Download Masses of Dwarf Satellites of the Milky Way

Masses of Dwarf
Satellites of the
Milky Way
Manoj Kaplinghat
Center for Cosmology
UC Irvine
Collaborators:
Greg Martinez
Quinn Minor
Joe Wolf
James Bullock
Evan Kirby
Marla Geha
Josh Simon
Louie Strigari
Beth Willman
CVnII
Milky Way circa 2008
LeoIV
Name
Year Discovered
LMC
1519
SMC
1519
Sculptor 1937
Fornax
1938
Leo II
1950
Leo I
1950
Ursa Minor 1954
Draco
1954
Carina
1977
Sextans
1990
Sagittarius 1994
Ursa Major I 2005
Willman I 2005
Ursa Major II 2006
Bootes
2006
Canes Venatici I 2006
Canes Venatici II 2006
Coma
2006
Segue I
2006
Leo IV
2006
Hercules 2006
Leo T
2007
Bootes II 2007
LeoIV
2008
UMaI
Sextans
Ursa Minor
BootesI/II
Coma
Draco
Herc
W1
Segue1
UMaII
Milky Way
Sag
LMC
Carina
SMC
Sculptor
Fornax
100,000 light years
J Bullock, M Geha and L Strigari
stellar velocities in dwarfs
Universal Mass Profile for dSphs
5
Observation: Line of sight
velocity of individual stars
Infer: Intrinsic dispersion
Estimate: Gravitational
potential well that results
in this dispersion
assuming equilibrium -mass of dark matter halo
There is a fundamental
degeneracy with the
ve l o c i t y d i s p e r s i o n
anisotropy of stars that
prevents the slope of
the mass profile from
being measured well.
Plots from Walker, Mateo, Olszewski,
Penarrubia, Evans, Gilmore ApJ 2009
Fig. 1.— Projected velocity dispersion profiles for eight bright dSphs, from Magellan/MMFS and MMT/Hectochelle data. Over-plotted are
profiles calculated from isothermal, power-law, NFW and cored halos considered as prospective “universal” dSph halos (Section 5). For each type
of halo we fit only for the anisotropy and normalization. All isothermal, NFW and cored profiles above have normalization Vmax ∼ 10 − 20 km
s−1 —see Table 3. All power-law profiles have normalization M300 ∼ [0.5 − 1.5] × 107 M" .
by α and γ. Thus the parameter Vmax sets the normalization of the mass profile.
3.4. Markov-Chain Monte Carlo Method
In order to evaluate a given halo model, we com-
What can we measure in the satellites
with line of sight velocity measurements?
Answer: Mass within the half-light
radius of stars. An excellent fit is:
M1/2
2
3r1/2 �σLOS
�
=
G
From
likelihood
analysis
r1/2 : 3D half-light radius
M1/2 : total mass within r1/2
No dependence on stellar
velocity anisotropy and this
was derived in Wolf et al 2010
Fit
Plot from Wolf, Martinez, Bullock,
Kaplinghat, Geha, Munoz, Simon, Avedo
MNRAS 2010
Also see Walker, Mateo, Olszewski,
Penarrubia, Evans, Gilmore ApJ 2009
What can we measure in the satellites
with line of sight velocity measurements?
Answer: Mass within the half-light
radius of stars. An excellent fit is:
M1/2
2
3r1/2 �σLOS
�
=
G
From
likelihood
analysis
r1/2 : 3D half-light radius
M1/2 : total mass within r1/2
No dependence on stellar
velocity anisotropy and this
was derived in Wolf et al 2010
The fact that mass within about
twice half-light radius is well
constrained was suggested in
Strigari, Bullock and Kaplinghat,
Astrophys.J.657:L1-L4,2007
Fit
Plot from Wolf, Martinez, Bullock,
Kaplinghat, Geha, Munoz, Simon, Avedo
MNRAS 2010
Also see Walker, Mateo, Olszewski,
Penarrubia, Evans, Gilmore ApJ 2009
Density of dark matter
at half light radius
Segue 1 estimate
1
Dashed line is
Einasto profile
with Vmax ~
18 km/s
0.1
0.01
!-2=0.007 Msun/pc
r-2=1kpc
100
3
1000
Half light radius in parsec
Wolf, Martinez, Bullock, Kaplinghat, Geha, Munoz, Simon, Avedo MNRAS 2010 (except Segue 1)
Also see Walker, Mateo, Olszewski, Penarrubia, Evans, Gilmore ApJ 2009
Density of dark matter
at half light radius
Segue 1 estimate
1
M1/2
Dashed line is
Einasto profile
with Vmax ~
18 km/s
2
3r1/2 �σLOS
�
=
G
0.1
0.01
!-2=0.007 Msun/pc
r-2=1kpc
100
3
1000
Half light radius in parsec
Wolf, Martinez, Bullock, Kaplinghat, Geha, Munoz, Simon, Avedo MNRAS 2010 (except Segue 1)
Also see Walker, Mateo, Olszewski, Penarrubia, Evans, Gilmore ApJ 2009
Density of dark matter
at half light radius
Segue 1 estimate
1
M1/2
Dashed line is
Einasto profile
with Vmax ~
18 km/s
0.1
0.01
!-2=0.007 Msun/pc
r-2=1kpc
100
2
3r1/2 �σLOS
�
=
G
Turn this into an
average density
within the stellar
half light radius
3
1000
Half light radius in parsec
Wolf, Martinez, Bullock, Kaplinghat, Geha, Munoz, Simon, Avedo MNRAS 2010 (except Segue 1)
Also see Walker, Mateo, Olszewski, Penarrubia, Evans, Gilmore ApJ 2009
Density of dark matter
at half light radius
Segue 1 estimate
1
Dashed line is
Einasto profile
with Vmax ~
18 km/s
Density
in solar
masses
per unit0.1
parsec
cube
0.01
M1/2
!-2=0.007 Msun/pc
r-2=1kpc
100
3
2
3r1/2 �σLOS
�
=
G
Turn this into an
average density
within the stellar
half light radius
2
�
�σLOS
ρ1/2 ∝ 2
r1/2
1000
Half light radius in parsec
Wolf, Martinez, Bullock, Kaplinghat, Geha, Munoz, Simon, Avedo MNRAS 2010 (except Segue 1)
Also see Walker, Mateo, Olszewski, Penarrubia, Evans, Gilmore ApJ 2009
Density of dark matter
at half light radius
Segue 1 estimate
1
Dashed line is
Einasto profile
with Vmax ~
18 km/s
Density
in solar
masses
per unit0.1
parsec
cube
0.01
M1/2
1/r
!-2=0.007 Msun/pc
r-2=1kpc
100
3
2
3r1/2 �σLOS
�
=
G
Turn this into an
average density
within the stellar
half light radius
2
�
�σLOS
ρ1/2 ∝ 2
r1/2
1000
Half light radius in parsec
Wolf, Martinez, Bullock, Kaplinghat, Geha, Munoz, Simon, Avedo MNRAS 2010 (except Segue 1)
Also see Walker, Mateo, Olszewski, Penarrubia, Evans, Gilmore ApJ 2009
Another way to see this: A Common Mass
Strigari, Bullock, Kaplinghat, Geha, Simon, Willman 2008
Density consistent with
basic LCDM predictions for
objects that collapse early
(before full reionization)
Another way to see this: A Common Mass
300 pc is a good radius
at which to compare the
ensemble to theory
( w h i c h d o e s n’t ye t
predict half-light radius)
Strigari, Bullock, Kaplinghat, Geha, Simon, Willman 2008
Density consistent with
basic LCDM predictions for
objects that collapse early
(before full reionization)
0.20
Another way to see this: A Common Mass
rmax [ kpc ]
0.15
1.00
0.10
0.10
0.05
1
Bullock,
109 Kaplinghat,
1010 Geha, Simon, Willman 2008
M [M ]
5. Quality of fits to subhalo density profiles, based on three different
ameter models, an NFW profile, a Moore profile and an Einasto
with α = 0.18. The circles show a measure for the mean deviation
Q, for 526 subhaloes in the main halo of the Aq-A-1 simulation.
haloes considered contain between 20 000 and nearly ∼10 million
. The lines in different colours show averages in logarithmic mass
each of the three profiles.
Density consistent with
basic LCDM predictions for
objects that collapse early
esults for Via Lactea II, as recently published by Diemand
2008) where a(before
cut-off offull
5 kmreionization)
s−1 was used. Interestingly,
haloes are substantially more concentrated than those in Via
II for the same lower cut-off. The Via Lactea II subhaloes
ually slightly less concentrated than our subhaloes selected
10 km s−1 . However, the origin of this offset may well lie in
nces in the adopted cosmologies (Macciò, Dutton & van den
2008).
subhalos
10
Vmax [ km s-1 ]
100
Springel et al. 08
10.00
rmax [ kpc ]
8
Strigari,
10
~tidal radius
0.00
107
0.01
300 pc is a good radius
at which to compare Aq-A-5
the
Aq-A-4
ensemble to theory
Aq-A-3
Aq-A-2
( w h i c h d o e s n’t ye
t
Aq-A-1
predict half-light radius)
1.00
Aq-A-5
0.10
Aq-A-4
Aq-A-3
Aq-A-2
Aq-A-1
0.01
1
10
Vmax [ km s-1 ]
100
Maximum rotation speed (km/s)
Figure 26. Relation between rmax and Vmax for main haloes (top) and
subhaloes (bottom) in the Aq-A series of simulations. We compare results
0.20
Another way to see this: A Common Mass
rmax [ kpc ]
0.15
1.00
0.10
0.10
0.05
1
Bullock,
109 Kaplinghat,
1010 Geha, Simon, Willman 2008
M [M ]
5. Quality of fits to subhalo density profiles, based on three different
ameter models, an NFW profile, a Moore profile and an Einasto
with α = 0.18. The circles show a measure for the mean deviation
Q, for 526 subhaloes in the main halo of the Aq-A-1 simulation.
haloes considered contain between 20 000 and nearly ∼10 million
. The lines in different colours show averages in logarithmic mass
each of the three profiles.
Density consistent with
basic LCDM predictions for
objects that collapse early
esults for Via Lactea II, as recently published by Diemand
2008) where a(before
cut-off offull
5 kmreionization)
s−1 was used. Interestingly,
haloes are substantially more concentrated than those in Via
II for the same lower cut-off. The Via Lactea II subhaloes
ually slightly less concentrated than our subhaloes selected
10 km s−1 . However, the origin of this offset may well lie in
nces in the adopted cosmologies (Macciò, Dutton & van den
2008).
subhalos
10
Vmax [ km s-1 ]
100
Springel et al. 08
10.00
rmax [ kpc ]
8
Strigari,
10
~tidal radius
0.00
107
0.01
300 pc is a good radius
at which to compare Aq-A-5
the
Aq-A-4
ensemble to theory
Aq-A-3
Aq-A-2
( w h i c h d o e s n’t ye
t
Aq-A-1
predict half-light radius)
1.00
300 pc
Aq-A-5
0.10
Aq-A-4
Aq-A-3
Aq-A-2
Aq-A-1
0.01
1
10
Vmax [ km s-1 ]
100
Maximum rotation speed (km/s)
Figure 26. Relation between rmax and Vmax for main haloes (top) and
subhaloes (bottom) in the Aq-A series of simulations. We compare results
What does M300 ~ 107 Msun tell you? ! Massive subhalos
VL2
subhalos
Slide from
James Bullock
What does M300 ~ 107 Msun tell you? ! Massive subhalos
VL2
subhalos
Vast majority of subhalos
in VL2 have M300 < 107
Msun!
Slide from
James Bullock
What does M300 ~ 107 Msun tell you? ! Massive subhalos
VL2
subhalos
Relation is steep for
Vmax < 10 km/s
M300 � 2 × 106 M⊙
Vast majority of subhalos
in VL2 have M300 < 107
Msun!
�
Vmax
5km/s
�2
Slide from
James Bullock
What does M300 ~ 107 Msun tell you? ! Massive subhalos
While massive
halos have weak
relation between
M300 and total
mass, we don’t care
about massive
(MW-size) halos!
VL2
subhalos
Relation is steep for
Vmax < 10 km/s
M300 � 2 × 106 M⊙
Vast majority of subhalos
in VL2 have M300 < 107
Msun!
�
Vmax
5km/s
�2
Slide from
James Bullock
What does M300 ~ 107 Msun tell you? ! Massive subhalos
While massive
halos have weak
relation between
M300 and total
mass, we don’t care
about massive
(MW-size) halos!
VL2
subhalos
Strigari plot
Relation is steep for
Vmax < 10 km/s
M300 � 2 × 106 M⊙
Vast majority of subhalos
in VL2 have M300 < 107
Msun!
�
Vmax
5km/s
�2
Slide from
James Bullock
What does M300 ~ 107 Msun tell you? ! Massive subhalos
While massive
halos have weak
relation between
M300 and total
mass, we don’t care
about massive
(MW-size) halos!
VL2
subhalos
Strigari plot
Relation is steep for
Vmax < 10 km/s
M300 � 2 × 106 M⊙
Vast majority of subhalos
in VL2 have M300 < 107
Msun!
�
Vmax
5km/s
�2
Slide from
James Bullock
preliminary remarks: case for dark matter in segue 1
(found in sdss at about 23 kpc from the sun)
Tidal radius without dark matter about the same as the radius that
contains half the light.
At relative velocity of 4 km/s, stars will move apart about 400 pc in the
time the dwarf takes to move about 20 kpc
Existence of extremely metal poor stars and large metallicity spread: not
found in star clusters
About 70 members
Measuring “mass” in Segue 1
with multi-epoch
measurements for
about half
2
3r1/2 �σLOS
�
M (r < r1/2 ) =
G
3 × 38pc × (3.8km/s)2
=
2
0.0043 pc M−1
⊙ (km/s)
!
= 3.8 × 105 M⊙
Intrinsic
dispersion
~3.8 km/s
M⊙
ρ(r < r1/2 ) = 1.7 3
pc
Simon, Geha et al, arxiv: 1007.4198
Martinez, Minor et al, in prep
suitably weighted by the measurement errors and this is
described
further
in section 3 new
(see eqs.
13 and 14).
Segue
1 analysis:
method
to For
eachhandle
star we define
r to be its projected
radius from the
membership
and binaries
center of Segue 1, which is already well constrained. Assuming
theremethod
are only
two stellar
populations,
themethod
Milky
A
fully Bayesian
that extends
the expectation
maximization
of
Walker,
Olszewski,
Sen, &the
Woodroofe,
M. 2009, AJ, 137,
Way
andMateo,
Segue
1 galaxies,
joint likelihood
for a3109
single
data point Di = {v, w, r} is
Stellar populations:
L(Di |M ) = F Lgal (Di |Mgal ) + (1 − F )LMW (Di |MMW ).
(1)
Here, Lgal and LMW are the individual probability distributions of Segue 1 and the Milky Way galaxies parameterized by the set Mgal,MW . The metallicity distribution
of the member and nonmember stars are each modeled
by Gaussians with mean metallicities w̄gal , w̄MW and
widths σw,gal , σw,MW respectively. The likelihood is assumed to be separable in velocity, position and metallicity, so that each individual probability distribution can
now be written as
Lgal,MW (v, w, r) = Lgal,MW (w)Lgal,MW (v|r)Lgal,MW (r)
(1)
suitably
by are
thethe
measurement
errors and this
is
and LMW
individual probability
distriHere,
Lgalweighted
described
section
3 new
(seeWay
eqs.
13 and 14).
For
Segue
1 analysis:
method
to
butions
of further
Segue
1inand
the Milky
galaxies
parameeachhandle
star
to be .its
projected
radius
from the
membership
and
binaries
The
metallicity
distribution
terized
by we
thedefine
set
Mrgal,MW
center
of Segue and
1, which
is already
wellare
constrained.
Asof
the member
nonmember
stars
each modeled
suming
theremethod
are
only
two metallicities
stellar
populations,
themethod
Milky
A
fully Bayesian
that extends
the expectation
maximization
,
w̄
and
by
Gaussians
with
mean
w̄
gal
MW
of
Walker,
Mateo,
Olszewski,
Sen,
&
Woodroofe,
M.
2009,
AJ,
137,
3109
Way and
Segue
galaxies,
the joint likelihood
for a single
, σ1w,MW
respectively.
The likelihood
is aswidths
σw,gal
r} velocity,
is
data point
Dseparable
i = {v, w, in
sumed
to
be
position and metallicStellar populations:
ity,
that
each
individual
probability
distribution
can
|M
)
=
F
L
(D
|M
)
+
(1
−
F
)L
(D
|M
L(Dso
i
gal
i
gal
MW
i
MW ).
now be written as
(1)
Separability:
and
LMW
are the (w)L
individual probability
distriHere, Lgal
(v,
w,
r)
=
L
(v|r)L
L
gal,MW
gal,MW
gal,MW
gal,MW (r)
butions of Segue 1 and the Milky Way galaxies parame(2)
.
The
metallicity
distribution
terized
by
the
set
M
gal,MW
where
of the member and nonmember stars
" are each modeled
#
w̄MW and
by Gaussians with mean
w̄gal
1 metallicities(w
− ,w̄gal,MW
)2
L
exp The
− likelihood
, σ!
is as- .
widths
gal,MWσ(w)
w,gal=
w,MW respectively.
2
2σand
2 in velocity, position
w,gal,MW
2πσw,gal,MW
sumed to be separable
metallicity, so that each individual probability distribution can
(3)
nowhave
be written
as
We
momentarily
dropped the model parameter notation
M(v,
. w,
The
factor (w)L
in equation
2
has
a
simple
r) last
= Lgal,MW
(v|r)L
Lgal,MW
gal,MW
gal,MW (r)
(1)
ensitysuitably
functions
of
relevant
model
parameters
(e.g.
dispersion,
weighted by the measurement errors and this is
LMW are In
thethe
individual
distriHere, Lbias
gal and
e selection
introduced.
classicalprobability
dSphs,
described
further
inand
section
3 new
(seeWay
eqs.
13 and 14).
For
Segue
1
analysis:
method
to
butions
of
Segue
1
the
Milky
galaxies
parameh contain hundreds to thousands of bright member
eachhandle
star
we
define
r to be .its
projected
radius
from
the
membership
and
binaries
The
metallicity
distribution
terized
by
the
set
M
the
selection
bias
introduced.
In
the
classical
dSphs,
the selection function gal,MW
may be difficult to quancenter
of Segue and
1, which
is already
wellare
constrained.
Asof
the
member
nonmember
stars
each
modeled
but
in
the
much
sparser
ultra-faints
it
is
frequently
ich Asuming
contain
hundreds
totwo
thousands
of bright
member
there
are
only
stellar
populations,
themethod
Milky
fully Bayesian
method
that extends
the expectation
maximization
,
w̄
and
by
Gaussians
with
mean
metallicities
w̄
straightforward
to
modelSen,
the&may
spectroscopic
selecgal
MW
rs,
the
selection
function
be
difficult
to
quanof
Walker,
Mateo,
Olszewski,
Woodroofe,
M.
2009,
AJ,
137,
3109
Way and
Segue
1 galaxies,
the joint likelihood
for a single
,
σ
respectively.
The
likelihood
is aswidths
σ
(Willman
et w,gal
al.much
2010;
Simon et
al. 2010). Here,
to
w,MW
y,
but
in
the
sparser
ultra-faints
it
is
frequently
r} velocity,
is
data point
Dseparable
i = {v, w, in
sumed
to
be
positionlikeand metallicspatial
selection
biases,
use the
conditional
Stellar
populations:
ore
straightforward
to we
model
the
spectroscopic
selecity,
so
that
each
individual
probability
distribution
can
dn L(v,
w|r)
=
L(v,
w,
r)/L(r).
From
the
previous
|M
)
=
F
L
(D
|M
)
+
(1
−
F
)L
(D
|M
).
L(D
i
gal
i Simon
gal
MW Here,
i
MW
(Willman
et
al.
2010;
et
al.
2010).
to
now
written as
ssion,
webe
have
(1)
oid spatial
selection
biases,
we
use
the
conditional
likeSeparability:
and
L
are
the
individual
probability
distriHere,
L
gal
MW
(w)L
(v|r)
L(v,
w|r)
=
f
(r)L
ood
L(v,
w|r)
=
L(v,
w,
r)/L(r).
From
the
previous
gal
gal
(v,
w,
r)
=
L
(w)L
(v|r)L
L
gal,MW
gal,MW
gal,MW
gal,MW (r)
butions of Segue 1 and the Milky Way galaxies parame(2)
cussion,
we
have
(w)L
(v|r)
(4)
+
(1
−
f
(r))
L
MW
MW
No
spatial
bias:
.
The
metallicity
distribution
terized
by
the
set
M
gal,MW
where
of isthe
member
and
nonmember
stars
each modeled
e f (r)
the
fraction
of gal
stars
that
dwarf
galaxy
(w)L
(v|r)
L(v,
w|r)
= f (r)L
" are
#
galare
2
,w̄gal,MW
w̄MW and
by position
Gaussians
metallicities(w
w̄gal
at the
r: with mean
1
−
)
(w)L
(v|r)
(4)
+
(1
−
f
(r))
L
MW
MW
!
L
=
exp
−
,
σ
respectively.
The
likelihood
is
as- .
widths
gal,MWσ(w)
w,gal
w,MW
2
2σ
ngal2(r)
3. BAYESIAN
M
he number
density
ofseparable
the 2πσ
dSph
stars
is
modeled
by
a
w,gal,MW
sumed
to
be
in
velocity,
position
and
metallicCan
now
constrain
halfw,gal,MW
.
(5)
f
(r)
=
ere fPlummer
(r) is the
fraction
of stars that are dwarf galaxy
ified
profile
of
the
form
ngal (r)
+ nMW (r)
light
radius independent
ity, so that each
individual
probability
distribution
can
Apart
from con
(3)
rs at the position
r:
of photometry
"−α/2.0
! as
velocity dispersio
now
be
written
2
Wethe
have
the model
no- mot
(r)momentarily
∝ 1 +bias
(r/raffects
(6) parameter
ngalselection
inciple
the Milky
Way
andnary orbital
s )dropped
(r)
nby
galeq.
tation
M
.
The
last
factor
in
equation
2
has
a
simple
distributions
equally,
so
that
5
any
spatial
sepersion
for
(v,
w,
r)
=
L
(w)L
(v|r)L
(r)binary
L
gal,MW
gal,MW
gal,MW
.
(5)
(r) = Plummer
re α = 5gal,MW
for the f
standard
profile. The
num-
L(vi |σ
a joint probability distribution in the measured velocities
vi and vcm , the velocity of the star system’s center-ofmass (which is unknown), and then integrate over vcm .
∝ (1
Segue
1
analysis
essentials:
binaries
The likelihood can be written as
Likelihood for each star assuming it is in Segue 1: where we have
L(vi |σi , ti , M ; σ, µ, B, P)
Binary orbital
! ∞
parametersJ(σ, µ, P)
=
P (vi , vcm |σi , ti , M ; σ, µ, B, P)dvcm
−∞
! ∞
=
P (vi |vcm , σi , ti , M ; B, P)P (vcm |σ, µ)dvcm(10)
−∞
Intrinsic R(vcm , P
The second factor in the integrand is the probabilitydispersion
Since the fac
distribution of the center-of-mass velocity of the stars,
eters, it is usu
which is Gaussian to good approximation:
aging velocities
only as a norm
−(vcm −µ)2 /2σ2
e
it is crucial to
√
P (vcm |σ, µ) =
(11)
tive normalizat
2πσ 2
Note that if a s
The first factor in the integrand of eq. 10 is the probpared to the m
ability of drawing a set of velocity measurements {vi }
N factor will b
given that it has center-of-mass velocity vcm . This probvariations are
ability distribution is determined by two factors, binarity
with binary be
L(vi |σ
= in the
P (vimeasured
|vcm , σi , ti , M
; B, P)P (vcm |σ, µ)dvcm(10)
a joint probability distribution
velocities
vi and vcm , the velocity of the−∞star system’s center-ofmass (which is unknown),The
andsecond
then factor
integrate
vcm . is the probability
in theover
integrand
∝ (1
Segue
1
analysis
essentials:
binaries
distribution
of
the
center-of-mass
velocity
of
the
stars,
The likelihood can be written as
which is Gaussian to good approximation:
Likelihood for each star assuming it is in Segue 1: where we have
−(v −µ) /2σ
2
2
e
L(vi |σi , ti , M ; σ, µ, B, P)
√ Binary orbital (11)
P
(v
|σ,
µ)
=
cm
! ∞
2πσ 2
parametersJ(σ, µ, P)
=
P (vi , vcm |σi , ti , M ; σ, µ, B, P)dvcm
The first factor in the integrand of eq. 10 is the prob−∞
! ∞
ability of drawing a set of velocity measurements {vi }
that
it has(v
center-of-mass
velocity
vcm . This prob=
P (vi |vcm , σi , tgiven
,
M
;
B,
P)P
|σ,
µ)dv
(10)
i
cm
cm
ability distribution is determined by two factors, binarity
−∞
Intrinsic R(vcm , P
cm
and measurement error. It can be written as follows:
The second factor in the integrand is the probabilitydispersion
Since the fac
distribution of the center-of-mass
velocity
of
the
stars,
P (vi |vcm , σi , ti , M ; B, P)
eters, it is usu
2
2
which is Gaussian to good approximation:
n
" e−(vi −vcm ) /2σi
aging velocities
#
= (1 − B)
+
BP
(v
|v
,
σi , ti , M ; P)
b
i
cm
2
only
as a norm
2πσi
i=1
−(vcm
−µ)2 /2σ2
e
it is crucial to
2
−(vcm −#v$)2 /2σ
√
m
P (vcm |σ, µ) =
(11)
e
tive normalizat
#
= (1 − B)N2πσ
(vi , 2σi )
2
2πσm
Note that if a s
%
The first factor in the integrand
the
pared to(12)
the m
+ of
BPeq.
− vis
, M ; P)
cm |σ
i , tiprobb (vi 10
ability of drawing a set of velocity
measurements {vi }
N factor will b
%
where Pbvelocity
(vi − vcm |σ
i , ti , M ; P) is the likelihood in the
. This probgiven that it has center-of-mass
v
variations are
cm
center-of-mass frame of the binary system, with the ve%
ability distribution is determined
bycenter-of-mass
two factors,frame
binarity
binary
=v −
v . be
locity in the
given by vwith
L(vi |σ
= in the
P (vimeasured
|vcm , σi , ti , M
; B, P)P (vcm |σ, µ)dvcm(10)
a joint probability distribution
velocities
vi and vcm , the velocity of the−∞star system’s center-ofmass (which is unknown),The
andsecond
then factor
integrate
vcm . is the probability
in theover
integrand
∝ (1
Segue
1
analysis
essentials:
binaries
distribution
of
the
center-of-mass
velocity
of
the
stars,
The likelihood can be written as
which is Gaussian to good approximation:
Likelihood for each star assuming it is in Segue 1: where we have
−(v −µ) /2σ
2
2
e
L(vi |σi , ti , M ; σ, µ, B, P)
√ Binary orbital (11)
P
(v
|σ,
µ)
=
cm
! ∞
2πσ 2
parametersJ(σ, µ, P)
=
P (vi , vcm |σi , ti , M ; σ, µ, B, P)dvcm
The first factor in the integrand of eq. 10 is the prob−∞
! ∞
ability of drawing a set of velocity measurements {vi }
that
it has(v
center-of-mass
velocity
vcm . This prob=
P (vi |vcm , σi , tgiven
,
M
;
B,
P)P
|σ,
µ)dv
(10)
i
cm
cm
ability distribution is determined by two factors, binarity
−∞
Intrinsic R(vcm , P
cm
and measurement error. It can be written as follows:
The second factor in the integrand is the probabilitydispersion
Since the fac
distribution of the center-of-mass
velocity
of
the
stars,
P (vi |vcm , σi , ti , M ; B, P)
eters, it is usu
2
2
which is Gaussian to good approximation:
n
" e−(vi −vcm ) /2σi
aging velocities
Mass ratio distribution
#
= (1 − B)
+
BP
(v
|v
,
σi , ti , M ; P)
b
i
cm
2
only
as a norm
Ellipticity distribution
2πσi
i=1
−(vcm
−µ)2 /2σ2
e
it is crucial to
Period distribution
2
−(vcm −#v$)2 /2σ
√
m
P (vcm |σ, µ) =
(11)
e
tive normalizat
(Mean period,
#
= (1 − B)N2πσ
(vi , 2σi )
2
2πσm
Note that if a s
Dispersion in period,
%
TheBinary
first factor
in the integrand
the
pared to(12)
the m
+ of
BPeq.
− vis
, M ; P)
fraction)
cm |σ
i , tiprobb (vi 10
ability of drawing a set of velocity
measurements {vi }
N factor will b
%
where Pbvelocity
(vi − vcm |σ
i , ti , M ; P) is the likelihood in the
. This probgiven that it has center-of-mass
v
variations are
cm
center-of-mass frame of the binary system, with the ve%
ability distribution is determined
bycenter-of-mass
two factors,frame
binarity
binary
=v −
v . be
locity in the
given by vwith
Test of binary likelihood code
intrinsic
dispersion
0.4km/s
(a) 0.4 km/s intrinsic dispersion, 10 year mean period
intrinsic
dispersion
3.7km/s
(c) 3.7 km/s intrinsic dispersion, 10 year mean period
intrinsic
dispersion
0.4km/s
(b) 0.4 km/s intrinsic dispersion, 10 year mean period
intrinsic
dispersion
3.7km/s
(d) 3.7 km/s intrinsic dispersion, 10 year mean period
Fig. 3.— Inferred probability distributions of the intrinsic dispersion and mean binary period for simulated Segue 1-like galaxies, using
our method of modeling the binary population (solid) compared to clipping 3σ velocity outliers (dashed). Each simulated galaxy uses the
same number of epochs, dates, velocity errors, and magnitudes as the stars in the actual Segue 1 member sample which consists of 69
stars. We generate the velocities from a Monte Carlo simulation and plot three different random realizations, all of which have a maximum
likelihood velocity dispersion of 4 ± 0.2 km/s after discarding 3σ outliers iteratively. For the intrinsic dispersion, we choose two cases: 0.4
km/s, which is the expected dispersion without dark matter, and 3.7 km/s which is our inferred most probable dispersion of the actual
Measuring dark matter mass in Segue 1:
effect of binary stars
Some part of the
measured velocity of a
star is due to orbital
motion
Repeat measurements at
about 1 year interval for many
stars needed to constrain
binary properties well enough
to estimate dark matter mass
Simon, Geha et al, arXiv: 1007.4198
Martinez, Minor et al, in prep
Measuring dark matter mass in Segue 1:
Measuring the period of binaries
Fig. 4.— (left) Inferred probability density of the velocity dispersion of Segue 1. Comparing the probability density with (solid
black line) and without (dotted blue line) the correction due to
binary motion, we see that correcting for binaries results in a lower
inferred dispersion and gives rise to a tail at low velocities, due
mainly to short-period binaries (section 5). Note that excluding
the star SDSSJ100704.35+160459.4, which is a 6-σ velocity outlier with a substantial membership probability, does not have a
significant impact on the inferred dispersion (dash-dotted green
line) since its possible membership and binarity is treated statistically. Exclusion of the red giants biases the probability distribution
(dashed red line) to higher dispersion values; this is primarily due
to their smaller measurement errors which give them a large relative weight in determining the velocity dispersion despite the small
number of probable members (six RGB stars in total).
effect on the general properties of the dispersion probability distribution—the spread, ≈ 4 km/sec peak, and
low velocity tail features are largely unaffected. This
is partly because its membership is treated in a statistical sense, and also because if the star is a member of Segue 1, the implied probability of being a binary is quite high ("pb # = 0.89). By comparison, the
inferred maximum-likelihood dispersion using the membership probabilities of Walker et al. (2009) (which is not
corrected for binaries) decreases from 5.5 km/s to 3.9
km/s when SDSSJ100704.35+160459.4 is removed from
the sample (Simon et al. 2010). On the other hand, excluding the giants from the sample does bias the result
Martinez,
MinorThis
et al,
in prepdue to the
to higher
dispersion values.
is primarily
smaller measurement errors in the red giant population
which give them a high relative weight in determining
the velocity dispersion despite their small numbers (six
RGB stars in total).
Fig. 5.— The probability density of the projected radius containing half the stars which are members of Segue 1. Plotted is
the probability density assuming a Plummer model (solid black),
a modified Plummer model (dashed green), and a Sersic model
(dash-dotted red) for the stellar density profile. Regardless of the
assumed stellar density profile, R1/2 is typically constrained to be
30 − 50 pc. When the full likelihood is used (dotted blue), R1/2 is
farther constrained to be 28+5
−4 which is in agreement with the best
photometric determination of R1/2 = 29+8
−5 pc (68% confidence
region denoted by vertical dotted lines) (Martin et al. 2008).
Fig. 6.— Probability density of the mean log-period of Segue
1’s binary population (solid curve). For comparison we plot our
fiducial prior on the mean period (dotted curve), which is deter-
sion. The prior on velocity dispersion was chosen to be
uniform since this is the parameter of interest.
After estimating the model parameters M , we can deMeasuring
dark matter
mass
in Segue
1:
rive membership
probabilities
for each
individual
star.
membership
The formula for the probability
of membership for the
i-th
is
Canstar
compute
probability density of membership:
f (ri )Lgal (wi , vi |ri )
.
pi =
f (ri )Lgal (wi , vi |ri ) + (1 − f (ri )) LMW (wi , vi |ri )
(9)
Mean values agree well
Because
infer aetprobability
distribution in the model
with we
the Walker
al 2009
parameters
pi willof also
method. M
The, power
the follow a probability distribution.present
Thus, here,
we will
method
is inquote the average membership
probability
#pi $.the model
expanding
space, discussing priors
and disentangling binaries
in the tail from members.
hood of
principl
the intr
it uses a
ties of t
in a con
In or
spheroid
Bayesia
binary s
and foc
dynami
hood of
ation 5; this is given by R = NM W . We therefore
ngalsion.
(0) The prior on velocity
ing
more
tha
in
terms
of
two
epochs,
it
can
be
better
dispersion
was
chosen
to
be
principl
therefore
eNR
as. aWe
model
parameter.
MW
al
more than
two epochs by aapproach
likelihood
uniform since this is the ing
parameter
of interest.
the
intr
lecting
binaries, the velocity distribution
of
Segue
chang
After estimating the model
parameters
M ,the
we can
de-locity
approach
also has
advantage
in
that a
it
uses
sumed
beMeasuring
Gaussian
with
dispersion
σ and
mean
distribution
of
Segue
dark
matter
mass
in
Segue
1:hence
rive to
membership
probabilities
for
each
individual
star.
is of
lestp
locity changes to characterize
the binary
ties
ty
µ.
principle
any ofvelocity
distri-for thestars in
spersion
σ and mean
membership
The Although
formula
forinthe
probability
membership
a con
than
if
hence is less affected by contamination
can
be
used,
there
is currently
nomembership:
evidence for
any
velocity
i-th
star
is distriCan
compute
probability
density
of
stars
than
if the velocities wereUnfortuna
used dire
deviations
from Gaussianity
in dSph velocity disntly no evidence
for f (r )LUnfortunately
present
sam
however, this
method
i
gal (wi , vi |ri )
ons.
section dis3, we discuss how this velocity
. several reaso
= velocity
pi In
in
dSph
present
samples
of
ultra-faint
galaxies
li
f
(r
)L
(w
,
v
|r
)
+
(1
−
f
(r
))
L
(w
,
v
|r
)
In
or
i
gal
i
i
i
i
MW
i
i
i
ution
is
modified
by
the
presence
of
binary
stars.
ss how this velocity
giants
insma
Se
several reasons. First, because
of the
(9)
spheroid
values
agree
e velocity
likelihood
of well
Milky Way stars, we use
sence
ofMean
binary
stars.
main
sequen
giants
in
Segue
1,
the
majority
of
the
sam
Because
we
infer
a
probability
distribution
in
the
model
Bayesia
with
the
Walker
et
al
2009
esancon
model
(Robin
et
al.
2003)
together
with
y Way
stars, M
we, use
arethe
consider
parameters
pi willof also
follow
a probability
distribubinary
main
sequence
stars
for
which
meass
method.
The
power
the
propriate
color-magnitude
cuts.
However,
to
allow
2003)
with
tion.together
Thus, here,
we will
quote
the average membership
and
foc
self.
For
th
are
considerable—of
the
same
order
as
present
method
is in
certainties
into
the
Besancon model, we allow the
ts.
However,
allow
probability
#p
dynami
expanding
determined
i $.the model
self.
For
this
reason,
the
threshold
fra
ty
distribution
to
be
shifted
by
a
small
amount
δ
model, space,
we allow
the priors
discussing
stars
with
m
determined
given
the
present
sample
size
retched
by disentangling
a factor δS,binaries
both of which will be wellby
a small
amount
and
tion between
stars
with
multi-epoch
measurements).
S
mined
by
the
data.
the tail
members.
of whichinwill
befrom
wellof the
degen
tion
between
threshold
fraction
and
dispe
refore our set of model parameters is
characterizi
the degeneracy of binary fraction
with o
Model space explored is of
large
meters
however,(e.g
th
α} binary
(7) population
= {R,isσ, µ, w̄, σw , w̄MW , σcharacterizing
w,MW , δ, S, rs , the
stars
than
f
however,
this
degeneracy
is
weaker
for
,
δ,
S,
r
,
α}
(7)
MW
s density of the model parameters M
probability
binary
stars than for red giants, so that
thecorre
un
= {vicorrection
}, and R =of{rMinor
the data
sets W =M
{wi }, Vbinary
i}
model
parameters
factor
etbyal.a (2010)
hood of
ation 5; this is given by R = NM W . We therefore
ngalsion.
(0) The prior on velocity
ing
more
tha
in
terms
of
two
epochs,
it
can
be
better
dispersion
was
chosen
to
be
principl
therefore
eNR
as. aWe
model
parameter.
MW
al
more than
two epochs by aapproach
likelihood
uniform since this is the ing
parameter
of interest.
the
intr
lecting
binaries, the velocity distribution
of
Segue
chang
After estimating the model
parameters
M ,the
we can
de-locity
approach
also has
advantage
in
that a
it
uses
sumed
beMeasuring
Gaussian
with
dispersion
σ and
mean
distribution
of
Segue
dark
matter
mass
in
Segue
1:hence
rive to
membership
probabilities
for
each
individual
star.
is of
lestp
locity changes to characterize
the binary
ties
ty
µ.
principle
any ofvelocity
distri-for thestars in
spersion
σ and mean
membership
The Although
formula
forinthe
probability
membership
a con
than
if
hence is less affected by contamination
can
be
used,
there
is currently
nomembership:
evidence for
any
velocity
i-th
star
is distriCan
compute
probability
density
of
stars
than
if the velocities wereUnfortuna
used dire
deviations
from Gaussianity
in dSph velocity disntly no evidence
for f (r )LUnfortunately
present
sam
however, this
method
i
gal (wi , vi |ri )
ons.
section dis3, we discuss how this velocity
. several reaso
= velocity
pi In
in
dSph
present
samples
of
ultra-faint
galaxies
li
f
(r
)L
(w
,
v
|r
)
+
(1
−
f
(r
))
L
(w
,
v
|r
)
In
or
i
gal
i
i
i
i
MW
i
i
i
ution
is
modified
by
the
presence
of
binary
stars.
ss how this velocity
9 Se
giants
insma
several reasons. First, because
of the
(9)
spheroid
values
agree
e velocity
likelihood
of well
Milky Way stars, we use
sence
ofMean
binary
stars.
main
sequen
giants
in
Segue
1,
the
majority
of
the
sam
Because
we
infer
a
probability
distribution
in
the
model
Bayesia
with
the
Walker
et
al
2009
esancon
model
(Robin
et
al.
2003)
together
with
y Way
stars, M
we, use
arethe
consider
parameters
pi willof also
follow
a probability
distribubinary
main
sequence
stars
for
which
meass
method.
The
power
the
propriate
color-magnitude
cuts.
However,
to
allow
2003)
with
tion.together
Thus, here,
we will
quote
the average membership
and
foc
self.
For
th
are
considerable—of
the
same
order
as
present
method
is in
certainties
into
the
Besancon model, we allow the
ts.
However,
allow
probability
#p
dynami
expanding
determined
i $.the model
self.
For
this
reason,
the
threshold
fra
ty
distribution
to
be
shifted
by
a
small
amount
δ
model, space,
we allow
the priors
discussing
stars
with
m
determined
given
the
present
sample
size
retched
by disentangling
a factor δS,binaries
both of which will be wellby
a small
amount
and
tion between
stars
with
multi-epoch
measurements).
S
mined
by
the
data.
the tail
members.
of whichinwill
befrom
wellof the
degen
tion
between
threshold
fraction
and
dispe
refore our set of model parameters is
characterizi
the degeneracy of binary fraction
with o
Model space explored is of
large
meters
however,(e.g
th
α} binary
(7) population
= {R,isσ, µ, w̄, σw , w̄MW , σcharacterizing
w,MW , δ, S, rs , the
stars
than
f
however,
this
degeneracy
is
weaker
for
,
δ,
S,
r
,
α}
(7)
MW
s density of the model parameters M
probability
binary
than
for red giants, so that
thecorre
un
— (left) Inferred probability density of the velocity stars
disFig. 5.— The probability density of the projected radius conSegue 1.data
Comparing
the probability
density
}, (solid
V to = {v
},halfand
Rwhich
=ofare
{rMinor
the
sets
W =M
{wiwith
taining
the stars
members
of Segue
1. aPlotted
is
icorrection
i}
model
parameters
factor
etbyal.
and without (dotted blue line) the correction duebinary
the probability density assuming a Plummer model
(solid(2010)
black),
Measuring dark matter mass in Segue 1:
further work
More epochs for a star help enormously in
constraining the binary orbit.
Estimate the effect of binaries on measuring
higher order moments like kurtosis. We already
know the contribution is large and this is important
if you are trying to extract the intrinsic kurtosis
(much more than for the attempt here to extract
the dispersion).
Gamma rays from DM annihilation in the
satellites: Fermi constraints
Our previous discussion has been somewhat divorced from CDM priors.
When interpreted in the context of CDM simulations, the estimates of
mass and flux are more constrained.
Segue 1 not included
Fermi/LAT collaboration, Bullock, Kaplinghat, Martinez 2010
Gamma-ray flux from annihilation
in segue 1
From Greg Martinez; Preliminary
Probing the small scale matter power spectrum is
interesting
0
MW
CMB
satellites
Cluster
Galaxy
lensing
Galaxies (OMEGA)
Lya
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
SUSY
Transfer Function (k)
Perturbations are
erased below the
free-streaming
length/damping
length
10
-7
10
-8
10
-9
10
CDM
50% from decay
100% from decay
1 keV sterile neutrino
-10
10
-3
10
-2
10
-1
10
k (h/Mpc)
0
10
1
10
2
10
conclusions
• Detailed Segue 1 analysis leads to the conclusion that it is a
highly dark matter dominated galaxy with an intrinsic dispersion
of about 3.7 (spread of about 1 km/s).
• Estimated central density within 40 pc has a mean value of
about 1 Msun/pc^3 -- the highest measured density in the
dwarfs. Interpreted in the context of LCDM, it should be among
the brightest sources of dark matter annihilation products.