Generation of toroidal magnetic fields in accretion disks

0.55 MB·

by  Mohammadtaher Safarzadeh

Mon.Not.R.Astron.Soc.000,000–000(2017) Printed17January2017 (MNLATEXstylefilev2.2)
Generation of toroidal magnetic fields in accretion disks
Mohammadtaher Safarzadeh1(cid:63), Smadar Naoz2, Alexander Sadowski3,
Lorenzo Sironi4, Ramesh Narayan5
1SchoolofEarthandSpaceExploration,ArizonaStateUniversity,Tempe,AZ85287-1404,USA;
2DepartmentofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles,CA90095,USA
3MITKavliInstituteforAstrophysicsandSpaceResearch,77MassachusettsAve,Cambridge,MA02139,USA
4DepartmentofAstronomy,ColumbiaUniversity,550W120thSt,NewYork,NY10027,USA
5Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
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0 17January2017
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n
ABSTRACT
a
Magnetic fields play an important role in the dynamics of accretion disks, however, the ori-
J
ginofthefieldsisoftenobscured.Hereweshowthatmagneticfieldscanbegeneratedinan
3
initiallynon-magnetizedaccretiondisksthroughtheBiermannbatterymechanism,wherethe
1
radialtemperatureprofileandtheverticaldensityprofileofthesesystemsprovidetheneces-

sarilyconditionsforthisprocesstooperatenaturally.Weconsiderthegenerationoffieldsina
]
E protoplanetarydisksanddisksaroundBlackHoles(BHs).Forprotoplanetaryaccretiondisks
H we find that the generated magnetic fields can be as strong as few Gauss over the lifetime
of the disk (5 Myr), for a solar mass star, and weakly dependent on the accretion rate. The
.
h generatedseedshavetoroidalstructurewithoppositesignintheupperandlowerhalfofthe
p disk. The same pattern exists in a thin accretion disk around a rotating BH, where the field
-
o generationrateincreasesforlargerBH’sspinparametersinaco-rotatingdiskandspinconfig-
r uration.Atafixedr/risco,whereristheradialdistancefromtheBH,thebatteryscalesasM−2
t forboththinaccretiondisksandAdvectionDominatedAccretionFlows(ADAF).ForADAFs
s
a afteronecharacteristicaccretiontimethebatterycangenerateafieldinthegalacticcenterat
[ theorderof5×10−6G.Inaddition,wetestthismechanisminGRMHDsimulationsandfind

ittobeinagoodagreementwithouranalyticalestimates.Thefactthatthebatteryworksin
1
v thelimitofzeroaccretionrate,itmakesthismechanismaviablecandidatetoprovidetheseed
0 fieldsfromaninitiallynon-magnetizedaccretiondisksuchthatlateronMagnetoRotational
0 Instability(MRI)takesover.
8
3 Keywords: accretion,accretiondiscs—dynamo—instabilities—magneticfields—MHD
0
.
1
0
7 1 INTRODUCTION turbulentdynamoduringthegravitationalcollapseforthefirstgen-
1 eration of protostars (e.g., Schleicher et al. 2010; Schober et al.
v: Accretiondisksareassociatedwithawiderangeofastrophysical 2012).Althoughthesemechanismscanplayaroleinprovidingthe
phenomena from disks around protostars to Black Holes (BHs).
i seed magnetic fields, here we propose that magnetic fields in ac-
X These disks are assumed to be magnetized and the accretion to
cretiondiskscanbenaturallyformedthroughtheBiermannbattery
happenthroughtheMagnetoRotationalInstability(MRI)process
r process(Biermann1950).
a (Balbus&Hawley1991).However,theoriginofthemagneticfield
inthediskisrarelydiscussed.Thereareafewapproachesthatare
Biermann(1950)showedthatifaplasmahasarotationalmo-
adoptedintheliteraturetoaddressthesourceofmagneticfieldin
tion, current must exists, which leads to the generation of mag-
accretiondisks.ForthestellarmassBHs,themagneticfieldistypi-
netic fields. This process has a wide range of applications from
callyassociatedwiththemagneticfieldoftheprogenitorstar(e.g.,
the generation of magnetic fields in stars (e.g., Biermann 1950;
Bisnovatyi-Kogan & Ruzmaikin 1974, 1976), while for a super-
Doi&Susa2011)togalacticscalemagneticfields(e.g.,Mestel&
massiveBHsthefieldisbelievedtobeaccretedfromthesurround-
Roxburgh1962;Widrow2002;Subramanianetal.1994;Subrama-
inggas(Narayanetal.2003;Eatoughetal.2013).Themagnetic
nian2010;Naoz&Narayan2013).Forexample,Naoz&Narayan
fieldsaroundthefirstgenerationofprotostarscanbetheresultof
(2013)showedthatmagneticfieldsarenaturallygeneratedinthe
largescalecascades,whereweakmagneticfieldsareamplifiedvia
earlyuniverse,atintergalacticscales,whichisconsistentwiththe
observedfields’strength.TheBiermannbatterywasalsoshownto
beabletogeneratemagneticfieldsinsidesupernovabubbles(e.g.,
(cid:63) E-mail:[e-mail] Hanayamaetal.2005).
(cid:13)c 2017RAS
2 Safarzadeh,Naozetal.
Forthegenerationofseedmagneticfieldsinthismechanism, asthetemperaturesofthedisksareestimatedfromfewthousands
anonalignedgradientofelectronsdensityandgradientofelectrons KelvinintheinnerregionsofthediskdowntotensofKelvininthe
pressureisneeded.Thissituationnaturallyarisesinthestructureof outerparts.However,asmentionedabove,theBiermannbatteryis
accretiondiskswherethepressuregradientismostlyinthevertical independentontheionizationfraction.Weadoptasimpledensity
directionandtemperaturehasaradialgradientinthedisk.These andtemperatureprofileforageometricallythin,Keplerianproto-
seedmagneticfieldscangrowinlineartimescalesandinadiffer- planetary disk, (for review of the derivation see Armitage 2013).
entially rotating disk can lead to a strong magnetohydrodynamic Theaxisymmetricdiskisdescribedbyitsverticalandradialstruc-
(MHD) instability through MRI that makes the field to grow ex- ture. Assuming a hydrostatic equilibrium the vertical structure is
ponentially with time to reach its equipartition value in the disk determinedbythepressuregradientequationinthezˆdirection
(e.g.,Bret2009).Herewefocusongeneratingseedmagneticfield
dP
in an initially non-magnetized accretion disk. We do not specify =−ρΩ2z, (5)
dz K
thenaturethatcausestheaccretionandweassumethataccretion
whereΩ istheKeplerianfrequencydefinedas:
takesplacewhilethediskisnotinitiallymagnetized(i.e.,MRIdoes K
notoperate).Turbulentviscositycouldbethoughtofaasplausible (cid:114)
GM
mechanismforourpurposes(e.g.,Shakura&Sunyaev1973).As ΩK = r3 (6)
wewillshowbelow,generationoftheseedmagneticfieldsthrough
Assuminganidealgasequationofstate,thedensityprofilein
Biermannbatterydependsonlyveryweaklyontheaccretionrate.
theverticaldirectionis
Therefore,aprioriwedonotrequireMRItobetakingplacetopro-
videaccretionforthebatterytowork. ρ= √Σ e2−Hz22 , (7)
Shiromoto et al. (2014) studied the generation of magnetic 2πH
fields in a first generation proto-circumstellar disk based on 2D WiththescaleheightdefinedasH=c /Ω andthespeedofsound
s K
radiation hydrodynamics simulations. Unlike their work, here we
hasonlyaradialdependency(e.g.,Armitage2013)
study a range of applications from generation of seed magnetic
(cid:115)
fieldsaroundaprotostar,tosupermassiverotatingblackholeatthe k T(r)
c = B . (8)
centerofthegalaxy. s µm
p
Thepaperisorganizedasfollow:In§2weintroducetheBier-
mannbatteryequation.In§3weapplytheBiermannbatteryequa- Thesurfacedensity,Σ,isgivenby
tion in protoplanetary disk setting. In §4 the battery is applied in  (cid:114) 
thinaccretiondiskaroundKerrblackholes.In§5wecomparethe Σ(r)= 3Mπ˙ν1− Rr∗ , (9)
ratespredictedforADAFswithonesobtainedforstate-of-the-art
generalrelativistic(GR)MHDsimulationsofradiativelyinefficient whereν = Hc α(Shakura&Sunyaev1973).αisadimensionless
s
accretionflowsandIn§6wesummarizeandgiveconclusions. parameterandourresultsdoesnotdependonitsspecificvalue.
Thetemperatureradialprofileisdeterminedbytheaccretion
rateandandthebalanceofheatingandcoolinginthediskwhich
2 BIERMANNBATTERY leadstoradialeffectivetemperatureprofileof
 (cid:114) 
Tnohne-vgaenniesrhaitniogncroofssmpargondeuticctofifeledlsecitnroBnienrummabnenrbdaetntesirtyyigsradduieentto, Te4ff(r)= 38GπMσrM3˙ 1− Rr∗ . (10)
∇n andelectronpressuregradient,∇P
e e whereσistheBoltzmannconstant.Wecomputethebatteryatz=
Hforallthecalculations.Tokeeptheanalysissimple,wesetT ∼
∂B ∇n ×∇P e
=∇×(u×B)−c e e . (1) Teff ineq(4)andT ∼Teff ineq(8).1Since∇ρand∇Tebothhave
∂t en2
e rˆandzˆcomponents,thefieldisgeneratedinthetoroidaldirection,
Assumingidealgasequationofstate, P = n k T andassuming i.e.,
e e B e
that the temperature of the electrons is equal to that of the gas, (cid:32) (cid:33)
∂B ck ∂T ∂ρ ∂ρ∂T
T =T wecanre-write∇n ×∇P as =− B e − e φˆ (11)
e disk e e ∂t eρ ∂r ∂z ∂r ∂z
∇n ×∇P =∇n ×(n ∇T +T ∇n )k , (2)
e e e e e e e B The Biermann battery will work so long as the density and
where k is Boltzman constant. Given n = χ ρ, where χ is the temperaturegradientsarenotparalleltoeachother.Thiscondition
B e e e
ionizationfraction,wefind willalmostalwaysbesatisfiedinadisk.Toavoidgettingintode-
∇n ×∇P =n k ∇n ×∇T =n k χ ∇ρ×∇T , (3) tails,weestimatetheorderofmagnitudeoftheBiermannbattery
e e e B e e e B e e
bycrossingtheverticalgradientofdensitywiththeradialgradient
Wherewithouttheinductiontermwehave oftemperature.ThereforeEquation(11)canbewrittenas:
∂B =−ckB∇ρ×∇Te (4) ∂B ck (cid:32)∂T ∂ρ(cid:33)
∂t e ρ =− B e φˆ, (12)
∂t eρ ∂r ∂z
Notethattheionizationfraction,χ ,cancelsoutandthustheequa-
e
which,canbewrittenas:
tiondependsonlyonthedensityandtemperatureofthesystem.
√
∂B cGMm µ7R +r( R /r−6)
= p ∗ ∗ zφˆ (13)
∂t 8e r4(r−R )
∗
3 PROTOPLANETARYDISK
Planetsareformedfromtheprotoplanetarydisksmadeofgasand 1 Thediskmid-planetemperatureisoftenjustafactorofafewlargerthan
dust surrounding young stars. These disks are not fully ionized, Teff.Wefeelthereforethatthisapproximationisreasonable.
(cid:13)c 2017RAS,MNRAS000,000–000
Generationoftoroidalmagneticfieldsinaccretiondisks 3
����-��
�=�
�=�(���������) ��-��
�=���
����-�� �=-�(���������)
�=����
(ϕ)(/)��� � |/∂(/)���� ��-��
∂| ��-��
-����-��
��-��
-����-��
��� ��� ��� ��� ��� ��� ���
�/�* ��� ��� ��� ��� ��� ��� ���
�
Figure1.Magnitudeoftherateofmagneticfieldgenerationasafunction
ofr/R∗fora1M(cid:12)staratz=Honthedisk.Generatedfieldschangetheir �����
signatrturn =49/36R∗.Theseedshaveanoppositesignontheupperand Figure 2. Shows the magnitude of magnetic field generation rate as a
lowerhalfofthedisk.TherateofseedgenerationthroughBiermannbattery function of r/risco for a 4 × 106M(cid:12) black hole with accretion rate of
fwoeraaklpyrodteoppelnandeetnatryondtihskeiasccprreotpioonrtiroanteal(∝toMt˙h1e/8a)c.cWreetinadgomptaµss=M2a.3ndinotnhliys Md˙er=so1f0m−1a0gMni(cid:12)t/uydeeasrtrfoonrgdeiffreinrean=t0sp.9in9pcaarsaemceotmerpsa.rTehdewbiathttearyScishawbaoruzstc2hoilrd-
calculation. blackholeatallradii.Theradiusatwhichthegeneratedseedfieldschange
signinazimuthaldirection(rturn)becomessmallerasthespinparameteris
Notethatthemagneticfieldchangesitsazimuthaldirectiongoing increased.rturn =2×riscoforaSchwarzschildblackhole.Wesetz=Hin
thiscalculation.Thefieldchangesitssignacrosstheequatorialplaneinφˆ
fromtheuppertothelowerhalfofthedisk.Thereisnodependence
direction.
onaccretionrateinthisequation,however,aswecomputethebat-
teryatz=H,itwouldmeanaweakdependenceonaccretiondate
(∝M˙1/8). which assuming a hydrostatic equilibrium leads to the following
The magnetic field generated over an accretion timescale densityprofile
(Trh/eurb)aitsterayt cthaenoorpdeerratoefov5e0rµtGhefolirfeatim1eMo(cid:12)fsthtaerdwisikthanRd∗c=an1reRac(cid:12)h. ρ= √Σ e2−Hz22BC (cid:114)C , (15)
aboutfewtotensofGaussinlargepartsofthedisk.InFigure1we 2πH B
showtheseedsgenerationrateinaproptoplanetarydisk.Theverti- where
ci.ael.,dzist=ancHe.adTohpetegdenineroauterdcasleceudlasticohnasnsgcealtehseiarsstihgenscfraolemhueipgphetr, Σ= M˙ (cid:32)1− (cid:114)risco(cid:33) , (16)
half to lower half of the disk and also at the characteristic radius 3πν r
rturn=49/36R∗. and H,ν,Σ are a function of r and a where a is specific angular
We note that the change of azimuthal sign of ∂B/∂t at rturn momentumoftheblackhole.
arisesbecauseTe hasamaximumatthisradius.Thisispredicted Thetemperatureprofilefromthebalanceofheatingandcool-
bythethindiskmodel,butwhetherthisexactshapeisfollowedby inginthediskis
realdisksisdisputed.
Theequipartitionfieldcanbeestimatedbyequatingthemag- T4(r)= 3GMM˙ (cid:32)1− (cid:114)risco(cid:33)D (17)
netic pressure ∼ B2/8π and the pressure of the ionized gas. For e 8πσr3 r B
protoplanetarydisksitisintheorderof103GatthevicinityofR ,
∗ A,B,CandDarerelativisticcorrection(Novikov&Thorne1973;
whichisordersofmagnitudelargerthantheBiermannseedfield.
Page&Thorne1974;Doerreretal.1996)definedas:
Furthermore,wealsoestimatewhenMRIstartstodominatedthe
accretionprocess(cid:112).Takingthemostunstablewavelength,∼2πvA/Ω, A = 1− 2GM + a2
where v = B/ 4πρ2 is the Alfven wave velocity, and equating c2r c2r2
A √
it to H = cs/Ω, for z ∼ H, we find magnetic field peaking at B = 1− 3GM + 2a GM
r/R(cid:12) ∼ 1.5, with strength of ∼ 200 G, which is order of magni- c2r c2r3/2
√
tudelargerthantheBiermannfieldatthisradiusafter5Myr,and 4a GM 3a2
staysinsimilarorderofmagnitudedifferenceatallradii. C = 1− +
c2r3/2 c2r2
√
1 (cid:90) r x2c2−6xGM+8a xGM−3a2
D = √ √ (cid:16) √ (cid:17) dx . (18)
4 SEEDSINDISKAROUNDKERRBLACKHOLES 2 r risco x x2c2−3xGM+2a xGM
Following Shakura & Sunyaev (1973) and Novikov & Thorne whereriscoistherootofthefollowingequationforthecaseof
(1973),theverticaldensityandradialtemperatureprofilesofthese aco-rotatingdisk:
disksisdescribedbysimilarequationsaspresentedabove,withthe (cid:114)
MGc a MGc a
followingrelativisticcorrections: 1−6 +8 −3 =0 (19)
r c r3 cr
dP GM C
=−ρ z , (14) Here we explore a simple example of a thin accretion disk
dz r3 B
withaconstantverticaltemperatureprofile(thus,Equation(12)is
applicable)inthegalacticcenter.Wenotethatthetheaccretionsys-
2 Forsimplicityweassumeacompletelyionizedgas. temaroundthesupermassiveblackholeatthecenterofourgalaxy
(cid:13)c 2017RAS,MNRAS000,000–000
4 Safarzadeh,Naozetal.
expressthediskprofileusingthefollowingscalingrelations:

��-�� �=������=��-��
 2 γ−1
�=������=��-� c2 = v2 (20)
 s 3γ−5/9 k
�=������=��-��
��-�� (cid:32) γ−1 (cid:33)
/)�� vr = − γ−5/9 αvk (21)
(
|/∂�� ��-�� M˙
∂| ρ = − , (22)
4πRHv
r
��-�� wherethescaleheightHisalongthezcoordinate,v istheradial
√ r
componentofthevelocityofthegas,v = GM/ristheKeplerian
k
velocity,andγisthespecificheatofthegasanditsvalueisbetween
��-��
��� ��� ��� ��� ��� ��� ��� 4/3to5/3,foraradiation-pressuredominatedandanon-relativistic
�
gas-pressuredominatedaccretionflow,respectively.Weassumea
����� gaspressuredominateddisk,i.e., P = c2ρ,andthedensityofthe
s
Figure3.Showstheabsolutemagnitudeofmagneticfieldgenerationrateas gasrelatestotheelectronnumberdensityvia:ne = χeρ/µ,where
afunctionofradiusfromthelaststablecircularorbit(risco)aSchwarzschild χeistheionizationfractionandµisthemeanmolecularmass.We
black hole with different values of M in M(cid:12) units and M˙ in (M(cid:12)/year) adoptµ=1.18mpandχe =1wherempisthemassoftheproton.
units.ThesolidlinecorrespondstoaSchwarzschildblackholewithM = UsingEquation(1)wefindthat
4×106M(cid:12) andM˙ = 10−10M(cid:12)/year.Thedashedlinecorrespondstothe
samemassbutwith10timeshigheraccretionrate.Thedottedlineisfora ∂B ∼ c ∂ρ ∂c2sφˆ ∼ µ6cGM(γ−1)φˆ, (23)
blackholewith10timessmallermassbutthesametheaccretionrate.At ∂t en ∂H ∂r er3(9γ−5)
e
afixedr/riscothebatteryisproportionaltoM−2andisweaklydependent whereinourlasttransitionwehavepluggedintherelationin
onaccretionrate(∝M˙1/8).Theradialturningpoint(rturn)isindependentof
bothmassandaccretionrate.Weassumez=Hinthiscalculation. Equations(20)-(22).NotethatinourADAFmodelthescaleheight
H describes the vertical component, and due to the “puffed up"
natureofthediskwesetH∼r.Thishassimplescalingof:
∂B (cid:32) r (cid:33)−3(cid:32) M (cid:33)−2 γ−1
∼3.6×106 G/sec, (24)
∂t r M (9γ−5)
sc (cid:12)
isprobablybestmodeledasanADAF(e.g.,Narayan&Yi1994;
Narayanetal.1995;Abramowiczetal.1988),whichareexpected wherer = 2GM/c2 istheSchwarzschildradius.Foracharacter-
sc
toresultinanevenlargerfieldstrength.WediscussADAFsinsec- isticaccretiontimet ∼r/v wefind:
acc r
tion5. (cid:32)M (cid:33)(cid:18)r (cid:19)3/2 γ−1
Figure2showsthemagneticfieldgrowthratefora4×106M B∼17.9 (cid:12) sc G, (25)
(cid:12) M r (9γ−5)
black hole with different spin parameters as a function of r/r .
isco
Thebatteryisstrongerforblackholeswithlargerspinparameterat whichisordersofmagnitudesmallerthantheequipartitionfieldin
afixedr/r . anADAFwhichcanbeestimatedas:
isco
We find that the Biermann battery in the thin accretion disk M (cid:18)r (cid:19)5/4
modelisweaklydependentonaccretionrate(∝M˙1/8atz=H)and Beq ∼ 7.8×107G M(cid:12) rsc ×
proportionalto M−2 atafixedr/risco.ThisisillustratedinFigure (cid:32) M˙ (cid:33)1/2(cid:32) 1−γ (cid:33)1/2
3.Asinthecaseofprotoplanetarydisk,thegeneratedseedshave . (26)
10−10M /yr α(5/9−γ)
oppositesignintheupperandlowerhalfofthedisk.Thechangeof (cid:12)
azimuthalsignofdB/dtatrturnarisesbecauseTehasamaximumat Fromtheserelationsweseethatforextremelylowaccretionrates
thisradiuswhichispredictedbythethindiskmodel(seeShafeeet (∼ 2.3×10−17(r/r )1/4 M yr−1)thetwomagneticfieldstrengths
sc (cid:12)
al.(2008);Nobleetal.(2009);Kulkarnietal.(2011)foradiscus- equate.
sionofthezero-torqueconditionattheISCOandtheeffectithas Wenowcomparetheestimatesofthegrowthrateoftheaz-
on the viscous heating profile in a thin accretion disk). However, imuthalcomponentofthemagneticfieldduetothebatteryobtained
Bφ changesitssignacrossthemidplaneandthisresultisarobust aboveundersimplifyingassumptions,withratescalculatedusinga
predictionofthetheory. moredetailedaccretionflowstructureproducedbyGRMHDsim-
ulations. For this purpose, we took the model of radiatively inef-
ficient,two-temperatureaccretionflowpresentedanddiscussedin
Sadowskietal.(2016)(modelRad8).Thisparticularsolutionde-
scribesanaccretionflowonanon-rotating,10M BHwithaverage
(cid:12)
accretionrateoftheorderof4×10−9M˙ .Tocalculatethebattery-
Edd
relatedgrowthofthemagneticfieldwetookthetime-averagedout-
5 ADVECTION-DOMINATEDACCRETIONFLOWS
putofthesimulationandappliedEq.(4).Resultsarepresentedin
DISKS
Fig4.
Inmanyaccretiondisksaroundblackholestheviscouslydissipated The top panel presents the growth rate multiplied by radius
accretionenergycangointoheatingtheaccretionflowratherthan cubed,i.e.,dB/dt(r/r )3,onthepoloidalplaneofthesimulation.
sc
beingradiatedaway.Thisisthemostimportantcharacterizationof The values are given in G/s. The most profound feature is the
advection-dominatedaccretionflows(ADAFs)disks(seeforrecent asymmetryagainsttheequatorialplane.Thisreflectsthefactthat
review: Yuan & Narayan 2014). Following Narayan & Yi (1994) thegradientsofdensityandtemperaturearesymmetricwithrespect
self-similarsolution,forα << 1andanaxis-symmetricflow,we to the equatorial plane, and therefore their cross product changes
(cid:13)c 2017RAS,MNRAS000,000–000
Generationoftoroidalmagneticfieldsinaccretiondisks 5
• Protoplanetary disks: Assuming a simple disk profile, we
foundthatatoroidalmagneticfieldscannaturallygrowfromini-
tiallynotmagnetizedaccretiondisksinprotoplanetarydisksover
thelifetimeofthedisk(attheorderoffewtotensofGauss,seeFig-
ure1).Thesefieldsarelinearlyproportionalwiththecentralmass
and weakly dependent of the accretion rate. The generated seeds
haveoppositesignsintheupperhalf(+φˆ)andlowerhalf(−φˆ)of
thedisk.Wealsofindthatthesefieldschangesignatacharacteris-
ticradiusofr =49/36R .Notethattheradialchangeofsignat
turn ∗
r arisesbecauseT ismaximumatthisradius,aspredictedbythe
turn e
thindiskmodel.However,thispictureisdisputedintheliterature.
• Thin accretion disk around a rotating BH: We adopted a
disk temperature and density profile, which includes relativistic
corrections (following Novikov & Thorne 1973; Page & Thorne
1974;Doerreretal.1996).Theseedshaveoppositesingintheup-
per and lower half of the disk and also change their sign in the
azimuthal direction at r = 2×r for a Schwarzschild black
turn isco
hole.LargerBHspinsyieldsmallervaluesofr andlargerrateof
turn
seedgeneration(seeFigure3).WenotethatthevanishingofT at
e
r=r ,(aspredictedbyShakura&Sunyaev1973)iscontrover-
ISCO
sialbecauseitcomesfromthezero-torqueboundaryconditionat
theISCOandfromignoringadvection.Thus,thebehaviorofsign
Fgiivgeunrein4.GT/hs,eignroawGthRrMatHeDofstihmeualzaitmiounthoaflamraagdniaettiivcefilyelidn,edffiBc/ident(tr/arcsccr)e3-, flipmaynotberealistic.Atafixedr/riscothebatteryscalesasM−2
andthebatteryisstrongerforBHswithlargerspecificangularmo-
tionflow(modelRad8fromSadowskietal.2016)fora10 M(cid:12) BH.The
toppanelshowsthegrowthrateinthepoloidalplane.dB/dtchangessigns mentum.Therelativisticdiskprofileleadstoaweakdependency
over the equatorial plane because of symmetry in gradients of density ontheblackholeaccretionrate∝M˙1/8(seeFigure3).
and temperature. The bottom panel shows the average absolute value of • ADAF: We also analyzed the generation of magnetic field
dB/dt(r/rsc)3averagedverticallyover−1<z/r<1. in an ADAF, where our analytic estimates, based on Narayan &
Yi (1994) self-similar solution, are in good agreement with with
GRMHDsimulations(seeFigure4).UsingEquation(25)wecan
sign.Forthesamereasonofdiskstructuresymmetry,thegrowth estimate the magnetic field generated after one characteristic ac-
rate at the equatorial plane is zero. Far from the plane, the field cretiontimeforthediskaroundthecentralBHinourgalacticcen-
does not change significantly and reaches values of the order of ter. We find that the Biermann battery can naturally give rise to
dB/dt(r/rsc)3≈1000G/s. ∼ 5×10−6 GforthesupermassiveBHatthecenterofourgalaxy
The bottom panel of Fig 4 shows the absolute value of the within one accretion timescale. After one year of operation this
growth rate averaged vertically over the region −1 < z/r < 1. It mechanismcannaturallyreachamagnitudeof7Gverynearthe
confirmsthetypicalvalueandshowsthatthegrowthratescalewell accretionradius(r ∼ r ),whichisattheorderofthefewGesti-
sc
withthethirdpowerofradius.Theaveragevalueshouldbecom- mationsofthemagneticfieldstrengthinthevicinityofthegalactic
pared to the analytical prediction based on Eq. (24) which gives, center(e.g.,Mos´cibrodzkaetal.2009).
for10M ,γ=5/3,andχ =1,
(cid:12) e
Wenotethatwhilethereisamplediscussionintheliterature
∂B(cid:32) r (cid:33)3=2400G/s. (27) abouttheroleofpoloidalfield(e.g.,Narayanetal.2003)insub-
∂t r sequentevolutionofaccretiondisks,thereislittlediscussionabout
sc
Bˆ .Ourstudyshowsthatthisfieldnaturallyevolvesindisksand
Havinginmindthatthisestimatewasderivedundersimplifiedas- φ
thereforeitwouldbeinterestingtosimulatetherolethatBiermann-
sumptions about the vertical and radial structure of ADAFs, one
generated Bˆ might play in the evolution of the accretion disks.
shouldconsidertheagreementverygood. φ
Thismayalsohelptobetterunderstandtheroleofazimuthalsign
reversalintheupperandlowerhalfofthediskandpotentiallyat
r=r .
turn
6 DISCUSSION
The generation of magnetic fields through the radiation pressure
7 ACKNOWLEDGEMENTS
on the electrons in the inner edge of the accretion disks has
beendiscussedinpreviousworks(Contopoulos&Kazanas1998; MSissupportedbytheNationalScienceFoundationundergrant
Bisnovatyi-Koganetal.2002;Contopoulosetal.2015).Inthislet- AST14-07835 and by NASA under theory grant NNX15AK82G.
terweshowedthatlargetoroidalmagneticfieldscouldnaturallybe SNacknowledgespartialsupportfromaSloanFoundationFellow-
generatedthroughBiermannbatteryprocessstartinginitiallyfrom ship.
zeromagneticfield.Theverticaldensityprofileofanaccretiondisk
andtheradialtemperatureprofilewhicharisesduetothebalance
between heating and cooling, lead to generation of toroidal seed
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