Mass modelling of dwarf spheroidal galaxies: the effect of unbound stars from tidal tails and the Milky Way

Abstract

We study the origin and properties of the population of unbound stars in the kinematic samples of dwarf spheroidal (dSph) galaxies. For this purpose we have run a high-resolution N-body simulation of a two-component dwarf galaxy orbiting in a Milky Way potential. In agreement with the tidal stirring scenario of Mayer et al., the dwarf is placed on a highly eccentric orbit, its initial stellar component is in the form of an exponential disc and it has a NFW-like dark matter (DM) halo. After 10 Gyr of evolution the dwarf produces a spheroidal stellar component and is strongly tidally stripped so that mass follows light and the stars are on almost isotropic orbits. From this final state, we create mock kinematic data sets for 200 stars by observing the dwarf in different directions. We find that when the dwarf is observed along the tidal tails the kinematic samples are strongly contaminated by unbound stars from the tails. We also study another source of possible contamination by adding stars from the Milky Way. We demonstrate that most of the unbound stars can be removed by the method of interloper rejection proposed by den Hartog & Katgert and recently tested on simulated DM haloes. We model the cleaned-up kinematic samples using solutions of the Jeans equation with constant mass-to-light ratio (M/L) and velocity anisotropy parameter. We show that even for such a strongly stripped dwarf the Jeans analysis, when applied to cleaned samples, allows us to reproduce the mass and M/L of the dwarf with accuracy typically better than 25 per cent and almost exactly in the case when the line of sight is perpendicular to the tidal tails. The analysis was applied to the new data for the Fornax dSph galaxy. We show that after careful removal of interlopers the velocity dispersion profile of Fornax can be reproduced by a model in which mass traces light with a M/L of 11 solar units and isotropic orbits. We demonstrate that most of the contamination in the kinematic sample of Fornax probably originates from the Milky Way.

Full text

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arXiv:astro-ph/0611296v2 30 Mar 2007
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 3 February 2008 (MN LATEX style file v2.2)
Mass modelling of dwarf spheroidal galaxies: the effect of
unbound stars from tidal tails and the Milky Way
Jaros? law Klimentowski,1Ewa L. ? Lokas,1Stelios Kazantzidis,2Francisco Prada,3
Lucio Mayer4,5and Gary A. Mamon6,7
1Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
2Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics, Stanford University, P.O. Box 20450, M/S 29,
Stanford, CA 94309, USA
3Instituto de Astrof´ ısica de Andalucia (CSIC), Apartado Correos 3005, E-18080 Granada, Spain
4Institute for Theoretical Physics, University of Z¨ urich, CH-8057 Z¨ urich, Switzerland
5Institute of Astronomy, Department of Physics, ETH Z¨ urich, Wolfgang-Pauli Strasse, CH-8093 Z¨ urich, Switzerland
6Institut d’Astrophysique de Paris (UMR 7095: CNRS and Universit´ e Pierre & Marie Curie), 98 bis Bd Arago, F-75014 Paris, France
7GEPI (UMR 8111: CNRS and Universit´ e Denis Diderot), Observatoire de Paris, F-92195 Meudon, France
3 February 2008
ABSTRACT
We study the origin and properties of the population of unbound stars in the kinematic
samples of dwarf spheroidal galaxies. For this purpose we have run a high resolution N-
body simulation of a two-component dwarf galaxy orbiting in a Milky Way potential.
In agreement with the tidal stirring scenario of Mayer et al., the dwarf is placed on a
highly eccentric orbit, its initial stellar component is in the form of an exponential disk
and it has a NFW-like dark matter halo. After 10 Gyrs of evolution the dwarf produces
a spheroidal stellar component and is strongly tidally stripped so that mass follows
light and the stars are on almost isotropic orbits. From this final state, we create mock
kinematic data sets for 200 stars by observing the dwarf in different directions. We find
that when the dwarf is observed along the tidal tails the kinematic samples are strongly
contaminated by unbound stars from the tails. We also study another source of possible
contamination by adding stars from the Milky Way. We demonstrate that most of
the unbound stars can be removed by the method of interloper rejection proposed
by den Hartog & Katgert and recently tested on simulated dark matter haloes. We
model the cleaned-up kinematic samples using solutions of the Jeans equation with
constant mass-to-light ratio and velocity anisotropy parameter. We show that even for
such a strongly stripped dwarf the Jeans analysis, when applied to cleaned samples,
allows us to reproduce the mass and mass-to-light ratio of the dwarf with accuracy
typically better than 25 percent and almost exactly in the case when the line of sight
is perpendicular to the tidal tails. The analysis was applied to the new data for the
Fornax dSph galaxy. We show that after careful removal of interlopers the velocity
dispersion profile of Fornax can be reproduced by a model in which mass traces light
with a mass-to-light ratio of 11 solar units and isotropic orbits. We demonstrate that
most of the contamination in the kinematic sample of Fornax probably originates from
the Milky Way.
Key words: galaxies: Local Group – galaxies: dwarf – galaxies: clusters: individual:
Fornax – galaxies: fundamental parameters – galaxies: kinematics and dynamics –
cosmology: dark matter
1 INTRODUCTION
Dwarf spheroidal (dSph) galaxies of the Local Group pro-
vide critical tests of theories of structure formation in the
Universe. The number density of dwarfs in the vicinity of
the Milky Way (MW) poses a problem for theories based on
Cold Dark Matter (Klypin et al. 1999; Moore et al. 1999).
In particular, ΛCDM N-body simulations predict a few hun-
dred dwarf galaxies in the Local Group while only a few tens
are observed. Their numbers, distribution, internal dynam-
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2
J. Klimentowski et al.
ics and density profiles have important implications also for
other issues like gravitational lensing and dark matter (DM)
detection experiments.
A number of solutions to the problem have been pro-
posed (for a recent review see Kravtsov, Gnedin & Klypin
2004), but since the degree of discrepancy with the models
depends on precise knowledge of dwarf galaxy masses one of
the important issues is to accurately determine the masses.
It is now generally believed that Galactic dSph galaxies are
dominated by DM, but it is still debated to what extent their
large velocity dispersions are due to their high DM content
and how much they can be contaminated by the presence of
unbound stars.
The unbound stars may originate from the dwarf it-
self due to tidal stripping in the potential of the giant host
galaxy around which the dwarf orbits or from the stellar
populations of the host galaxy. The latter source of contam-
ination can be dealt with by careful photometric analysis.
Recent studies have shown that restricting the target stars
for spectroscopic observations by the colour-magnitude dia-
gram of the dwarf galaxy does not solve the problem (Kleyna
et al. 2002) but additional colour-colour diagrams can help
distinguish between the dwarf galaxy giant star population
and the MW dwarf stars (Majewski et al. 2000a; Mu˜ noz et
al. 2006).
The contamination due to tidally stripped stars however
remains poorly understood although convincing evidence for
the presence of tidal tails exists for some dSph galaxies. Ma-
jewski et al. (2000b) and Mu˜ noz et al. (2006) find a signifi-
cant excess of Carina member stars above the nominal King
radius while Coleman et al. (2005) find excess number den-
sity of stars around Fornax depending on the direction of the
measurements. Mart´ ınez-Delgado et al. (2001) detected the
tidal extension in the Ursa Minor dwarf and G´ omez-Flechoso
& Mart´ ınez-Delgado (2003) modelled the properties of tidal
tails in this galaxy to estimate its mass-to-light ratio. Detec-
tion of tidal tails remains extremely hard though due to very
low surface brightness of the tails and critical dependence
on the assumed background level which is very difficult to
estimate reliably.
Modelling of the dSph kinematics by the Jeans analysis
is still the major tool to determine their mass content. These
are usually performed on stellar samples selected by a sim-
ple cut-off in velocity (with respect to the mean systemic
velocity of the dwarf) or on stars with velocities within 3
standard deviations from the mean (e.g. Wilkinson et al.
2004). In the recent study of the Draco dSph ? Lokas, Mamon
& Prada (2005) using the sample selected by Wilkinson et
al. and modelling both the dispersion and kurtosis of the
line-of-sight velocity distribution showed that no realistic,
consistent solution to the Jeans equations can be found un-
less some stars are rejected from the sample. Along the same
lines, Mu˜ noz et al. (2006) argued that for all of the detected
Carina stars to be bound the dwarf would have to possess an
enormous mass-to-light ratio of a few thousand solar units.
These arguments strongly suggests the presence of unbound
stars in the traditionally selected samples.
The purpose of this work was to study the origin of
unbound stars and test the methods of dealing with them
using N-body simulations. The paper is organized as follows.
In section 2 we describe the simulation details. Section 3
presents the main properties of the simulated dwarf at its
final stage. In section 4 we describe the creation of the mock
kinematic data and apply a method for interloper removal to
data sets contaminated by the tidally stripped stars and the
stellar populations of the MW. Results of fitting the models
to the cleaned data are presented in section 5. In section 6
we apply the developed procedure to the real data for the
Fornax dSph galaxy. The discussion follows in section 7.
2 THE SIMULATION
The ‘tidal stirring’ model originally proposed by Mayer et
al. (2001) constitutes a viable mechanism for producing the
majority of dSphs in the Local Group. These authors per-
formed N-body simulations of the dynamical evolution of
dwarf galaxies in a MW potential and concluded that the
dSph progenitors were rotationally supported low surface
brightness dwarf systems that were accreted by the DM halo
of the dominant spiral galaxy at early times. Strong gravita-
tional interactions with the primary host potential at peri-
centre heat the stellar disks and produce objects whose stel-
lar structure and kinematics resemble those of dSph galaxies.
Very recently, variants of this model have been shown to suc-
cessfully explain properties such as the morphology-density
relation and the extreme mass-to-light ratios inferred for
some of the dwarfs (Mayer et al. 2006, 2007).
In this investigation, we adopt the tidal stirring model of
Mayer et al. (2001) and perform a high-resolution N-body
simulation of a dwarf disk-like system orbiting within the
static host potential of a MW-sized galaxy. Our approach
thus neglects the effects of dynamical friction and the re-
sponse of the primary to the presence of the satellite. How-
ever, this choice is justified because orbital decay times are
expected to be longer than the Hubble time given the dif-
ference in mass between the two systems and the additional
mass loss due to tidal stripping (Mayer et al. 2001; Hayashi
et al. 2003; Kazantzidis et al. 2004).
Live dwarf galaxy models are constructed using the
technique by Hernquist (1993) and consist of an exponential
stellar disk embedded in a spherical and isotropic Navarro et
al. (1997, hereafter NFW) DM halo. The structural proper-
ties of dark haloes and disks are motivated by the currently
favoured concordance ΛCDM (ΩM = 0.3, ΩΛ = 0.7, h = 0.7)
cosmological model (Mo, Mao & White 1998). The density
distribution of the NFW profile is given by
ρs
(r/rs)(1 + r/rs)2,
ρ(r) =
(1)
where ρs is a characteristic inner density and rs denotes the
scale radius of the system. The NFW density law has a cu-
mulative mass profile that diverges at large radii. In order to
keep the total mass finite, we model the halo density profile
beyond the virial radius of the system, rvir, using an expo-
nential cutoff (Kazantzidis et al. 2004). The concentration
parameter c = rvir/rs controls the shape of the halo density
profile.
The stellar disk follows an exponential distribution in
cylindrical radius R and its vertical structure is modelled by
isothermal sheets
Md
4πR2
Rd
ρd(R,z) =
dzdexp
?
−R
?
sech2?z
zd
?
, (2)
where Md, Rd and zd denote the mass, radial scale length,
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Mass modelling of dSph galaxies
3
and vertical scale height of the disk, respectively. In our
modelling, we parametrize the disk mass to be a fraction
md of the halo virial mass, Md = mdMvir, and we specify
the disk vertical scale height in units of the radial disk scale
length.
Once a set of cosmological parameters is adopted, the
virial quantities Mvir and rvir are uniquely determined by
the halo circular velocity at the virial radius, Vvir (Mo et al.
1998). Furthermore, the DM halo carries some net angular
momentum specified by the dimensionless spin parameter,
λ = J/G?
lar momentum and energy, respectively. We follow Springel
& White (1999) and distribute the angular momentum of
the DM halo by setting the halo streaming velocity to be a
fixed fraction of the local total circular velocity. Our models
implicitly assume that the disk forms out of collapsed gas
which started with the same specific angular momentum as
the halo and that such angular momentum was conserved
during infall (Mo et al. 1998). Finally, the DM halo is adia-
batically contracted in response to the growth of the stellar
disk under the assumptions of spherical symmetry, homol-
ogous contraction, circular DM particle orbits, and angular
momentum conservation (Blumenthal et al. 1986). Adiabatic
contraction allows to construct nearly self-consistent disk
galaxy models (see Kazantzidis et al. 2006 for a complete
discussion). For further details on the adopted modelling we
refer the reader to Hernquist (1993) and Kazantzidis et al.
(2006).
The dwarf galaxy halo has a virial velocity of Vvir =
20 kms−1, implying a virial mass and radius of Mvir ≃
3.7×109M⊙ and rvir ≃ 40.2 kpc respectively, and a concen-
tration of c = 15, resulting in a scale radius of rs ≃ 2.7 kpc
and a maximum circular velocity of Vpeak≃ 30 kms−1. The
latter value is in the range of those inferred for some of the
progenitors of dSphs from high resolution N-body simula-
tions (Kazantzidis et al. 2004; Kravtsov et al. 2004). After
entering the host potential, Vpeak will decrease due to mass
loss processes. Moreover, once the scatter in halo concentra-
tions reflecting the different formation epochs is considered,
the adopted value of c is consistent with the typical values
expected at these mass scales at z ∼ 2 (Col´ ın et al. 2004).
We fix the disk mass fraction to md = 0.04 which is typ-
ical of mass models that reproduce dwarf and low surface
brightness galaxies (Jimenez, Verde & Oh 2003). The expo-
nential disk scale length is then fixed by the adopted spin
parameter, λ = 0.08, which is chosen to be larger than the
average value of halo spins of ∼ 0.045 found in cosmologi-
cal N-body simulations (e.g. Vitvitska et al. 2002) for two
reasons. First, modelling of rotation curves of dwarf galax-
ies suggests that they have an average spin larger than the
mean value of the galaxy population as a whole (Jimenez
et al. 2003). Second, using a large spin parameter enables
us to construct a low surface brightness model as required
by the tidal stirring scenario. Finally, we adopt a constant
vertical scale height across the disk equal to zd = 0.1Rd.
The radial disk scale length Rd is uniquely determined for
a given set of parameters Mvir, c, λ, and mdand is equal to
Rd∼ 1.3 kpc. The setup of the stellar disk is complete once
the Toomre parameter, Q, is set (Toomre 1964). The mean
Toomre parameter of the disk was about 1.5.
The external, spherically symmetric static tidal field is
based on the dynamical mass model A1 for the MW pre-
|E|/M5
vir, where J and E are the total halo angu-
Table 1. Parameters of galaxy models.
SystemParameterValue
Dwarf:Vvir
c
λ
md
Mvir
rvir
rs
Vpeak
Rd
zd
Q
NDM
ǫDM
Nd
ǫd
20 kms−1
15
0.031
0.04
3.7 × 109M⊙
40.2 kpc
2.7 kpc
30 kms−1
1.3 kpc
0.13 kpc
1.5
4 × 106
100 pc
106
50 pc
Milky Way:Mvir
c
λ
Md
Rd
zd
Mb
ab
1012M⊙
12
0.031
4 × 1010M⊙
3.5 kpc
0.35 kpc
8 × 109M⊙
700 pc
sented in Klypin, Zhao & Somerville (2002). Specifically,
the DM halo has a virial mass of Mvir = 1012M⊙, a concen-
tration parameter of c = 12, and a dimensionless spin pa-
rameter of λ = 0.031. The halo was adiabatically contracted
to respond to the growth of the stellar disk and bulge. The
mass and thickness of the stellar disk were Md = 0.04Mvir
and zd = 0.1Rd, respectively, and Rd = 3.5 kpc was the
resulting disk scale length. The mass and scale radius of the
bulge are specified by Mb= 0.008Mvir and ab= 0.2Rd.
The orbits of the MW dSphs are currently poorly
constrained observationally. Nevertheless, their current dis-
tances, which give an indication of the apocentre of their
orbits, coupled with studies of the orbital properties of cos-
mological haloes, can be used to constrain the orbital pa-
rameters of dSphs. Using this information we placed the
dwarf galaxy on an orbit bound and coplanar with respect
to the MW disk with the apocentre radius of rapo = 120 kpc
and rapo/rper = 5/1, close to the median ratio of apocen-
tric to pericentric radii found in high resolution cosmological
N-body simulations (Ghigna et al. 1998). We begin the sim-
ulation by randomly orienting the disk of the satellite and
placing it at apocentre, and we integrate the orbit forward
for 10 Gyr. This timescale corresponds to about five orbital
periods (Torb∼ 2 Gyr) in the chosen orbit and represents a
significant fraction of cosmic time.
The simulation was performed using PKDGRAV, a
multi-stepping, parallel, tree N-body code (Stadel 2001). We
sampled the dwarf galaxy with N = 4×106DM particles and
N = 106stellar particles, and employed a gravitational soft-
ening length of 100 and 50 pc, respectively. Numerical and
structural parameters of the static primary and live dwarf
galaxy model are summarized in Table 1.
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J. Klimentowski et al.
Figure 1. Close-up of the stellar component of the simulated
dSph galaxy. Only every tenth star was plotted.
3PROPERTIES OF THE SIMULATED DWARF
In this section we summarize the properties of the dSph
galaxy formed in the simulation. Fig. 1 shows the stellar
component of the dwarf galaxy and its neighbourhood in the
final output of the simulation. The spheroidal component
and the tidal tails are clearly visible. We start by measuring
the density distribution of stars and DM in the remnant and
comparing them with the initial distributions. The centre of
the mass distribution was determined iteratively by calcu-
lating the centre of mass of particles enclosed in a sphere
of a given size, choosing a new smaller sphere around thus
determined centre and repeating the procedure until rea-
sonable convergence is reached. The velocities used in the
following analysis were calculated with respect to the mean
velocity of particles within 0.5 kpc from the dwarf centre.
All the quantities were plotted starting from r = 200 pc, i.e.
twice the softening scale for DM particles so that they are
not affected by resolution.
We consider both the distributions of all particles found
in the vicinity of the dwarf and of particles bound to the
dwarf that can be actually counted as belonging to it. In
order to determine which particles are bound we treat the
dwarf as an isolated object and calculate the escape veloc-
ities in the standard way. The potential energy of a given
particle is calculated by summing up the contribution from
all the other particles in a large box of 20 kpc on a side.
This is a straightforward, although computationally expen-
sive, procedure which needs to be applied in the case of
a non-spherical object. The particles identified as unbound
are then removed from the sample and the calculation is
repeated until no more particles are removed.
The distribution of bound DM particles is shown in
the upper panel of Fig. 2 with open circles. For compari-
son we also plot the distribution of all DM particles found
in the vicinity of the dwarf with open triangles. We can see
that both distributions preserve the cusp of the initial dis-
tribution at small radii: fitting a few inner data points with
Figure 2. Upper panel: the density distribution of DM particles
in the final stage of the dwarf. Open circles show the results for
bound particles, open triangles for all dark particles in the vicinity
of the dwarf. The dashed line is the best fit with formula (3). Mid-
dle panel: the same as the upper panel, but for stellar particles.
The dashed line is the best fit obtained with the S´ ersic profile
(4). Lower panel: mass-to-light density ratios measured for the
bound (open circles) and for all particles (triangles). The dashed
line shows the best-fitting constant value for r < 2.5 kpc. In each
panel the solid line plots the initial profile.
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Mass modelling of dSph galaxies
5
Figure 3. Ratios of the number of bound particles to the total
number of particles for stars (solid line) and DM (dotted line) as
a function of the distance from the dwarf centre.
ρ ∝ rαwe obtain α ≈ −1.2 both for the initial and final DM
distribution in the dwarf in agreement with Kazantzidis et
al. (2004) who found that the cusp is robust against tidal
disruption. At larger radii the final DM profiles steepen
strongly and flatten at around r = 2 − 3 kpc due to the
presence of tidal tails. Although the effect is weaker for the
bound particles, it is clearly visible, i.e. some of the parti-
cles in the tails are bound to the dwarf. Rejecting the last
three data points for the bound particles we fitted to the
distribution an analytic formula proposed by Kazantzidis et
al. (2004)
ρd(r) = Cr−1exp
?
−r
rb
?
,(3)
which was found to fit a density distribution of a stripped
DM halo in a similar simulation but with only DM. The
dashed line shows the fitted density profile with parameters
C = 1.2×107M⊙/kpc2and rb= 0.53 kpc. For comparison
we also show with a solid line the initial NFW-like density
profile of the progenitor.
A similar comparison for the stellar particles is shown
in the middle panel of Fig. 2. Again the symbols indicate
respectively bound stars (open circles) and all stars (open
triangles). The dashed line is the deprojected S´ ersic profile
(S´ ersic 1968; Prugniel & Simien 1997)
ν(r) = ν0
?r
RS
?−p
exp
?
−
?r
RS
?1/m?
(4)
describing the 3D density distribution of the stars, where
the constants are related to the usual surface density distri-
bution
Σ(R) = Σ0exp[−(R/RS)1/m]
by ν0 = Σ0Γ(2m)/{2RSΓ[(3 − p)m]} and p = 1.0 −
0.6097/m+0.05463/m2(see Lima Neto, Gerbal & M´ arquez
1999; ? Lokas 2002). Our best-fitting parameters for the den-
sity profile of bound stars are listed in the first row of Ta-
(5)
Figure 4. Profiles of the anisotropy parameter β for bound stellar
(solid line) and dark particles (dotted line).
ble 3. The solid line in the middle panel of Fig. 2 shows the
spherically averaged exponential profile of the initial stellar
disk.
The density distributions shown in the two upper panels
of Fig. 2 indicate that the dwarf lost a significant part of
both its mass components. Counting the bound particles
in the final stage, we find the final dark component has a
mass of 3.3 × 107M⊙and the stellar one 1.3 × 107M⊙.
These should be compared to the initial masses which were
respectively 4.1 × 109M⊙and 1.5 × 108M⊙. We therefore
find that during the evolution the dwarf lost 99 percent of
its DM and 91 percent of the stellar mass.
In the lower panel of Fig. 2 we plot the mass-to-light
density ratios measured for the bound and for all particles
(again with open circles and triangles respectively). The val-
ues were expressed in solar units assuming a mass-to-light
ratio for the stars of Υ = 3 M⊙/L⊙typical for stellar pop-
ulations of dSph galaxies (Schulz et al. 2002) so that the
distribution of light is l(r) = ν(r)/Υ with ν(r) given by
equation (4). We note that within the main body of the
dwarf (r < 2.5 kpc) the mass-to-light density ratio is al-
most constant with a mean value of 10.3 M⊙/L⊙. Again,
for comparison we also show with a solid line the initial ratio
of mass and light densities.
It is also interesting to look at the ratios of the number
of bound particles to the total number of particles for the
two components as a function of radius which are plotted in
Fig. 3. Note, that although tidal tails start to dominate the
density at about 2.5 kpc from the centre of the galaxy (see
Fig. 1), up to about 5 kpc most of the tidal tail particles
are still bound. The fact that some of the material in the
tails is bound to the satellite was first noticed by Johnston
(1998). The magnitude of this effect will however depend on
the mass of the dwarf and its position on the orbit. The sit-
uation poses a potential difficulty in dynamical modelling:
even though some stars were stripped from the dwarf and
formed the tidal tails, they still have velocities not very dif-
ferent from the dwarf’s mean and cannot be distinguished
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