A&A 431, 517-521 (2005)
DOI: 10.1051/0004-6361:20041122
P. Kroupa1,2, -
C. Theis1,3 -
C. M. Boily4
1 - Institut für Theoretische Physik und Astrophysik der
Universität Kiel, 24098 Kiel, Germany
2 -
Sternwarte Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
3 -
Institut für Astronomie der Univ. Wien,
Türkenschanzstr. 17, 1180 Vienna, Austria
4 -
Observatoire Astronomique de Strasbourg, 11 rue de
l'Université, 67000 Strasbourg, France
Received 20 April 2004 / Accepted 13 October 2004
Abstract
We show that the shape of the observed distribution of
Milky Way (MW) satellites is inconsistent with their being drawn from
a cosmological sub-structure population with a confidence of
99.5 per cent. Most of the MW satellites therefore cannot be
related to dark-matter dominated satellites.
Key words: Galaxy: evolution - Galaxy: halo - galaxies: dwarf - galaxies: kinematics and dynamics - galaxies: Local Group - Galaxy: formation
Calculations of structure formation within the framework of cold dark
matter (CDM) cosmology show that Milky-Way-type (MW) systems have the
same scaled theoretical distribution of sub-haloes as rich galaxy
clusters, and within 500 kpc they should contain about 500 sub-haloes
with masses
(Moore et al. 1999;
Klypin et al. 1999; Governato et al. 2004). However, only
13 dwarves have been found within a distance of 500 kpc around the MW.
The observed dwarves may only comprise a sub-set of the actually present
CDM sub-structures (Stoehr et al. 2002; Hayashi et al. 2003;
Bullock et al. 2000; Susa & Umemura 2004; Kravtsov et al. 2004).
Such biasing could be the result of complex early baryonic physics
that cannot, at present, be treated theoretically in sufficient
detail, but Kazantzidis et al. (2004) point out that this cannot be
the entire solution.
An additional path to testing predictions of CDM cosmology is to
compare the shape of the observed satellite distribution to the
theoretical shapes (Zaritsky & Gonzalez 1999; Hartwick 2000; Sales &
Lambas 2004). The sub-structures fall inwards from filaments that are
spatially thicker than the virialised regions of the hosts. However,
within its virialised region, the number distribution of sub-structure
in a theoretical host halo follows that of its dark-matter (DM)
distribution. CDM models predict the host DM haloes to be oblate with
flattening increasing with increasing mass and radius (Combes 2002;
Merrifield 2002). The ratio of minor to major axis of the DM density
distribution has the value
for MW-sized haloes
within the virial radius. The intermediate-to-major-axis ratio is
(Bullock 2002). When dissipative baryonic
physics is taken into account the haloes become more axis-symmetric
(larger
)
and more flattened,
within the virial radius. The minor axis is co-linear with the
angular momentum of the baryonic disk (Dubinski 1994). Prolate haloes
do not emerge. The empirical evidence is that the MW dark halo is
somewhat flattened (oblate) with
within
kpc (Olling & Merrifield 2000, 2001; Ibata et al.
2001; Majewski et al. 2003; Martínez-Delgado et al. 2004).
Beyond this distance the shape is likely to be more oblate (Bullock
2002), but invoking continuity shows that the axis ratio
cannot change drastically. The theoretical sub-structure distribution
of MW-type hosts must therefore be essentially isotropic (Ghigna et al.
1998; Zentner & Bullock 2003; Diemand et al. 2004; Kravtsov et al. 2004; Aubert et al. 2004).
If the MW dwarves do indeed constitute the shining fraction of DM sub-structures, then their number-density distribution should be consistent with an isotropic (i.e. spherical) or oblate power-law radial parent distribution. This is assumed to be the case by most researchers, given the relatively small number of satellites. In this paper we show that, despite its smallness, the MW satellite sample is inconsistent with a cosmological sub-structure population. We do this by concentrating on the most elementary facts, namely the positions of the satellites.
Table 1 lists distances and coordinates of the N=16dwarves closest to the MW. Given these data, Galactocentric coordinates
are calculated,
,
with uncertainties
derived from the uncertainties in D.
Table 1:
Dwarf galaxies
within the vicinity of the MW. The first column is a running number
used throughout this text; the parentheses contain the running
number used in Sect. 4 after excluding the SMC and UMi.
D and eD are the distance and its uncertainty, respectively.
l, b are the Galactic longitude and latitude, respectively, as
seen from the Sun and defined such that l=0, b=0 points towards
the Galactic centre which is assumed to lie at a distance
kpc from the Sun, and l increases in anticlockwise
direction. The Galactocentric distance of the dwarf is given by R.
The name of the dwarf is given in the 7th column. The data
are from Mateo (1998, Table 2), except that for the LMC
D and eD are taken from Salaris et al. (2003) and
Clementini et al. (2003), and likewise for the SMC
from Dolphin et al. (2001). The remaining columns
contain the plane-fitting results for the innermost N dwarves
(Sect. 3):
is the largest distance to the
Galactic centre of this sample, and the fitted plane has a root-mean
square height
and a distance to the Galactic centre
.
For comparison, the final column lists the root-mean-square
height
for samples of
theoretical dwarves with an isotropic isothermal radial number
density profile (p=2) and radial cutoff
.
The data are plotted in Fig. 1 after clockwise rotation
by an angle
about the Z-axis,
and likewise for the
uncertainties. The distribution is highly anisotropic and planar. It
is the aim of this contribution to quantify the significance of this
anisotropy. A rotation of Fig. 1 by
shows
the distribution to be approximately disk-like (Fig. 2).
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Figure 1:
The position of the innermost 11 MW satellites
(Table 1) as viewed from a point located at
infinity and
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Figure 2:
As Fig. 1 but viewed from
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A plane can be described by the HESSE form,
,
where
is the normal
vector,
a vector pointing from the origin (the centre of the
MW) to a point in the plane, and
an arbitrary vector from
the origin to the plane. With
and
being the coordinates of the galaxies,
,
becomes
identical to the Hesse form if d(i)=0; d(i) being the distance of
the ith dwarf to the plane.
is
the shortest distance of the plane to the origin. The problem of
finding the plane can thus be reduced to a least-squares linear
regression problem, where the aim is to find the coefficients,
with the condition
,
that minimises
.
To achieve this the method of normal equations
using Gauss-Jordan elimination is employed to solve the set of linear
equations (Press et al. 1992). For each fitted plane the
root-mean square height of the resulting disk distribution is
calculated,
.
Note that the applied minimisation does not include the location of
the Galactic centre as a constraint. Thus, in principle the fitted
plane to a small number of dwarves (
)
could lie far from
the Galactic centre. The weights that do enter the regression are
merely given by the uncertainties in distance. The direction of the
normal vector, or the location of the pole of the plane,
,
follows from
.
As no kinematical information is included the
direction of the pole is ambiguous,
,
or
.
Table 1 lists some results of the fitted plane for a
decreasing number of dwarves. The empirical disk height, ,
is
always much smaller than the theoretical height,
,
for an
isothermal and isotropic model number density distribution centred on
the origin of the MW. The MW dwarves thus appear to be
distributed as a great disk with a ratio of height to radius
0.15.
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Figure 3:
The position on the Galactic sky of the poles of the planes fitted to
the dwarves of Table 1. Plotted are
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The poles of the planes and the orbital poles of the dwarves LMC, SMC,
Draco and UMi agree remarkably well (Fig. 3). This
is surprising because the results are obtained using completely
different methods. The position of the poles of the planes found
here depend only on the spatial distribution of the dwarves. In
contrast, an orbital pole is the direction of the orbital angular
momentum and relies on the direction of the measured proper motion of
the respective object. Sgr is on a polar orbit but has a kinematical
pole (
,
Palma et al. 2002) lying
approximately at a right angle to the great disk
and to the MW disk. On the basis of the weakly bound core of
Sgr which makes it difficult for Sgr to survive the many orbits
implied by its current angular momentum, Zhao (1998) proposed
that it may have been scattered into its present low-pericenter orbit
by an encounter with the LMC/SMC about 2-3 Gyr ago. Sgr contributes
the most deviant cos
value in the sample because it is
closest to the MW centre and thus high above the great disk. Taking
Sgr out of the sample would increase the discrepancy, quantified in
Sect. 4, between the dwarf sample and the hypothesis that
they are the visible cosmological sub-halo population.
The null hypothesis is that the N observed dwarves are drawn from a cosmological population. We therefore need to establish the probability that the observed distribution is drawn from a spherical parent distribution.
The vector pointing from the Galactic centre to the closest point,
P,
on the plane is
,
and
the vector from this point P
to a dwarf is
.
The angle,
,
between the normal
vector and the dwarf as viewed from P
is then given by
.
The
cumulative distribution of cos
about the fitted plane is
calculated for the observed sample using the innermost N dwarves,
and also for
model dwarves distributed according to
the theoretical parent radial power-law distribution which is centred
on and isotropic about the Galactic centre. The KS test quantifies
the confidence that can be placed in the hypothesis that the observed
sample stems from this parent distribution. The results, plotted in
Fig. 4, show that this hypothesis can be rejected with a
confidence of better than 98 per cent, and even 99.6 per cent for
.
This comes about because the real sample is deficient near
the poles of the great disk.
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Figure 4:
The probability,
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Orbital pole analyses have shown that the SMC, UMi and the LMC form a
kinematical family (Palma et al. 2002). Taking these two objects out
of the sample, kinematically-linked dwarves are removed with the
expectation that the remaining dwarves should be more consistent with
an isotropic parent distribution. As the thin curves in
Fig. 4 show this is not the case. Instead, the
probabilities that the N=(9) sample without the SMC and UMi stems
from an isotropic parent distribution is reduced (as compared to the
N=9 sample). This comes about because the two dwarves are relatively
close to the Galactic centre thus adding relatively large angles when they are included.
The disk-like distribution of the dwarves lying near to the MW noted in Fig. 1 is therefore highly significant. The local dwarves do not stem from an isotropic distribution. Their distribution is therefore severely at odds with the sphericity of the MW dark matter halo, and even more at odds with an oblate halo having the same orientation as the MW disk.
Cosmological models can be tested, among other ways, by comparing the
theoretical sub-structure distribution with observed satellite
distributions. The theoretical distribution contains about 500 sub-haloes
within approximately 500 kpc of a MW-type galaxy,
follows an approximately power-law radial distribution with
,
and is essentially isotropic. The well-known MW distribution
contains only a dozen dwarves, is indeed consistent with the
theoretical radial distribution but is highly anisotropic. The
anisotropy is such that the MW dwarves form a disk-like structure with
a root-mean-square height of 10-30 kpc which lies nearly
perpendicularly to the plane of the MW. The pole of this great disk
lies close to the orbital poles of the LMC, the SMC, Draco and Ursa
Minor. The distance of closest approach of the plane to the Galactic
centre,
kpc, is much smaller than the radial extent
of the Galactic disk (
20 kpc) or even the root-mean square
height,
,
of the disk of satellites (
).
This is a strong indication that the sample of dwarves within about
250 kpc is relaxed in the Galactic potential. Their orbits must be
confined within the great disk because the likelihood of obtaining
such a disk-like dwarf distribution given a true underlying isotropic
distribution (that ought to match the sphericity of the MW DM halo) is
less than 0.5 per cent. This result persists even after removing the
kinematically related SMC and UMi from the analysis. A distribution
of polar orbits with arbitrary eccentricities and orientation of
orbital planes is also excluded with the same confidence because it
leads to an isotropic distribution of dwarves. An oblate MW dark
matter halo would yield an even larger discrepancy with the disk of
satellites.
An alternative approach is taken by Hartwick (2000) who argues that
the 10 satellites within 400 kpc (the LMC and SMC are combined into
one satellite) map the MW DM halo shape and form a highly inclined and
highly prolate system with minor/major axis ratio
.
However, the extreme triaxiality derived in this way is
completely inconsistent with the observational and theoretical shapes
of CDM host-haloes and sub-structure distributions
(Sect. 1).
The approach taken here differs by noting the very significant mismatch between (i) the disk-like satellite distribution; (ii) the independent empirical constraints on the shape of the MW dark matter halo; and (iii) the theoretical shapes of CDM host haloes (Sect. 1). In the view presented here, the mismatch between the number and spatial distribution of MW dwarves compared to the theoretical distribution challenges the claim that the MW dwarves are cosmological sub-structures that ought to populate the MW halo.
A more natural and more conservative (by not resorting to exotic physics) explanation for the MW dwarf distribution in a great disk with a ratio of height to radius of 0.1-0.2 would appear to be in terms of a causal connection between most of them. This could be the case if most of the dwarves stem from one initial gas-rich parent satellite on an eccentric near-polar orbit that interacted with the young MW, perhaps a number of times, forming tidal arms semi-periodically as its orbit shrank. The early gas-rich tidal arms may have condensed in regions to tidal dwarf galaxies, as is observed in present-day interacting gas-rich galaxies (e.g. Knierman et al. 2003; Weilbacher et al. 2003). The LMC may be the most massive remnant of this larger satellite, while the lesser dwarves may be its old children (Lynden-Bell 1976). The Magellanic Stream may be just such a newly formed but meagre tidal feature (Kunkel 1979), and the alignment of the disk of satellites with the surrounding matter distribution (Hartwick 2000) may simply result from the gas-rich parent satellite having come in from that direction. The different chemical enrichment and star-formation histories of the various dwarves (e.g. Ikuta & Arimoto 2002; Grebel et al. 2003) may in this case be a result of their different initial masses, which will have been significantly larger than their present-day baryonic masses (Kroupa 1997) and the complex interplay between stellar evolution, tides, gaseous stripping and gas accretion during the orbits within the MW halo, none of which are presently understood in much detail. The simulations of Kroupa have shown that ancient tidal dwarf galaxies may appear similar to some of the observed dSph satellites.
The sub-structure under-abundance problem extends to fossil galaxy groups where early photo-evaporation could not have removed baryons from the sub-structures (D'Onghia & Lake 2004), and a sub-structure overabundance is evident for rich clusters (Diemand et al. 2004). CDM cosmology thus faces a sub-structure challenge on all mass scales.