Initial Magnetization of Galaxies by Exploding, Magnetized Stars

arXiv:0712.1540v2 [astro-ph] 14 Dec 2007

K. Kowalik

1

, M. Hanasz

1

1

TorunCentreforAstronomyofNi olausCoperni usUniversity Ka per.Kowalikastri.umk.pl,Mi hal.Hanaszastri.umk.pl

We ondu t a series of magnetohydrodynami al (MHD) simulations of magnetized interstellar medium (ISM)disturbedbyexplodingstars. Ea hstardepositsarandomlyoriented,dipolarmagneti eldintoISM. The simulations areperformedina Cartesian box, ina referen eframe that is orotatingwith thegala ti disk. Themedium isstratiedbyverti al gala ti gravity. Theresulting turbulent stateof ISMmagnetized bythestellarexplosionsispro essedwiththeaidofFourieranalysis. Theresultsleadstothe on lusionthat the input of magneti energy from exploding stars is additionally multiplied by dierential rotation. The resulting magneti eldappearsto growupinsmall-s ale omponent,while thetotal magneti uxremains limited. Our results indi ate thatmagneti eld originating from explodingstars an be asour e of initial magneti eldsfor a subsequent dynamo pro ess.

Introdu tion

There is a strong observational eviden e that magneti elds are present in virtually all galaxies. It is ommonlybelievedthatthose eldsaregenerateddue toan

αω

dynamo pro ess,wheredierential rotation

(ω)

and heli alturbulen e

(α)

areresponsible for reatinga strong,larges alemagneti eld froma weak, smalls ale initial one [12℄. The dynamo an amplify and restru ture the magneti eld (see i.e. [16 ℄ for a re ent review of gala ti dynamo theory), yet it annot reate a new one, thus a seed eld is required. Although,the originoftheseed, magneti eldis a mysteryyetto be solved,a few theories on erning the problem exist. One of these theories [14 ℄ points to the very rst generation of stars asa possible sour e of theseed, magneti elds.

Even if a star is born without any primordial magneti eld, any nonparallelism between the gradient of pressure and the gradients of thermodynami al quantities like density or temperature, results in non vanishing

∇ ×

E

(more details in[11 ℄). That, a ordingto Faraday'slaw, implies timedependent magneti eld. This ee t is known as Biermann battery pro ess [1 ℄. Eventually, the newly reated magneti eld is amplied by a stellar dynamo. If, during its evolution, the star explodes as a supernova or undergoes a signi ant massloss, the frozeninplasma magneti eld is spread throughout the ISM, initiating the

αω

dynamo. The aim of this paper is to verify experimentally the hypothesis presentedby Rees [14 ℄ that young galaxieshave beenmagnetized bypro essesofstellar origin. Assuggested in[14 ℄, ifwe onsiderthat supernova remnant (like the Crab Nebula) depositsuxof order

10

34

G m

2

,then

N

su h remnantswould in rease the net ux in galaxy by a fa tor

N

x

,where

x ∈ [1/3, 1/2]

. As far as the authors know, nobody hasevertried to test this hypothesis ina numeri al experiment (however, a paper on erning quitesimilar problem was re ently published [5℄). Although this quantitative estimation seems to be onrmed by the results ofthis paper,some qualitatively newee ts arebeing found.

Physi al setup and numeri al model

We assume that gas forming gala ti disk is ompletely ionized, and apply the standard set of MHD equations (see[9℄),supplementedwiththeverti al gravitationala elerationandrotational pseudo-for esin

theequationof gasmotion

∂v

∂t

+ (v · ∇)v = −

1

̺

∇



p +

B

2

8π



−

_{2Ω × v + Ω}

2

_xˆ

_{x − g}

_z

₊

(B · ∇)B

4π̺

,

(1)

∂̺

∂t

+ ∇ · (̺v) = 0,

(2)

̺

 ∂ǫ

∂t

+ v · ∇ǫ



+ p∇v = 0,

(3)

∂B

∂t

= ∇ × (v × B),

(4)

∇ ·

_B

_{= 0,}

(5)

j

=

c

4π

∇ ×

B,

(6)

withadditionof the adiabati equation ofstate

p = (γ − 1)̺ǫ

(7)

with

(γ = 5/3)

.

Tosolvethesetofpartialdierential equationsnumeri allywe applyourownparallelized3DMHD ode based on the relaxing TVD s heme [10 ℄, whi h is des ribed in details by Tra and Pen [15℄ and extended for MHDsystem of equations by Pen et al. [13℄. The algorithm of magneti eld evolution, based on the onstrainttransport(CT)algorithm[2℄,preservesthedivergen e-freemagneti eldatthema hinea ura y. We hoseareferen eframe orotating withthedisk,at the

R

0

= R

⊙

(where

R

⊙

isSun'sgala ti radius) and use, inaddition to rotational pseudo-for esthe shearing-periodi boundary onditions [8℄,whi h area modi ation ofperiodi boundary onditions,thatis designed to modeldierentially rotatingastrophysi al disks. For further details on erning shearing box see Gressel and Ziegler [4℄. We introdu e the lo al referen e frame by adding the terms of Coriolis for e

2Ω × v

and the tidal expansion of the ombined, ee tive entrifugal and gravitational potential about

R

0



Ω

2

_x

,to the equation of motion (1). Following Ferriere [3℄ theverti al omponent of gala ti , gravitational a eleration (

g

z

term in(1 )) an be expressed as

−

_g

_z

_(R

_⊙

_{, z) = (4.4 · 10}

−

9

ms

−

2

₎

z

pz

2

_{+ (0.2}

kp

)

2

+ (1.7 · 10

−

9

ms

−

2

₎

z

1

k

pc

(8) Numeri al simulations

Weperform numeri al simulations oftheinterstellar medium des ribed above,perturbed withrandomly distributedmagnetizedsupernova(SN)explosions. Ea hstellar explosiondepositsthedipolarmagneti eld within a spheri al region of radius 10 p . The omputational domain represents a re tangular region of

0.5kpc × 0.5kpc × 1.5kpc

in

x

,

y

and

z

dire tions respe tively, and the grid resolution is

125 × 125 × 375

ells. Weassumethatstellar explosionsareuniformlydistributeda rossthegala ti plane, whereasverti al distribution is normal, with

σ = 100

p . Inthe present lo alapproximation we negle t the ee t of spiral arms, sin e our omputational domain overs only a small volume of the gala ti disks. In this approa h one ould onsider a time modulation of the supernova rate, orresponding to the passages of spiral arms throughthe omputationalvolume,however,wedonotexpe tasigni antee tsofthismodulationonlong times ales. Ea h explosion is realized by adding thermal energy to the gas insphere of radius

10

p . The explosionenergyiss aleddownbyseveralordersofmagnitudewithrespe ttotherealSNenergyoutput,due to limitations ofthe present versionof our ode. Furthermore, ea h explosiondeposits arandomly oriented (dire tions distributeduniformlyonsphere) dipolarmagneti eld

B

dip

= ∇ × A

,where

A(r, ϕ, θ) = A

0

r sin θ

(l

2

_{+ r}

2

_{+ 2rl sin θ)}

3/2

e

ϕ

(9)

where

A(r, ϕ, θ)

is a ve tor potential of a dipolar magneti eld reated by an ele tri urrent in toroidal ir uitofnaldiameter

l

[9℄, orrespondingtotheabovementionnedsizeofSNremnant. Theassumeddensity inthegala ti planeis

̺

0

= 0.32564

M

⊙

/

p

3

_∼

₁₃

atom/ m

3

andstarexplosionsrateis

σ = 20

kp

−

2

Myr

−

1

. Bothquantities

ρ

0

, σ

are derived fromre ent observationaldata [3℄.

Results

Inthis se tionwe dis usstheevolutionoftheinterstellar mediumwhi hissubje ttoa gradual magneti-zation byexplodingstars. A typi alsnapshotsdisplaying greys ale oded gasdensityand magneti ve tors

B

at

t = 30

Myrareshown inFig.1. A ording to expe tations, magneti eldinthe diskvolume displays a random onguration, whi h results asa superpositionof randomly oriented small-s aledipolar magneti elds. Theu tuationsofgasdensityresultfromtheinputofmagneti andthermalenergyinea hexplosion region.

Aswe anseeinFig.2,the exponential growthofthe meanmagneti uxisvisibleduringtherst phase of the simulation. However, after roughly

60

Myr magneti ux ease to grow, whereas magneti energy ontinues to grow due to the ongoing SN explosions a tivity. We show in Fig.3 a plot of total magneti energy s aled to the suppliedmagneti energy,and spe trumofmagneti energy u tuations, as afun tion of time.

As it is apparent in Fig.3, the total magneti energy grows faster than one would expe t from simple summation of magneti energies fromindividualexplosionevents. Thegrowth ofmagneti energy is appar-entlyenhan edbydierentialrotation, whi hamplies thetoroidalmagneti eld omponent viastret hing theradialmagneti eld. Thisee t isdes ribed bythe indu tion equation(4), whi h impliesthefollowing approximated equation fortheazimuthal magneti eld

∂B

ϕ

∂t

≃

GB

r

,

(10)

where

G = rdΩ/dr ≃ Ω

isthe measureofdierential rotation. Sin ethegala ti angularvelo ityapplied in oursimulationis

Ω = 0.05

,thetoroidalmagneti eldisgeneratedona times aleof20Myr. Therespe tive growthtimeofmagneti energyshouldbetwi eshorter. Sin ethedipolarmagneti eld,issuppliedintothe initiallyunmagnetizedmedium,thegrowthofmagneti energyisinitiallyslow,butlaterontheampli ation of magneti eldbydierential rotationspeeds up. Asit is apparent inFig.3, the observed growth timeof magneti energyis onsistent withour estimation.

In theright panel of Fig.3 we show the spe trum of magneti energy at

t = 25

Myr and

t = 119

Myr, along

x

and

y

dire tions,obtained bymeansofFourieranalysis. Thetwo straight lines orrespondingto the slope-5/3, shownfor omparison, represent the Kolmogorov's spe trum. The spe tralanalysisof magneti energy intwodierenttimeinstantsshows aweektenden yof steepeningof thespe trumofmagneti eld u tuationsalongthe

x

-dire tionandatteningin

y

dire tion. TheresultspresentedinFig.3meanthatthe spe trumofmagneti u tuations, whi hisstronglyanisotropi at thebeginningoftheexperimentbe omes more and more isotropi in ourse of time. The overall spe trum of magneti u tuations at the end of our simulation remains relatively at withrespe t to Kolmogorov's spe trum. Magneti energy umulated on small spatial s ales remains large in omparison to the energy on large s ales. Although the simulation periodof 120Myr isstillshort withrespe tto thegala ti rotation periodoftheorder of200 Myr,one an saythattheevolutionof magneti spe trumisrather weak.

Con lusions

Our results indi atethatthe stru ture of stellarorigin gala ti magneti elddoesnot tto the urrent pi ture of polarimetri radio-observations of disk galaxies (see [16 ℄ for referen es on observational results of gala ti magneti elds). The resulting magneti eld onguration an serve asan initial ondition for further explorationof

αω

dynamo pro ess.

In thenext step we plan to extend our physi al setup withthe osmi ray omponent, des ribed by the diusion-adve tionequation, asithasbeendone byHanaszandLes h[7℄. Thepresen eof osmi raysleads inevitably to the Parkerinstability, anda very e ient

αω

-dynamo pro ess [6℄. We also planto extend the timeofthesimulationsto atleast

1

Gyr,and tosimulate thewholegala ti diskinfull3D,insteadofusing lo al approximation. Summarising, notethe following ee ts of random, magnetized supernova explosions inthedierentiallyrotating interstellar medium:

•

limitedgrowth ofmagneti uxa ompaniedwith umulation ofenergy insmall-s alemagneti elds;

•

an additional ee t ofmagneti eldampli ation bydierential rotation;

•

arelativelyslowevolutionofmagneti spe trum,indi atingthatasubsequentdynamopro essinvolving

A knowledgements

This work wassupported by the Ministryof S ien e and Higher Edu ation of Poland through thegrant 1/P03D/004/26. Thepresented omputationshavebeenperformedontheHYDRAbeowulf lusterinTorun Centre for Astronomy.

Referen es

[1℄ BiermannL.,S hlúterA.Phys.Rev.,V.82,p.863(1951)

[2℄ EvansC.R.,HawleyJ.F.ApJ,V.332, p.659(1988)

[3℄ FerriereK.ApJ,V.497,p.759(1998)

[4℄ GresselO.,Ziegler,U.,CoPhC,176,652,(2007)

[5℄ GresselO.,ZieglerU.IAUS,V.237,p.415(2007)

[6℄ HanaszM.,KowalG.,Otmianowska-MazurK.,Les hH.ApJ,V.605,p.L33(2004)

[7℄ HanaszM.,Les hH.A&A,V.412,p.331(2003)

[8℄ HawleyJ.F.,GammieC.F.,BalbusS.A.ApJ,V.440,p.742(1995)

[9℄ Ja ksonJ.D.,"Classi alEle trodynami s"(1999)

[10℄ JinS.,XinZ.Comm.PureAppl.Math.,V.48,p.235(1995)

[11℄ KempJ.C.PASP,V.94,p.627(1982)

[12℄ ParkerE.N."Cosmi almagneti elds: Theirorigin andtheira tivity"(1979)

[13℄ PenU.-L.,ArrasP.,WongS.ApJS,V.149,p.447(2003)

[14℄ ReesM.J.QJRAS,V.28, p.197(1987)

[15℄ Tra H.,PenU.-L.PASP,V.115, p.303(2003)

Figure 1. Sli esthroughthe omputational domainshowing densitymagneti eldat

T = 30

Myr. The leftpanelshowsaverti alsli ethroughthedomainat

y = 0

p ,whiletherightpanelrepresentsahorizontal sli eat

z = 0

p .

Figure 2. Thetotalmagneti energy(leftpanel) andevolution ofthemeanmagneti uxintime(right panel).

Figure 3. Temporal evolutionofthetotalmagneti energy s aled tomagneti energy suppliedin super-nova remnants(left panel)and magneti energy spe trumanalyzedalong

x

and

y

dire tions(right panel).