lecture 1 - uni-frankfurt.de
TRANSCRIPT
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.