Foreground effect on the J-factor estimation of ultra-faintdwarf spheroidal galaxies

Journal:Manuscript ID

Monthly Notices of the Royal Astronomical Society
MN-17-2091-MJ

Manuscript type:

Main Journal

Date Submitted by the Author:

17-Jun-2017

Complete List of Authors:

Keywords:

Ichikawa, Koji; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo
Horigome, Shun-ichi; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo
Ishigaki, Miho; The University of Tokyo, Kavli Institute for the Physics andMathematics of the Universe
Matsumoto, Shigeki; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo
Ibe, Masahiro; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo; Institute for Cosmic Ray Research, The
University of Tokyo
Sugai, Hajime; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo
Hayashi, Kohei; Kavli Institute for the Physics and Mathematics of the
Universe, The University of Tokyo
galaxies: dwarf < Galaxies, galaxies: kinematics and dynamics < Galaxies,
gamma-rays: galaxies < Resolved and unresolved sources as a function of
wavelength, instrumentation: spectrographs < Astronomical
instrumentation, methods, and techniques, (cosmology:) dark matter <Cosmology, astroparticle physics < Physical Data and Processes

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MNRAS 000, 1–11 (2017)

Preprint 16 June 2017

Compiled using MNRAS LATEX style file v3.0

Foreground eﬀect on the J-factor estimationof ultra-faint dwarf spheroidal galaxies
Koji Ichikawa,(a) Shun-ichi Horigome(a) Miho N. Ishigaki,(a) Shigeki Matsumoto,(a)Masahiro
Ibe,(a,b) Hajime Sugai,(a) and Kohei Hayashi,(a,c)(a)
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, 277-8583, Japan(b) ICRR, The University of Tokyo, Kashiwa, 277-8582, Japan(c) Kavli Institute for Astronomy and Astrohysics, Peking University, Beijing 100871, China

Accepted XXX. Received YYY; in original form ZZZABSTRACT

Dwarf spheroidal galaxies (dSphs) are promising targets for the gamma-ray dark matter (DM) search. In particular, DM annihilation signal is expected to be strong in someof the recently discovered nearby ultra-faint dSphs, which potentially give stringent
constraints on the O(1) TeV WIMP DM. However, the various non-negligible systematic uncertainties complicate the estimation of the astrophysical factors relevant for
the DM search in these objects. Among them, the eﬀects of foreground stars particularly attract attention because the contamination is unavoidable even for the future
kinematical survey. In this article, we assess the eﬀects of the foreground contamination on the astrophysical J-factor estimation by generating mock samples of stars in
the four ultra-faint dSphs and using a model of future spectrographs. We investigate
various data cuts to optimize the quality of the data and apply a likelihood analysis
which takes member and foreground stellar distributions into account. We show thatthe foreground star contaminations in the signal region (the region of interest) can be
estimated with statistical uncertainty by interpolating the foreground star distribution in the control region where the foreground stars dominate the member stars. Suchregions can be secured at future spectroscopic observations utilizing a multiple object
spectrograph with a large field of view; e.g. the Prime Focus Spectrograph mounted
on Subaru Telescope. The above estimation has several advantages: The data-driven
estimation of the contamination makes the analysis of the astrophysical factor stable against the complicated foreground distribution. Besides, the uncertainties of theastrophysical factor are treated statistically.
Key words: galaxies: dwarf – galaxies: kinematics and dynamics – γ-rays: galaxies– instrumentation: spectrographs – dark matter – astroparticle physics

1

INTRODUCTION

Various astrophysical observations such as the dynamics of galaxy clusters (Zwicky 1933), rotation curvesofspiralgalaxies (Rubin, Thonnard & Ford
1978;
Rubin, Ford & Thonnard 1980), and gravitational lensing (McLaughlin 1999; Lokas & Mamon 2003; Clowe et al.
2006; Bradac et al. 2006), strongly indicate the existenceof dark matter (DM) in the astronomical objects. A recent
global fit of the Cosmic Microwave Background (CMB),
Large Scale Structure (LSS), and Supernovae (SNe) observations (Ade et al. 2016) reveal that quarter of the totalenergy of the universe consists of DM. One of the most
attractive candidates of DM is weakly interacting massive
particle (WIMP), which naturally explains the observed
dark matter density with its annihilation channels into
© 2017 The Authors

lighter standard model particles. Particularly, the WIMP
dark matter with ≲ O(1) TeV has drawn attention in thecontext of the physics beyond the standard model such as
supersymmetry (see e.g. Jungman, Kamionkowski & Griest
1996 also Murayama 2007; Feng 2010).
Gamma-ray indirect detection experiment, which aims
to observe gamma-rays induced by the DM annihilation, hasa strong sensitivity to this O(1) TeV WIMP. Among various astronomical objects, dwarf spheroidal satellite galaxies(dSphs) associated with the Milky Way are the ideal targets due to its small distance (∼ 10 − a few hundred kpc
from the solar system) and dense DM environment withlow astrophysical background. However, recent studies
show that expected signal flux coming from the dSphs
is significantly aﬀected by various uncertainties such as

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2

K. Ichikawa et al.

the statistical procedure (Martinez et al. 2009), DM distribution (Geringer-Sameth et al. 2015; Bonnivard et al.
2015; Hayashi et al. 2016), stellar distribution (Ullio & Valli
2016), unresolved binary stars ( Mateo, Olszewski & Walker
2008; Koch et al. 2007; Minor 2013; Simon & Geha
2007a; Simon et al. 2011a; McConnachie & Cote 2010;
Koposov et al. 2011; Kirby et al. 2013; Simon et al. 2015)
and foreground contamination (Bonnivard et al. 2016;
Ichikawa et al. 2017).
Although future deep spectroscopic surveys would mitigate most of these systematic uncertainties, the foregroundcontamination remains problematic because the fraction of
the foreground would not be suppressed or even become
worse in the future observation. In Ichikawa et al. (2017)(hereafter KI17), we have investigated the eﬀects of the foreground contamination for classical dSphs and found thateven for the case of classical dSphs, in which the foregroundfraction is less than 5 %, the contamination can lead to an
overestimation of the signal flux by a factor of ∼ 3. In KI17,
we resolve this foreground eﬀect by introducing a multicomponent fit in which the distributions of the member stars
and foregrounds are mixed.
The situation is more problematic for ultra-faint dSphs(UFDs). The UFDs were discovered after SDSS II and contains smaller number of the stars inside the system. Although recent kinematical analyses (Bonnivard et al. 2015;
Ackermann et al. 2015; Geringer-Sameth et al. 2015) indicate that the signal flux coming from the UFDs can be muchstronger than that from the classical dSphs, the uncertainties of these signal fluxes are much larger due to the lack of
the knowledge of the kinematics inside the system. In particular, a recent study (Bonnivard et al. 2016) reveals thatthe foreground contamination can significantly aﬀect the estimation of the signal flux by two orders of magnitude at
most. Therefore, precise analysis of the foreground eﬀect for
the UFDs is required and will play an essential role in the
future deeper spectroscopic surveys.
In this paper, we test the foreground eﬀect for the UFDs
by generating realistic stellar mock data and applying the
likelihood analysis developed in KI17. We also compare theresults with those obtained by the other conventional analyses. The organization of this paper is as follows. In Sec. 2, we
review the formula of the gamma-ray signal flux and defines
the so-called J-factor. In Sec. 3, we provide the procedureof our analysis. The results of the fits are given in Sec. 4.
Finally, we summarize our discussion in Sec 5.

2

SIGNAL FLUX AND J-FACTOR

The gamma-ray signal flux of DM annihilation stemmed
from the dSphs can be expressed by the following formula:()  C⟨σv⟩ ∑dNγ  × J(∆Ω) .
b
Φ(E, ∆Ω) = f
2
dE f 
 4πmDMf

(1)

The coeﬃcient C is 1/2 for Majorana and 1/4 for Dirac dark
matter. Dark matter mass is defined by mDM . The productof the total annihilation cross section σ and the relative velocity v is averaged with the velocity distribution function(represented by ⟨. . . ⟩). The branching fraction of the annihilation channel f is denoted by b f , while the diﬀerential

number density of photons from a given final-state f is givenby (dNγ /dE) f .
The factor J after the parenthesis in the right-hand side(so-called J-factor) reflects the amount of the squared DM
density inside the cone with a solid angle ∆Ω:[∫]∫2J(∆Ω) =dΩdl ρ (l, Ω) .
(2)∆Ω

l.o.s.

Here we define the dark matter profile at a distance l andangle Ω by ρ(l, Ω). The integration of l is performed along
the line-of-sight.
As we have discussed in KI17, the dominant uncertaintyof the signal flux comes from the J-factor. This is because
while the parenthesis in Eq. (1) is well controlled by the calculation of particle physics, the estimation of the J-factor
is limited by the number of the kinematical stellar data ofthe dSphs. Although the size of the uncertainty of the Jfactor is still under discussion,1 the error bar can be a few
orders of magnitude larger for the UFDs. To suppress boththe statistical and systematical uncertainties, future deep
spectroscopic observation is mandatory.
Currently, velocity along the line of sight (v) and the
projected distance from the centre of the dSph (R) are used
for the DM profile estimation. This information can be utilized to construct a velocity dispersion curve of the stars in
the dSph. This observed dispersion curve can be obtainedby projecting the 3-dimensional dispersion curve along theline-of-sight. Under the assumption of the spherical symmetry, the projected velocity dispersion at a projected radiusof R can be written by)∫ ∞ (R2 ν∗ (r)σr2 (r)22
σl.o.sdr 1 − βani (r) 2 √
, (3)
(R) =
Σ∗ (R) Rr
1 − R2 /r 2
where r denotes the un-projected distance from the centre of the dSph, and Σ∗ (R) is the projected spatial stellar
distribution obtained by integrating the stellar distributionν∗ (r) along the projected direction. The anisotropy parameter βani is defined by βani = 1 − (σθ2 /σr2 ) where we define the radial, azimuthal, and polar components of the 3dimensional dispersion curve as σr , σθ , and σϕ , respectively,
in a spherical coordinate and take σθ = σϕ for the sphericalsymmetry.
The dispersion curve is related to the gravitational potential (i.e. the dark matter profile) of the dSph by thesecond moment of the Boltzmann equation of the stellar phase-space distribution, which is called Jeans equation (Binney & Tremaine 2008). Under the assumption of
constant βani , the radial velocity dispersion σr (r) can be
expressed as (van der Marel 1994; Mamon & Lokas 2005)( ′ ) 2βani∫ ∞1r
GM(r ′ ) ′σr2 (r) =dr .
(4)ν∗ (r ′ )ν∗ (r) rrr ′2Here G is the gravitational constant, and M(r) is theenclosedmass of the dark matter halo. Note M(r) ≡
∫r
′2 ρ
4πr
(r ′ )dr ′ under the spherical assumption. FromDM
0
Eq. (3) and Eq. (4), we can estimate the DM profile ρDM by
1

This is due to the various biases in the estimation: the statistical procedure, DM halo model, stellar distribution, unresolved
binaries and foreground contamination, as reviewed in KI17.
MNRAS 000, 1–11 (2017)

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Foreground eﬀect on the J-factor estimation
constructing the dispersion curve σl.o.s. (R) from the datasetof observed {v, R}.

3

ANALYSIS

In this section, we introduce the mock-based analysis developed in KI17. In our analysis, we first generate realistic
mock dSph stellar data including foreground stars. We sample this stellar data by accounting for a spectroscopic capability, which provides realistic mock samples of a future
observation. We next attempt to decrease the foregroundfraction by imposing a selection rule. In this paper, we consider two approaches: naive cuts and selection by using themembership probability. Finally, we perform the halo profile estimation by using two types of the likelihood functions,which have single and mixed component(s) in their distribution function respectively. In Sec. 4, we will provide theresults of the analyses by three combinations of the selections and fits: naive cut + mixed component fit, membership selection + single component fit, and naive cut + single
component fit. They correspond to the KI17, conventional
and the most naive approaches, respectively.
3.1

Mock dSphs

As models of the mock dSphs, we consider thefour UFDs (Ursa Major II, Coma Berenices, Segue 1, andUrsa Major I), in which the observation suggests abundant DM (Hayashi et al. 2016; Bonnivard et al. 2015;
Ackermann et al. 2015; Geringer-Sameth et al. 2015). We
estimate their dark matter halo profiles by the method ofGeringer-Sameth et al. (2015) based on the data provided
by the kinematical observations (Simon & Geha 2007b;
Simon et al. 2011b) 2 and use the obtained DM profiles forthe inputs of the dSph mocks.
In our analysis, the generalized dark matter halo density
profile (Hernquist 1990; Dehnen 1993; Zhao 1996) is adopted
as the input dark matter profile for the mock data and fit of
the likelihood analysis:ρDM (r) = ρs (r/rs )−γ (1 + (r/rs )α )−(β−γ)/α ,(5)where r denotes the (un-projected) distance from the centre of the dSphs, and parameters ρs, rs represent the typicaldensity and scale of the halo respectively, while parametersα, β, γ determine the shape of the halo density profile. We
also assume Plummer profile (Plummer 1911) for the member stellar distribution:ν∗ (r) = (3/4πre3 ) (1 + (r/re )2 )−5/2 .

(6)Here re denotes the projected half-light radius of the dSph
∫and we normalize the stellar distribution ν∗ (r) to satisfy
4πr 2 ν∗ (r)dr = 1. The input parameters are shown in Table 1.3
2

The data of Simon & Geha (2007b) was kindly provided by
Josh Simon (private communication).
3
In order to construct the kinematical data by using themethod of Cuddeford (1991) consistently, we set the range of theanisotropy βani to be βani < 0. See, e.g., Ciotti & Morganti (2010)
for the limitation of the halo parameters in the analytical solutionof the Jeans equation.
MNRAS 000, 1–11 (2017)3

The mock stellar data of each dSph is constructed by assigning the colour, chemical abundance, andkinematical information. Synthetic colour-magnitude diagrams are generated by utilizing the PARSEC stellar
isochrones (Bressan et al. 2012) to represent observed properties of each dSph. In detail, we first randomly draw astellar initial mass from the Salpeter initial mass function. For that mock star, the age is drawn from an uniform distribution in the range 1010.10 -1010.12 years, motivated by the fact that the UFDs analysed in this work havebeen reported to be dominated by an old stellar population (de Jong et al. 2008). Similarly, the value of metallicity ([Fe/H]) is drawn from a Gaussian distribution with themean and dispersion approximately consistent with thoseestimated by Kirby et al. (2011) and Norris et al. (2010).
Based on a theoretical isochrone for the given age and [Fe/H]
values obtained above, the absolute magnitude, colour andsurface gravity corresponding to the stellar initial mass are
assigned. The apparent magnitude and observed colour arethen calculated by adopting the distance modulus from
McConnachie (2012b) and adding typical photometric errors as well as the Galactic extinction. At this point, the staris discarded if it is fainter than the i-band limiting magnitude of 22.5. The mock stars are repeatedly generated until the number of member stars brighter than the limiting
magnitude estimated by Martin et al. (2008) is reached. An
example of the resulting CMD is shown in Fig 1. To build
50 mock data for each dSph, the whole process is repeated
50 times by adding a Gaussian noise consistent with theuncertainty in the number of member stars estimated byMartin et al. (2008). The position and velocity of each star
are assigned consistently with the input dark matter potential using the method of Cuddeford (1991) with the assumptions of the constant velocity anisotropy and spherical
distribution. The non-member stars belonging to the Milky
Way galaxy are also included, which are generated by the
Besançon model (Robin et al. 2003).
3.2

SpectrographIn our analysis, we adopt the same detector capability
in KI17 (see Table 3 in KI17). The observing parameters are based on the capability of the Prime Focus
Spectrograph (PFS) attached to 8.2 m Subaru telescope.
PFS is the next generation spectrograph of the SuMiRe
project (Takada et al. 2014; Sugai et al. 2015; Tamura et al.
2016) and the science operation is planned to start around2019 − 2020. The key advantages of PFS are its large fieldof-view (∼ 1.38◦ diameter), 2394 fibers, and the wide wavelength coverage (380 − 1260 nm) mounted on the large aperture telescope. One of the main targets is the classicaldSphs (Fornax, Sculptor, Draco, Ursa Minor, and Sextans),
for which line-of-sight velocities of stars are measured witha precision dv of ∼ 3 km/s down to magnitudes deeper thani ∼ 21 covering a wide area well beyond their tidal radii.
The unique capability of PFS has also an advantage in observing ultra-faint dwarf galaxies, increasing the sample sizeby a factor of 2 or more and simultaneously covering thetarget galaxy and the foreground/background Milky Waystars. The latter aspect is crucial in eﬃciently taking theeﬀect of contaminating stars into account as in the analysis
presented later in this paper.

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4

K. Ichikawa et al.
(
Model dSph

d [kpc]re [pc]Ursa Major II
Coma Berenices
Segue 1
Ursa Major I

32
44
36
97

149
64
29
732

log10

ρs
[M⊙ /pc3 ]-0.370-0.283
0.306
0.587

)

(
log10

rs
[pc]

2.62
2.27
1.93
1.97

)

α

β

γ

− log10 (1 − βani )

2.36
2.87
0.973
2.89

3.28
6.79
3.94
8.04

0.0328
0.178
1.15
0.302

-0.975-0.894-0.00155-0.625(log10JInput)[GeV2 /cm5 ]

19.70
18.74
19.66
18.52

Table 1. The input parameters of each dSph. The distances from the earth and projected half-light radii are shown by d andre (McConnachie 2012a). The DM halo and kinematical parameters ρs , rs , α, β, γ, and βani are determined by fitting the stellar
data provided by Simon & Geha (2007b) for Ursa Major II, Coma Berenices, Ursa Major I and Simon et al. (2011b) for Segue 1 under thesame procedure as Geringer-Sameth et al. (2015). The JInput shows the J-factors calculated within an angular radius of 0.5 degree under
the input DM halo parameters and distance.

To take the spectroscopic capabilities into account, wesmear the mock velocity, surface gravity (log g) and metallicity ([Fe/H]) data with widths corresponding to the expected measurement errors, 3 km/s, 0.5 dex and 0.5 dex, respectively and select the stars which locate at r < d sin θ ROI ,reflecting the limitation of the region of interest. Here d denotes the distance of each dSph and θ ROI is the angularradius of the region of interest.
The depth of the survey depends on the exposure time.
We adopt three cases of the upper bound of the magnitude(i max = 21, 21.5 and 22). In the first case, we demonstrate
the current sensitivity reach.4 The second case (i max = 21.5)
is for a deeper survey with an integration time of several
nights. The third case is for an ultimate reach.

3.3

Data selection

Before the likelihood analysis, the foreground contaminationin the mock data can be largely reduced by using the information of its position, velocity, surface gravity, metallicity,and colour-magnitude. We here adopt two approaches to thedata reduction: naive cut approach and more sophisticatedmembership selection.

3.3.1

3.3.2

Membership selection

The latter strategy utilizes the membership probability ofeach star. The membership probability is defined by theprobability to find a member star at a given position, velocity, surface gravity, and, metallicity. We calculate thismembership probability by a conventional approach given
by Walker et al. (2009). In the calculation, the distributionsof the foreground stars are also taken into account. The distributions of the velocity, surface gravity, and metallicity
except for the foreground velocity distribution are modeledby R-independent single Gaussians, while the foreground velocity distribution is fixed without free parameters and thespatial distributions are more generally parametrized. The
detail of this process is given in Walker et al. (2009) and theappendix of KI17. We select stars within 95% confidence
level of the membership probability. We provide the numbers of the member and foreground stars after this selectionin the ‘Membership selection’ column in Table 3. Comparedwith the case of the ‘Naive cut’, much higher purities of thedata is obtained by this procedure, while some fraction of themember star is eliminated. We here stress that this approachassumes the constant velocity dispersion in the membershipprobability assignment. The member stars eliminated in theselection are mostly due to the constant velocity-dispersion
bias and therefore aﬀect the estimation of the J-factors.

Naive cutIn this approach, we impose the cuts of the velocity, surface
gravity, metallicity, and colour-magnitude on the dataset
and optimize them by (roughly) tuning the boundaries of
the cuts by eye. The velocity cut is a ±60 [km/s] range fromeach bulk velocity vdSph . The lower and upper bounds ofthe surface gravity g and metallicity [Fe/H] are given in Table 2 for each dSph, while the region of the colour-magnitude
diagram is shown in Fig. 1. Note that we choose these boundaries to include most of the stars in clumps. Although hardercuts can be imposed to reduce the fraction of the foregroundstars, the cut eliminates scattered member stars and the reconstructed velocity distribution can be distorted and derivea bias of the halo estimation. We provide the numbers of themember and foreground stars after the cuts in the ‘Naive cut’column in Table 3.

3.4

Kinematical fit

In this section, we provide two types of the analysis for the
kinematical fit. The first one is the single component fit, inwhich all the data is regarded as member star. The second
one is the mixed component approach developed in KI17,
in which the member and foreground distribution are simultaneously fitted. For both the fits, we apply the unbinned
likelihood analysis to the halo estimation.
3.4.1

Single Component fit

The single component fit is performed by assuming that thedata used for the fit contains only member stars, which implies that the likelihood function is given by∑−2 ln Ls = −2ln( fMem (vi, Ri )) ,
(7)
i

4

Since the size of the UFDs is smaller than that of the classicaldSphs, the kinematical data provided by the current observations
is deeper than the classical dSphs (i ∼ 19.5).

where fMem (v, R) is the distribution function of the member
stars. The index i runs all the stars in the mock data set. We
assume that the velocity distributions of the member stars
MNRAS 000, 1–11 (2017)

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Foreground eﬀect on the J-factor estimation
dSph

vdSph [km/s]

rmax [pc]log10 (g/[cm/s2 ])lower

log10 (g/[cm/s2 ])upper[Fe/H]lower[Fe/H]upper

Ursa Major II
Coma Berenices
Segue 1
Ursa Major I-116.5
98.1
208.5
-55.3

294
238
139
732

0.2
0.1
0.9
0.0

4.9
4.7
5.1
3.7

−4.5
−4.3
−6.1
−4.9

−1.5
−1.5
−1.2
−0.9

5

Table 2. The bulk velocity, the truncation radius and cut conditions for each dSph. The status of the mock dSphs. The bulk velocity ofUrsa Major II, Coma Berenices, and Ursa Major I is from Simon & Geha (2007a) and Segue 1 from Simon et al. (2011a). The truncation
radii are from Geringer-Sameth et al. (2015). See the text for more details.

Condition

Raw

Naive cutMembership selectiondSph

θROI [degree]

imax [mag]

NMem

NFG

NMem

NFG

NMem

NFG

Ursa Major II

0.65

21
21.5
22

80
150
233

829
988
1149

76
141
214

75
103
132

54
89
131

5
4
4

Coma Berenices

0.65

21
21.5
22

35
58
92

579
743
898

34
55
85

58
85
110

29
44
66

2
2
1

Segue 1

0.65

21
21.5
22

26
46
66

585
704
922

23
40
58

65
86
130

23
41
57

1
1
1

Ursa Major I

0.65

21
21.5
22

42
55
63

680
831
953

37
48
56

32
39
44

26
34
38

1
1
1

Table 3. The averaged numbers of the member (foreground) stars are given by NMem (NFG ). The Raw column shows the numbers of
the stars after the colour-magnitude cut and the cut of the region of interest. The details of the naive cuts and membership selection are
given in the text.

can be approximated by a single Gaussian and hence the
distribution functions can be expressed asfMem (v, R) = 2πRΣ∗ (R) CMem G[v; vMem, σl.o.s (R)] .

(8)

Here G[x; µ, σ] denotes the Gaussian distribution of a variable x with a mean value µ and a standard deviation σ.
We note that the parameter vMem represents the bulk velocity of the dSph and mostly converges to the input bulk
velocity vdSph . The distribution functions are normalized∫r∫v
by CMem to satisfy 0 ROI dR v upper dv fMem (v, R) = 1 where
lower
rROI ≡ d sin θ ROI .

3.4.2

Mixed Component fit

In the mixed component fit, the stellar distribution is considered to be the sum of the foreground and member star
distribution. The likelihood function Lm is defined by introducing the membership fraction parameter s as follows∑
ln(s fMem (vi, Ri ) + (1 − s) fFG (vi, Ri )) , (9)−2 ln Lm = −2iwhere fFG (v, R) is the distribution function of the foregroundstars. We model the foreground distribution function by the
production of the three Gaussians, corresponding to the foreground thin disc, thick disc, and halo components:
fFG (v, R) = 2πR CFG

3
∏
j=1

MNRAS 000, 1–11 (2017)

G[v; vFGj , σFGj ] ,(10)with vFGj , σFGj ( j = 1, 2, 3) being parameters of the distribution. Here we assume that the parameters σFGj are
independent of R in contrast to the dispersion of the member star. The∫ r constant∫ v CFG denotes the normalization factor
to satisfy 0 ROI dR v upper dv fFG (v, R) = 1.
lower
For the sake of the convergence of the mixed component
fit, we constrain the parameters vFGj , and σFGj by using
the data in the control region (i.e., the region in which thenumber of the member stars is negligible). In KI17, we deduce the foreground velocity distribution by using the dataout of the region of the velocity cut and interpolate it to
the signal region. For the UFD case, on the other hand,since the bulk velocities of these dSphs are not as large asthat of classical dSphs, the foreground estimation by using
the control region in the velocity distribution does not efficiently work. Instead, we define the control regions in the
distribution of the spatial position by setting an annulus
centred at each dSph galaxy from the radius of the signal
region to the PFS threshold, θ = 0.65◦ . Here, the radii ofthe signal regions are chosen to be 2re, 4re, 4re , and re for
Ursa Major II, Coma Berenices, Segue 1, and Ursa Major I respectively, based on their half-light radii re .
When we perform a fit to the control region, we take into
account the eﬀect of the thin and thick disc components ofthe foreground stars in addition to the halo component, because the disc components remain after surface gravity andmetallicity cuts in case of UFDs. This contrasts to the case ofclassical dSphs, where the foreground stars mainly belong to
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and standard deviations of vFGj , σFGj , which are defined
as vFG0j , σFG0j , dvFGj , and dσFGj respectively. Finally we
use this information as a prior for vFGj , σFGj by multiply∏ing
j=1,2,3 G[vFGj ; vFG0j , dvFGj ] G[σFGj ; σFG0j , dσFGj ]to the likelihood function L in Eq.(9).

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Fit algorithm

The likelihood function (multiplied by the foreground priors
for the mixed component fit) are searched by performing
the Metropolis-Hastings algorithm (Metropolis et al. 1953;
Hastings 1970) of the Markov Chain Monte Carlo (MCMC)
method. The parameter set of the single component fit consists of the five free parameters of the dark matter halo(ρs, rs, α, β, γ), one velocity anisotropy parameter βani and
one nuisance parameters (vMem ), while the mixed component fit also has the other seven nuisance parameters (s,vFGj , σFGj ). In the MCMC method, the halo parameters
are searched under the flat/log-flat priors within the rangeof −4 < log10 (ρs /[M⊙ /pc3 ]) < 4, −2 < log10 (rs /[kpc]) < 5,
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Using 50 mocks for each case (i max = 21, 21.5, and 22),
we test three types of the J-factor estimation: the methodof KI17 (naive cut + mixed component fit), Conventional
analysis (membership selection + single component fit), andContaminated fit (naive cut + single component fit). We
here stress that in the KI17 approach, the velocity distribution of the foreground is parametrized by the fit and therefore the error bar of the J-factor involves the uncertainty of
the foreground distribution, while we fix the spatial stellar
distributions of member and foreground stars in the likelihood. This contrasts with the Conventional approach inwhich a fixed model of the foreground velocity distribution
and parametrized spatial distributions are used in the selection.

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Figure 1. The colour-magnitude map for each dSph. We impose the colour-magnitude cut by the blue shaded region. The
red (blue) dots show the members (foreground) stars. The stars
on the map are residuals after the cuts of the ROI, velocity, andlog g.

the three foreground components we assume the foreground
distribution can be expressed by a sum of three Gaussianfunctions. We first perform fits by the three Gaussian model
for control region data on which colour-magnitude, i max andROI cuts are imposed, and obtain the best-fitting values andstandard deviations of each Gaussian. Then we perform secondary fits for the control region data with all naive cuts(colour-magnitude, i max , ROI, surface gravity and [Fe/H])
imposed on, using the best-fitting Gaussians achieved in thefirst fit as the priors. Here we obtain the best-fitting values

4
4.1

RESULTS
J-factor and velocity dispersion curves

Fig. 2 shows the results of these three approaches, namely,
the method of KI17, the Conventional analysis, and theContaminated fit by blue, orange, and green bars, respectively. Here we give the averaged median values oflog10 (J/[GeV2 /cm5 ]) for each fit by the dots. The lighter
error bars show the averages of the widths of the 68 % quantiles, while the darker ones show the square roots of the
68 % quantiles and the standard deviations of the medianvalues, written in an additional way to the lighter ones. The
grey dashed lines show the input values. For each dSph,
three bars with the same colours correspond to the case ofi max = 21, 21.5, and 22 with θ ROI = 0.65 respectively, fromthe left. All J-factors are calculated within an angular radius of 0.5 degree (i.e., ∆Ω = 2.4 × 10−4 sr), which is the standard size for the J-factor calculation. We here choose themost conservative radius, given by Geringer-Sameth et al.
(2015).
MNRAS 000, 1–11 (2017)

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Figure 2. The J-factors obtained by the fits are plotted. The blue, orange, and green dots show the J-factor estimations of KI17,
Conventional and Contaminated analysis. The lighter error bars of each point show the average of the 68 % quantile, while the darker
ones show the square root of the 68 % quantiles and the standard deviation of the median values. The grey dashed lines show the input
values. For each dSph, three bars with the same colours correspond to the case of imax = 21, 21.5, and 22 with θROI = 0.65 respectively,from the left.

In the Contaminated analysis (green bars in Fig 2), theoverestimation of the J-factor becomes more than an order ofthe magnitude. This is because the dispersion curve inflates
due to the foreground contamination which mainly locate at
the outer region with a large velocity dispersion (∼ 30 − 40km/s). Since the fraction of the foreground contaminatingstars is more than 50 %, the overestimation is much larger
than that of the classical case.
On the other hand, the J-factors seem to be successfully reproduced by the Conventional approach (orange bars
in Fig 2). However, since this approach assumes a constantvelocity dispersion in its membership calculation, the dispersion curve after the selection is more or less flattened
than the input one, which lead to a small bias to the Jfactor estimation. To elucidate this bias, we pick up mocksamples with noticeable bias eﬀects and show these velocity
dispersion curves in the left column of Fig. 3. We also provide the typical uncertainty in the dispersion curve and its
sample-to-sample scatter for the 50 mocks in the right column of the same figure. The red lines show the median valueof the dispersion curves obtained by the fit of the Conventional approach (averaged by the 50 mocks), while the green
band shows the (averaged) 68 % quantile. The median values of the dispersion curves also fluctuate sample by sample,
reflecting the quality of the sample. We show this fluctuation by the orange shaded regions which are obtained by thesquare root sum of the standard deviation of the median values of the 50 mocks and the 68 % quantiles. The input dispersion curves are also shown by the grey dashed lines. Especially for Ursa Major II, one can see from the figures thatthe constant velocity-dispersion bias non-negligibly flattens
the shape of the curve and derive an underestimation of the
J-factor by a factor of five at most. Meanwhile, since thechanges of the dispersions curves of the other dSphs are not
MNRAS 000, 1–11 (2017)as large as the Ursa Major II case, the flattened biases are
not obvious. We here stress that this bias becomes stronger
for a larger size of the stellar data, as can be seen in thethree orange bars for Ursa Major II in Fig 2.
We also note the results for the Ursa Major I case. Although the number of the stars in Ursa Major I does not
significantly diﬀer from the other dSphs (see Table 3), both
the Contaminated and KI17 approaches cannot determine
the J-factor as precisely as those of the other dSphs. Especially, in the Contaminated approach in spite of the factthat the outer region of the Ursa Major I is more precisely
determined than the other dSphs, the J-factor obtained bythis fit poorly converges. It implies that the relation between
the input DM halo shape and the stellar distribution of the
Ursa Major I realizes a diﬃcult situation for the J-factor estimation. Though it might be because the most of the stars
exist at a distance R over the peak of the input dispersion
curve,5 the condition is not solidly determined and we left
this analysis to future work.
KI17 approach (blue bars in Fig 2) also provides successful J-factor estimations. Interestingly, their error bars
except for the Ursa Major I case are smaller and the medianvalues of the J-factors are closer to the input values thanthose obtained by the Conventional approach. The systematical bias appearing at the Ursa Major II of the Conventional
procedure also vanishes in this approach. These results imply that the KI17 method eﬀectively uses larger statistics
without any bias even facing 50 − 100 % foreground contamination.
As a demonstration of the mixed fit, we choose suitable
5

Although this relation is the same as Segue 1, the fluctuationin the Ursa Major I case can be obvious due to the larger positionof the peak (∼ 100 pc).

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Figure 3. Left panel: The examples of the dispersion curve obtained by the Conventional approach for the case of θROI = 0.65, imax =21.5 for Ursa Major II, Coma Berenices, Segue 1, and Ursa Major I from top to bottom). To demonstrate the flattening bias discussed in
the text, we choose representative examples in the mock samples and show these velocity dispersion curves. The binned dispersions of themock data after the membership selection are shown by the red dots with error bars. The orange line shows the dispersion curve of thebest-fitting. The dashed line is obtained from the input parameter of the dSph dark matter halo. Right panel: The distribution of thedispersion curve obtained by the fit of the Conventional approach. The red lines show the median value of the dispersion curves (averagedby the 50 mocks). The green bands show the (averaged) 68 % quantile. The median value of the dispersion curve also fluctuate sample
by sample, reflecting the quality of the sample. The fluctuations by the quality of the samples are shown by the orange shaded regions,which are obtained by the square root sum of the standard deviation of the median values of the 50 mocks and the 68 % quantiles. The
input dispersion curves are also shown by the grey dashed lines.

examples which reproduces the input dispersion curves welland show them in the left column of Fig. 4. Since the dataconsist of the mixture of the member and foreground stars,
the dispersion curve largely inflates at the outer region by
the foreground stars, while the inner most bins are mostly
dominated by the member stars and give dispersions witha small error bars. Thanks to the foreground prior, the fit
is not too much aﬀected by these outer and inner bins andbalanced to lead a valid estimation.
We also give the distribution of the dispersion curve
obtained by the fit in the right column of the same figure.
Compared with the distribution of the Contaminated approach, the width of the 68 % quantile is larger in the outerregion, while it becomes smaller in the inner region (except
for the Ursa Major I case). The results of the J-factor estima-tion implies that the width in the inner region preferentiallyaﬀects the uncertainties of the J-factors. We also note thatthe median dispersion curve of the Ursa Major II case successfully follows the input curve at R ∼ 250 pc, in contrast
with that of the Contaminated approach.
4.2

Implication to gamma-ray detections

We finally demonstrate the foreground eﬀect to the sensitivity lines of the gamma-ray indirect detection. For thesensitivity lines, we consider the observation of the the nextgeneration telescope, Cherenkov Telescope Array (CTA). We
here assume 50 hours observation of line plus continuum signal from each UFD as a benchmark, following the assumption used by Lefranc et al. (2016). Through the calculation
MNRAS 000, 1–11 (2017)

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Figure 4. Left panels: Examples of the dispersion curve of θROI = 0.65, imax = 21.5 case for the mixed fit (Ursa Major II, Coma Berenices,
Segue 1, Ursa Major I, from top to bottom). The binned dispersions of a mock data after the naive cut are shown by the blue dots witherror bars. The blue lines show the mixed-component dispersion curve obtained by the best-fitting, while the orange lines show thedispersion curves of the member stars of the dSphs, calculated from Eq. (4) by inputting the best-fitting parameters. The dashed orange
lines are obtained by using the mock parameter of the dSph dark matter halo in Table 1 and the foreground prior parameters vFG0 j ,σFG0 j . Right panels: The same figures as the right panels of Fig 3 but obtained by the fit of the KI17 approach.

we assume Wino DM, promising candidate of triplet DM. We
calculate the signal photon flux in Eq. (1) by the branchingratios of Wino DM b f , extracted from Lefranc et al. (2016),
and the continuum spectrum (dNγ /dE) f of each branching, calculated by using PPPC 4 DM ID (Cirelli et al.
(2011)). The signal photon flux and the instrument response functions (IRFs) of CTA are merged to construct
the likelihood for each UFD. Here we assume the obi
served photon number in ith bin Nobs
are equal to estii
mated mean background Nbkg . in order to achieve expectedimean sensitivities. The IRFs and Nbkg
are calculated by
Cherenkov Telescope Array Observatory gGmbH (2017).
The upper panel of Fig.5 shows sensitivity lines of photon cross section ⟨σv⟩γγ achieved by combined likelihood
analysis of 50 hours observation for each UFD. 6 Here weassume the 50 hours observation for four UFDs. The sensitivity line obtained by the Contaminated approach is about
100 times severe than other methods, as expected from Jfactors in Fig. 2. It clearly shows the importance of a careful
estimation of the J-factors, since otherwise the dark mattermodel will be constrained too aggressively. Conventional and
KI17 approaches avoid such a problem and there is no significant diﬀerence between them at the level of the present
observational depth. The diﬀerence will appear when the Jfactors are estimated at more deeper observation, as can be
expected from the J-factors in Fig. 2 . In the lower panel of
Fig.5 we show the improvement of the sensitivity with the
KI17 method by increasing the observation depth. We notepure

sensitivity ⟨σv ⟩γγ
Wino⟨σv ⟩γγ6pure

For the ”pure” photon model where bγγ

MNRAS 000, 1–11 (2017)= 1, for instance, its

Wino )⟨σv ⟩ Wino ,is obtained by (1 + bγWino
/2bγγ
γγ
Z

is the sensitivity of Wino DM in Fig.5. This
where
reinterpretation is verified because the continuum spectrum of
Wino DM barely aﬀect to its sensitivity lines.

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Figure 5. Sensitivity lines of Wino DM annihilating into two
photons ⟨σv ⟩γγ . All coloured lines are achieved by combined
likelihood analysis of 50 hours observation for each UFD. The
black line is the photon cross section of Wino DM extracted by
Lefranc et al. (2016). In particular, Upper panel: Sensitivity
lines at imax = 21.5. Each line assumes the J-factor values reproduced by the KI17, Conventional and Contaminated analysis(blue, orange and green). Lower panel: Sensitivity lines achievedby the KI17 analysis at imax = 21, 21.5, and 22 (blue, purple andred).

tor, using the mock kinematical data of the four representative ultra-faint dwarf spheroidal galaxies. This is because wecannot completely distinguish the foreground stars from thedSph’s member stars even if imposing several data cuts. We
have adopted our developed fitting analysis, KI17, utilizing
the future spectroscopic survey, PFS. Such a multi-object
spectrograph with large field of view enables us to observe
numerous number of stellar spectra required the KI17 analysis.
For comparison, we have performed three types of the Jfactor estimation: the KI17 methods, the Conventional analysis and the Contaminated fit. As the result of the analysis,
the J-factor value estimated by the Contaminated analysis
is up to a few hundred times larger than the input value
and its confidence interval is significantly small, because allstellar data after naive cut are regarded as member stareven including the foreground contamination. On he other
hand, the KI17 and Conventional analysis can reproduce theinput J-factor value within 1σ confidence levels except for
Ursa Major II.
For the case of Ursa Major II, the Conventional approach underestimates the J-factor value with respect to the
input value. This is because the Conventional approach assumes the line-of-sight velocity dispersion profile to be constant over the whole radius, although the observed velocity
dispersion curve of Ursa Major II is non-negligibly flat andflares up at larger radii. It leads the member stars in theouter region to be neglected for the consistency with theassumption. Therefore we stress that the constant velocitydispersion bias may have a large impact on the J-factor estimation.
The likelihood function of the KI17 method includes theinformation of both the foreground stars and the member
ones together with the parameters describing their distribution functions and the properties of the foreground stars areroughly determined by the photometric and spectroscopic
observations of the stars in the control region. It allows thismethod to treat correctly and statistically the eﬀect of theforeground contamination for the observational data. Therefore, our statistical method should become powerful tool for
the J-factor estimate of the MW dSphs in the PFS-era andit is worthwhile to calculate the conservative sensitivity of
WIMP DM.
Acknowledgmentsthat the sensitivity lines become lower as the J-factor values
converge to the inputs due to good convergence propertyof the KI17 method and the improvement of sensitivity becomes gentle around i max = 21.5. From the view point of thethermal Wino dark matter search where its mass is predicted
to be about 3 TeV (Hisano et al. 2007), it will be crucial tochoose the observational depth at around i max = 21.5, as
can be seen in the panel.

We would like to give special thanks to Josh Simon andMarla Geha for giving us the kinematic data of UFD
galaxies. This work is supported in part by the Ministryof Education, Culture, Sports, Science, and Technology(MEXT), Japan, No. 16H01090 (for K. H.), No. 15H05889,
No. 25105011 (for M. I.), No. 16H02176, No. 26104009 (for S.
M.), No. 17H02878 (for S. M. and M. I.), and No. 17K14249(for M. N. I.). The work of K.I. is also supported by the
JSPS Research Fellowships for Young Scientists. Finally,
Kavli IPMU is supported by World Premier International
Research Center Initiative (WPI), MEXT, Japan.

5

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