Magnetic fields and the outer rotation curve of M31

Abstract

Recent observations of the rotation curve of M31 show a rise of the outer
part that can not be understood in terms of standard dark matter models or
perturbations of the galactic disc by M31's satellites. Here, we propose an
explanation of this dynamical feature based on the influence of the magnetic
field within the thin disc. We have considered standard mass models for the
luminous mass distribution, a NFW model to describe the dark halo, and we have
added up the contribution to the rotation curve of a magnetic field in the
disc, which is described by an axisymmetric pattern. Our conclusion is that a
significant improvement of the fit in the outer part is obtained when magnetic
effects are considered. The best-fit solution requires an amplitude of ~4
microG with a weak radial dependence between 10 and 38 kpc.

Full text

arXiv:1010.0270v1 [astro-ph.CO] 1 Oct 2010
Magnetic fields and the outer rotation curve of M31
B. Ruiz-Granados and J.A. Rubi˜no-Mart´ın
Instituto de Astrof´ısica de Canarias (IAC), E-38200, La Laguna, Tenerife (Spain)
and
Departamento de Astrof´ısica, Universidad de La Laguna, E-38205, La Laguna, Tenerife (Spain)
E. Florido and E. Battaner
Departamento F´ısica Te´orica y del Cosmos. Universidad de Granada, Granada(Spain)
and
Instituto de F´ısica Te´orica y Computacional Carlos I, Granada(Spain)
ABSTRACT
Recent observations of the rotation curve of M31 show a rise of the outer part that can not
be understood in terms of standard dark matter models or perturbations of the galactic disc
by M31’s satellites. Here, we propose an explanation of this dynamical feature based on the
influence of the magnetic field within the thin disc. We have considered standard mass models
for the luminous mass distribution, a NFW model to describe the dark halo, and we have added
up the contribution to the rotation curve of a magnetic field in the disc, which is described by
an axisymmetric pattern. Our conclusion is that a significant improvement of the fit in the outer
part is obtained when magnetic effects are considered. The best-fit solution requires an amplitude
of ∼4µG with a weak radial dependence between 10 and 38 kpc.
Subject headings: galaxies: individual (M31) , galaxies: magnetic fields
1. Introduction
Recent high sensitivity measurements of the ro-
tation curve of M31 (Chemin et al. 2009; Corbelli et al.
2010, hereafter C09 and C10 respectively) suggest
that it highly rises at the outermost part of the
disc of M31 (r&25 −30 kpc) (see Figure 1).
This behaviour cannot be considered an exception.
Similar outer rising rotation curves can be found in
other galaxies. Noordermeer et al. (2007) stated
that “in some cases, such as UGC 2953, UGC 3993
or UGC 11670 there are indications that the rota-
tion curves start to rise again at the outer edges
of the HI discs”, suggesting follow-up observa-
tions of higher sensitivity to investigate this fact
in more detail. Other examples could be found
in some rotation curves provided by Spano et al.
(2008), in particular the curves of UGC 6537
(SAB(r)c), UGC 7699 (SBcd/LSB), UGC 11707
(Sadm/LSB) or UGC 11914 (SA(r)ab) measured
by means of Fabry-Perot spectroscopy. Other
potential examples could be NGC 1832 and NGC
2841 (Kassin et al. 2006), even with a larger error,
and ESO 576-G51 and ESO 583-G7 (Seigar et al.
2005), even if the outer rising is small. Potential
nearby candidates could be NGC 3198, DDO 154
or NGC 7331 (de Blok et al. 2008). Therefore,
outer rising rotation curves are not uncommon
in spirals, being the case of M31 the most repre-
sentative. This puzzling dynamic feature requires
a theoretical explanation. A detailed study by
C09 showed that the NFW, Einasto or pseudo-
isothermal dark matter halos fail to reproduce the
exact shape of the rotation curve of M31 in the
outer region and found no differences between the
various halo shapes. Moreover, they found new HI
structures as an external arm and thin HI spurs
in the outskirts of the disc. These spurs have
been also observed by Braun et al. (2009). Thus,
a primarily explanation of this gas perturbations
1

could be the interactions with the M31’s satellites
as NGC 205 (Geehan et al. 2006; Corbelli et al.
2010). The main problem of matching the prop-
erties of the giant stellar stream observed in the
south of M31 (Ibata et al. 2001) is that the orbital
of the companion that produce the stream is not
constrained satisfactorily (Fardal et al. 2006) or
high velocities for the radial orbits are found (see
e.g. Howley et al. 2008). Although the northeast-
ern (NE) and southwestern (SW) parts of M31
show different kinematical properties, both sug-
gest a rise in the outer part of the rotation curve
(see Figure 5 and 6 in C09 and C10, respectively).
In this work, we show that magnetic dynamic
effects constitute a clear and simple basis to
interpret this feature. Some particular mod-
els of magnetically driven rotation curves have
been presented (Nelson 1988; Battaner et al. 1992;
Battaner & Florido 1995, 2000, 2007; Kutschera & Jalocha
2004; Tsiklauri 2008). Although those models
were originally developed to explain flat rotation
curves without dark matter, our purpose here is
more conventional and we will consider the con-
tribution of both a NFW dark matter halo and a
magnetic field added to match the velocity in the
outermost region.
Magnetic fields are known to slowly decrease
with radius (see e.g. Beck et al. 1996; Han et al.
1998; Beck 2001, 2004, 2005), and therefore they
become increasingly important at the rim. For
large radii, an asymptotic 1/r-profile for the field
strength provides an asymptotic vanishing mag-
netic force. This 1/r-profile has been found to
match the polarized synchrotron emission of the
Milky Way (Ruiz-Granados et al. 2010) and NGC
6946 (Beck 2007), and will be considered in this
work.
2. Mass models and the magnetic field of
M31
In this section, we briefly present the luminous
and dark mass models used to describe the ob-
servational rotation curve of M31. As we are in-
terested in the outer region (i.e. distances higher
than 10 kpc), we follow C10 and we neglect the
bulge contribution. We will also adopt from C09
and C10 the parameters describing stellar disc and
gaseous distribution, respectively. Therefore, we
will only leave as free parameters those describing
two physical components: the dark matter halo
and the magnetic field.
2.1. Disc model
The stellar disc is assumed to be exponen-
tial (Freeman 1970), being the surface mass den-
sity
Σ = Σ0exp −r
Rd,(1)
where Σ0is the central density of stars and Rd,
the radial scale factor.
The contribution to the circular velocity of the
stellar disc is (see Binney & Tremaine 1987)
vd(r) = 4πGΣ0Rdy2[I0(y)K0(y)−I1(y)K1(y)],
(2)
where y=r/2Rdand I0, K0, I1, K1are the modi-
fied Bessel’s functions for first and second order.
The gaseous disc mainly contains HI, molecular
gas and helium. The estimated gaseous distribu-
tion of M31 represents approximately a 9% (for
C09) and 10% (for C10) of the total mass of the
stellar disc (see Table 1).
For the spatial distribution of the gas, we again
assume the same exponential law given by Equa-
tion (1).
2.2. Halo model
The dark matter halo is described here by a
NFW profile (Navarro et al. 1996)
ρh(r) = δcρc
r
Rhh1 + r
Rhi2,(3)
where δcis a characteristic density contrast,
ρc= 3H2
0/8πG is the critical density and Rh
is the radial scale factor. The contribution to
the circular velocity due to this profile is given
by (Navarro et al. 1997)
vh(x) = V200s1
x
ln(1 + cx)−cx
1+cx
ln(1 + c)−c
1+c
,(4)
where V200 is the velocity at the virial radius R200
which is assumed ∼160 kpc for C09 and R200 ∼
200 kpc for C10, cis the concentration parameter
of the halo which is defined as c=R200 /Rhand
xis r/R200 . According to C09 and C10, the total
dark halo mass at the respective virial radius is
2

M200 ∼1012 M⊙. Our free parameters are V200
and c. Values from C09 are c= 20.1±2.0 and
V200 = 146.2±3.9km/s. Values from C10 are
c= 12 and V200 ∼140 km/s.
2.3. The regular magnetic field of M31
The first measurements of the magnetic field of
M31 were obtained by Beck (1982) using the po-
larized synchrotron emission at 2700 MHz, show-
ing that a magnetic pattern was aligned with HI
structures and formed a ring at r≈10 kpc.
They obtained a strength of Breg = 4.5±1.2µG.
Berkhuijsen et al. (2003) used observations of po-
larized emission to show that M31 hosts an ax-
isymmetric field. Han et al. (1998) used Faraday
rotation measurements to show that the regular
field could extend at least up to galactocentric
distances of r∼25 kpc without significant de-
crease of the strength and at least with a z∼1
kpc of height above the galactic plane. However,
recently, Stepanov et al. (2008) have shown that
these results contained a significant contribution
of Galactic foregrounds and so, it is difficult to in-
fer the model from these measurements. In any
case, as we show below, the detailed structure of
the field is not relevant for the rotation curve.
Fletcher et al. (2004) presented a detailed
study of the regular field of M31 based on multi-
wavelength polarized radio observations in the re-
gion between 6 and 14 kpc. By fitting the observed
azimuthal distribution of polarization angles, they
found that the regular magnetic field follows an
axisymmetric pattern in the radial range from 8 to
14 kpc. The pitch angle decreases with the radial
distance, being p∼ −16◦for distances r < 8 kpc,
and p∼ −7◦for r < 14 kpc. This fact implies
that the field becomes more tightly wound with
increasing galactocentric distance. They found
a total field strength (i.e. regular and turbulent
components) of ≈5µG. For the regular field they
found that it became slowly lower, reaching at
r∼14 kpc a strength of ∼4.6µG.
In this paper, our basic assumption is that the
regular magnetic field of M31 is described by an
axisymmetric model, that extends up to 38 kpc.
For this model, the components in cylindrical co-
ordinates are given by
Br=B0(r) sin(p) (5)
Bφ=B0(r) cos(p),(6)
where pis the pitch angle and B0(r) is the field
strength as a function of the radial distance. As
shown below, from the point of view of the descrip-
tion of the rotation curve, the relevant component
is Bφ. Here, we will assume that p= 0◦, as we
are mostly interested in the outer region where p
is very low. Several previous probes also indicates
low values of p. In any case, this is not a strong
assumption in the sense that a non-zero pvalue
can be absorbed into the field strength (B1) as a
different amplitude. For the field strength, as a
baseline computation, we shall consider a radial
dependence of B0(r) or equivalently Bφgiven by
Bφ(r) = B1
1 + r
r1
,(7)
where r1represents the characteristic scale at
which B0(r) decreases to half its value at the galac-
tic centre and B1is an amplitude in which we
are absorbing the cos(p) factor. This expression
has an appropriate asymptotic behaviour, in the
sense that we obtain a finite value when ris close
to the galactic center (r→0), and asymptoti-
cally tends to ∝1/r when r→ ∞, as suggested
Battaner & Florido (2007). Observations carried
out by Fletcher et al. (2004) established that
Bφ(r= 14kpc)≈4.6µG. (8)
This observational value at this radius was con-
sidered as fixed in our baseline computation. By
substituting into Equation (7), we can find a rela-
tion between B1and r1that allows us to re-write
Equation (7) in terms of a single free-parameter
(r1)
Bφ(r) = 4.6r1+ 64.4
r1+r,(9)
where ris given in kpc and Bφ(r) in µG.
3. Methodology
3.1. Observational rotation curve of M31
We have considered the two datasets from C09
and C10. They consist on a set of 98 and 29 mea-
surements of the circular velocity (and their as-
sociated error bars) respectively, which were ob-
tained with the high-resolution observations per-
formed with the Synthesis Telescope and the 26-
m antenna at the Dominion Radio Astrophys-
ical Observatory (C09) and with the wide-field
3

and high-resolution HI mosaic survey done with
the help of the Westerbork Synthesis Radiotele-
scope and the Robert C. Byrd Green Bank Tele-
scope (GBT) (Braun et al. 2009). For our pur-
poses, we consider only distances higher than
r > 10 kpc to illustrate the effects of the mag-
netic field. The actual data points on the rota-
tion curve were obtained after fitting a tilted ring
model to the data, and assuming a distance to
M31 of 785 kpc. C09 derived a value of the incli-
nation angle of i∼74◦, which is lower than that
derived from optical surface photometry measure-
ments (Walterbos & Kennicutt 1987) and by C10
(i∼77◦). The data points are plotted in Figure 1,
in two separate panels.
3.2. Influence of the magnetic field on the
gas distribution
The presence of a regular magnetic field af-
fects the gas distribution (Battaner et al. 1992;
Battaner & Florido 1995, 2000). The fluid mo-
tion equation can be written as (see e.g. Battaner
1996)
ρ∂~v0
∂t +ρ~v0·~
∇~v0+~
∇P=n~
F+1
4π~
B·~
∇~
B−∇ B2
8π,
(10)
where ρis the gas density; ~v0, the velocity of the
fluid; P, the pressure; ~
F, the total force due to
gravity and ~
B, the magnetic field. We assume
standard MHD conditions i.e. infinite conduc-
tivity. Equation (10) is simplified by assuming
axisymmetry and assuming pure rotation, where
~v0= (v0r, v0φ, v0z) = (0, θ, 0) even if these condi-
tions are not necessary regarding the dynamic ef-
fects in the radial direction. Taking into account
all these facts, the motion equation in the radial
cylindrical coordinate is
ρ−dΦ(r)
dr +θ2
r−dP
dr −Fmag
r= 0,(11)
where Φ(r) is the gravitational potential; Fmag
r,
the radial component of the magnetic force, and
Pthe pressure of the fluid. We can assume that
pressure gradients in the radial direction are neg-
ligible (Battaner & Florido 2000). In this case,
then the radial component of the magnetic force
is given by
Fmag
r=1
4π B2
φ
r+1
2
dB2
φ
dr !,(12)
and the contribution of the magnetic field to the
circular velocity is given by
v2
mag =r
4πρ B2
φ
r+1
2
dB2
φ
dr !.(13)
3.3. Modelling the rotation curve
The rotation curve is obtained, as usual, by
quadratic summation of the different contributions
θ(r)2=vb(r)2+vd(r)2+vh(r)2+vmag(r)2,(14)
where we explicitly set vb(r) = 0 as mentioned
above.
3.4. Model selection
For the luminous mass models, the different pa-
rameters are considered as fixed values in our anal-
ysis (see Table 1). For the NFW dark halo, we
constrain the V200 and cparameters, allowing a
range for V200 between 100 and 220 km/s with
steps of 0.5 km/s and c∈[5,30] with steps of 0.3.
The contribution of the magnetic field to the ro-
tation curve is fitted through one free parameter,
r1, that we are considering which is equivalent to
fit Bφas we discussed above. For this parameter,
we have explored values in the range from 1 to
1000 kpc. Our analysis is based on a reduced-χ2
as the goodness-of-fit statistic. Thus, the best-fit
parameters are obtained by minimizing this func-
tion
χ2=1
N−M
n
X
i=1
(θobs
i−θmodel
i)2
σ2
i
,(15)
where Nis the total number of points to which
we have measured the rotational velocity and de-
pends on the considered dataset (N= 74 for C09
and N= 27 for C10) and Mis the number of free
parameters. The sum runs over the observational
data points, being θobs
ithe observed velocity and
θmodel
ithe modelled velocity, which depends on
the particular model. We shall consider two mod-
els: one without magnetic contribution (DM) and
another with the magnetic field (DM+MAG). Fi-
nally, σiis the observational error bar associated
to each data point.
4. Results and discussion
Our main results are summarized in Figure 1.
The dotted line shows our best-fit rotation curve
4

Dataset Disc parameters
C09 Rd= 5.6 kpc
Md= 7.1×1010 M⊙
Mgas ∼6.6×109M⊙
C10 Rd= 4.5 kpc
Md= 8.0×1010 M⊙
Mgas ∼7.7×109M⊙
Table 1: Fixed parameters for bulge and disc.
for C09 and C10 when considering only the usual
dynamical components (the stellar component and
the dark halo at this range of distances), while the
solid line shows the result when adding up also the
magnetic contribution.
The most important result is that, as shown
in Battaner & Florido (2000), the effects of mag-
netism on the rotation curve are only relevant at
large radii (in this case of M31, at distances larger
than about 25 kpc). In this sense, the radial range
of the datasets of C09 and C10 are optimal to ob-
serve the magnetic effects.
Table 2 summarizes our results for the best-
fit solutions without (labelled as DM model)
and with magnetic field influence (DM + MAG
model). As shown, magnetic effects on the gaseous
disc significantly decrease the value of the reduced
χ2statistic for both datasets. Specially for C09,
the fit is significantly improved when taking into
account a new parameter (∆χ2= 6.5). The radial
scale factor of the magnetic field (r1) is uncon-
strained in both cases, but shows clear preference
for high values, which means that the best-fit so-
lution for the field slowly decreases with the radial
distance in the considered interval (i.e. between
10 and 38 kpc). For example, at r∼38 kpc, the
field is found to be Bφ&4.4µG for C09 and
Bφ&4.0µG for C10 for this best-fit solution.
Both values are compatible with the strength of
the field obtained by Fletcher et al. (2004) who
found a nearly constant strength of the regular
field of about ∼5µG between 6 kpc and 14 kpc.
Moreover, when no radial variation of the strength
is considered, (i.e. if r1→ ∞), and we fit for the
amplitude Bφ, we obtain that Bφ= 4.7+0.6
−0.7µG
which it is again compatible with results discussed
above. This suggests that the data do not require
an important radial variation of the strength of the
field for the considered range of distances, or in
other words, the contribution of the second term
in the r.h.s. of Equation (12) is negligible (i.e.
dB2
φ/dr ≪B2
φ/r). Therefore, if we had consid-
ered another radial profile (e.g. exponential), we
would have found a large radial scale factor too.
The azimuthal component of the field is practi-
cally constant between 10 and 38 kpc.
Our results imply large magnetic fields at large
radii. How these are produced lies beyond our
scope. On the other hand, we would expect the
extragalactic field to be also of this order of mag-
nitude. Theoretical predictions (Dar & de R´ujula
2005) suggest values of the level of few µG for the
intergalactic magnetic fields (hereafter IGMF).
Kronberg (1994); Govoni & Feretti (2004); Kronberg
(2005) have reviewed observations of µG level in
rich clusters, though no direct measurements have
been reported for the IGMF in the Local Group
near M31. The observational evidence for IGMF
is still weak, but quite strong IGMF near galaxies
cannot be disregarded.
We finally note the apparent discrepancy be-
tween our derived parameters for the DM model
and those obtained by C09 and C10 (see Sect. 2.2).
However, it is important to stress that we are re-
stricting the fit to the outer region of M31 (r >
10 kpc). In this outer range, there is a weaker de-
pendence of the shape of the rotation curve on the
concentration parameter c, and thus lower values
of care found because the fit tries to compensate
the rising behaviour in the outer part. The inclu-
sion of the magnetic field corrects this apparent
discrepancy, and in this case the values of care
now compatible with those obtained by C09 and
C10.
5

5. Conclusions
The rotation curve of M31 (Chemin et al. 2009;
Corbelli et al. 2010) is rather representative of the
standard rotation curves of spirals for r < 30 kpc,
but it seems to rise out to, at least 38 kpc, the
limiting distance of the observations. Indeed, this
behaviour is not restricted to M31, and we are
probably dealing with a common dynamical fea-
ture of many other spirals. Therefore, the out-
ermost rising rotation curve is a very important
theoretical challenge.
It is certainly a challenge as the standard dark
matter halo models, in particular the universal
NFW profiles, do not account for this dynamical
unexpected behaviour.
A conventional galactic model, with bulge, disc
and dark matter halo, has been shown to provide
good fit to the data in the range r < 20 kpc (C09,
C10). Here, we take advantage of these results
and we do not fit any of the luminous components,
which are taken to be exactly the same as those
proposed by C09 and C10. We have restricted our
study to the region r > 10 kpc, and we only fitted
the parameters describing the dark matter halo,
and the magnetic field contribution. Our main
conclusion is that magnetic fields are not ignor-
able for explaining large-scale dynamic phenom-
ena in M31, producing a significant improvement
of the fit of the rotation curve at large distances.
Moreover, the required field strength of the reg-
ular component (Bφ∼4µG) is fully consistent
with the measured magnetic field in M31 at least
up to r∼15 kpc.
This conclusion seems very reasonable, as mag-
Model Parameters C09 C10
DM V200 (km/s) 160.2±2.0 132.1+5.7
−5.4
c12.3±0.6 19.1+2.4
−2.2
χ219.8 1.1
DM + MAG V200 (km/s) 133.8+1.7
−1.3120.0+4.7
−4.0
c22.7+1.2
−1.125.0+2.8
−2.9
r1(kpc) >888.0>185.0
χ213.3 0.6
Table 2: Best-fit for the rotation curve with and
without the contribution of magnetic fields for r&
10 kpc.
netic fields are amplified and act “in situ”, and
therefore they become increasingly important at
the rim, where gravity becomes weaker.
The best-fit model of the magnetic fields re-
quires a field strength slowly decreasing with
radius. This slow decrease is compatible with
present values of the strength derived from ob-
servations of the polarized synchrotron emission
of the disc, but clearly we need measurements of
Faraday rotation of extragalactic sources at this
large radii to confirm that the magnetic field is
present up to this distance and to trace unam-
biguously the regular component. Hence, future
experiments such as LOFAR1and SKA2(Beck
2009), will be extremely important, allowing a de-
tailed explorations on the galactic edge as well as
in the intergalactic medium.
This work was partially supported by projects
AYA2007-68058-C03-01 of Spanish Ministry of
Science and Innovation (MICINN), by Junta de
Andaluc´ıa Grant FQM108 and by Spanish MEC
Grant AYA 2007-67625-C02-02. JAR-M is a
Ram´on y Cajal fellow of the MICINN.
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Fig. 1.— Best-fit solutions for the rotation curve
of M31, with and without including the contri-
bution of a regular magnetic field component.
Top shows the C09 dataset and bottom the C10
dataset. Asterisks and rombhus represent the ob-
servational data with the associated error bars.
The solid line is the best fit derived in this paper,
including the contribution of the regular magnetic
field over the gaseous disc. The dotted line is the
best-fit model obtained without the contribution
of the magnetic field.
8