
-1.05

12

0.34
0.17
-0.05
0.07
0.92
0.03
0.35
0.04
0.94
-0.07
0.59
-0.44
-0.20
-0.05
0.04
1.43
0.11
-0.04
0.17
-0.27
0.21
-0.02
0.11
-0.01
0.61
0.05
-0.62

1.13
0.09
-0.16
0.34
2.08
-0.35
1.75
0.24
1.53
-0.40
1.06
1.25
-0.89
-0.93
0.24
0.84
1.62
-0.07
1.13
1.36
0.26
-0.52
-0.09
-2.82
1.84
1.67
1.71

-0.35
0.10
0.04
-0.01
-0.70
0.30
-0.52
-0.09
-0.03
0.19
-0.06
-0.58
0.09
0.15
-0.15
0.47
-0.98
-0.01
-0.82
-1.01
-0.20
0.37
0.12
2.68
-0.45
-0.63
-1.26

0.05
0.21
0.01
-0.31
0.15
0.29
0.07
-0.05
0.05
0.10
-0.07
-0.03
-0.03
0.01
0.31
0.08
-0.06
-0.11
-0.12
-0.05
0.04
0.08
0.16
-0.08
-0.08
-0.21
-0.37

0.29
0.11
-0.01
-0.03
0.71
-0.19
0.64
-0.01
0.78
-0.07
0.15
0.25
-0.85
-0.72
0.45
-0.61
0.25
-0.28
0.13
-0.41
-0.10
0.16
-0.06
0.08
0.80
0.37
-0.29

Tucana2
Tucana2

Triangulum2
Triangulum2

Segue2
Segue2

Segue1
Segue1

Reticulum2
Reticulum2

Pisces2
Pisces2

Pegasus3
Pegasus3

LeoT
LeoT

Horologium1
Horologium1

HerculesHercules

Grus1
Grus1

Eridanus2
Eridanus2

Draco2
Draco2ComaBerenices
ComaBerenices

Carina2
Carina2

CanesVenatici2
CanesVenatici2

CanesVenatici1

CanesVenatici1

Bootes2

Bootes2

Bootes1

Bootes1

Aquarius2

Aquarius2

log10J/[GeV2 cm 5]

log10J/[GeV2 cm 5]

Leo5

Leo4

Hydrus1

V50 = 18 km/s

Tucana3
Tucana3

Leo5

Leo4
Hydrus1

sat.
sat.+PB13
sat.+BM13
sat.+PB19
sat.+BM18
sat. (convl.)

sat.
sat.+PB13
sat.+BM13
sat.+PB19
sat.+BM18
sat. (convl.)

analysis, while gray ones show the results of conventional analysis [11]. Blue, orange, green and red bars correspond to analyses with the satellite prior and
PB13, BM13, PB19 and BM18 priors, respectively. The upper panel shows results when V50 = 10.5 km/s, while the lower panel is for V50 = 18 km/s.

Figure 5: Estimated J-factors of UFDs in Table 4 (dots) and their 68% credible intervals (error bars). Black bars are the results with satellite prior only

20.0
19.5
19.0
18.5
18.0
17.5
17.0
16.5
16.0
15.5

UrsaMajor1
UrsaMajor1

V50 = 10.5 km/s

UrsaMajor2
UrsaMajor2

20.0
19.5
19.0
18.5
18.0
17.5
17.0
16.5
16.0
15.5
Willman1
Willman1

13

19.0
18.8

log10J/[GeV2 cm 5]

18.6
18.4
18.2
18.0
sat.
sat.+PB13
sat.+BM13
sat.+PB19
sat.+BM18
sat. (convl.)

17.8
17.6
17.4

CarinaDraco

Fornax

Leo_1

Leo_2

Sculptor Sextans UrsaMinor

Figure 6: Same figure as Fig. 5 but for the classical dSphs listed in Table 5.
Tables 4 and 5 show the median values of J-factor posteriors with 68% (∼ 1σ) credibleintervals. Left three columns show results without SHMR priors. In particular, “flat” column
denotes those without any cosmological priors (only with the likelihood and the photometryprior). Following columns are those with SHMR priors, PB13, BM13, PB19, and BM18,
respectively. These results are also shown in Fig. 5. In Fig. 5, we also show the results of
conventional analysis [11] as gray bars for comparison.
We show the Bayes factor of each model in Table 6. Column 1 shows the Bayes factor ofsat18 to a reference model sat10.5 for each dSphs. Columns 2-5 shows the Bayes factors of thesatellite prior sat10.5 and SHMR analyses to the satellite prior only analysis as a reference.
Columns 6-9 are same as Columns 2-5 but for sat18 cases. Here a positive (negative) value
indicates that the corresponding model is more (less) credible than sat10.5 .

4Discussion

4.1

Posterior

For Boötes 2, Draco 2, Leo 4, Pegesus 3, Pisces 2, Segue 2, Triangulum 2 and Tucana 3, theirposterior distributions of rs -ρs without satellite priors (likelihood) are broadly distributed(Figs. 7 and 8). This is because observational errors of spectroscopic data set are too large
to exclude small rs -ρs region (dSph without dark matter). In such a case, the GS15-like cut
excludes heavier halo mass region but estimated J-factor is still distributed broadly towards
the small rs -ρs region, thus the choice of scanning range of rs and ρs strongly affects theresult of estimation. This problem is solved by introducing the satellite prior because it
excludes the small rs -ρs region based on the formation history of dSphs.
For the other ultra faint dSphs, posterior distribution becomes more ridgy thanks to alarge amount of kinematical data. In contrast with those obtained in SA20, the likelihoodedges becomes narrow towards the upper left (compact) or lower right (faint) regions,which indicates that the height of likelihood peak varies from the upper left to the lower

14

right. This is thanks to the radial dependence of the velocity dispersion σ(R); even though
σ can be constant by varying rs and ρs properly, σ(R) cannot be kept to fit observed stellar
velocity distribution at all radii.
Figure 4 shows that introducing R-dependence in the likelihood function mitigate thedegeneracy between ρs and rs in SA20. Since certain combinations of ρs and rs gives same
2
value of mean σlos
, R-independent likelihood as used in SA20 has a degeneracy problem.
2
In contrast, the function σlos
is not equivalent even for such a combination, hence it allows
us to distinguish these parameter sets. Introducing R-dependence however causes anotherissue, namely, arbitrariness of anisotropy function β(r), which is just assumed to be constant for simplicity in this study. In order to remove unexpected bias, this arbitrarinessshould be carefully treated in the further study as well as other arbitrariness such as theaxisymmetrisity.

4.2

J-factor and Bayes factor

Figure 5 shows that, in the satellite prior only analysis, our estimates of log-J-factor arelarger by ∼ O (0.1) than those estimated in SA20. This is because the R-dependence of ourlikelihood function weakens the ρs -rs degeneracy, as mentioned in the previous section, andexcludes too compact (small rs , large ρs ) or faint (large rs , small ρs ) dark matter halo withsmall J-factor value.
Figure 5 also shows that SHMR priors decrease the uncertainty of J-factor by up to about
50%, but estimated median values have SHMR model dependence and some estimations
are not consistent with each other. For instance, the PB13 prior tends to predict smaller
J-factor than other priors for dSphs with large J-factors such as Segue 1. Conversely, theBM18 prior gives larger J-factor than other priors for small J-factor dSphs such as Leo 4.
These features come from the difference of SHMR models. As shown in Fig. 2, SHMR models
have different slopes for small Mh region around the mass scale of dSph halos. In particular,
PB13 model has large m∗ /Mh ratio, while BM18 one smaller m∗ /Mh than others. Once m∗is fixed by observations, large m∗ /Mh gives small Mh , and vice versa. We note that Ref. [67]reported M /L ∼ 1.6, thus our estimates of stellar mass obtained by assuming M /L = 1 arepotentially smaller than actual values. However, this discrepancy has no significant effecton our estimation because of the scatter of SHMR models.
Bayes factors help us understand the model dependence of the estimated J-factors.
Tables 4 and 6 show that models whose estimate is deviated from the result of satellite
prior only analysis tend to have small Bayes factors. For instance, the PB13 model showsln BF ® −3 for Segue 1 and Willman 1, and the BM18 shows ln BF ® −1 for Leo 4. It means
that, in terms of the Bayesian analysis, the results of PB13 for Segue 1 and Willman 1 arevery strongly less reliable than those of the satellite prior only analysis, and the results of
BM18 for Leo 4 are substantially less reliable, respectively. We can understand this feature
through posteriors in Figs. 7 and 8. For these dSphs, posteriors obtained by PB13 or BM18
are significantly deviated from the contour of the satellite prior only analysis, which meansthat these SHMR models and the satellite prior are incompatible. In contrast, models having
comparable J-factors to the satellite prior only analysis have Bayes factors almost equal toor larger than the satellite prior only analysis. This tendency of the Bayes factors indicatesthat the estimated J-factor values with the satellite prior only analysis are stable even when
considering SHMRs.
We can utilize this tendency in the opposite direction; not evaluating dark matter profiles
15

by using SHMRs, but evaluating SHMRs by using dark matter profiles. The relation between
J-factors and Bayes factors suggest some possibilities that there are some unknown biases
in the observation of these dSphs or that some SHMR models having small Bayes factors
are invalid for certain ultra faint dSphs. The latter possibility could originate from the difference of the construction of these models; the PB13 model predicts larger m∗ /Mh values
than the others around small halo mass region, while those of the BM18 model is smaller
than the others (see Fig. 2). In particular, Ref. [36] indicated that PB13 assumed a strong
surface-brightness incompleteness correction for faint galaxies that is no longer observationally supported [68], which causes the overestimate of the SHMR around low halo mass
region. For BM18, Ref. [37] pointed out that the underestimate of the BM18 model aroundlow-halo mass region occurs to compensate the overestimation of the number of massive
galaxies caused by the Eddington bias. Further investigation of these features would help
us to improve and calibrate these SHMR models using dSph observation or reveal some
unknown nature of dSphs.
Since the J-factor values of the ultra faint dSphs obtained in this work are not significantly different from conventional values, there are no significant updates for the current
dark matter constraints of the indirect detection experiment. The detection sensitivity depends on the lower bounds of J-factors. Because J-factors of dSphs with the largest J-factors
such as Segue 1 and Ursa Major 2 do not change significantly even when considering cosmological priors having largest Bayes factors, constraints on dark matter parameters do also
not show significant difference. The constraints however could be updated when only weselect a part of dSphs as detection targets, where J-factor lower bound of each dSph matters.
In contrast, from Table 5 and Fig. 6, the J-factor uncertainty of classical dSphs obtainedin this work are reduced up to about 20% of the results of SA20 due to the considerationof R-dependence of the velocity dispersion. In particular, the Draco dSph, having one of
the largest J-factors, has larger lower bound by about 0.25 in logarithmic scale than SA20.
Since indirect detection sensitivity reflects the lower bound of J-factor, the sensitivity might
be stronger than the results of SA20. Here we should note that our results has implicit biasof dSph model construction. In other conventional works such as Ref. [8] the uncertainty
of dark matter profile is taken into account by introducing more general dark matter profile
models and they indicate the deviation of dark matter profile from the simple NFW profile.
In this paper, however, we neglect the uncertainty of dark matter profile by fixing it to be
the NFW profile and also the that of the anisotropy profile by assuming constant model,thus our results have an implicit bias based on the model construction. In order to calculate
more reliable J-factor values, we need further investigation to implement the flexibility ofdark matter profile into the cosmological prior analysis.

5

Summary and conclusion

In this paper, we utilized two cosmological priors (satellite and SHMR) and a likelihood
function with radial dependence to obtain better constraints on the dark matter halo profile
of dSphs through the kinematical fitting using spherical Jeans equation. We prepared somedifferent setups for the cosmological priors and estimate the posterior probability density
function and J-factor. We compared these models and showed that our J-factor estimatesobtained by using the satellite prior are stable in terms of their Bayes factors even when
considering another cosmological prior, the SHMR prior. Cosmological priors and the Rdependence of the likelihood mitigates the degeneracy between parameters and decrease16

the uncertainty of J-factor values upto about 50% for ultrafaint dSphs and about 20% forclassical dSphs. These estimates would be updated by introducing the flexibility of dSph
models (e.g. anitotropy, halo profile and non-sphericity).

Acknowledgements
This research made use of Astropy,7 a community-developed core Python package for Astronomy [69, 70], NumPy,8 the fundamental package for scientific computing with Python [71],and pandas,9 a fast, powerful, flexible and easy to use open source data analysis and manipulation tool, built on top of the Python programming language [72, 73]. This work was
supported by NEXT KAKENHI Grant Numbers, JP20H05850 (S.H. and S.A.), JP20H05861(S.A.), JSPS KAKENHI Grant Numbers, JP20H01895, JP21K13909 and JP21H05447 (K.H.).
Kavli IPMU is supported by World Premier International Research Centre Initiative (WPI),
MEXT, Japan.

A

Posteriors

Posterior probability density distribution projected onto rs -ρs plane are shown in Figs. 7
to 9. Here Figs. 7 and 8 are for ultra faint dSphs, while Fig. 9 is for classical dSphs.

7

http://www.astropy.org
https://numpy.org/
9
https://pandas.pydata.org/
817

Figure 7: Posterior probability density distribution projected onto rs -ρs plain with sat.10.5 prior. The

gray shaded area shows the cosmological constraint adopted in Ref. [8]. The gray contours shows theprobability density distribution of is the satellite prior sat.10.5 . The blue dots and contours illustrate
the shape of the likelihood function (flat prior). Colored contours shows the posterior probability
density distribution assuming our cosmological priors.

18

Figure 7 (cont.):
19

Figure 8: Same figure as Fig. 7 but using the satellite prior sat.18 .

20

Figure 8 (cont.):
21

Figure 9: Same figure as Figs. 7 and 8 but for classical dSphs.
22

References[1] N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys., 641:A6, 2020. [Erratum: Astron.Astrophys. 652, C4 (2021)].
[2] Jonathan L. Feng. Dark Matter Candidates from Particle Physics and Methods of Detection. Ann. Rev. Astron. Astrophys., 48:495–545, 2010.
[3] Biplob Bhattacherjee, Masahiro Ibe, Koji Ichikawa, Shigeki Matsumoto, and Kohei
Nishiyama. Wino Dark Matter and Future dSph Observations. JHEP, 07:080, 2014.
[4] Shin’ichiro Ando and Koji Ishiwata. Sommerfeld-enhanced dark matter searches withdwarf spheroidal galaxies. Phys. Rev. D, 104(2):023016, 2021.
[5] Junji Hisano, Shigeki Matsumoto, and Mihoko M. Nojiri. Explosive dark matter annihilation. Phys. Rev. Lett., 92:031303, 2004.
[6] Junji Hisano, Shigeki. Matsumoto, Mihoko M. Nojiri, and Osamu Saito. Nonperturbative effect on dark matter annihilation and gamma ray signature from galactic
center. Phys. Rev. D, 71:063528, 2005.
[7] James Binney and Scott Tremaine. Galactic dynamics. 1987.
[8] Alex Geringer-Sameth, Savvas M. Koushiappas, and Matthew Walker. Dwarf galaxy
annihilation and decay emission profiles for dark matter experiments. Astrophys. J.,
801(2):74, 2015.
[9] Nagisa Hiroshima, Shin’ichiro Ando, and Tomoaki Ishiyama. Modeling evolution ofdark matter substructure and annihilation boost. Phys. Rev. D, 97(12):123002, 2018.
[10] Shin’ichiro Ando, Tomoaki Ishiyama, and Nagisa Hiroshima. Halo Substructure Boosts
to the Signatures of Dark Matter Annihilation. Galaxies, 7(3):68, 2019.
[11] Shin’ichiro Ando, Alex Geringer-Sameth, Nagisa Hiroshima, Sebastian Hoof, Roberto
Trotta, and Matthew G. Walker. Structure formation models weaken limits on WIMP
dark matter from dwarf spheroidal galaxies. Phys. Rev. D, 102(6):061302, 2020.
[12] Julio F. Navarro, Carlos S. Frenk, and Simon D. M. White. A Universal Density Profile
from Hierarchical Clustering. ApJ, 490(2):493–508, December 1997.
[13] Risa H. Wechsler and Jeremy L. Tinker. The Connection between Galaxies and their
Dark Matter Halos. Ann. Rev. Astron. Astrophys., 56:435–487, 2018.
[14] Kohei Hayashi, Koji Ichikawa, Shigeki Matsumoto, Masahiro Ibe, Miho N. Ishigaki,
and Hajime Sugai. Dark matter annihilation and decay from non-spherical dark halos
in galactic dwarf satellites. Mon. Not. Roy. Astron. Soc., 461(3):2914–2928, 2016.
[15] V. Bonnivard et al. Dark matter annihilation and decay in dwarf spheroidal galaxies:
The classical and ultrafaint dSphs. Mon. Not. Roy. Astron. Soc., 453(1):849–867, 2015.
[16] H. C. Plummer. On the problem of distribution in globular star clusters. MNRAS,
71:460–470, March 1911.

23

[17] Kohei Hayashi, Michele Fabrizio, Ewa L. Łokas, Giuseppe Bono, Matteo Monelli, Massimo Dall’Ora, and Peter B. Stetson. Dark halo structure in the Carina dwarf spheroidal
galaxy: joint analysis of multiple stellar components. Mon. Not. Roy. Astron. Soc.,
481(1):250–261, 2018.
[18] Andrew B. Pace, Manoj Kaplinghat, Evan Kirby, Joshua D. Simon, Erik Tollerud, Ricardo R. Muñoz, Patrick Côté, S. G. Djorgovski, and Marla Geha. Multiple chemodynamic stellar populations of the Ursa Minor dwarf spheroidal galaxy. Mon. Not. Roy.
Astron. Soc., 495(3):3022–3040, 2020.
[19] W. J. G. de Blok. The Core-Cusp Problem. Advances in Astronomy, 2010:789293,
January 2010.
[20] G. A. Mamon and E. L. Łokas. Dark matter in elliptical galaxies - II. Estimating themass within the virial radius. MNRAS, 363:705–722, November 2005.
[21] Xiaohu Yang, H. J. Mo, Youcai Zhang, and Frank C. van den Bosch. An Analytical
Model for the Accretion of Dark Matter Subhalos. ApJ, 741(1):13, November 2011.
[22] Camila A. Correa, J. Stuart B. Wyithe, Joop Schaye, and Alan R. Duffy. The accretion history of dark matter haloes - III. A physical model for the concentration-mass
relation. MNRAS, 452(2):1217–1232, September 2015.
[23] Tomoaki Ishiyama, Steven Rieder, Junichiro Makino, Simon Portegies Zwart, Derek
Groen, Keigo Nitadori, Cees de Laat, Stephen McMillan, Kei Hiraki, and Stefan Harfst.
The Cosmogrid Simulation: Statistical Properties of Small Dark Matter Halos. ApJ,
767(2):146, April 2013.
[24] Fangzhou Jiang and Frank C. van den Bosch. Statistics of dark matter substructure I. Model and universal fitting functions. MNRAS, 458(3):2848–2869, May 2016.
[25] G. Efstathiou. Suppressing the formation of dwarf galaxies via photoionization. MNRAS, 256(2):43P–47P, May 1992.
[26] Andrew S. Graus, James S. Bullock, Tyler Kelley, Michael Boylan-Kolchin, Shea
Garrison-Kimmel, and Yuewen Qi. How low does it go? Too few Galactic satellites
with standard reionization quenching. MNRAS, 488(4):4585–4595, October 2019.
[27] Anne A. Thoul and David H. Weinberg. Hydrodynamic Simulations of Galaxy Formation. II. Photoionization and the Formation of Low-Mass Galaxies. ApJ, 465:608, July
1996.
[28] Nickolay Y. Gnedin. Effect of Reionization on Structure Formation in the Universe.
ApJ, 542(2):535–541, October 2000.
[29] Matthias Hoeft, Gustavo Yepes, Stefan Gottlöber, and Volker Springel. Dwarf galaxies
in voids: suppressing star formation with photoheating. MNRAS, 371(1):401–414,
September 2006.
[30] Takashi Okamoto, Liang Gao, and Tom Theuns. Mass loss of galaxies due to an ultraviolet background. MNRAS, 390(3):920–928, November 2008.

24

[31] Jonathan R. Hargis, Beth Willman, and Annika H. G. Peter. Too Many, Too Few, orJust Right? The Predicted Number and Distribution of Milky Way Dwarf Galaxies.
ApJ, 795(1):L13, November 2014.
[32] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David
Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright,
Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay
Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan
Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, RobertCimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors.
SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261–272, 2020.
[33] Risa H. Wechsler and Jeremy L. Tinker. The Connection Between Galaxies and Their
Dark Matter Halos. ARA&A, 56:435–487, September 2018.
[34] Peter S. Behroozi, Risa H. Wechsler, and Charlie Conroy. The Average Star Formation
Histories of Galaxies in Dark Matter Halos from z = 0-8. ApJ, 770(1):57, June 2013.
[35] Benjamin P. Moster, Thorsten Naab, and Simon D. M. White. Galactic star formation and accretion histories from matching galaxies to dark matter haloes. MNRAS,
428(4):3121–3138, February 2013.
[36] Peter Behroozi, Risa H. Wechsler, Andrew P. Hearin, and Charlie Conroy. UNIVERSEMACHINE: The correlation between galaxy growth and dark matter halo assembly from z = 0-10. MNRAS, 488(3):3143–3194, September 2019.
[37] Benjamin P. Moster, Thorsten Naab, and Simon D. M. White. EMERGE - an empiricalmodel for the formation of galaxies since z ∼ 10. MNRAS, 477(2):1822–1852, June
2018.
[38] Sumio Watanabe. A Widely Applicable Bayesian Information Criterion. Journal of
Machine Learning Research, 14:867–897, 8 2012.
[39] Harold Jeffreys. The Theory of Probability. Oxford Classic Texts in the Physical Sciences.
1939.
[40] Alan W. McConnachie. The Observed Properties of Dwarf Galaxies in and around the
Local Group. AJ, 144(1):4, July 2012.
[41] G. Torrealba, S. E. Koposov, V. Belokurov, M. Irwin, M. Collins, M. Spencer, R. Ibata,M. Mateo, A. Bonaca, and P. Jethwa. At the survey limits: discovery of the Aquarius 2 dwarf galaxy in the VST ATLAS and the SDSS data. MNRAS, 463(1):712–722,
November 2016.
[42] Sergey E. Koposov, G. Gilmore, M. G. Walker, V. Belokurov, N. Wyn Evans, M. Fellhauer,
W. Gieren, D. Geisler, L. Monaco, J. E. Norris, S. Okamoto, J. Peñarrubia, M. Wilkinson, R. F. G. Wyse, and D. B. Zucker. Accurate Stellar Kinematics at Faint Magnitudes:
Application to the Boötes I Dwarf Spheroidal Galaxy. ApJ, 736(2):146, August 2011.
[43] Andreas Koch, Mark I. Wilkinson, Jan T. Kleyna, Mike Irwin, Daniel B. Zucker, Vasily
Belokurov, Gerard F. Gilmore, Michael Fellhauer, and N. Wyn Evans. A Spectroscopic
Confirmation of the Bootes II Dwarf Spheroidal. ApJ, 690(1):453–462, January 2009.
25

[44] Alexander P. Ji, Anna Frebel, Joshua D. Simon, and Marla Geha. High-resolution Spectroscopy of Extremely Metal-poor Stars in the Least-evolved Galaxies: Bootes II. ApJ,
817(1):41, January 2016.
[45] Joshua D. Simon and Marla Geha. The Kinematics of the Ultra-faint Milky Way Satellites: Solving the Missing Satellite Problem. ApJ, 670(1):313–331, November 2007.
[46] T. S. Li, J. D. Simon, A. B. Pace, G. Torrealba, K. Kuehn, A. Drlica-Wagner, K. Bechtol,
A. K. Vivas, R. P. van der Marel, M. Wood, B. Yanny, V. Belokurov, P. Jethwa, D. B.
Zucker, G. Lewis, R. Kron, D. L. Nidever, M. A. Sánchez-Conde, A. P. Ji, B. C. Conn, D. J.
James, N. F. Martin, D. Martinez-Delgado, N. E. D. Noël, and MagLiteS Collaboration.
Ships Passing in the Night: Spectroscopic Analysis of Two Ultra-faint Satellites in theConstellation Carina. ApJ, 857(2):145, April 2018.
[47] Nicolas F. Martin, Marla Geha, Rodrigo A. Ibata, Michelle L. M. Collins, Benjamin P. M.
Laevens, Eric F. Bell, Hans-Walter Rix, Annette M. N. Ferguson, Kenneth C. Chambers,
Richard J. Wainscoat, and Christopher Waters. Is Draco II one of the faintest dwarf
galaxies? First study from Keck/DEIMOS spectroscopy. MNRAS, 458(1):L59–L63,
May 2016.
[48] T. S. Li, J. D. Simon, A. Drlica-Wagner, K. Bechtol, M. Y. Wang, J. García-Bellido,
J. Frieman, J. L. Marshall, D. J. James, L. Strigari, A. B. Pace, E. Balbinot, Y. Zhang,
T. M. C. Abbott, S. Allam, A. Benoit-Lévy, G. M. Bernstein, E. Bertin, D. Brooks, D. L.
Burke, A. Carnero Rosell, M. Carrasco Kind, J. Carretero, C. E. Cunha, C. B. D’Andrea,
L. N. da Costa, D. L. DePoy, S. Desai, H. T. Diehl, T. F. Eifler, B. Flaugher, D. A.
Goldstein, D. Gruen, R. A. Gruendl, J. Gschwend, G. Gutierrez, E. Krause, K. Kuehn,
H. Lin, M. A. G. Maia, M. March, F. Menanteau, R. Miquel, A. A. Plazas, A. K. Romer,
E. Sanchez, B. Santiago, M. Schubnell, I. Sevilla-Noarbe, R. C. Smith, F. Sobreira,E. Suchyta, G. Tarle, D. Thomas, D. L. Tucker, A. R. Walker, R. H. Wechsler, W. Wester,B. Yanny, and DES Collaboration. Farthest Neighbor: The Distant Milky Way Satellite
Eridanus II. ApJ, 838(1):8, March 2017.
[49] Matthew G. Walker, Mario Mateo, Edward W. Olszewski, Sergey Koposov, Vasily Belokurov, Prashin Jethwa, David L. Nidever, Vincent Bonnivard, III Bailey, John I., Eric F.
Bell, and Sarah R. Loebman. Magellan/M2FS Spectroscopy of Tucana 2 and Grus 1.
ApJ, 819(1):53, March 2016.
[50] Sergey E. Koposov, Andrew R. Casey, Vasily Belokurov, James R. Lewis, Gerard
Gilmore, Clare Worley, Anna Hourihane, S. Randich, T. Bensby, A. Bragaglia, M. Bergemann, G. Carraro, M. T. Costado, E. Flaccomio, P. Francois, U. Heiter, V. Hill, P. Jofre,
C. Lando, A. C. Lanzafame, P. de Laverny, L. Monaco, L. Morbidelli, L. Sbordone,
Š. Mikolaitis, and N. Ryde. Kinematics and Chemistry of Recently Discovered Reticulum 2 and Horologium 1 Dwarf Galaxies. ApJ, 811(1):62, September 2015.
[51] Sergey E. Koposov, Matthew G. Walker, Vasily Belokurov, Andrew R. Casey, Alex
Geringer-Sameth, Dougal Mackey, Gary Da Costa, Denis Erkal, Prashin Jethwa, Mario
Mateo, Edward W. Olszewski, and John I. Bailey. Snake in the Clouds: a new nearby
dwarf galaxy in the Magellanic bridge*. MNRAS, 479(4):5343–5361, October 2018.
[52] Sydney A. Jenkins, Ting S. Li, Andrew B. Pace, Alexander P. Ji, Sergey E. Koposov, andBurçin Mutlu-Pakdil. Very Large Telescope Spectroscopy of Ultra-faint Dwarf Galaxies.
I. Boötes I, Leo IV, and Leo V. ApJ, 920(2):92, October 2021.
26

[53] Dongwon Kim, Helmut Jerjen, Marla Geha, Anirudh Chiti, Antonino P. Milone, Gary
Da Costa, Dougal Mackey, Anna Frebel, and Blair Conn. Portrait of a Dark Horse: a
Photometric and Spectroscopic Study of the Ultra-faint Milky Way Satellite Pegasus
III. ApJ, 833(1):16, December 2016.
[54] Evan N. Kirby, Joshua D. Simon, and Judith G. Cohen. Spectroscopic Confirmation of
the Dwarf Galaxies Hydra II and Pisces II and the Globular Cluster Laevens 1. ApJ,
810(1):56, September 2015.
[55] J. D. Simon, A. Drlica-Wagner, T. S. Li, B. Nord, M. Geha, K. Bechtol, E. Balbinot,
E. Buckley-Geer, H. Lin, J. Marshall, B. Santiago, L. Strigari, M. Wang, R. H. Wechsler,
B. Yanny, T. Abbott, A. H. Bauer, G. M. Bernstein, E. Bertin, D. Brooks, D. L. Burke,
D. Capozzi, A. Carnero Rosell, M. Carrasco Kind, C. B. D’Andrea, L. N. da Costa, D. L.
DePoy, S. Desai, H. T. Diehl, S. Dodelson, C. E. Cunha, J. Estrada, A. E. Evrard, A. Fausti
Neto, E. Fernandez, D. A. Finley, B. Flaugher, J. Frieman, E. Gaztanaga, D. Gerdes,
D. Gruen, R. A. Gruendl, K. Honscheid, D. James, S. Kent, K. Kuehn, N. Kuropatkin,
O. Lahav, M. A. G. Maia, M. March, P. Martini, C. J. Miller, R. Miquel, R. Ogando, A. K.
Romer, A. Roodman, E. S. Rykoff, M. Sako, E. Sanchez, M. Schubnell, I. Sevilla, R. C.
Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle, J. Thaler,
D. Tucker, V. Vikram, A. R. Walker, W. Wester, and DES Collaboration. Stellar Kinematics and Metallicities in the Ultra-faint Dwarf Galaxy Reticulum II. ApJ, 808(1):95,
July 2015.
[56] Joshua D. Simon, Marla Geha, Quinn E. Minor, Gregory D. Martinez, Evan N. Kirby,
James S. Bullock, Manoj Kaplinghat, Louis E. Strigari, Beth Willman, Philip I. Choi,
Erik J. Tollerud, and Joe Wolf. A Complete Spectroscopic Survey of the Milky Way
Satellite Segue 1: The Darkest Galaxy. ApJ, 733(1):46, May 2011.
[57] Evan N. Kirby, Michael Boylan-Kolchin, Judith G. Cohen, Marla Geha, James S. Bullock, and Manoj Kaplinghat. Segue 2: The Least Massive Galaxy. ApJ, 770(1):16,
June 2013.
[58] Evan N. Kirby, Judith G. Cohen, Joshua D. Simon, Puragra Guhathakurta, Anders O.
Thygesen, and Gina E. Duggan. Triangulum II. Not Especially Dense After All. ApJ,
838(2):83, April 2017.
[59] J. D. Simon, T. S. Li, A. Drlica-Wagner, K. Bechtol, J. L. Marshall, D. J. James, M. Y.
Wang, L. Strigari, E. Balbinot, K. Kuehn, A. R. Walker, T. M. C. Abbott, S. Allam,
J. Annis, A. Benoit-Lévy, D. Brooks, E. Buckley-Geer, D. L. Burke, A. Carnero Rosell,
M. Carrasco Kind, J. Carretero, C. E. Cunha, C. B. D’Andrea, L. N. da Costa, D. L. DePoy, S. Desai, P. Doel, E. Fernandez, B. Flaugher, J. Frieman, J. García-Bellido, E. Gaztanaga, D. A. Goldstein, D. Gruen, G. Gutierrez, N. Kuropatkin, M. A. G. Maia, P. Martini, F. Menanteau, C. J. Miller, R. Miquel, E. Neilsen, B. Nord, R. Ogando, A. A. Plazas,
A. K. Romer, E. S. Rykoff, E. Sanchez, B. Santiago, V. Scarpine, M. Schubnell, I. SevillaNoarbe, R. C. Smith, F. Sobreira, E. Suchyta, M. E. C. Swanson, G. Tarle, L. Whiteway,
B. Yanny, and DES Collaboration. Nearest Neighbor: The Low-mass Milky Way Satellite Tucana III. ApJ, 838(1):11, March 2017.
[60] Beth Willman, Marla Geha, Jay Strader, Louis E. Strigari, Joshua D. Simon, Evan Kirby,
Nhung Ho, and Alex Warres. Willman 1—A Probable Dwarf Galaxy with an Irregular
Kinematic Distribution. AJ, 142(4):128, October 2011.
27

[61] Matthew G. Walker, Mario Mateo, and Edward W. Olszewski. Stellar Velocities in the
Carina, Fornax, Sculptor, and Sextans dSph Galaxies: Data From the Magellan/MMFS
Survey. AJ, 137(2):3100–3108, February 2009.
[62] Matthew G. Walker, Edward W. Olszewski, and Mario Mateo. Bayesian analysis ofresolved stellar spectra: application to MMT/Hectochelle observations of the Draco
dwarf spheroidal. MNRAS, 448(3):2717–2732, April 2015.
[63] Mario Mateo, Edward W. Olszewski, and Matthew G. Walker. The Velocity Dispersion
Profile of the Remote Dwarf Spheroidal Galaxy Leo I: A Tidal Hit and Run? ApJ,
675(1):201–233, March 2008.
[64] Meghin E. Spencer, Mario Mateo, Matthew G. Walker, and Edward W. Olszewski. A
Multi-epoch Kinematic Study of the Remote Dwarf Spheroidal Galaxy Leo II. ApJ,
836(2):202, February 2017.
[65] Meghin E. Spencer, Mario Mateo, Edward W. Olszewski, Matthew G. Walker, Alan W.
McConnachie, and Evan N. Kirby. The Binary Fraction of Stars in Dwarf Galaxies: The
Cases of Draco and Ursa Minor. AJ, 156(6):257, December 2018.
[66] Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, and Jonathan Goodman. emcee: The MCMC Hammer. PASP, 125(925):306, March 2013.
[67] Joanna Woo, Stéphane Courteau, and Avishai Dekel. Scaling relations and the fundamental line of the local group dwarf galaxies. MNRAS, 390(4):1453–1469, November
2008.
[68] Richard P. Williams, I. K. Baldry, L. S. Kelvin, P. A. James, S. P. Driver, M. Prescott,
S. Brough, M. J. I. Brown, L. J. M. Davies, B. W. Holwerda, J. Liske, P. Norberg, A. J.
Moffett, and A. H. Wright. Galaxy And Mass Assembly (GAMA): detection of lowsurface-brightness galaxies from SDSS data. MNRAS, 463(3):2746–2755, December
2016.
[69] Thomas P Robitaille, Erik J Tollerud, Perry Greenfield, Michael Droettboom, Erik
Bray, Tom Aldcroft, Matt Davis, Adam Ginsburg, Adrian M Price-Whelan, Wolfgang E
Kerzendorf, Alexander Conley, Neil Crighton, Kyle Barbary, Demitri Muna, Henry
Ferguson, Frédéric Grollier, Madhura M Parikh, Prasanth H Nair, Hans M Günther,
Christoph Deil, Julien Woillez, Simon Conseil, Roban Kramer, James E H Turner, Leo
Singer, Ryan Fox, Benjamin A Weaver, Victor Zabalza, Zachary I Edwards, K. Azalee Bostroem, D J Burke, Andrew R Casey, Steven M Crawford, Nadia Dencheva,
Justin Ely, Tim Jenness, Kathleen Labrie, Pey Lian Lim, Francesco Pierfederici, Andrew
Pontzen, Andy Ptak, Brian Refsdal, Mathieu Servillat, and Ole Streicher. Astropy: A
community Python package for astronomy. A&A, 558:A33, 10 2013.
[70] A. M. Price-Whelan, B M Sipőcz, H M Günther, P L Lim, S M Crawford, S Conseil, D L
Shupe, M W Craig, N Dencheva, A Ginsburg, J. T. VanderPlas, L D Bradley, D PérezSuárez, M. de Val-Borro, T L Aldcroft, K L Cruz, T P Robitaille, E J Tollerud, C Ardelean,
T Babej, Y P Bach, M Bachetti, A V Bakanov, S P Bamford, G Barentsen, P Barmby,
A Baumbach, K L Berry, F Biscani, M Boquien, K A Bostroem, L G Bouma, G B Brammer, E M Bray, H Breytenbach, H Buddelmeijer, D J Burke, G Calderone, J. L. Cano
Rodríguez, M Cara, J V M Cardoso, S. Cheedella, Y. Copin, L. Corrales, D Crichton,

28

D. D’Avella, C Deil, É Depagne, J P Dietrich, A Donath, M Droettboom, N Earl, T Erben, S Fabbro, L A Ferreira, T Finethy, R T Fox, L H Garrison, S L J Gibbons, D A Goldstein, R Gommers, J P Greco, P Greenfield, A M Groener, F Grollier, A Hagen, P Hirst,
D Homeier, A J Horton, G Hosseinzadeh, L Hu, J S Hunkeler, Ž Ivezić, A Jain, T Jenness, G Kanarek, S Kendrew, N S Kern, W E Kerzendorf, A Khvalko, J King, D Kirkby,
A M Kulkarni, A Kumar, A Lee, D Lenz, S P Littlefair, Z Ma, D M Macleod, M Mastropietro, C. McCully, S Montagnac, B M Morris, M Mueller, S J Mumford, D Muna,
N A Murphy, S Nelson, G H Nguyen, J P Ninan, M Nöthe, S Ogaz, S Oh, J K Parejko,
N Parley, S Pascual, R. Patil, A A Patil, A L Plunkett, J X Prochaska, T Rastogi, V. Reddy
Janga, J Sabater, P Sakurikar, M Seifert, L E Sherbert, H Sherwood-Taylor, A Y Shih,
J Sick, M T Silbiger, S Singanamalla, L P Singer, P H Sladen, K A Sooley, S Sornarajah,
O Streicher, P Teuben, S W Thomas, G R Tremblay, J E H Turner, V Terrón, M. H. van
Kerkwijk, A. de la Vega, L L Watkins, B A Weaver, J B Whitmore, J Woillez, and V Zabalza. The Astropy Project: Building an Open-science Project and Status of the v2.0
Core Package. AJ, 156(3):123, 8 2018.
[71] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli
Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J.
Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew
Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre
Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi,
Christoph Gohlke, and Travis E. Oliphant. Array programming with NumPy. Nature,585(7825):357–362, September 2020.
[72] The pandas development team. pandas-dev/pandas: Pandas 1.2.4, April 2021.
[73] Wes McKinney. Data Structures for Statistical Computing in Python. In Stéfan van derWalt and Jarrod Millman, editors, Proceedings of the 9th Python in Science Conference,
pages 56 – 61, 2010.

29