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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

Rhi2,(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