Magnetic spiral arms in galaxies

Anvar ShukurovDepartment of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU

Accepted 1998 July 1. Received 1998 June 15; in original form 1998 March 30

A B S T R A C TLarge-scale magnetic fields in spiral galaxies are strongest in spiral-shaped regions, themagnetic arms. It was recently discovered for the galaxy NGC 6946 that magnetic arms can beinterlaced with the gaseous arms, rather than coinciding with them. There are indications thatthe magnetic and gaseous arms may cross in some other galaxies. We suggest that magneticarms can be located away from regions of higher gas density (gaseous arms) becauseinterstellar turbulence is stronger in the latter. We predict magnetic arms interlaced withgaseous arms in galaxies with weak dynamos, whereas the two spiral structures should overlapin galaxies with strong dynamo action; in an intermediate case, the magnetic and gaseousspiral structures overlap in the inner galaxy and are interlaced in the outer parts (as, possibly, inM51). Another plausible mechanism to produce displaced magnetic and gaseous spiralpatterns results from a delay in the dynamo response to the enhancement of turbulence inthe gaseous arms. This should lead to the magnetic and gaseous arms crossing at the corotationradius, as possibly observed in the galaxies IC 342 and M83. We also argue that spiral armsonly weakly affect the local scaleheight of the galactic gas layer.

Key words: magnetic fields – MHD – ISM: magnetic fields – galaxies: ISM – galaxies:magnetic fields – radio continuum: galaxies.

1 I N T RO D U C T I O N

A traditional understanding of the interaction between interstellarmagnetic fields and galactic spiral arms considers the fields as apassive component of the interstellar medium. The magneticdiffusion time owing to the molecular diffusivity by far exceedsthe galactic lifetime. Therefore magnetic field is frozen into theinterstellar gas, and responds instantaneously and directly to anychanges of the gas density, being controlled by magnetic fluxconservation. Because of this, galactic magnetic fields wereexpected to be stronger in gaseous spiral arms where the plasmadensity is enhanced (Roberts & Yuan 1970). Interstellar turbulencedramatically changes this picture, e.g. by an enhancement of themagnetic diffusivity at large scales. With the observed scaleof interstellar turbulence l . 100 pc and its rms velocity v . 10–30 km s¹1, the turbulent magnetic diffusivity isb . 1

3 lv . 1026 cm2 s¹1. The turbulent magnetic diffusion timeh2=b . ð5–30Þ × 108 yr based on the scaleheight of the ionized

galactic disc, h ¼ 400–1000 pc, is significantly shorter than thegalactic lifetime. Hence magnetic fields can behave differently atlarge and small scales: they are frozen into the gas at scales smallerthan l (but larger than a fraction of a parsec), whereas at scales largerthan l their behaviour is more independent of the gas densityvariations.

A striking example of the departure of large-scale galacticmagnetic fields from a tight correlation with the ambient plasma

density is given by magnetic spiral arms located between thegaseous arms, as discovered in the spiral galaxy NGC 6946 (Beck& Hoernes 1996). A similar phenomenon was observed previouslyin the galaxy IC 342 (Krause, Hummel & Beck 1989a; Krause1993) and suspected for M81 (Krause, Beck & Hummel 1989b) andM83 (Sukumar & Allen 1989). In these galaxies, polarized syn-chrotron emission, a tracer of the large-scale (regular) magneticfield, is (or is suspected) to be strongest in spiral-shaped regions,called magnetic arms, displaced from the gaseous arms. It is stillunclear how widespread this phenomenon is, and which propertiesof the galaxies control the location of magnetic arms with respect tothe gaseous ones.

The distinction between the scalings of the large-scale and small-scale magnetic fields with density is highlighted by the fact that thetotal equipartition magnetic fields are well correlated with the gasdensity at scales as large as about 1 kpc (Berkhuijsen 1997).Therefore the total magnetic field (regularþturbulent) has mildmaxima in the gaseous arms even if the regular magnetic field isdisplaced from them.

In this Letter we propose a simple explanation of galacticmagnetic arms based on the natural assumption that the generationof the large-scale galactic magnetic field is a threshold phenom-enon, i.e. that the field can be maintained only provided that someparameter exceeds a certain critical value. In the case of mean-fielddynamo theory, this parameter is the dynamo number.

Our arguments use explicitly the language of mean-field dynamo

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theory. However, we do not rely on any detailed properties of thedynamo, but rather appeal to generic properties of self-excitationsystems, where the magnetic field arises in a simple bifurcationwhen a certain control parameter exceeds the generation threshold.It is only natural to assume that the control parameter decreaseswith v, since v determines the turbulent magnetic diffusivity. Non-linear behaviour of mean-field dynamos is a matter of debate now(see recent reviews by Childress & Gilbert 1995 and Beck et al.1996), but our conclusions are independent of the details of themean-field dynamo theory.

2 T H E S T E A DY- S TAT E M AG N E T I C F I E L D

The regeneration of the large-scale (regular) magnetic field in aturbulent medium is a result of competition between the conversionof turbulent kinetic energy into magnetic energy and the turbulentmagnetic diffusion that destroys the field. In the framework of themean-field dynamo theory (Moffatt 1978; Parker 1979; Krause &Radler 1980), the control parameter is the dynamo numberD ¼ a0Gh3b¹2, where a0 . l2Q=h is a measure of the deviationsof the turbulent motions from mirror symmetry, G ¼ r dQ=dr is therotational shear, Q is the angular velocity of rotation and r is thegalactocentric distance. Using the above expressions for a0 and b

we obtain (Ruzmaikin, Shukurov & Sokoloff 1988; Beck et al.1996)

D . 10hv

� �2

GQ : ð1Þ

The dynamo can maintain the large-scale magnetic field againstturbulent magnetic diffusion provided that jDj $ Dcr with Dcr < 10.For jDj < Dcr , the averaged induction equation has only the trivialsolution B ¼ 0 for the large-scale magnetic field B. At jDj ¼ Dcr, anon-trivial non-linear solution bifurcates. For jDj slightly aboveDcr, the steady-state strength of the magnetic field is given by

B < B0jDj

Dcr¹ 1

� �1=2

for jDj $ Dcr : ð2Þ

Here B0 is the characteristic regular field strength which is plausiblydetermined by equipartition between magnetic and turbulentkinetic energies,

B0 . Kð4prv2Þ1=2; ð3Þ

where r is the total gas density and K is some number of order unity.Equation (2) describes the generic behaviour of the non-linearsolution near the bifurcation. More specifically, it can be obtainedfrom a widespread parametrization of the back-reaction of theLorentz force (which is quadratic in magnetic field) on the turbu-lence in terms of the quenched dynamo number

DB ¼ D 1 þ B2=B0

2ÿ �¹1ð4Þ

(a particular case of this parametrization is alpha-quenching –Krause & Radler 1980). A steady state is reached when jDBj ¼ Dcr,which immediately yields (2). This simple description of magneticfield generation embraces the main features and, as we shall see, itprovides insight into the origin of magnetic spiral arms.

3 T H E A R M – I N T E R A R M C O N T R A S T I N T H EL A R G E - S C A L E M AG N E T I C F I E L D

According to simulations of spiral density waves in a cloudyinterstellar medium (Levinson & Roberts 1981; Roberts & Haus-man 1984), a typical arm–interarm density contrast for H I clouds is

ra=ri . 4, where subscripts ‘a’ and ‘i’ henceforth refer to regionsinside the gaseous arms and their interarm regions, respectively. Inresponse to the enhanced gas density, young massive stars and H II

regions concentrate in the gaseous arms as star formation is moreactive there. Because of the enhanced supernova rate and morevigorous gravitational interactions between interstellar clouds, theturbulent velocity is larger in the gaseous arms, typically va=vi . 2(Roberts & Hausman 1984). Observational evidence for strongerturbulence in the arms (Rohlfs & Kreitschmann 1987; Garcıa-Burillo, Combes & Gerin 1993) is less straightforward, albeitcompatible with the above estimate of va=vi. The resulting ratiobetween the dynamo numbers is

Da

Di.

vahi

viha

� �¹2

.14; ð5Þ

where we have assumed for numerical estimation that ha ¼ hi (seeSection 4). Thus the dynamo action in the spiral arms is weaker thanbetween them, provided that v=h in the arms is larger than in theinterarm regions.

The arm–interarm contrast in the large-scale magnetic fieldsfollows from equations (2) and (5) as

Ba

Bi.

ra

ri

� �1=2va

vi

jDaj=Dcr ¹ 1jDij=Dcr ¹ 1

� �1=2

ð6Þ

. 4jDaj=Dcr ¹ 14jDaj=Dcr ¹ 1

� �1=2

:

Together with equation (2), this relation is applicable whenjDj $ Dcr, and it shows that the field in the arms is weaker thanbetween them, Ba=Bi < 1, if

jDaj <1 ¹ B0

2i =B0

2a

1 ¹ ri=raDcr . 1:25 Dcr : ð7Þ

The coefficient of Dcr exceeds unity wherever vi < va andB0i Þ B0a :

The conclusion is straightforward: magnetic arms must occurbetween the gaseous arms in galaxies with weak dynamos, i.e. jDj

small enough to satisfy the inequality (7). A mild enhancement ofthe turbulent velocity suppresses significantly the dynamo action inthe arms of such galaxies. In contrast, galaxies with a strongdynamo, where equation (7) is not satisfied, must have the strongestlarge-scale magnetic field within the arms.

The suppression of the dynamo efficiency in the arms byenhanced turbulent velocity can make the large-scale magneticfields stronger between the gaseous arms even at the kinematicstage of the dynamo (Schreiber & Schmitt 1997). As shown by ournon-linear arguments, this localization of enhanced magnetic fieldsis continued at the non-linear stage even though B0 is maximum inthe gaseous arms. Thus interlaced magnetic and gaseous arms maydevelop at very early stages of galactic evolution, and persist inmature galaxies.

In the case jDaj; jDij q Dcr, we have from equations (5) and (6)Ba=Bi . ðra=riÞ

1=2ha=hi > 1 independently of va=vi. This limitingcase may occur in some galaxies and in the central parts of manygalaxies. Although equation (2) is applicable only near the gen-eration threshold this asymptotic result may still be viable, as weonly need that B does not decrease with jDj in the steady state. Note,however, that a steady state with Ba=Bi > 1 may require a time ofseveral Gyr to be established.

In those galaxies where the dynamo number (1) decreases withradius, magnetic arms can coincide with the gaseous ones in theinner parts of the galaxy and be interlaced in the outer galaxy.Possibly this occurs in M51, where strong large-scale magnetic

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fields occur both in the gas spiral arms and in the interarm regions(Berkhuijsen et al. 1997). Observations with higher resolution andmore precise determinations of relevant galactic parameters arerequired to verify this possibility.

The main limitation of the arguments presented above is that theyare of a local nature and do not account for the effects of advectionby the azimuthal velocity. The magnetic arms supported by the localdynamo action can be wound up by differential rotation. However,this difficulty does not arise near the corotation radius. Furthermore,interlaced magnetic and gaseous arms are expected to occur ingalaxies with weak dynamos, i.e. with weak differential rotation,where the effects of azimuthal advection are minimized.

Another, independent, possibility is that magnetic arms arisebecause of a certain time lag t between turbulence enhancement inthe density arms and the response in dynamo coefficients,especially a. The azimuthal phase shift between the gaseous andmagnetic arms is then

Df ¼ ðQ ¹ QpÞt ;

where Qp is the angular velocity of the gaseous arms (densitywaves). This would lead to magnetic arms that cross the gaseousarms at the corotation radius where Q ¼ Qp. There are no signs ofsuch a crossing in NGC 6946, where the azimuthal phase shiftbetween the field and density spiral patterns is more or less constantwith radius for 4 & r & 12 kpc (Frick et al. 1998). We emphasize,however, that polarized arms cross the optical arms in IC 342, andthis possibly occurs near the corotation radius (M. Krause, privatecommunication). It is thus possible that there are at least two effectsresponsible for magnetic arms in galaxies – the suppression ofdynamo activity in the arms as discussed above, and a delayedresponse of the dynamo to the galactic density waves. In the lattercase, the magnetic and gaseous arms should cross, and the positionof their crossing is an indicator of the corotation radius.

The enhancement of the mean helicity of interstellar turbulencetriggered in the arms may be delayed by a time t because the meanhelicity is an inviscid integral of motion and so can evolve only on along diffusive time-scale. Therefore t can well exceed the correla-tion time of the turbulence. In order to have a phase shift ofDf ¼ 1

2 p at a radius r ¼ R, which would correspond to magneticarms located halfway between the gas arms, we need

t .pR

2V0ð1 ¹ R=rcÞ;

where we have assumed that Q ¼ V0=r and Qp ¼ V0=rc, with rc thecorotation radius and V0 the linear velocity of rotation. Settingtentatively V0 ¼ 200 km s¹1, R ¼ 5 kpc and rc ¼ 2R, we obtaint < 8 × 107 yr. This is about 10 times the eddy turnover time.

4 T H E S C A L E H E I G H T O F T H E G A S L AY E R

We have assumed that h is not much affected by the spiral arms. Ifthe gas layer is in hydrostatic equilibrium, the gas scaleheight ish . p=ðrgÞ ; where p is the total pressure and g is the verticalgravitational acceleration. The total pressure has four main additivecomponents: the turbulent pressure pt ¼ 1

2 rv2, the thermal pressurepth, the magnetic pressure pm ¼ H2

=8p with H being the totalmagnetic field, and the cosmic ray pressure pcr. Assuming thatpth ¼ pt and pcr ¼ pm, we obtain

h . ðv2 þ V2AÞ=g ; ð8Þ

where VA is the Alfven speed.It is usually assumed that there is equipartition between magnetic

fields and interstellar turbulence, that is v . VA. If, in addition,

hydrostatic equilibrium can be established instantaneously, then weobtain ha=hi . 4. However, a slightly more careful analysis showsthat the energy equipartition between magnetic fields and turbu-lence and/or hydrostatic equilibrium is more plausibly maintainedonly on average over the galactic disc, but not locally; as a result, theazimuthal variations in the scaleheight are significantly weaker thansuggested by the above estimate.

The arm–interarm contrast in the total magnetic field can beestimated from the intensity of the total synchrotron emission,« ~ ngH2

', where ng is the number density of relativistic electronsand H' is the total magnetic field perpendicular to the line of sight.Under energy equipartition or pressure balance between relativisticelectrons and cosmic rays, we have « ~ H2H2

' . H4. Accordingto Beck & Hoernes (1996), the arm–interarm contrast in « is about 2in NGC 6946, so Ha=Hi . 1:2 or at most 1.4 if « ~ H2. Withra=ri ¼ 4, we obtain VA;a=VA; i . 0:7. Now, assuming that theaverage energy densities in the total magnetic field and turbulenceare equal to each other, rav2

a þ riv2i ¼ raV2

A;a þ riV2A; i (assuming

that the arms and interarm regions have equal widths), we obtainv2

i . 0:2V2A;i and v2

a . 1:4V2A;a.

Now, using equation (8), we obtain ha=hi . 1. The scaleheightonly weakly responds to the spiral arms because azimuthal varia-tions in the turbulent velocity are compensated by those in theAlfven velocity.

Deviations from the local equipartition, v ¼ VA, can be due to thefact that the passage time of a spiral arm, 108 yr (for an arm width of2 kpc, a pitch angle of 15◦ and a relative velocity of 100 km s¹1), isonly slightly longer than the regeneration time of the randommagnetic field, which exceeds the turbulence correlation time of107 yr by an uncertain factor of order unity. Therefore the magneticfield may not have enough time to adjust itself to the varyingturbulent intensity in the travelling density wave far from thecorotation radius. In addition, it cannot be excluded that thefluctuation dynamo can generate random magnetic fields at alevel above equipartition (Belyanin, Sokoloff & Shukurov 1993)in the interarm regions.

The time-scale for the hydrostatic equilibrium to be established isdetermined by the sound travel time across the scaleheight,h=cs . 3 × 107 yr, where cs ¼ 10–30 km s¹1 is the sound speed.Since this time-scale is not much shorter than the passage time of aspiral arm, this also suppresses arm–interarm variations of the discscaleheight (Shukurov & Sokoloff 1998). (Note, however, that thesituation may be different near the corotation radius where the spiralarms do not move with respect to the gas.)

The low arm–interarm variations in the disc scaleheight areconsistent with determinations of the scaleheight of neutral hydro-gen in the Milky Way. For example, the scaleheights of H I

determined at different galactocentric radii by Malhotra (1995,fig. 6) show regular localized variations and random scatter at alevel of about 20 per cent, and this is a plausible upper limit on thearm–interarm variations. We conclude that h can hardly change bya factor of 2 as v does, so h=v is a minimum in the gaseous arms asassumed in Section 3.

5 D I S C U S S I O N

Our conclusions only weakly depend on the assumed form of non-linearity (4): we only require that DB be a monotonically decreasingfunction of B. If DB decreases with B more rapidly than inequation (4), then the range of dynamo numbers near D ¼ Dcr,where the interarm fields are stronger, becomes narrower.

The theory of spiral density waves in a cloudy interstellar

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medium (Roberts & Hausman 1984) predicts that maxima in therms velocity of clouds, which can be identified with the maxima ofthe turbulent velocity, are displaced from the maxima of the gasdensity. Then the turbulent pressure 1

2 rv2 is affected by the spiralpattern to a lesser extent than the density and velocity individuallyand B0a=B0i ¼ 4 , as adopted above for numerical estimates, is anoverestimate.

Recent numerical simulations of non-linear mean-field dynamosin a disc with v enhanced in the arms (Rohde & Elstner 1997)support our explanation of interlaced magnetic arms, and confirmequation (6) by showing that Ba=Bi indeed decreases when thedynamo number decreases (see fig. 4 of Rohde & Elstner 1997).However, this numerical model incorporates many physical effectsof very different degrees of importance, so it is difficult to decidewhat parameters control different aspects of the numerical solution.In a similar numerical model, Moss (1998) assumes that theturbulent magnetic diffusivity is modulated by the spiral pattern,also resulting in interlaced spiral patterns in gas density andmagnetic field.

It has been suggested that magnetic arms can be a manifestationof slow magnetohydrodynamic (MHD) density waves (Fan & Lou1996; Lou & Fan 1998). This explanation requires that the galacticrotation velocity deviates from the solid-body law by not more thanVA . 10–30 km s¹1. However, galaxies do not rotate rigidly, evenin their inner parts. The magnetic arms in NGC 6946 are locatedbetween the optically visible ones over a radial range4 & r & 12 kpc (Beck & Hoernes 1996; Frick et al. 1998). Devia-tions from solid-body rotation are as strong as 50–100 km s¹1 overthis range of radii (Carignan et al. 1990). The magnetic arms are stillpresent at r . 12 kpc, where solid-body rotation is not evensuspected. Nevertheless, MHD density waves can be importantfor the global magnetic patterns in galaxies, and they should beincorporated into galactic dynamo theory.

ACKNOWLEDGMENTS

I am grateful to R. Beck, E. M. Berkhuijsen, A. Brandenburg,M. Krause, D. Moss, D. Schmitt, D. D. Sokoloff andK. Subramanian for numerous helpful discussions and criticalreading of the manuscript.

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This paper has been typeset from a TEX=LATEX file prepared by the author.

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