Rela%vis%cAstrophysicsandMagnetohydrodynamics

YosukeMizunoITP,GoetheUniversityFrankfurt

Spaciallecture“GRMHD”,August13th-17th,USP-IAG,SaoPaulo,Brazil

Lecture 1:

RelativisticRegime• Kineticenergy>>rest-massenergy

– Fluidvelocity~lightspeed(Lorentzfactorγ>>1)– Relativisticjets/ejecta/wind/blastwaves(shocks)inAGNs,GRBs,Pulsars

• Thermalenergy>>rest-massenergy– Plasmatemperature>>ionrestmassenergy(p/ρc2~kBT/mc2>>1)– GRBs,magnetarflare?,Pulsarwindnebulae

• Magneticenergy>>rest-massenergy– Magnetizationparameterσ>>1– σ=Poynitingtokineticenergyratio=B2/4πρc2γ2– Pulsarsmagnetosphere,Magnetars

• Gravitationalenergy>>rest-massenergy– GMm/rmc2=rg/r>1– Blackhole,Neutronstar

• Radiationenergy>>rest-massenergy– E’r/ρc2>>1– Supercriticalaccretionflow

Applica%onsofRela%vis%cAstrophysics

• BlackHoles:• high,lowaccre%onrateAGN• %daldisrup%onevent• X-raybinaries• long-soaGRBs• BH-BHmergerforGWsources

• Neutronstars:• pulsarmagnetosphere• core-collapsesupernova• short-hardGRBs• NS-NSmergerforGWsources

• Jets/rela%vis%cwind:• extra-galac%cjets/ouelows• pulsarjet/wind• microquasars• gamma-raybursts

• Laboratoryphysics:• rela%vis%cheavy-ioncollision• plasmalaboratoryexperiments

RelativisticJets• Relativisticjets:outflowofhighlycollimatedplasma• Microquasars,ActiveGalacticNuclei,Gamma-RayBursts,Jetvelocity～c

• Genericsystems:Compactobject（White

Dwarf,NeutronStar,BlackHole）+Accretion

Disk

• KeyIssuesofRelativisticJets• Acceleration&Collimation• Propagation&Stability

• ModelingforJetProduction• Magnetohydrodynamics(MHD)• Relativity(SRorGR)

• ModelingofJetEmission• ParticleAcceleration• Radiationmechanism

RadioobservationofM87jet

RelativisticJetsinUniverse

Mirabel&Rodoriguez1998

PlasmaDynamicsvicinityofBHandShadowPorthetal.(2017)

•Initial:Accretiontorus+weaksinglemagneticfieldloop•InsidetorusbecomesturbulentbyMRI•Poyntingfluxdominatedjetisdevelopedneartheaxis

CalculatedRadiationimagebyGRRTcode(Thermalsynchrotrontotalintensity)

• WecanobtainBHshadowimage,spectrum,lightcurve(+polarization)via3DGRMHDsimulations

density

EventHorizonTelescopeInternationalcollaborationprojectofVeryLongBaselineInterferometry(VLBI)atmm(sub-mm)wavelength

Createavirtualradiotelescopethesizeoftheearth,usingtheshortestwavelength

λ = 1.3 mm (ν = 230 GHz)D ~ 10,000 km => λ/D ~ 25 µas

Twomaintargets:SgrA*&M87

Animation:C.Fromm

Moscibrodzka + (2011)

EventHorizonTelescopein2017

• AtacamaLargeMillimeterArray(ALMA),Chile

• ALMAPathfinderExperiment(APEX),Chile

• JamesClerkMaxwellTelescope(JCMT),Hawaii

• LargeMillimeterTelescope(LMT),Mexico

• IRAM30-meterTelescope,Spain

• SouthPoleTelescope(SPT),SouthPole

• SubmillimeterArray(SMA),Hawaii

• SubmillimeterTelescope(SMT),Arizona

M. J

ohns

on/S

AO

EventHorizonTelescopein2018

SMA/JCMT

SMT

SMT

IRAM 30m

IRAM 30m

GLT

GLT

GLT

ALMA

ALMA/ APEX

ALMA/ APEX

LMT

LMT

LMT

D. M

arro

ne/U

ofA

SPT

SPT

SPT

SMA/JCMT

SMA/JCMT

FluidDynamics• Fluiddynamicsdealswiththebehaviourofmaserinthelarge(averagequan%%esperunitvolume),onamacroscopicscalelargecomparedwiththedistancebetweenmolecules,l>>d0~3-4x10-8 cm,nottakingintoaccountthemolecularstructureoffluids.

• Macroscopicbehaviouroffluidsassumedtobecon%nuousinstructure,andphysicalquan%%essuchasmass,density,ormomentumcontainedwithinagivensmallvolumeareregardedasuniformlyspreadoverthatvolume.

• Thequan%%esthatcharacterizeafluid(inthecon%nuumlimit)arefunc%onsof%meandposi%on:

density(scalarfield)velocity(vectorfield)pressuretensor(tensorfield)

FluidApproachtoPlasmas• Fluidapproachdescribesbulkpropertiesofplasma.Wedonotattemptto

solveuniquetrajectoriesofallparticlesinplasma.Thissimplificationworksverywellformajorityofplasma.

• FluidtheoryfollowsdirectlyfrommomentsoftheBoltzmannequation.

• EachofmomentsofBoltzmann(Vlasov)equationisatransportequationdescribingthedynamicsofaquantityassociatedwithagivenpowerofv

Continuityofmassorchargetransport

Momentumtransport

Energytransport

Single-FluidTheory:MHD• Undercertaincircumstances,appropriatetoconsiderentireplasmaasa

singlefluid.• Donothaveanydifferencebetweenionsandelectrons.• Approachiscalledmagnetohydrodynamics(MHD).

• Generalmethodformodelinghighlyconductivefluids,includinglow-densityastrophysicalplasmas.

• Single-fluidapproachappropriatewhendealingwithslowlyvaryingconditions.

• MHDisusefulwhenplasmaishighlyionizedandelectronsandionsareforcedtoactinunison,eitherbecauseoffrequentcollisionsorbytheactionofastrongexternalmagneticfield.

ApplicabilityofHydrodynamicApproximation

• Toapplyhydrodynamicapproximation,weneedthecondition:• Spatialscale>>meanfreepath• Timescale>>collisiontime

• Thesearenotnecessarilysatisfiedinmanyastrophysicalplasmas• E.g.,solarcorona,galactichalo,clusterofgalaxiesetc.

• Butinmagnetizedplasmas,theeffectivemeanfreepathisgivenbytheionLarmorradius.

• HenceifthesizeofphenomenonismuchlargerthantheionLarmorradius,hydrodynamicapproximationcanbeused.

ApplicabilityofMHDApproximation

• Magnetohydrodynamics(MHD)describemacroscopicbehaviorofplasmasif• Spatialscale>>ionLarmorradius• Timescale>>ionLarmorperiod

• MHDcannottreat• Particleacceleration• Originofresistivity• Electromagneticwaves• etc

FluidMotion• Themotionoffluidisdescribedbyavectorvelocityfieldv(r),(whichis

meanvelocityofallindividualparticleswhichmakeupthefluidatrandparticledensityn(r).

• Wediscussthemotionoffluidofasingletypeofparticleofmass/charge,m/q,sochargeandmassdensityareqnandmn

• Theparticleconservationequation(continuityequation):

• Expandthetoget:• Significanceisthatfirsttwotermsareconvectivederivativeofn

• Socontinuityequationcanbewritten:

Lagrangian&EulerianViewpoint• Lagrangian:sitonafluidelementandmovewithitasfluidmoves

• Eulerian:sitatafixedpointinspaceandwatchfluidmovethroughyourvolumeelement:identityoffluidinvolumecontinuallychanging• :rateofchangeatfixedpoint(Euler)• :rateofchangeatmovingpoint(Lagrange)

• :changeduetomotion

Lagrangianviewpoint

Eulerianviewpoint

Single-FluidEquationsforFullyIonizedPlasma

• Cancombinemultiple-fluidequationsintoasetofequationsforasinglefluid.

• Assumingtwo-specialsplasmaofelectronsandions(j = eori):

• Forafullyionizedtwo-speciesplasma,totalmomentummustbeconserved:

• As mi >> methetime-scalesincontinuityandmomentumequationsforionsandelectronsareverydifferent.Thecharacteristicfrequenciesofaplasma,suchasplasmafrequencyorcyclotronfrequencyaremuchlargerforelectrons.

• Whenplasmaphenomenaarelarge-scale(L >> λD)andhaverelativelylowfrequencies(ω << ωplasmaandω << ωcyclotron),onaverageplasmaiselectricallyneutral(ni ~ ne).Independentmotionofelectronsandionscanthenbeneglected.

• Canthereforetreatplasmaassingleconductingfluid,whoseinertiaisprovidedbymassofions.

• Governingequationsareobtainedbycombiningtwoequations(electron+ions)

• First,definemacroscopicparametersofplasmafluid:Massdensity

ElectriccurrentCenterofMassVelocity

Totalpressuretensor

Chargedensity

Single-FluidEquationsforFullyIonizedPlasma

MHDMassandChargeConservation• Usingcontinuityeq:

• Multiplybyqiandqe andaddcontinuityequationstoget:

• whereJistheelectriccurrentdensity:andtheelectriccharge:

• Multiplyeqbymiandme,

• whereisthesingle-fluidmassdensityandvisthefluidmassvelocity

Chargeconservation

Massconservation/continuityequation

MHDEquationofMotion• Equationofmotionforbulkplasmacanbeobtainedbyaddingindividual

momentumtransportequationsforionsandelectrons.

• LHSofmomentumtransporteq:

• Difficultyisthatconvectivetermisnon-linear.• Butnotethatsinceme << micontributionofelectronmomentumismuchless

thanthatfromion.Soweignoreitinequation

• Approximation:Centerofmassvelocityisionvelocity:• LHSofmomentumtransporteq:

• RHSofmomentumtransporteq:

• Ingeneral,secondterm(Electricbodyforce)ismuchsmallerthanJ x Bterm.Soweignored.

• Therefore,LHS+RHS:

• Foranisotropicplasma,wheretotalpressureisp = pe + pi and

Equationofmotion

Equationofmotion

MHDEquationofMotion

GeneralizedOhm’sLaw• Thefinalsingle-fluidMHDequationdescribesthevariationofcurrentdensity

J.• Considerthemomentumequationsforelectronandions:

• Multipleelectronequationbyqe/meandionequationbyqi/mi andadd:

(WeignoresecondtermofLHSaswedealingwithsmallperturbation)

• Foranelectricallyneutralplasmaandusingand,wecanwrite

• Asand.Inthermalequilibrium,kineticpressuresofelectronsissimilartoionpressure(Pe ~ Pi)

GeneralizedOhm’sLaw

• Thecollisionaltermcanbewritten:whereηisthespecificresistivity,q2relatestofactthatcollisionsresultfromCoulombforcebetweenions(qi)andelectrons(qe)andtotalmomentumtransferredtoelectronsinanelasticcollisionwithanionisvi – ve .

• Nowqi= - qeandne = niandJ=neqe(ve-vi),=>• Theequationcanbewrittenas

• Whereηisatensor.ThisisgeneralizedOhm’slaw(4.3)

GeneralizedOhm’sLaw

• ForasteadycurrentinauniformE,andB = 0sothat

• Ingeneralform,theelectricfieldEcanbefound:

• Considerrighthandsideofthisequation:• Firstterm:E associatedwithplasmamotion• Secondterm:Halleffect• Thirdterm:AmbipolardiffusionfromE-fieldgeneratedbypressuregradients• Fourthterm:Ohmiclosses/Jouleheatingbyresistivity• Fifthterm:Electroninertia

GeneralizedOhm’sLaw

OneFluidMHDOhm’sLaw• GeneralizedOhm’slaw

• Nowassumeplasmaisisotropic,sothatAlsoweneglectHalleffectandAmbipolardiffusioningeneralizedOhm’slawsincenotimportantinone-fluidMHD.Forslowvariations,J=constant,socanwritegeneralizedOhm’slawas:

• Rearranginggives,

• Whereσ=1/ηiselectricalconductivity

One-fluidMHDOhm’slaw

SimplifiedMHDEquations• AsetofsimplifiedMHDequationscanbewritten:

• FluidequationsmustbesolvedwithreducedMaxwellequations

• Herewehaveassumedthatthereisnoaccumulationofcharge(i.e.,ρe = 0)

• Completesetofequationsonlywhenequationofstateforrelationshipbetweenpandn (ρ)isspecified.

(displacementcurrenttermisignoredforlowfrequencyphenomena)

TheInductionEquation• Takingthecurlofone-fluidMHDOhm’slaw:

• Assumingσ=const.Substitutingfor　　fromAmpere’slawandusingthelawofinductionequations(Faraday’slaw):

• Thedoublecurlcanbeexpandingfromvectoridentity

• ThesecondterminR.H.S.iszerobyGauss’slaw(　).So

MHDinductionequation

• TheMHDinductionequation,togetherwithfluidmass,momentum,andenergyequations(EoS),aclosesetofequationsforMHDstatevariables(ρm,v,p,B)

Here,

TheInductionEquation

IdealMHD• Inthecasewheretheconductivityisveryhigh(),theelectric

fieldisE=-vxB(motionalelectricfieldonly).ItisknownasidealMagnetohydrodynamics.

• Asetofequations:

• ThisisthemostsimplestassumptionforMHD.ButthisiscommonlyusedinAstrophysics.

MagneticFieldBehaviorinMHD• MHDinductionequation:

• Dominant:convection• Infiniteconductivitylimit:idealMHD.• Flowandfieldareintimatelyconnected.Fieldlinesconvectwiththeflow.(fluxfleezing)

• TheflowresponsetothefieldmotionviaJxBforce• Dominant:Diffusion

• Inductionequationtakestheformofadiffusionequation.• Fieldlinesdiffusethroughtheplasmadownanyfieldgradient• Nocouplingbetweenmagneticfieldandfluidflow• CharacteristicDiffusiontime:

• Ratiooftheconvectiontermtothediffusionterm:

MagneticReynold’snumber

Hereusing

• Rewritecontinuityequation:

• firsttermdescribescompression(fluidcontractsorexpansion)• Secondtermdescribesadvection

• Theinductionequation(idealMHD)canbewrittenas,usingstandardvectoridentities:

• Equationissimilartocontinuityequation.• Firstterm:compression• Secondterm:advection• Thirdterm:newtermdescribesstretching.Itisrelatedmagneticfieldamplification

MagneticFieldBehaviorinMHD

MomentumEquation• Fromequationofmotionandcontinuityequations

• Usingdefinitionofmagneticstresstensor,themomentumequationis

Momentumdensity

Stresstensor

Iisthree-dimensionalidentitytensor

ConservationFormofIdealMHDEqs

Massconservation

Momentumconservation

Energyconservation

inductionequation

Idealequationofstate

Neglectinggravityforce.Thisformisoftenusedinnumericalsimulation.

PoyntingFlux• Fromenergyconservationequation,energyfluxis

• Thiscomposehydrodynamicpartandmagneticpart.• Themagneticpartcanbetransformed:

• ThisiscalledPoyntingflux(Poyntingvector),whichrepresentstheflowofelectromagneticenergy

Entropyconservationequation• ThebestrepresentationoftheconservationformofMHD

equationisintermsofthevariables,ρ, v, eandB.• Apeculiaradditionalvariableisthespecificentropys• Foradiabaticprocessofidealgas,conservationofentropyis

• Butthisisnotinconservationform(butexpressestheconservationofspecificentropyco-movingwiththefluid)

• AgenuineconservationformisobtainedbyvariableρmS,theentropyperunitvolume

Entropyconservationequation

HydrovsMHDNewtonianMHDequationisshownthecouplingofhydrodynamicswithmagneticfield

MHDequationisrecoveredhydrodynamicequationswhenB=0.

HydrovsMHD• ConservationformofNewtonianhydrodynamicequations

Summary• Singlefluidapproachofplasmaiscalledmagnetohydrodynamics(MHD).

• Inthecasewheretheconductivityisveryhigh,theelectricfieldisE= -v x B.ItisknownasidealMHD.

• InidealMHD,magneticfieldisfrozenintothefluid

• Lorentzforcedividestwodifferentforces:magneticpressure&curvatureforce

• TheinductionequationinidealMHDshowsevolutionofmagneticfield.Itisincludingcompression,advectionandstretching

• TheinductionequationinresistiveMHDincludesdiffusionofmagneticfield.

• Fromenergyconservationequation,energyfluxcomposeshydrodynamicpartandmagneticpart.MagneticpartiscalledPoyntingflux.