Tidal evolution of discy dwarf galaxies in the Milky Way potential: the formation of dwarf spheroidals

Abstract

We conduct high-resolution collisionless N-body simulations to investigate the tidal evolution of dwarf galaxies on an eccentric orbit in the Milky Way (MW) potential. The dwarfs originally consist of a low surface brightness stellar disc embedded in a cosmologically motivated dark matter halo. During 10 Gyr of dynamical evolution and after five pericentre passages, the dwarfs suffer substantial mass loss and their stellar component undergoes a major morphological transformation from a disc to a bar and finally to a spheroid. The bar is preserved for most of the time as the angular momentum is transferred outside the galaxy. A dwarf spheroidal (dSph) galaxy is formed via gradual shortening of the bar. This work thus provides a comprehensive quantitative explanation of a potentially crucial morphological transformation mechanism for dwarf galaxies that operates in groups as well as in clusters. We compare three cases with different initial inclinations of the disc and find that the evolution is fastest when the disc is coplanar with the orbit. Despite the strong tidal perturbations and mass loss, the dwarfs remain dark matter dominated. For most of the time, the one-dimensional stellar velocity dispersion, σ, follows the maximum circular velocity, Vmax, and they are both good tracers of the bound mass. Specifically, we find that Mbound∝V3.5max and in agreement with earlier studies based on pure dark matter simulations. The latter relation is based on directly measuring the stellar kinematics of the simulated dwarf, and may thus be reliably used to map the observed stellar velocity dispersions of dSphs to halo circular velocities when addressing the missing satellites problem.

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arXiv:0803.2464v2 [astro-ph] 14 May 2009
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 14 May 2009(MN LATEX style file v2.2)
Tidal evolution of disky dwarf galaxies in the Milky Way
potential: the formation of dwarf spheroidals
Jaros? law Klimentowski,1Ewa L. ? Lokas,1Stelios Kazantzidis,2Lucio Mayer3,4
and Gary A. Mamon5,6
1Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
2Center for Cosmology and Astro-Particle Physics; and Department of Physics; and Department of Astronomy,
The Ohio State University, Physics Research Building, 191 West Woodruff Avenue, Columbus, OH 43210, USA
3Institute for Theoretical Physics, University of Z¨ urich, CH-8057 Z¨ urich, Switzerland
4Institute of Astronomy, Department of Physics, ETH Z¨ urich, Wolfgang-Pauli Strasse, CH-8093 Z¨ urich, Switzerland
5Institut d’Astrophysique de Paris (UMR 7095: CNRS and Universit´ e Pierre & Marie Curie), 98 bis Bd Arago, F-75014 Paris, France
6GEPI (UMR 8111: CNRS and Universit´ e Denis Diderot), Observatoire de Paris, F-92195 Meudon, France
14 May 2009
ABSTRACT
We conduct high-resolution collisionless N-body simulations to investigate the tidal
evolution of dwarf galaxies on an eccentric orbit in the Milky Way (MW) potential.
The dwarfs originally consist of a low surface brightness stellar disk embedded in a
cosmologically motivated dark matter halo. During 10 Gyr of dynamical evolution
and after 5 pericentre passages the dwarfs suffer substantial mass loss and their stellar
component undergoes a major morphological transformation from a disk to a bar and
finally to a spheroid. The bar is preserved for most of the time as the angular mo-
mentum is transferred outside the galaxy. A dwarf spheroidal (dSph) galaxy is formed
via gradual shortening of the bar. This work thus provides a comprehensive quanti-
tative explanation of a potentially crucial morphological transformation mechanism
for dwarf galaxies that operates in groups as well as in clusters. We compare three
cases with different initial inclinations of the disk and find that the evolution is fastest
when the disk is coplanar with the orbit. Despite the strong tidal perturbations and
mass loss the dwarfs remain dark matter dominated. For most of the time the 1D
stellar velocity dispersion, σ, follows the maximum circular velocity, Vmax, and they
are both good tracers of the bound mass. Specifically, we find that Mbound∝ V3.5
and Vmax∼√3σ in agreement with earlier studies based on pure dark matter simula-
tions. The latter relation is based on directly measuring the stellar kinematics of the
simulated dwarf and may thus be reliably used to map the observed stellar velocity
dispersions of dSphs to halo circular velocities when addressing the missing satellites
problem.
max
Key words: galaxies: Local Group – galaxies: dwarf – galaxies: fundamental param-
eters – galaxies: kinematics and dynamics – cosmology: dark matter
1 INTRODUCTION
In the currently favoured cold dark matter (CDM) paradigm
of hierarchical structure formation, structure develops from
the ‘bottom-up’ as small, dense dark matter clumps collapse
first and subsequently undergo a series of mergers that re-
sult in the hierarchical formation of large, massive dark mat-
ter haloes. According to this model, dwarf galaxies consti-
tute the building blocks of larger galaxies and those that
have survived until now are expected to be among the old-
est structures in the Universe. Cosmological N-body sim-
ulations set within the CDM paradigm predict too many
substructures around galaxy-sized systems compared to the
number of dwarf galaxy satellites of the Milky Way (MW)
and M31 (Klypin et al. 1999; Moore et al. 1999), giving rise
to the so-called ‘missing satellites’ problem. It is still unclear
whether this problem lies in the theory or in the inadequacy
of current observations for very faint galaxies (for a review
see Kravtsov, Gnedin & Klypin 2004a; Simon & Geha 2007).
Among the dwarf galaxies of the Local Group (see Ma-
teo 1998 for a review), dwarf spheroidals (dSph) are the
most numerous. Owing to their proximity to the primary
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J. Klimentowski et al.
galaxies it is entirely plausible that dSphs have been affected
by tidal interactions. Their stellar distribution is supported
by velocity dispersion and under the assumption that they
are in dynamical equilibrium, dSphs are often characterized
by very high dark matter contents and mass-to-light ratios.
Though it has been suggested that such high velocity dis-
persions may result from lack of virial equilibrium (Kuhn &
Miller 1989) or even non-Newtonian dynamics like MOND
(Milgrom 1995; ? Lokas 2001), the general consensus is that
dSphs are the most dark matter dominated galaxies in the
Universe (Gilmore et al. 2007).
With increasing accuracy of observational data more
precise studies of dSph dynamics can be performed. Re-
cently, their nearly flat velocity dispersion profiles have been
modelled using the Jeans formalism assuming a compact
stellar component embedded in an extended dark matter
halo (? Lokas 2002; Kleyna et al. 2002). However, there are
still uncertainties concerning the exact form of the dark mat-
ter density distribution. For example Walker et al. (2006)
exclude that dark matter follows light in the Fornax dSph,
while Klimentowski et al. (2007) argue that it is possible
to fit a constant mass-to-light ratio if unbound stars are re-
moved from the kinematic sample. Similar conclusions were
reached by ? Lokas (2009) using much larger data samples for
Fornax, Carina, Sculptor and Sextans dwarfs.
In recent years, two types of numerical studies have
been performed to elucidate the origin and evolution of
dSphs. The first is based on cosmological N-body simula-
tions that follow only the dark matter component. A region
of the size of the Local Group or of a MW-sized halo is se-
lected at z = 0, traced back to the initial conditions and
then resimulated with higher resolution (Klypin et al. 2001;
see also Kravtsov et al. 2004a; Warnick & Knebe 2006; Die-
mand, Kuhlen & Madau 2007; Martinez-Vaquero, Yepes &
Hoffman 2007 for examples of this approach). This strategy
allows to accurately study the formation and evolution of
host haloes and their substructure. However, because the
baryonic component is neglected, it cannot address the in-
ternal structure and kinematics of dwarfs.
The second approach is based on evolving a high-
resolution numerical model of a single dwarf galaxy placed
on a representative orbit around its host. In most cases, the
host galaxy is represented by a static external potential (e.g.
Piatek & Pryor 1995; Johnston, Sigurdsson & Hernquist
1999; Mayer et al. 2001; Hayashi et al. 2003; Kazantzidis,
Magorrian & Moore 2004a; Kazantzidis et al. 2004b; Helmi
2004; Klimentowski et al. 2007; Pe˜ narrubia, Navarro & Mc-
Connachie 2008). With this method one readily isolates the
effects of tidal interactions on the internal structure and
kinematics of the dwarf. However, this approach requires
making assumptions regarding the initial conditions and ne-
glects the effects of dynamical friction, the evolution of the
host galaxy’s potential, or interactions between dwarfs.
In this paper we adopt the second approach. Our initial
conditions are based on the ‘tidal stirring’ model (Mayer et
al. 2001). In particular, the progenitor dwarfs comprise a
low surface brightness stellar disk embedded in a dark mat-
ter halo, and they are placed on an eccentric orbit within a
static potential representing the MW. We aim at elucidat-
ing the dynamical mechanisms whereby disky dwarfs can
be transformed into spheroidal galaxies and how the evo-
lution occurs. In addition, we investigate the relation be-
Figure 1. The orbit of the simulated dwarf galaxy projected on
the plane of the MW disk. The simulation starts at the right-hand
side of the Figure. Numbers indicate the time from the start of
the simulation in Gyr. The circle in the middle shows the position
of the host galaxy.
tween the stellar velocity dispersion, the maximum circular
velocity and the bound mass of the dwarfs as a function of
time. Knowing these relations is crucial for interpreting cor-
rectly the missing satellites problem as they reflect the link
between dark halo properties and observable properties of
dSphs.
The paper is organized as follows. In section 2 we sum-
marize the simulation details. In section 3 we study an ex-
ample of galaxy evolution in terms of the mass loss, mor-
phological transformation and changes in internal kinemat-
ics. Section 4 discusses the dependence of the results on the
initial orientation of the dwarf galaxy disk and section 5 is
devoted to observational consequences of the tidal stirring
scenario. The discussion follows in section 6.
2THE SIMULATIONS
In this section we summarize the most important features of
the simulations used for this study (see Klimentowski et al.
2007 for details). The simulations were designed according to
the ‘tidal stirring’ scenario proposed by Mayer et al. (2001)
which suggests that the progenitors of the majority of dSph
galaxies were rotationally supported low surface brightness
disky dwarfs similar to present-day dwarf irregulars.
The dwarf progenitors consisted of a stellar disk embed-
ded in a dark matter halo. The disk is modelled by N = 106
particles of mass 149.5 M⊙. Its density drops exponentially
with radius in the rotation plane and its vertical structure
is modelled by isothermal sheets. The dark matter halo con-
sists of N = 4 × 106particles of mass 1035.7 M⊙ which are
distributed according to the NFW profile. The concentration
parameter of the NFW halo is c = 15. Beyond the virial ra-
dius the density profile was truncated with an exponential
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Tidal evolution of disky dwarf galaxies
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Figure 2. The modulus of the velocity vector of the dwarf galaxy (solid line, left axis) and its acceleration (dashed line, right axis) as a
function of time. Vertical dotted lines indicate pericentre passages.
function to keep the total mass finite (Kazantzidis et al.
2004a). The total mass of the progenitor was M = 4 × 109
M⊙ (the virial mass is Mvir = 3.7 × 109M⊙) which cor-
responds to the virial velocity of 20 km s−1. The models
include adiabatic contraction of the halo in response to the
baryons (Blumenthal et al. 1986). The peak velocity Vmax is
initially equal to 30 km s−1.
The dwarf galaxies evolve on their orbit around the host
galaxy which is modelled by a static gravitational potential.
This approach neglects several effects like dynamical fric-
tion or evolution of the host galaxy itself. However, for our
setup the timescale of the dynamical friction should signifi-
cantly exceed the orbital timescale (Colpi, Mayer & Gover-
nato 1999), while the host halo should be already in place
by the time the dwarf enters it. Indeed cosmological hydro-
dynamical simulations that model the formation of the MW
suggest that it had already accreted most of its dark matter
and baryonic mass between z = 1 and z = 2, i.e. between
8 and 10 Gyr ago (Governato et al. 2007). For simplicity,
the potential of the host galaxy is thus assumed to have the
present-day properties of the MW. We use a static NFW
halo with a virial mass of Mvir = 1012M⊙. We also add the
potential of a stellar disk with mass MD = 4 × 1010M⊙.
We choose a single, typical cosmological orbit with
apocentre to pericentre ratio of ra/rp ≈ 5 (e.g. Ghigna et
al. 1998). With a fairly small pericentre of 22 kpc the or-
bit should be quite typical among those of subhaloes falling
into the MW halo quite early, around z = 2 (Diemand et al.
2007; Mayer et al. 2007). Our default simulation (described
in Klimentowski et al. 2007) had the disk of the dwarf galaxy
initially oriented perpendicular to the orbital plane. For the
purpose of the present study we ran two additional simu-
lations with the disk lying in the orbital plane and at the
inclination of 45 deg (and all the other simulation parame-
ters, including the orbit, unchanged). In both new cases the
motion of stars in the disk was prograde with respect to the
orbital motion of the dwarf.
The simulations were performed using PKDGRAV, a
multi-stepping, parallel, tree N-body code (Stadel 2001).
The gravitational softening length was 50 pc for stars and
100 pc for dark matter. We used 200 outputs of each simu-
lation saved at equal snapshots of 0.05 Gyr.
3GALAXY EVOLUTION
3.1The orbit
In this section we discuss the results of our default or ref-
erence simulation where the disk of the dwarf was initially
oriented perpendicular to the orbital plane. The dependence
on the initial orientation of the disk is discussed in section
4.
Fig. 1 shows the orbit of the simulated dwarf galaxy dur-
ing its whole evolution projected on the orbital xy plane, the
plane of the stellar disk of the host galaxy. The simulation
started at the right-hand side of the Figure, at the apocen-
tre, 120 kpc away from the host galaxy. Numbers correspond
to the evolution time in Gyr since the start of the simulation.
The simulation ends after 10 Gyr of evolution at the bottom
of the Figure. There are 5 pericentres during the evolution.
Due to the lack of dynamical friction all orbits have very
similar parameters, except for the in-plane precession which
amounts to about 90 deg per orbit. Small changes are due to
the fact that as the dwarf is stripped the matter around it
induces small fluctuations on its orbital dynamics. The peri-
centre distance rpvaries from 21.7 kpc to 23.7 kpc, while the
apocentre ra from 110.9 kpc to 113.6 kpc. The ratio remains
ra/rp ≈ 5 as for typical cosmological haloes (e.g. Ghigna et
al. 1998). Figure 2 shows the total velocity of the galaxy and
its acceleration. All orbit passages are very similar in terms
of these parameters, as expected based on the absence of
dynamical friction.
3.2Mass loss
In order to transform a stellar disk into a spheroidal com-
ponent the galaxy needs to be strongly affected by tides,
and several pericentre passages are required in order for
that to happen (Mayer et al. 2001, 2007). Prolonged mass
loss is unavoidable since each subsequent tidal shock weak-
ens the potential well of the dwarf. The amount of mat-
ter lost depends strongly on the orbital parameters and the
structural properties of the dwarf galaxy (e.g. Hayashi et al.
2003; Kazantzidis et al. 2004a,b; Pe˜ narrubia, McConnachie
& Navarro 2007). As consecutive orbits in the simulation are
very similar, the dwarf is affected by a similar tidal field in
each orbit. Thus, we can study different stages of the evo-
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J. Klimentowski et al.
Figure 3. The evolution of the mass and mass-to-light ratio of the dwarf galaxy in time. In the upper panel we plot the mass of all
bound particles (solid line), dark matter (dashed line) and stars (dotted line). The middle panel shows the relative mass loss. The lower
panel plots the mass-to-light ratio assuming the stellar mass-to-light ratio of 3 solar units. The solid line presents the results for all bound
particles, the dashed line plots the same quantity measured inside a sphere containing 90 percent of bound stars. In all panels vertical
dotted lines indicate pericentre passages.
lution of the dwarf galaxy without the complication of an
orbit decaying because of dynamical friction.
In order to calculate the bound mass we need to de-
fine the bound particle. For simplicity we decided to treat
the dwarf galaxy as an isolated object and define unbound
particles as the ones with velocity greater than the escape
velocity from the gravitational potential of the dwarf galaxy.
In order to speed up the calculations we used the treecode
to estimate gravitational potentials for all particles in all
snapshots.
The upper panel of Fig. 3 shows the mass of the galaxy
as a function of time, while the middle panel of this Figure
shows the relative change of the total mass. Mass loss in-
creases dramatically at the pericentres due to tidal shocking.
For example, during the second pericentre passage in a very
short time three times more matter is lost than during the
rest of the whole second orbit. The first pericentre passage
is even more devastating, stripping more than two-thirds of
initial galaxy mass. Eventually, after 10 Gyr of evolution 99
percent of the initial mass is lost from the dwarf. This is
consistent with analytical models of the initial tidal trun-
cation and subsequent tidal shocks (Taylor & Babul 2001;
Taffoni et al. 2003).
The lower panel of Fig. 3 plots the mass-to-light ratio
calculated assuming the stellar mass-to-light of 3 M⊙/L⊙.
This should be true for the present time, but not necessar-
ily for the whole evolution since the dwarf might undergo
periodic bursts of star formation (Mayer et al. 2001). How-
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Tidal evolution of disky dwarf galaxies
5
Figure 4. Circular velocity profiles of the dwarf galaxy at apoc-
entres. The highest curve is for the first apocentre (beginning of
the simulation), the lowest for the last (end of the simulation).
Dots indicate the maximum values Vmax.
ever, for dwarfs falling in early, 10 Gyr ago as assumed here,
most of the gas will be quickly stripped by ram pressure
aided by the cosmic ionizing background radiation (Mayer
et al. 2007), so that star formation will cease soon after the
dwarf approaches the MW. We present the values obtained
by dividing the total bound mass by the total light as well as
the values measured inside the radius containing 90 percent
of the bound stars. We can see that in both cases the ratio
decreases rapidly in the early stages, as the extended dark
matter halo is disrupted easily while not many stars are lost.
Note however that during the subsequent evolution the stel-
lar mass loss traces the mass loss of dark matter. Starting
from the second pericentre passage the mass-to-light ratio
decreases slowly and relaxes at the value of about 10, re-
maining constant till the end of the simulation.
3.3The circular velocity
Although the bound mass is a real physical parameter it is
more convenient to express the mass in terms of the maxi-
mum circular velocity of the galaxy Vmax, which, as we show
below, is also closely related to the stellar velocity disper-
sion. This is the standard method used in cosmological sim-
ulations (e.g. Moore et al. 1999; Stoehr et al. 2002; Kravtsov
et al. 2004a; Diemand et al. 2007). It is based on the assump-
tion that Vmax changes less than other subhalo parameters
during its evolution, making it a good choice for tracking
subhaloes themselves. Figure 4 shows the profiles of the cir-
cular velocity Vc = [GM(r)/r]1/2of the dwarf at the six
apocentres with dots marking the maximum values Vmax.
The upper panel of Fig. 5 shows the evolution of Vmax in
time and the radius rmax at which this velocity occurs.
It is interesting to note the main difference between
Vmax and the total bound mass. While the bound mass
decreases during the whole evolution, Vmax remains rather
constant between the pericentre passages, decreasing only
shortly after the passage, at the time when the tidal tails
are formed. This suggests that Vmax may be a better mea-
sure of the mass as it is not affected by the formation of
tidal tails farther out. On the other end, Vmax decreases by
almost a factor of 3 during the orbital evolution, similar to
the case of the most heavily stripped dark matter subhaloes
in Kazantzidis et al. (2004b) and Kravtsov et al. (2004a).
This happens despite the presence of the baryons which tend
to moderate the effect of tidal shocks by making the poten-
tial well deeper, especially once bar formation occurs (see
below and Mayer et al. 2007). Hence, for subhaloes that
have fallen in early and completed several orbits with fairly
small pericentre passages, Vmax is expected to evolve signif-
icantly. Therefore although haloes of present-day dSphs are
consistent with having Vmax in the range 15 − 30 km s−1
(Kazantzidis et al. 2004b; Strigari et al. 2007) they could be
much more massive in the past when they entered the MW
halo. Such massive dwarfs were likely not affected by reion-
ization, contrary to some suggestions (e.g. Grebel, Gallagher
& Harbeck 2003).
The left panel of Fig. 6 shows the relation between Vmax
and the total bound mass Mboundduring the entire course of
dynamical evolution. The solid line presents the whole evolu-
tionary path and the dotted one the relation Mbound∝ Vα
where α = 3.5. Remarkably, although during the 10 Gyr of
evolution the dwarf galaxy loses ∼ 99 percent of its mass
and its Vmax decreases by a factor of 3, the galaxy moves
roughly along the Mbound ∝ V3.5
largest (smallest) deviations from this relation occur at or-
bital pericentres (apocentres). Our findings are in agreement
with Hayashi et al. (2003) who found that Mbound∝ V3
numerical simulations of tidal stripping of pure NFW sub-
haloes and Kravtsov et al. (2004a) who reported a value of
α = 3.3 for stripped dark matter subhaloes in cosmological
N-body simulations.
A similar comparison can be made in terms of the stel-
lar mass instead of the total bound mass. The right panel
of Fig. 6 shows the relation between the total luminosity of
the stars and Vmax, L ∝ Vα
using the bound stellar mass and assuming a mass-to-light
ratio of the stellar component of 3 M⊙/L⊙ as before. This
time the power-law relation is followed only in the interme-
diate stages with α = 3, while it flattens in the beginning
and at the end of the simulation. A similar comparison was
done by Kravtsov et al. (2004b) for dark matter haloes in
their N-body simulations. They used SDSS luminosity func-
tion to predict stellar luminosities in the R band for their
haloes. In the range 100 km s−1< Vmax < 200 km s−1their
relation also follows a power-law with α = 3. It would be
difficult to compare exact fits of the relation as one would
need to apply proper bolometric corrections, which differ for
different types of stars.
max,
max line. As expected, the
maxin
max. The luminosity was calculated
3.4Internal kinematics and shape
For all kinematic measurements instead of using all bound
stars we restricted the analysis to stars within the maximum
radii rmax corresponding to the maximum circular velocity
Vmax. In the final output this value corresponds to the radius
r = 1.1 kpc which is well within the main stellar body of
the galaxy. In this way we avoid the contamination by stars
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