Issue |
A&A
Volume 511, February 2010
|
|
---|---|---|
Article Number | A89 | |
Number of page(s) | 16 | |
Section | Extragalactic astronomy | |
DOI | https://doi.org/10.1051/0004-6361/200913297 | |
Published online | 17 March 2010 |
A wide-field H I mosaic of Messier 31
II. The disk warp, rotation, and the dark matter halo
E. Corbelli1 - S. Lorenzoni1 - R. Walterbos2 - R. Braun3 - D. Thilker4
1 -
INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5,
50125 Firenze, Italy
2 -
Department of Astronomy, New Mexico State University,
PO Box 30001, MSC 4500, Las Cruces, NM 88003, USA
3 -
CSIRO-ATNF, PO Box 76, Epping, NSW 2121, Australia
4 -
Center for Astrophysical Sciences, Johns Hopkins
University, 3400 North Charles Street, Baltimore, MD
21218, USA
Received 15 September 2009 / Accepted 19 December 2009
Abstract
Aims. We test cosmological models of structure formation
using the rotation curve of the nearest spiral galaxy, M 31,
determined using a recent deep, full-disk 21-cm imaging survey smoothed
to 466 pc resolution.
Methods. We fit a tilted ring model to the HI data from 8
to 37 kpc and establish conclusively the presence of a dark halo
and its density distribution via dynamical analysis of the rotation
curve.
Results. The disk of M 31 warps from 25 kpc outwards
and becomes more inclined with respect to our line of sight. Newtonian
dynamics without a dark matter halo provide a very poor fit to the
rotation curve. In the framework of modified Newtonian dynamic (MOND)
however the 21-cm rotation curve is well fitted by the gravitational
potential traced by the baryonic matter density alone. The inclusion of
a dark matter halo with a density profile as predicted by hierarchical
clustering and structure formation in a CDM
cosmology makes the mass model in newtonian dynamic compatible with the
rotation curve data. The dark halo concentration parameter for the best
fit is C=12 and its total mass is
.
If a dark halo model with a constant-density core is considered, the
core radius has to be larger than 20 kpc in order for the model to
provide a good fit to the data. We extrapolate the best-fit
CDM
and constant-density core mass models to very large galactocentric
radii, comparable to the size of the dark matter halo. A comparison of
the predicted mass with the M 31 mass determined at such large
radii using other dynamical tracers, confirms the validity of our
results. In particular the
dark halo model which best fits the 21-cm data well reproduces the mass
of M 31 traced out to 560 kpc. Our best estimate for the
total mass of M 31 is
,
with 12
baryonic fraction and only 6
of the baryons in the neutral gas phase.
Key words: galaxies: ISM - galaxies: individual M 31 - galaxies: kinematics and dynamics - dark matter - radio lines: galaxies
1 Introduction
Rotation curves of spiral galaxies are fundamental tools to study the
visible mass distributions in galaxies and to infer the properties of
any associated dark matter halos. These can then be used to constrain
cosmological models of galaxy formation and evolution. Great effort
has been devoted in recent years to test theoretical predictions of
cosmological models regarding the detailed structure of dark matter halos via
observations on galactic and sub-galactic scales. Knowledge of the
halo density profile from the center to the outskirts of galaxies is
essential for solving crucial issues at the heart of galaxy formation
theories, including the nature of the dark matter itself. Numerical
simulations of structure formation in the flat cold dark matter
cosmological scenario (hereafter CDM) predict a well defined
radial density profile for the collisionless particles in virialized
structures, the NFW profile (Navarro et al. 1996).
While there
is a general consensus that hierarchical assembly of
CDM
halos yields ``universal'' mass profiles on a large scale
(i.e. independent of mass and cosmology aside from a simple two
parameter scaling), there is still some controversy on the central
density profile, and on the relative scaling parameters. For example
Navarro et al. (2004) proposed the ``Einasto profile'', a
three-parameter formulation, to improve the accuracy of the fits to cuspy
inner density profiles of simulated halos. The two parameters of the
``universal'' NFW density profile are the halo overdensity and the
scale radius, or (in a more useful parameterization) the halo
concentration and its virial mass. For hierarchical structure
formation, small galaxies should show the highest halo concentrations
and massive galaxies the lowest ones (Macciò et al. 2007). Do
the observations confirm these predictions? Dwarf galaxies with
extended rotation curves have often contradicted this theory since
central regions show shallow density cores, i.e. very low dark matter
concentrations
(Rhee et al. 2004; Gentile et al. 2007,2005).
This has given new insights on the nature of dark matter and lead to
discussion on how the halo structure might have been altered by the
galaxy formation process (e.g. Gnedin & Zhao 2002). Recent
hydrodynamical simulations in
framework have shown that
strong outflows in dwarf galaxies inhibit the formation of bulges and
decrease the dark matter density, thus reconciling dwarf galaxies with
theoretical predictions (Governato et al. 2010).
Shallow density cores have often been supported also by high
resolution analysis of rotation curves of spirals and low surface
brightness galaxies (de Blok et al. 2001; Gentile et al. 2004).
However, uncertainties related to the presence of non-circular motion (often
related to the presence of small bars and bulges), observational
uncertainties on the observed velocities, and the possibility of dark matter
compression during the baryonic collapse leave still open the question
on how dark matter is effectively distributed in today's galaxies
(Corbelli 2003; Corbelli & Walterbos 2007; van den Bosch et al. 2000).
Bright galaxies, because of the large fraction of visible to dark
matter, do not offer the possibility to trace dark matter very
accurately in the center. Uncertainties related to distance estimates,
to the disk-bulge light decompositions, and the typically limited
extention of the gaseous disks beyond the bright star forming disks
further limit the ability to derive accurate dynamical mass models.
These difficulties can be alleviated in the case of the spiral galaxy
M 31 (Andromeda). Owing to its size, proximity, well known distance,
and to constraints on its structural parameters from the long history
of observations at all wavelengths, Andromeda (the nearest giant
spiral galaxy) offers a unique opportunity to analyze in detail
the mass distribution and the dark halo properties in bright disk
galaxies. Massive galaxies like M 31 can probe dark matter on mass
scales much larger than that of the dwarfs, of order
1012 .
The Milky Way and Andromeda are the most massive members of the Local Group. Any estimate of their total mass and of the structure of their dark matter halos is a requirement for any study of the dynamics of the Local Group, its formation, evolution, and ultimate fate of its members (e.g. Li & White 2008). Difficulties in the determination of the Milky Way's mass components, related to the fact that our solar system is deeply embedded in its disk, can be overcome in the case of M 31. M 31 is known to have a complex merging history. Its multiple nucleus (Lauer et al. 1993) and the extended stellar stream and halo (e.g. Chapman et al. 2008; Irwin et al. 2001) are clear signs of a tumultuous life. According to the hierarchical models of galaxy formation it is conceivable that M 31 has grown by accretion of numerous small galaxies. It is likely the most massive member of the Local Group (e.g. Klypin et al. 2002). It is therefore of great interest to test the other predictions of hierarchical models such as the presence and structure of a dark matter halo around it. Contrary to dwarf galaxies, luminous high surface brightness galaxies such as M 31, cannot be used to test the central dark matter distribution, not only because of the large surface density of baryons which makes it difficult to constrain dark matter, but also because of possible adiabatic contraction (Seigar et al. 2008; Klypin et al. 2002). An extended and well defined rotation curve can instead be complemented by the extensive information now available on the M 31 stellar disk, stellar stream, globular clusters and orbits of Andromeda's small satellite galaxies to establish the dark matter density profile at large galactocentric distances. And this is one of our goals.
M 31 was one of the first galaxies where Slipher (in 1914) found evidence of rotation and also the first galaxy to have a published velocity field (Sofue & Rubin 2001, and references therein). Using the M 31 rotation curve, Babcock (1939) was the first person to advocate unseen mass at large radii in a galaxy. Since then much effort has been devoted to study the rotation curve of M 31 and to understand the relation between the light and the mass distribution. Despite a century of dedicated work, there are still many unsettled questions concerning the shape of the M 31 rotation curve, the contribution of visible and dark matter to it, and the changing orientation of the M 31 disk. Detailed HI surveys with single dish or synthesis observations (e.g. Newton & Emerson 1977; Cram et al. 1980; Unwin 1983; Bajaja & Shane 1982; Brinks & Burton 1984; Braun 1991) have been analyzed to find local kinematical signatures of spiral arm segments, of the interaction with M 32 or of a warped disk or ring. Even though many authors have pointed out the presence of a warp in M 31, i.e. of a systematic deviation of the matter distribution from equatorial symmetry, a complete quantitative analysis of the parameters of such a distortion at large galactocentric radii is still missing. Previous modelling of the HI warp has been based on a combination of high resolution inner disk HI data and much lower resolution and sensitivity outer disk data. The models assumed a rotation curve but no independent fit of rotation curve and warping of the disk has been attempted (e.g. Henderson 1979; Brinks & Burton 1984). Some papers (e.g. Carignan et al. 2006) analyze the extended rotation curve of M 31 using only HI data along the direction of the optical major axis, without considering the possibility of a warped disk. Only very recently Chemin et al. (2009) use deep 21-cm survey of the M 31 based on high resolution synthesis observations to model the warp and the rotation curve simultaneously.
Our first aim is to use the new WRST HI survey of M 31 (Braun et al. 2009) to define the amplitude and orientation of the warp using a tilted ring model. A set of free rings will be considered for which the following parameters need to be determined: the orbital center, the systemic velocity, the inclination and position angle with respect to our line of sight, and the rotational velocity. The geometric properties of the best fitting tilted ring model will then be used to derive the rotation curve from the 21-cm line observed velocities. The final goal will be to use the rotation curve for constraining the baryonic content of the M 31 disk and the presence and distribution of dark matter in its halo through the dynamical analysis.
Our recent deep wide-field HI imaging survey reaches a maximum
resolution of about 50 pc and 2 km s-1 across a
kpc2 region. This makes our database the most detailed
ever made of the neutral medium of any complete galaxy disk, including
our own. Observations and data reduction are described in
Braun et al. (2009) (hereafter Paper I). In Paper I we
analyzed HI self-absorption features and find opaque atomic gas
organized into filamentary complexes. While the gas is not the
dominant baryonic component in M 31, we take these opacity
corrections into account in determining the dynamical contributions of
the various mass components to the M 31 rotation curve. In this paper
we use the data at a resolution of 2 arcmin (457 kpc) in order to gain sensitivity
in the outermost regions. At this spatial resolution, we reach a brightness
sensitivity of 0.25 K. Considering a typical signal width of
20 km s-1 our sensitivity should be appropriate for detecting HI gas at column densities as low as 1019 cm-2.
In Sect. 2, we describe the modeling
procedures for determining the disk warp in M 31 and discuss the resulting
disk orientation. In Sect. 3, we determine the rotation curve
and the uncertainties associated with it. Various dynamical mass models for
the rotation curve fit are introduced and discussed in Sect. 4.
We determine the total baryonic and dark mass of this galaxy.
Together with complementary data at very large galactocentric radii, we
confirm the predictions of CDM cosmological models.
Section 5 summarizes the main results of this paper.
We assume a distance to M 31 of 785 kpc throughout, as derived by
McConnachie et al. (2005) (
kpc).
2 Tilted rings: modeling procedures
For a dynamical mass model of a disk galaxy it is necessary to reconstruct the tri-dimensional velocity field from the velocities observed along the line of sight. If velocities are circular and confined to a disk one needs to establish the disk orientation for deriving the rotation curve, i.e. the position angle of the major axis (PA), and the inclination of the disk with respect to the line of sight (i). If the disk exhibits a warp these parameters vary with galactocentric radius. This is often the case for gaseous disks which extend outside the optical radius and which often show a different orientation than the inner one. Our attempt to understand the kinematics of M 31 is done performing a tilted ring model fit to the data, under the assumption of circular motion. Because of this assumption we will use the tilted ring model outside the inner 8 kpc region, i.e. where deviations of gas motion from circular orbits are expected to be small. We will not consider local perturbations to the circular velocity field such as those due to spiral arms. The comparison between the velocities predicted by a tilted ring model and the data is done over all azimuthal angles and not only around the major axis. This will average out spiral arm perturbations.
![]() |
Figure 1:
The first moment map. The intensity-weighted mean velocity has been
computed
from the 120 arcsec data cube at a spectral resolution of 2 km s-1,
using a 4- |
Open with DEXTER |
In Fig. 1 we show the first moment map (i.e. the intensity-weighted
mean velocity along the line of sight) of our 21-cm Andromeda survey
(see Braun et al. 2009, for more details).
Because of the large angular extent of Andromeda, it is necessary to
consider the correct transformation between angular and cartesian
coordinates in order to derive galactocentric distances. A detailed
description of the spherical trigonometry involved can be found in the
literature (e.g. van der Marel & Cioni 2001). We shall summarize
here the most important formulae. Note that these take into account
the variations in the distance between us and the Andromeda regions
that do not lie along the line of nodes (the disk on the near side
(West) half is closer to us than on the far side (East) half).
Consider a point at a given right ascension and declination
(
)
in the Andromeda disk having inclination i and
position angle
.
The distance between the observer and
this point is D. The center of the galaxy has right ascension,
declination (
)
and distance D0 from the
observer. Consider
as the angle between the tangent to the
great circle on the celestial sphere through (
)
and
(
)
and the circle of constant declination
measured counterclockwise starting from the axis that runs
in the direction of decreasing RA at constant declination
(in practice
). The value of
along the
line of nodes (for points along the major axis) is
(
). The angular distance between
(
)
and (
)
is the second angular
coordinate called
.
We shall work in a Cartesian coordinate
system (x,y,z) which has its origin at the galaxy center, the
x-axis antiparallel to RA, the y-axis parallel to declination
axis and the z-axis toward the observer. The coordinates of the
observed pixels in this system will be:
![]() |
(1) |
where
![]() |
(2) |
Because of the warp, i and




The tilted ring model is (as usual) based on a number of geometrical
and kinematical parameters relative to the HI disk. To represent the
HI distribution in M 31 we consider 110 rings
between 10 and 35 kpc in radius. Each ring is characterized
by its radius R and by 6 additional parameters: the circular velocity ,
the inclination i, the position angle
,
the systemic velocity
and
the position shifts of the orbital center with respect to the galaxy center
(
). These last 3 parameters allow the rings to
be non concentric and the gas at each radius to have a
velocity component perpendicular to the disk (such as that given by
gas outflowing or infalling into the disk). If the average value of this
velocity component is uniform and non zero over the whole disk it implies
that effectively the systemic velocity determined from the gas
velocity field is different from the assumed one. We shall also consider
the case of a tilted
ring model with uniform values of
.
To test the model each ring is subdivided into segments
of equal area smaller than the spatial resolution of the dataset in
use. We convolve the emission from various pieces
with the beam pattern at each location to compute the velocity along
the line of sight
and its coordinates in a rest frame, defined above. In the same rest
frame we consider the 21-cm dataset. The telescope beam shape for the
dataset we use (a smoothed version of the original data)
can be well described by a Gaussian function with full width half
maximum (FWHM) of 2 arcmin. With this method we naturally account for
the possibility that the intensity-weighted mean velocity in a pixel
might result from
the superposition along the line of sight of emission from different
rings. For each position in the map we compare the observed
velocity
with the velocity predicted by the model
.
The observed velocity is the mean
velocity of the HI gas at each position. The assignment of a
measure of the goodness of the fit in this modeling procedure is done
using a
test on the difference between
and
.
The noise is uniform in our map
(see Braun et al. 2009) and we assign uniform uncertainties
to the observed values
.
In order to keep the model
sensitive to variations of parameters
of the outermost rings, each pixel in the map is assigned equal
weight. Pixels with higher or lower 21-cm surface brightness
contribute equally to determine the global goodness of the model fit.
Since the velocity channel
width of our survey is about 2 km s-1, we arbitrarily set
equal to the width of 3 channels (6 km s-1).
This is simply a scaling factor.
The equation below defines the reduced
which we will use
throughout this paper
![]() |
(3) |
In our definition of





Given the numerous free parameters ()
it is still unreasonable
to search for the minimal solution scanning a multidimensional grid
(of 66 dimensions). We use two procedures to determine the best
fitting model. In the first procedure (hereafter P1) we apply the
technique of partial minima. We evaluate
by varying each
parameter separately and interpolating to estimate the value for a
minimum
for each parameter. We then repeat the procedure over
a smaller parameter interval around the new solution. We start the
minimization with the following initial set of free parameters:
,
,
km s-1,
for all free rings. As an initial guess of
we
give the average rotational velocity derived in each free ring by
deprojecting the observed velocities within a 20
cone of the
optical line of nodes,
(using
and
for the deprojection). We carried out several
optimization attempts under a variety of initial conditions to avoid
that our partial minima technique carries the solution towards a
relative minimum.
The best fitting ring model is shown in Figure 2 by the
heavy continuous line. It gives a
.
We display only 3 of
the 6 free parameters for each free ring, namely the inclination i, the
position angle
and the systemic velocity
.
The
shifts of the orbital centers are small and shown in
Fig. 5. The minimum
is obtained for an
inclination which radially decreases by a few degrees between 10 and
25 kpc in radius and then increases out to 85
for the
outermost ring. The position angle shows marginal variations out to
25 kpc, then it decreases by about 10
in the outskirts.
However, while from 10 to 28 kpc low
values are obtained for
combinations of i and
not very different than the best
fitting model, beyond 28 kpc we can find tilted ring models which give
acceptable fits (with a
value within 20
of the minimum)
with different combinations of i and
.
The short dashed
line in Fig. 2 shows one model for which the inclination
of the outermost rings does not increase and their position angle instead varies
between 30
and 45
.
Clearly the outermost rings are not
very stable, due to a lack of 21-cm emission at these large radii.
Hence, some care is required when the model is extrapolated to even
larger radii.
The single constant value of systemic velocity which minimizes the
is -306 km s-1.
The errorbars displayed for the best fitting model
indicate parameter variations for which the
of the model
is within 20
of the minimum. This is done varying
just one parameter at a time with respect to the combination of parameters
which gives the minimum
.
The rotational velocities
obtained from the fit will
be discussed in the next Section where we compare them to the
rotation curve of Andromeda derived from the first moment map
using the geometrical
parameters and
of the best tilted ring model.
![]() |
Figure 2:
Assortment of ``free ring'' models fit to the HI data according
to modeling procedure P1. We display 33 of the 66 free parameters:
the inclination i, position angle |
Open with DEXTER |
In a second procedure (hereafter P2) we searched for a minimum by
neglecting the radial variations of
.
We set
and
km s-1. We have determined
this value of
as described below. In M 31, the integrated 21-cm
profile shows that the North-East receding side has more gas than the
South-West approaching side (their ratio being 1.13), at intermediate
velocities between the central one and the velocity of maximum emission.
This is
likely due to the disturbance caused by interaction with M 32.
Hence we cannot simply compute the systemic velocity by averaging the
observed mean velocities and weighting these with the 21-cm surface
brightness. Considering the galaxy disk between 10 and 25 kpc with an
average inclination of 77.7
and position angle 38
we
compute the average systemic velocity in rings. We weight the data
with the surface brightness intensity. For points south of the minor axis we
multiply their weight by the ratio of the HI mass in the northern side
to that of the southern side in that the ring. The resulting
is
shown in Fig. 3. Averaging the systemic velocities over all
rings we thus obtain a value of -306 km s-1. We searched for a minimal
solution scanning a multidimensional grid (12 dimensions)
corresponding to parameter variations of the outermost 4 free
rings. The initial grid of variations for the inclination and position
angle considered are
,
around
the standard values of the inner disk (
,
). The maximum velocity shifts considered with respect to
the initial values of
are
25,15,10,5 km s-1.
We then consider finer grids for the outermost 4 rings. We have
compared the observed velocities to the velocities predicted by tilted
ring models using all possible combinations of parameters for the
outermost 4 free rings. For the 8 innermost
free rings, parameter variations have been consider only in a
3-dimensional space, namely the minimum
has been found for
one ring at a time, considering all combinations of
for
that particular ring. As shown in Fig. 3 the
resulting free parameters for the best fitting tilted ring model are
very similar to those obtained using the first method.
![]() |
Figure 3:
Best fitting P2 solution for i and |
Open with DEXTER |
We define the residual velocities as
![]() |
(4) |
where



Residual velocities are generally smaller than 10 km s-1 and no
characteristic patterns are visible across the galaxy.
It is worth noticing that, if we trace the major axis position angle by
a close inspection of the first moment map, the position angle seems
about constant at 38
out to galactocentric distances
of 28 kpc, then it decreases to about 32
at 32 kpc and finally
it increases again back to 38
at 38 kpc. Given the
consistency between the best fitting model for the first and second
procedure (Figs. 2 and 3) and also between these
tilted ring models and the inspection of the database, we will not
consider models in which the position angle increases much above 38
for the outermost rings in the rest of this paper.
2.1 The NEMO results
To check our results for the best fitting tilted ring model we also
used the standard least-square fitting technique developed by
Begeman (1987), as implemented in the ROTCUR task within
the NEMO software package (Teuben 1995)
(hereafter P3). The galactic disk is subdivided into rings, each
of which is described by the usual 6 parameters. Starting with the
initial estimate for the fitting parameters, these are then adjusted
iteratively for each ring independently until convergence is achieved.
We run NEMO considering or neglecting the variations of the orbital
centers and systemic velocity for each ring. The ROTCUR task gives less
accurate results
than our P1 method since it does not take into account the
distance variations between the observer and the far/near side of the
galaxy and minimizes the free parameters of one ring at a time without
subsequent iterations. We ran ROTCUR with 24 free rings
from 8 to 34 kpc. For the last ring the solutions did not
converge. We show in Fig. 4 the resulting i,
and
for 3 cases. In one case the fit is done over all data points with uniform
weight, while varying
and the orbital centers
and
.
In a
second attempt, we fitted the data weighted with the cosine function
of the galactic angle away from the major axis and excluding data
within a 20
angle around minor axis. In a third attempt, we
fitted the data without considering possible orbital center shifts and
setting
to the average value found by ROTCUR when
was allowed to vary. This method confirms that the average systemic
velocity of the disk of Andromeda is slightly more negative than
previously thought and that there is a warp in outer disk which
brings the orbits closer to be edge-on.
We conclude that the 3 fitting methods largely produce similar results and will make use of the results of most accurate procedure, P1, in the rest of the paper.
![]() |
Figure 4:
Best fitting solutions obtained with the task ROTCUR in the
NEMO package (P3). The resulting position shifts of orbital centers
are very small and hence are not shown. The open circles indicate the
best fitting parameters when we exclude data within a 20 |
Open with DEXTER |
![]() |
Figure 5:
Orbital center shifts in arcmin as derived from tilted ring
model fits. Positive values of
|
Open with DEXTER |
2.2 Comparison with previous results
As we mentioned in the Introduction, only recently
a 21-cm survey made with a synthesis telescope has been used to model
the warp and the rotation curve of M 31 simultaneously
(Chemin et al. 2009). The conclusion is similar
to ours, namely that the warp of M 31 is such that the outer disk is at
higher inclination and lower position angle than the inner disk. However a
close inspection reveals
some differences. First, the average value of the disk inclination between 6 and 27 kpc that Chemin et al. (2009) find is 74
while
what we find between 10 and 28 kpc
is 77
,
more consistent with that derived from optical
surface photometry Walterbos & Kennicutt (1987). Secondly, the
inclination of the outermost fitted radius is higher for our model compared to what
Chemin et al. (2009) find, while the position angle is slightly lower.
We list below three main differences in the two fitting methods which can help
explaining the variations in the resulting tilted ring model parameters.
- We model the bulk rotational velocity of M 31 using the mean rotational velocity
observed at each position.
That is, we use the first moment map made from the 120 arcsec by 2 km s-1
data cube, following 4-
blanking in the
120 arcsec by 18.5 km s-1 data cube to determine what is included in the
integral. Chemin et al. (2009) use the peak velocity and if there are
multiple components they use the peak velocity of the ``main component''.
The main component is defined
to be the one which has the largest velocity relative to the systemic velocity
of the galaxy. Their choice is based on the observational evidence of multiple
peak profiles, very prominent in the inner regions. As we explain fully in
the next subsection we did not include the innermost region
(from 0 to 8 kpc in radius) in our analysis because it is dominated by
non-circular motions.
Even though taking the mean velocity we might be systematically biased to
lower apparent rotational velocities, the bias effects
are minimal at large radii where we are most interested to the dynamics.
In general, we find that assuming the peak velocity of the main component
as the best approximation to the rotation curve, as in Chemin et al. (2009),
is very model dependent. Moreover, to determine it in a
robust way one has to assume certain observational conditions
(i.e. to which level is a high velocity feature accepted in terms
of signal-to-noise; also, what happens if the highest velocity
feature is blended inside a bright component so
that only a ``shoulder'' but not a secondary peak can be seen in the spectra).
Even though it is true that observations
carried out with a low resolution and projection effects bias the mean
velocity towards lower velocities, we feel that
results might be more robust when considering the
mean velocity than other estimates if the system is as complex, asymmetric
and disordered as M 31 (and observed with a high spatial resolution).
Accretion events, which M 31 is experiencing, produce several distinctive
morphological and dynamical
signatures in the disk, including long-lived ring-like features, significant
flares, bars and faint filamentary structures above the disk plane.
In M 31, the likely
non-negligible disk thickness coupled with a complex, asymmetric
warping and with the presence of non-circular motion related to
multiple spiral arm segments intersecting along the line
of sight, makes difficult to assess the reliability of velocity indicators
different than the mean for tracing the bulk circular motion of the disk.
To prove the complexity of the system, we would like to point out
that across the disk of M 31 and especially along the bright ring-like
structure at 10-15 kpc in radius, double peak profiles
are often present. Sometime peaks are separated by more than 100 km s-1.
There is however not a systematic azimuthal or
radial pattern as to whether is the fainter or the brighter peak to show the
most extreme velocity from systemic. If it is just the warped outer disk
superimposed with the main disk to cause the multiple peak profiles we
should have found a more systematic behavior since the neutral hydrogen
surface density decreases considerably beyond 15 kpc.
High resolution IR maps of M 31 (Gordon et al. 2006)
show that the northern half the 15 kpc
arm, distinct on the major axis, merges into other
spiral arms (or ring like structure) at 10 kpc. Peak velocities might more
closely trace non-circular motion of the arms and produce wiggles in the rotation
curve which do not average out when additional perturbations are
present. This is the case for M 31, in which the southern
half is more strongly tidally perturbed than the northern half. Such
a curve cannot be reproduced without modelling the spiral arms locally
in term of mass condensation and non-circular motion.
- As we mentioned earlier in this section we believe that it is relevant for M 31, a very extended and nearby galaxy, to derive galactocentric distances in the frame of a tilted ring model using appropriate spherical geometry. This takes into account the fact that the near side of the galaxy is effectively closer than the far side of the galaxy. Procedures often available for deriving the kinematical parameters are built for the more numerous more distant galaxies and do not account for this effect. Moreover, our routine works with 66 free parameters simultaneously since the best fitting values of the parameters for each ring cannot be found independently from those relative to other rings when a warp is present. In fact the warp makes two or more rings overlap along the line of sight and in the overlapping region the expected velocity depend on all parameters of the rings which overlap.
- In fitting the tilted ring model to the data we do not apply any angular
dependent weight i.e. the same weight is given to pixels close to minor
axis than to major axis. This is because we would like to minimize the
risk of amplifying the kinematical signatures of sporadic features which
happens to lie close to major axis. As we will see in the next Section,
our model produces a stable rotation curve, not very sensitive to the
choice of the opening angle
around the major axis.
2.3 Why exclude the innermost region from fitting
Opposite to Chemin et al. (2009) we do not include the inner regions in our analysis. This is because after the pioneer work of Lindblad (1956) several other papers have pointed out in the inner region of M 31 the presence of morphological structures, such as a bar and a bulge, associated with streaming motion and non-circular orbits (e.g. Beaton et al. 2007; Athanassoula & Beaton 2006; Gordon et al. 2006; Stark & Binney 1994). In particular Berman & Loinard (2002); Berman (2001) have shown that the anomalous velocities observed in the inner region of M 31 can be explained as the response of the gas to the potential of a triaxial rotating bulge. Using a bulge effective radius of 10 arcmin they have derived which family of periodic elliptical orbits exist. They find that the bulge gives a non-negligible contribution to the galaxy potential out to about 7 kpc, and only at larger radii circular motion related to the disk gravitational potential dominates. The model well reproduced the velocities observed through the CO J=1-0 line emission. Since in this paper we are only modelling the large scale circular motion of the disk we use the mean velocity of the HI gas as tracer of the circular velocity from 8 kpc outwards.
![]() |
Figure 6:
Rotational velocities derived for the northern and southern
halves of M 31 using the geometrical parameters of the best fitting tilted
ring model as shown in Fig. 2. For this figure
|
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3 The rotation curve and the radial distribution of the baryons
We now apply the geometrical parameters of the best fitting tilted ring model
to derive the rotation curve of the galaxy. We set the inclination,
position angle and systemic velocity to the values shown by the heavy
continuous line in Fig. 2, and consider also the small shifts
of the orbital centers obtained by our minimization procedure P1.
We derive the rotation curve by averaging the rotational velocities of
data points in radial bins 1 kpc wide. For radii
between 8 and 10 kpc, we extrapolate
the model parameters of the innermost ring centered at 10 kpc. For radii larger
than 35 kpc, we extrapolate the model parameters of the outermost ring
at 35 kpc. Just for curiosity, we also checked what would be the rotation curve of
the inner regions for our dataset if we assume circular motion and
inside 8 kpc we set the disk inclination equal to the inclination
of the ring at 10 kpc and we consider 28
as position angle
(this is the average value we derive from an inspection of the mom-1 map). For
this innermost region we consider zero shifts for the orbit centers and
-306 km s-1 as systemic velocity. Figures 6 and 7
show the large dispersion in the velocities relative the central region
due to the presence of multiple components and to the uncertainties related
to orbital eccentricities inside 8 kpc, discussed in the previous section.
To complement 21-cm data in the inner regions we show in Fig. 7
the peak brightness velocities of CO lines (Loinard et al. 1995).
These have been observed along the major axis assumed to be at PA = 38
and with
.
However, notice that this is shown not just
to point out the consistency of the molecular and atomic gas
velocities, but to emphasize one has to consider non-circular motion to
properly trace the rotation curve in the inner region (Berman & Loinard 2002; Berman 2001). Hence we shall not use the CO data as well as
the 21-cm data at radii smaller than 8 kpc.
In the rest of the paper, we will only analyze the rotation curve
between 8 and 37 kpc. Beyond 37 kpc the northern and southern
halves do not give consistent rotation curves for any of the deconvolution
models. This is likely due to highly perturbed orbits in the outermost
regions, especially for the southern half which is closer to M 32.
![]() |
Figure 7:
The bottom panel shows the rotation curves obtained for the
northern and southern halves of the galaxy by averaging the data shown
in Fig. 6 in radial bins 1 kpc wide. Errorbars refer to the
dispersion around the mean. Filled triangles are
for the northern half, open squares for
the southern half of the galaxy.
Rotational velocities in the inner region derived from CO data
by Loinard et al. (1995) are shown with filled and open circles
for the northern and southern major axis, assumed to be at PA = 38 |
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We consider points
which lie within an opening angle
on either side of the major axis.
We first derive the rotation curve in the northern and
southern halves separately. We check the consistency of the rotation curves
for different values of
,
with the value of the rotational
velocities determined by the tilted ring model over the whole galaxy,
and between the northern and the southern halves.
When we vary
between 15
and 75
,
we obtain
the same rotation curves consistently (variations are less than
1 km s-1). Only the dispersion around the mean
increases slightly as we increase
.
We shall use
for the rest of the paper.
The rotation velocities in the two halves are consistent (within 3-
).
At many radii, mean velocities in the northern and southern halves
differ by less than 1-
corresponding to
only a few km s-1
at most, as shown in Figs. 6 and 7.
In Fig. 6 we also
display the rotational velocity parameter of the best tilted ring model.
The best fitting ring model gives the most consistent rotation curves.
We also tried to deconvolve the observed velocities using other tilted
ring models shown in Fig. 2, but they give less consistent
rotation curves. Average velocities are derived in the two halves by
assigning a weight to each pixel equal to the integrated brightness intensity
i.e. to the HI column density along the line of sight in the limit of
optically thin 21-cm line. The global rotation curve is the arithmetic
mean of the average rotational velocities in the two galaxy halves.
The errorbars of the global rotation curve shown in the upper panel
of Fig. 7 are computed as
![]() |
(5) |
where




In Table 1 we give the parameters of the best fitting tilted ring model
and of the average rotation curve of M 31 in the radial interval which
will be used in the next Section for the dynamical mass model.
The position shifts of the orbital centers are rather small and
can be neglected. The systemic velocity shifts,
,
are computed respect to the nominal value
km s-1.
Table 1: The HI rotation curve of M 31 and the parameters of the best fitting tilted ring model.
3.1 A comparison with other rotation curve measures
In Fig. 8 we show for comparison the rotation curves
derived in this paper (continuous line) and some previously determined
ones. In panel (a) we display data for north-east galaxy side, in panel (b) for
the south-west side and in panel (c) the average values.
Original data in (a) and (b) have been scaled according to an
assumed distance D=785 kpc and systemic velocity
km s-1.
We show both optical data from Kent (1989a)
and 21-cm data from Newton & Emerson (1977); Emerson (1976).
Unfortunately most of the previous determinations rely on
an assumed fixed inclination for the disk and on data along
the major axis alone. That implies that possible local velocity
perturbations will not be averaged out. These are clearly visible
especially in the literature data between 8 and 20 kpc
relative to the south-west galaxy half
(panel (b)), strongly perturbed by the M 32 tidal interaction.
In the northern side we derive a somewhat lower rotational
velocity, perhaps due to the presence of the warp which implies a higher
disk inclination, not accounted for by previous data analysis.
Taking into account these limitations, the general
agreement seems good.
The asterisk symbols are used in (c) to display the
average rotation curve of Chemin et al. (2009) which
lies above ours, due especially to somewhat lower inclination the
authors derive for the tilted ring model. Also, despite their
analysis masking some perturbations such as the
external arm, their curve is less smooth than ours. As discussed already
in the previous section, this might
be due to different choices of velocity components to extract
the rotation curve or to their use of weighting function which
gives more weight to data points close to major axis. A side
effect of this choice is to
retain any local velocity perturbation present along major axis.
A position velocity plot made along the photometric major axis,
at a position angle of 38,
is shown in Fig. 9.
Our adopted average rotation curve, projected back along major axis,
has been superimposed to it (diamonds).
The average value of the disk inclination and systemic velocity,
as derived in our best fitting tilted ring model in the radial interval
of interest, has been used. For a comparison
we display also the average rotation curve of Chemin et al. (2009)
after applying the disk inclination and systemic velocity
corrections adopted by the authors (asterisk symbols). The figure clearly
shows that
despite the different gas velocities adopted by the two teams to trace the
rotation curve, as explained in detail
in Sect. 2, differences in the rotational velocities at large radii
before deprojection are marginal.
The difference between the two measurements becomes
more significant when rotation curves (deprojected) are directly compared.
This illustrates the relevance of the differences in the
parameters of the best tilted ring model found
by the two teams. In particular Chemin et al. (2009) derive lower
inclination angles for the M 31 disk and hence higher rotational
velocities. The agreement between the PV diagram
along the major axis and the component of the rotational velocities
measured along the line
of sight improves, as it should, when using only data in
a very small opening angle around the major axis and data in the two
galaxy halves are kept separate. Notice, for example, the higher
velocity present along the major axis around 0.7 angular offset
respect to the average rotation curve. That is also visible
in the optical data shown in the middle panel of Fig. 8.
This anomalous velocity present along the major axis is averaged
out when data from other directions is taken into account.
![]() |
Figure 8:
The rotation curves derived in this
paper (continuous line) and some previously determined ones. In panel
a) we display data for north-east galaxy side, in panel b) for
the south-west side and in panel c) the average values. Filled
circles are for Kent (1989a) optical data along the
major axis,
open squares and open triangles are for Emerson (1976)
and Newton & Emerson (1977) 21-cm data along the
major axis. Asterisk symbols are used in c) to display the
average rotation curve of Chemin et al. (2009). Original
data in a) and b) have been scaled according to an
assumed distance D=785 kpc and systemic velocity
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![]() |
Figure 9:
Position velocity diagram along the photometric major axis,
at a position angle of 38 |
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3.2 The bulge-disk decomposition and the stellar distribution
Radial profiles of stellar disk surface brightness can usually be represented
by an exponential distribution. The mass-to-light ratio is
determined by the dynamical analysis of the rotation curve or by the optical
colors once the radial exponential disk scale length is known.
The disk scale length can be established from surface brightness profiles in
various bands and it might be wavelength dependent. In Andromeda, it varies from
7.7 kpc in the U-band to 5.9 in the R-band (Walterbos & Kennicutt 1988) to 4.5 kpc
in the K-band (Hiromoto et al. 1983; Battaner et al. 1986). The images used at optical wavelengths
(Walterbos & Kennicutt 1988) can trace the disk surface brightness out to
galactocentric radii of about 25 kpc,
while the available K-band images (including that obtained through the 2MASS
survey) loose sensitivity beyond 20 kpc. The discrepancy
between the optical and infrared scalelength can be easily explained in terms
of a radially varying star formation history or of an extinction
gradient. Light in the K-band traces the
mass distribution better than optical light because of the reduced extinction
and because most of the stellar mass in galaxy disks is due to old, low mass
stars. Therefore for the dynamical analysis, K-band scale lengths are usually
preferred despite the reduced sensitivity of infrared images.
Recent mid-infrared observations of Andromeda obtained with the
Infrared Array Camera on board of the Spitzer Space Telescope
(Barmby et al. 2006) show a disk scale length of 6.1 kpc at 3.6 m
measured on the same radial interval as measured by Battaner et al. (1986)
using the K-band light.
A larger scale length at 3.6
m is also found in M 33 by comparing
the Spitzer image with the 2MASS K-band image (Verley et al. 2009).
This might imply that intermediate age, cool supergiants contribute a
substantial fraction of the NIR emission at 3.6
m (Mould et al. 2008).
The determination of the disk scale length depends also on the bulge-disk
light decomposition and K-band images might not be deep enough to obtain an
accurate bulge-disk decomposition. Nevertheless we feel that the shorter disk
scalelength measured in the K-band images compared to 3.6 m images
is not affected by this limitation.
M 33 in fact has no bulge and indeed the scale length at 2.2
m is shorter than
at 3.6
m.
A spherical model for the bulge of M 31 is a simplification. Detailed modelling of
the surface brightness shows that at very least the bulge is an oblate spheroid with
axis ratio of 0.8 (Kent 1983) but most likely it is a triaxial
bulge (e.g. Berman 2001, and references therein).
For the purpose of this paper, which is the dynamical mass modelling
of the disk rotational velocities (beyond 8 kpc), we can neglect the
asphericity of the bulge which affects the orbits in the innermost few kpc.
Using a R1/4 de Vaucouleurs law (i.e. a Sersic index of 4) for the light
distribution it is customary to characterize the bulge by its
effective radius which encloses half of its total light.
At optical wavelengths (in the R- and V-band) Walterbos & Kennicutt (1988)
found a bulge effective radius of 2.3 kpc and a bulge to disk luminosity
ratio B/D = 0.82. Using the same dataset in the R-band
combined with Hubble Space Telescope data of Lauer et al. (1993)
and data from Kent (1983) for the innermost regions
Geehan et al. (2006) derived a much smaller bulge scale radius:
0.61 kpc (similar to what can be inferred by inspecting the 2MASS images).
Using the 3.6
m Spitzer image
Barmby et al. (2006) modelled the bulge light distribution with a
R1/4 de Vaucouleurs law and found a 1.7 kpc effective radius and a bulge
to disk light ratio of B/D = 0.78. Using the same data Seigar et al. (2008)
obtain a bulge effective radius of 1.93 kpc for a Sersic index n=1.7,
and a disk scalelength of 5.91 kpc
with B/D = 0.57. Fits to the bulge light distribution using smaller Sersic
indexes have also been done by Worthey et al. (2005) using a
brightness profile from an
I-band image out to 24 kpc: for a Sersic model with n=1.6 the bulge effective
radius was found to be 0.89 kpc and the disk scale length 5.7 kpc, even though
a steepening of the scale length is clearly visible beyond 15 kpc.
An faint stellar disk which extends as far as 40 kpc from the Andromeda
nucleus has been recently pointed out by Ibata et al. (2005) with
an exponential scale length of 5.1 kpc in the I-band.
Given the uncertainties in the disk and bulge mass distribution we will attempt to fit the rotation curve using a varying disk scalelength between 4.5 and 6.1 kpc in steps of 0.2 kpc. For the bulge we shall use 4 possible parameter combinations, namely an effective radius of 2.0 and 0.7 kpc and a Sersic index n=4, and n=1.6. Given the fact that we cannot constrain the dynamical contribution of the bulge since we are not fitting the motion in the inner regions, our purpose will be only to see if the dynamical fit to the disk improves considerably when using any of the four combinations.
3.3 Stellar mass-to-light ratios
Analysis of the star formation histories of the bulge and disk of M 31 suggest
that there is no age difference between the bulge and the disk
(Olsen et al. 2006). However previous attempts to fit former rotation
curves of this galaxy found a higher mass-to-light ratio for the disk than for the
bulge (Widrow et al. 2003), which was unexpected given the older age
of bulges relative to disks.
Hydrodynamic simulations of the triaxial bulge of M 31 by
Berman (2001) found a B-band mass-to-light ratio of 6.5
for the bulge i.e. a stellar mass of 1010 .
In the disk of Andromeda there is also a color
gradient visible in the disk (Seigar et al. 2008; Walterbos & Kennicutt 1988)
since B-R varies between 1.7 in the inner regions to 1.3 in the outer regions (this can be due to
changes in metallicity, age or extinction). According
to Bell & de Jong (2001) the mass-to-light ratio in the K-band expected
from B-R colors can vary from the value of 1 in the central regions to 0.65 in
the outer disk if extinction does not change radially.
The mass-to-light ratio in the B-band can vary between 8 and
2.5
and we will consider these two values as the extreme
acceptable disk mass-to-light ratio values. Since we fit the rotation curve beyond
8 kpc we cannot constrain the bulge mass-to-light ratio and hence we will
consider also models with equal mass-to-light ratios for the bulge and the
stellar disk.
The bulge and disk blue luminosities which we shall use
to compute the mass-to-luminosity ratio are
and
respectively.
These have been derived
using the integrated B-band magnitude measured by Walterbos & Kennicutt (1988),
corrected for absorption, assuming that the bulge contribution is 30
of
the total emission in the B-band. As we mentioned in the previous subsection,
the decomposition
of the light profile into a bulge and disk component is somewhat uncertain
and especially the bulge integrated luminosity is not firmly estabished yet
(see also Kent 1989b).
3.4 The gas surface density
For the gaseous disk we shall consider the atomic hydrogen
and the molecular gas surface density. These will be multiplied
by 1.33 to account for the presence of helium.
Using the best fitting tilted ring model (P1) we derive the
radial distribution of neutral atomic gas, perpendicular to the galactic plane,
in the optically thin approximation. This is shown in Fig. 10
as a function of the
galactocentric radius. The corresponding total HI mass is
.
To consider the possible presence of opaque
clumps we can multiply the HI surface brightness by 1.3 since
this is the correction inferred by Braun et al. (2009). This correction
has however no noticeable effect for the dynamical analysis carried out in the
next section.
The continuous line in Fig. 10 is the
log of the function
.
We shall use
to compute the dynamical contribution of the
atomic gas mass to the rotation curve. It has the following expression
![]() |
(6) |
and it provides a good fit to the atomic surface density distribution perpendicular to the galactic plane. The sharp decline of the HI beyond 27-30 kpc is likely due to the ionization of the atomic hydrogen by the local extragalactic UV background radiation (Corbelli & Salpeter 1993). Hence the fitting function which approximates the total atomic gas distribution is shallower than the HI distribution in the outer region. There is also ionized atomic gas in the inner regions, as can be inferred from the H

For the molecular gas fraction we considered the average radial variation (North+South) as shown in Fig. 10 of Nieten et al. (2006).
![]() |
Figure 10: The neutral atomic hydrogen column density perpendicular to the galactic plane is shown as a function of galactocentric distance. The best fitting tilted ring model P1 is used for the deconvolution of the 21-cm line brightness image. The optically thin line approximation has been used to convert the surface brightness in HI gas column density. The continuous line shows the total atomic gas surface density of the M 31 disk as modelled for the dynamical analysis. |
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4 Dynamical analysis
In this section we will analyze the mass distribution in the Andromeda
galaxy. We first attempt to fit the rotation curve of Andromeda,
derived in the previous section, from 8 to 37 kpc without a dark matter
halo. Then we carry on the dynamical analysis considering two different
models for the radial distribution of the dark matter
density distribution. We will discuss at the end
the resulting total mass of Andromeda in the context also of data at very
large radii from observations of the Andromeda satellites and other objects.
We shall consider both models in which the the disk and the bulge have
the same mass-to-light ratio
and models in which these are two independent variables. Unless stated
differently we shall consider the HI gas in the optically thin approximation.
The half thickness of the disk is assumed to be 0.5 kpc and the contribution
of the disk mass components to the rotation curve is computed according to
Casertano (1983). We use the reduced chi-square statistic,
,
to judge the goodness of a model fit.
We consider first 2 free parameters: the disk and the bulge mass-to-light
ratio
(M/L)d,b,
which we vary continuously. For the stellar disk we vary the
exponential scalelength hd using steps of 0.2 kpc in the interval
kpc. For the bulge we consider 2 possible effective radii
hb and Sersic index n=4 and n=1.7.
4.1 Newtonian and non-Newtonian dynamics without dark matter
We first attempt to fit the rotation curve in the framework of Newtonian
dynamics without considering a dark matter halo.
The best fit is obtained for a disk
scalelength of 6.1 kpc with
,
which
is the maximum allowed value. The fit is slightly better when a bulge effective
radius of 2 kpc is used. This is shown in the top panel of
Fig. 11. However the fit is
generally poor being the reduced
for all possible combinations
of parameters (see Table 2). Hence the model with no dark matter
fails to fit the data under the assumption of Newtonian gravity.
The fit stays poor even if unreasonably high values for the stellar mass-to-light
ratio are considered (>
).
The evident failure is due to the declining rotation curve
predicted by the baryonic mass distribution beyond 26 kpc.
An alternative explanation for the mass discrepancy has been proposed by Milgrom
by means of the modified Newtonian dynamics or MOND (Milgrom 1983).
Outside the bulk of the mass distribution, MOND predicts a much slower decrease of
the (effective) gravitational potential, with respect to the Newtonian case. This
is often sufficient to explain the observed non-keplerian behavior of RCs
(Sanders & McGaugh 2002). According to this theory the dynamics becomes
non-Newtonian below a limiting acceleration value,
cm s-2,
where the effective gravitational acceleration takes the value
,
with gn the acceleration in Newtonian dynamics
and x=g/a0. Here we shall use the critical acceleration value a0
derived from the analysis of a sample of rotation curves
cm s-2 (Sanders & McGaugh 2002).
We have tested MOND for different choices of the interpolating function
(see Famaey & Binney 2005, for details). In particular we have used the
``standard'' and the ``simple'' interpolation function and found that the ``simple''
interpolation function provides slightly better fits to the M 31 data. Hence
we shall use the ``simple'' interpolating function
in the rest of
the paper.
Corbelli & Salucci (2007) have already tested the former M 31 rotation curve in the MOND framework. They concluded that, in M 31, MOND fails to fit the falling part of the rotation curve at intermediate radii. However, this assessment was made using lower quality data and in the absence of an appropriate knowledge of the tilted ring model parameters.
![]() |
Figure 11:
The 21-cm rotation curve for M 31 and the
best fitting mass model with no
dark halo in the Newtonian case ( top panel) and for MOND
( middle panel).
The short-dashed lines indicate the Newtonian stellar bulge and disc
contribution to the rotation curve. The dot-dashed line is the gas contribution.
The figure displays the the best fitting mass model which is obtained for the
Newtonian case when
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Figure 11 shows the MOND best-fit model curve when
(M/L)d and (M/L)b are two independent free parameters (bottom panel). The
mass-to-light ratio best fitting
values are
(M/L)d = 2.5,
(M/L)b = 5.2, and hd = 4.5 kpc. The value of
is 1.3, hence MOND provide a good fit to the data.
If we reduce the number of free parameters to just one by setting
(M/L)d = (M/L)b
the fit is still reasonably good: the lowest
is 1.37 corresponding to
.
The rotational velocities
predicted by MOND are only slightly higher than observed for
kpc
and slightly lower than the data for
kpc.
As shown in Table 2, the goodness of the fits are not very sensible to the
bulge mass-to-light ratio, since we are excluding the central regions. It is
however to be noticed that MOND best fits to the actual data require a higher
mass-to-light ratio for the bulge than for the disk, in agreement with the
expected older age of the bulge stellar component.
In M 31 it is around
10 kpc that
and non-Newtonian corrections start to be important and
force a falling Newtonian RC into a flat one, more consistent with the data.
We notice that for MOND the value of the disk scalelength used to fit the
baryonic matter distribution is very important. In fact the fit
becomes poor if hd=6.1 kpc, as shown in Fig. 11.
If future photometric studies will
confirm a disk scalelength of 6.1 kpc then one will have to consider
possible variations of the assumed distance to M 31 to make MOND predictions
more consistent with the kinematics traced by 21-cm data.
4.2 Dark matter halo models
In the previous subsection we have seen that Newtonian dynamic fits to
the rotation curve without considering a dark matter halo are rather poor.
We will now use the M 31 rotation curve presented in this paper
to test in detail the consistency of a possible halo density profile
with theoretical models which predict a well defined dark matter
distribution around galaxies. Namely, we
shall consider a spherical halo with a dark matter density profile as
originally derived by Navarro et al. (1996,1997)
for galaxies forming in a Cold Dark Matter scenario. We
consider also the Burkert dark matter density profile
(Burkert 1995) since this successfully fitted the rotation
curve of dark matter dominated dwarf galaxies (e.g. Gentile et al. 2007, and references
therein).
Both models can describe the dark matter halo density profile using two
parameters.
The density profile proposed by Burkert (1995) has a
constant-density core and is given by:
![]() |
(7) |
A strong correlation between the two parameters

![]() |
(8) |
Using this relation, we show in Fig. 12 the best fitting mass model when a Burkert model for the dark halo is considered. The values of the free parameters are: RB=77 kpc, (M/L)d=8.0





















![]() |
Figure 12:
The M 31 rotation curve (points) and the best-fitting
mass models (solid line) using a Burkert dark halo profile with
hd=5.1 kpc, hb=2 kpc and n=4. Also shown are
the dark halo contribution (dot-dashed line), the stellar disk
and bulge (short-dashed line) and the gas contribution
(long-dashed line). In the top panel, we show the best fit mass
model (
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![]() |
Figure 13:
The 68 |
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The NFW density profile is:
![]() |
(9) |
Numerical simulations of galaxy formation find a correlation between















![]() |
(10) |
where





We now fit the M 31 rotation curve using two free parameters: the
halo concentration C and (M/L)d, the disk mass-to-light ratio.
We shall consider first no dispersion around the relation given by the
above equation and a bulge mass-to-light ratio equal to that of the disk.
The 1-,
2-
and 3-
ranges are determined using the
reduced
and assuming Gaussian statistics. They are computed by
determining in the free parameter space the projection ranges, along each
axis, of the hypersurfaces corresponding to the 68.3
and 95.4
confidence levels.
The best fitting mass model is obtained for a disk scalelength hd=4.5 kpc.
The bulge parameters are not of much relevance for the goodness of the fit in
the region of interest to this paper.
The stellar mass-to-light ratio for the best fit is
(M/L)d=4.2 /
,
if we assume that it does not
vary between the bulge and the disk component, and the value of the reduced
is 1.12. The total dark halo mass is
(corresponding to C=11.9).
If we allow variations
between the disk and the bulge mass-to-light ratio, the best fit mass model gives
(M/L)d=5.0
/
and
(M/L)b=2.7
/
and a minimum
value of 1.08. These combination of M/L ratios is not realistic since we don't expect the bulge to have a lower
M/L ratio than the disk. But higher values of the bulge M/L ratio increases
only slightly the
value.
In Fig. 14 we show the modelled rotation curve according to
the best fitting mass model when hb=2.0 kpc and n=4. The
dark halo mass is
corresponding to C=12.
As we increase the disk scalelength the
increases slightly,
the dark halo mass
decreases and the stellar mass-to-light ratio increases
(for example for a change in the disk
scalelength from 4.5 to 6.1 kpc the minimum
changes from 1.12 to 1.34,
decreases from
to
and
(M/L)b,d=5.7
/
).
![]() |
Figure 14:
The M 31 rotation curve (points) and the best-fitting
mass model (solid line) using the NFW dark halo profile with C=12
in the frame of
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We have computed the confidence area in the (M/L)d-C plane and shown
them in Fig. 15. When the mass-to-light ratio
of the disk and the bulge are two independent variables the best fitting
(M/L)b value is unrealistically low but the value of the concentration
parameters does not change. Figure 15 shows also the wider
confidence areas obtained when twice the dispersion of 0.13, as estimated
from numerical simulations, is considered around
in the
relation.
![]() |
Figure 15:
The 68 |
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Table 2 summarizes the main results of the mass models
considered for fitting the rotation curve of M 31. In Col. (1) we give
the short name of the mass model: No-DM is simply a Newtonian
dynamic fit with no dark matter. For MOND we use the no dark matter and
the ``simple'' interpolation function.
The scalelengths of the bulge and of the disc are labeled with the symbol
hb and hd respectively. We consider hd as a free parameter
in the interval 4.5-6.1 kpc. The best fitting value of a possible additional
free parameter of each model, P+ is given in Col. (7). For the No-DM and
MOND models there is no additional free parameter. For the Burkert halo model
P+ is the core radius in kpc, for the CDM mass model P+ is
the concentration parameter C. For the bulge and disk mass-to-light
ratio, the range of possible values considered are: 2.5
(M/L)
in units of (M/L)
.
In the first half of the Table, we consider both (M/L)b and (M/L)das free parameters while in the second part we use (M/L)b = (M/L)d.
Since, for
the simple Newtonian dynamic fit with no dark matter, the bulge and disk
mass-to-light ratios come out equal to the highest allowed value,
the parameters are listed only in the first part of the table.
We can see clearly in Table 2 that
the goodness of the fit does not
depend on the bulge mass distribution, as expected, since we are only
fitting the rotation curve for R>8 kpc. It also shows that only a
pure Newtonian mass model with no dark matter halo fails to fit the data.
Both MOND, a Burkert halo model with a large constant density
core, and the NFW profile in the framework
of
provide good mass model fits to the rotation curve
of Andromeda derived in this paper. In the
next subsection we shall use data from other sources to test the
rotation curve at larger radii than those provided by the HI dataset.
Table 2: Parameters of the best fitting mass model.
4.3 Baryonic and total mass of Andromeda
We consider now the mass estimate of Andromeda from different sources at galactocentric radii from 30 to 560 kpc. We derive a rotation curve by computing the expected circular rotational velocity given the mass estimate within a radius R. We consider data from planetary nebulae, globular clusters, stellar streams and Andromeda satellites to constrain the Andromeda total mass at large radii. Since each paper considers an ensemble of objects and describes in detail the method used to derive the mass, we give in Table 3 the resulting mass estimate and reference the original papers where a more detailed description of the database and analysis can be found. If the authors have assumed a different distance to Andromeda than what we use in this paper, we have scaled their radius and mass estimate accordingly. In Fig. 16 we plot the rotational velocities at large distances from the center of Andromeda derived from the mass estimates given in Table 3.
Table 3: Andromeda mass estimated at large galactocentric radii.
We plot also the velocities at large radii predicted by the
mass models
which best fits our rotation curve i.e.
a NFW dark halo profile with concentration C=12 (continuous line). A Burkert halo
model with a very large core radius, such as that required by the best fitting mass
model to the 21-cm rotation curve data, predicts higher velocities than observed at
distances shown in Fig. 16. But a Burkert halo profile with a core
radius RB=28 kpc,
compatible with the 21-cm rotation curve data, is also compatible with data at
large galactocentric distances
(short dashed line in Fig. 16). For both halo
models we have considered the values of the free parameters hd, (M/L)d and
(M/L)b, which best fit 21-cm rotation curve.
Given the fact that the
and Burkert halo
were constrained using only HI data between 8 and 37 kpc
it is quite remarkable how well the two halo models fit the data
out to about 560 kpc. We did not test MOND predictions in this framework
because Andromeda masses in the original paper have been derived using
Newtonian dynamic.
Since the best fit Burkert halo model was essentially
a constant density profile in the region traced by the 21-cm rotation curve,
we show also the predicted velocities of this model
in case the dark halo is truncated beyond 37 kpc.
A simple keplerian fall off of the observed velocities outside the
region covered by our 21-cm dataset however fails to fit the data at larger
galactocentric radii. Only the outermost 3 data points are compatible
with a keplerian fall-off regime. For the
mass model, the virial
radius corresponding to a concentration C=12 is 270 kpc, hence the predicted
rotational velocities of the outermost data points in Fig. 16 are in
the keplerian regime.
The best fitting
mass model implies a stellar (disk+bulge) mass of
and a dark matter halo mass of
.
Adding to the stellar mass the cold gas
mass (neutral and molecular hydrogen plus helium) of
we estimate a total baryonic mass of Andromeda
of
.
This adds
to the dark matter halo mass, giving
![]() |
(11) |
as our best estimate of the total mass of the Andromeda galaxy. The associated baryonic fraction is 0.12, very similar to the cosmic inferred value of 0.14.
![]() |
Figure 16: Rotational velocities predicted at large distances from the center of Andromeda according to several datasets analyzed by previous papers (see Table 3 for details). The continuous line shows the rotational velocities predicted by an extrapolation at large radii of the best fitting mass model with a NFW dark halo profile (C=12). The short dashed line shows the predicted velocities of a Burkert halo model with RB=28 kpc. The long dashed line is for a constant halo dark matter density profile which is truncated outside 37 kpc and gives a Keplerian fall off of the velocity at larger distances. |
Open with DEXTER |
5 Summary
We determine the rotation curve of the M 31 disk from 8 to 37 kpc using a
tilted ring model fit to the velocity field mapped in the
full-disk, 21-cm
imaging survey of Braun et al. (2009). The orientation of
the rings have been determined using three different
techniques which give rather similar results. For our most accurate modelling
method (P1), we use 11 equally spaced
free rings, which cover galactocentric distances between 9 and 36 kpc, whose
parameters are varied independently. Each free ring has 6 degrees of
freedom, since we allowed the systemic velocity and center position of
each ring to vary (in addition to the circular rotation, inclination
and position angle). This implies a total of 66 degrees of freedom in our
model. Between two consecutive free rings, parameters are
determined by a linear interpolation. We find that the disk of M 31 warps
from 25 kpc outwards by lowering its position angle and becoming more inclined
with respect to our line of sight. The disk reaches an inclination of 86
at 35 kpc. The geometry of the outermost
two rings has somewhat larger uncertainties, but the tilted ring model
which gives the best fit to the data also produces consistent
rotation curves in the two separate halves of the galaxy. Furthermore,
these rotation curves do not depend on the value of limiting
angle around the major axis chosen for selecting the data. We find
-306 km s-1 as the average value of the systemic velocity of the
gaseous disk of M 31. The rotation curve of M 31 is consistent with
being flat beyond 20 kpc and we carry on a dynamical analysis to
determine the baryonic and non-baryonic mass distribution of the
nearest spiral galaxy.
The M 31 rotation curve cannot be reproduced using Newtonian dynamics
and only the stellar and gaseous mass components. Without a dark matter halo
however, MOND provide a good fit to the galaxy gravitational
potential in the region considered. We test the density
profile and mass predictions of hierarchical clustering and structure
formation in a CDM cosmology, together with a dark halo
model having a constant density core. Both models are able
to reproduce the rotation curve of M 31 to a high level of accuracy.
The constant density core model which fits M 31 has a core radius
comparable to the size of the disk of M 31 and therefore is in practice a
constant dark matter density model.
Using the relation between the concentration parameter C and the dark halo
mass
as for a NFW density profile in a flat
CDM cosmology, we
find a best fit halo concentration parameter C=12 which implies a dark
matter halo mass
.
If we assume that the stellar disk and the bulge have the same mass-to-light
ratio we find
/
.
If the
mass-to-light ratio of the disk and the bulge are two independent variables then
the best fit gives a slightly higher value for the disk,
(M/L)d=5.0+0.6-1.0.
We are unable to constrain the bulge mass-to-light ratio
since we discarded the innermost rotation curve in our fit
(derived without considering elliptical orbits).
A wider range of C and (M/L)d values are
found when a dispersion is considered around the average
log C - log
relation, as suggested by numerical simulations of
structure formation in a
cosmology.
An interesting result is that
the extrapolation of a constant core dark halo model,
as well as of the best fit
dark halo model, beyond the region
traced by the rotation curve are in good agreement
with the Andromeda mass traced by other dynamical indicators
(globular clusters, streams, satellites) at larger radii, out to 560 kpc.
The constant-core best fitting halo model has a very large core radius (77 kpc) and a
high virial mass, not consistent with the data at large galactocentric radii. However
models with a somewhat smaller core radius, provide acceptable fit to the 21-cm
rotation curve and to data at larger galactocentric radii.
The total estimated mass of M 31 from our
mass model fit to the 21-cm rotation curve in the framework of
cosmology
is 1.3
,
with a 0.12 baryonic fraction.
This is similar to the cosmic inferred baryonic fraction of 0.14 and implies a formation
redshift zf=1.2 for the Andromeda galaxy.
R.W. acknowledges support from Research Corporation for the Advancement of Science. We acknowledge the anonymous referee for his/her criticism to an earlier version of the paper.
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All Tables
Table 1: The HI rotation curve of M 31 and the parameters of the best fitting tilted ring model.
Table 2: Parameters of the best fitting mass model.
Table 3: Andromeda mass estimated at large galactocentric radii.
All Figures
![]() |
Figure 1:
The first moment map. The intensity-weighted mean velocity has been
computed
from the 120 arcsec data cube at a spectral resolution of 2 km s-1,
using a 4- |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Assortment of ``free ring'' models fit to the HI data according
to modeling procedure P1. We display 33 of the 66 free parameters:
the inclination i, position angle |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Best fitting P2 solution for i and |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Best fitting solutions obtained with the task ROTCUR in the
NEMO package (P3). The resulting position shifts of orbital centers
are very small and hence are not shown. The open circles indicate the
best fitting parameters when we exclude data within a 20 |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Orbital center shifts in arcmin as derived from tilted ring
model fits. Positive values of
|
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Rotational velocities derived for the northern and southern
halves of M 31 using the geometrical parameters of the best fitting tilted
ring model as shown in Fig. 2. For this figure
|
Open with DEXTER | |
In the text |
![]() |
Figure 7:
The bottom panel shows the rotation curves obtained for the
northern and southern halves of the galaxy by averaging the data shown
in Fig. 6 in radial bins 1 kpc wide. Errorbars refer to the
dispersion around the mean. Filled triangles are
for the northern half, open squares for
the southern half of the galaxy.
Rotational velocities in the inner region derived from CO data
by Loinard et al. (1995) are shown with filled and open circles
for the northern and southern major axis, assumed to be at PA = 38 |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
The rotation curves derived in this
paper (continuous line) and some previously determined ones. In panel
a) we display data for north-east galaxy side, in panel b) for
the south-west side and in panel c) the average values. Filled
circles are for Kent (1989a) optical data along the
major axis,
open squares and open triangles are for Emerson (1976)
and Newton & Emerson (1977) 21-cm data along the
major axis. Asterisk symbols are used in c) to display the
average rotation curve of Chemin et al. (2009). Original
data in a) and b) have been scaled according to an
assumed distance D=785 kpc and systemic velocity
|
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Position velocity diagram along the photometric major axis,
at a position angle of 38 |
Open with DEXTER | |
In the text |
![]() |
Figure 10: The neutral atomic hydrogen column density perpendicular to the galactic plane is shown as a function of galactocentric distance. The best fitting tilted ring model P1 is used for the deconvolution of the 21-cm line brightness image. The optically thin line approximation has been used to convert the surface brightness in HI gas column density. The continuous line shows the total atomic gas surface density of the M 31 disk as modelled for the dynamical analysis. |
Open with DEXTER | |
In the text |
![]() |
Figure 11:
The 21-cm rotation curve for M 31 and the
best fitting mass model with no
dark halo in the Newtonian case ( top panel) and for MOND
( middle panel).
The short-dashed lines indicate the Newtonian stellar bulge and disc
contribution to the rotation curve. The dot-dashed line is the gas contribution.
The figure displays the the best fitting mass model which is obtained for the
Newtonian case when
|
Open with DEXTER | |
In the text |
![]() |
Figure 12:
The M 31 rotation curve (points) and the best-fitting
mass models (solid line) using a Burkert dark halo profile with
hd=5.1 kpc, hb=2 kpc and n=4. Also shown are
the dark halo contribution (dot-dashed line), the stellar disk
and bulge (short-dashed line) and the gas contribution
(long-dashed line). In the top panel, we show the best fit mass
model (
|
Open with DEXTER | |
In the text |
![]() |
Figure 13:
The 68 |
Open with DEXTER | |
In the text |
![]() |
Figure 14:
The M 31 rotation curve (points) and the best-fitting
mass model (solid line) using the NFW dark halo profile with C=12
in the frame of
|
Open with DEXTER | |
In the text |
![]() |
Figure 15:
The 68 |
Open with DEXTER | |
In the text |
![]() |
Figure 16: Rotational velocities predicted at large distances from the center of Andromeda according to several datasets analyzed by previous papers (see Table 3 for details). The continuous line shows the rotational velocities predicted by an extrapolation at large radii of the best fitting mass model with a NFW dark halo profile (C=12). The short dashed line shows the predicted velocities of a Burkert halo model with RB=28 kpc. The long dashed line is for a constant halo dark matter density profile which is truncated outside 37 kpc and gives a Keplerian fall off of the velocity at larger distances. |
Open with DEXTER | |
In the text |
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