denedsolved

PlanetaryMagnetospheres:SolvedProblemsandProblemSet1SolvedProblems1.
Inclassicalelectromagnetictheory,themagneticmomentLassociatedwithacircularcurrent'loop'ofradiusRwhichcarriesacurrentIisgivenbytheproductofcurrentandlooparea:L=IπR2.
ApplythisdenitiontothecurrentcarriedbyaparticleofchargeqandmassmgyratinginasingleplaneaboutamagneticeldofstrengthB.
Theparticlethusmovesonacircularorbitwithspeedv⊥andradiusrg=mv⊥/(qB).
Showthatthemagneticmomentassociatedwiththecurrentrepresentedbytheparticle'smotionisequaltotherstadiabaticinvariantdiscussedinlectures,i.
e.
=W⊥/B,theratioofgyrationalkineticenergytoeldstrength.
SolutionCurrentischargeperunittimewhichpassesaxedpoint.
Fortheparticle,thismaybewrittenI=q/T,whereTisthegyroperiod,i.
e.
I=q2B/(2πm).
TheareaoftheorbitalcircleisA=πr2g=πm2v2⊥/(q2B2).
HenceIA=12mv2⊥/B=W⊥/B.
2.
Foranidealcollisionlessplasmaofbulkvelocityu,Ohm'sLawreducestoE=u*B,whereEistheconvectiveelectriceld.
ShowthatthevelocitycomponentperpendiculartoBisgivenbyu⊥=E*B/B2.
SolutionUsingthegivendenitionofE,wemaywriteE*B/B2=(u*B)*B/B2.
Now,(u*B)*B/B2=(B2u(B·u)B)/B2.
Ifwedeneaunitvectorb=B/B,wehaveE*B/B2=u(b·u)b=uu||=u⊥.
3.
FollowingonfromQuestion2,ageneralplasmaowuissometimesdescribedbyitscorrespondingpatternofconvectiveelectriceldE.
IfEcanbedescribedasthegradientofascalarpotentialthroughE=φE,thenwehaveu⊥=φE*B/B2.
Assumeforsimplicitythatu||=0.
Considerplasmamotionina'magnetosphericequatorial'planewhichcontainstheEarth-Sunlineandisper-pendiculartotheEarth'smagneticdipoleaxis.
Explainwhythe'streamlines'oftheplasmaowinthisplane(curveswhichhavealocaltangentvectorparalleltou)arealsocurvesofconstantφE(i.
e.
equipotentialcurves).
Inthisequatorialplane,wemaywriteφEasthesumoftwoterms:φE=φCR+φCONV.
Thersttermisthecorotationpotentialanddominatesclosetotheplanet.
Itisgivenby:φCR=EBER3E/r,whereEistheEarth'sangularvelocityofrotation,BEistheequatorialeldstrengthattheEarth'ssurface,REistheEarth'sradiusandrisradialdistancefromtheplanet'scentre.
Thesecondtermistheconvectionpotentialanddescribessunwardows(associatedwithmagnetotailreconnec-tion)whichcarryplasmafromthemagnetotailtowardsthedayside:φCONV=Eoy,1whereEoistheconvectionelectriceld(assumedconstant)andyistheCartesiancoordinatemeasuredalonganaxis(lyingintheequatorialplane)whichpassesthroughtheEarth'scentre(theorigin)andisperpendiculartotheupstreamsolarwinddirection(solarwindowsalongthenegativexdirection).
yispositivetowardsdusk.
Thereisa'stagnation'pointintheow,lyingonthepositiveyaxis,whoselocationmaybeestimatedasthepointwherethemagnitudesofthetwopotentialtermsareequal.
Showthattheradialdistanceofthestagnationpointisgiven(inunitsofEarthradii)by:rsp/RE=(EBERE/EO)1/2UsingreasonablevaluesfortheEarthparameters,andavalueEO=1mV/m,calculatersp/REfortheEarth'smagnetosphere.
HowdoesvariabilityinEOaffectthisdistanceForJupiter,theplanet'sverystrongeld,sizeandrotationratecausersptolieoutsidetheactualmagnetosphere-whatisthephysicalmeaningofthisresultSolutionSettingthemagnitudesofφCONVandφCRtobeequal,andusingthefactthattheradialdistancerisequaltoyforapointonthepositiveyaxis,weobtain:EBER3E/r=EOr→(r/RE)=EBERE/EOUsingtheEOvaluegiven(andtransformingtoMKSunits),arotationperiodof24hoursfortheEarth,aradiusof6370kmfortheEarth,andBE=3*105T,weobtain:(rsp/RE)=EBERE/EO=(2π/(24*3600))*3*105*6730*103/103=3.
83AnincreaseinEOrepresentsastrongerowassociatedwiththeDungeycycle,andaconsequentlysmallerstagnationdistance,whichapproximatelyrepresentsthetransitiondistancefromsunwardowintheoutermag-netospheretocorotationalowintheplasmasphere.
Jupiter'sstagnationpointlyingoutsideitsmagnetospheremeansthatthedaysideequatorialmagnetosphereofJupiterisdominatedbyrotationalows(morecorrectly,(sub)corotationalwithrespecttotheplanet-seethelecturenotes).
4.
ThemagneticeldstrengthBduetotheEarth'sdipoleeldmaybeexpressedas:B=(BER3E/r3)(3cos2θ+1)1/2,(1)whereBEistheequatorialeldstrengthattheEarth'ssurface,REistheEarth'sradiusandrisradialdistancefromtheplanet'scentre.
θdenotesmagneticcolatitude(themagneticequatorisdenedbyθ=π/2).
ThefollowingformulaisforthepitchangleαcassociatedwiththelossconeatapointPwheretheeldstrengthisBP:sin2αc=BP/BS,(2)whereBSisthemagneticeldatthesurfaceoftheplanetwhichismagneticallyconnectedtothepointPalongthesameeldline.
Calculatethevalueαcasafunctionofdistanceforlocationsinthemagneticequatorialplane,usingthedipoleapproximation.
Youmayndthefollowingformulafortheshapeofadipolemagneticeldlineuseful:r=LREsin2θ,(3)whereLREistheequatorialcrossingdistanceoftheeldline.
2SolutionForanymagneticequatorialpointatdistanceLRE,adipoleeldlinepassingthroughthatpointwillintersecttheEarth'ssurfaceatacolatitudeθigivenby:RE=LREsin2θi→sinθi=1/L→cosθi=±(L1)/L(4)HencethemagneticeldmagnitudeBSisgivenby:BS=(BER3E/R3E)(3cos2θi+1)1/2=BE(3(11/L)+1)1/2.
(5)Wecanalsoevaluatethedipoleformulaatθ=π/2,r=LREtoobtainBP:BP=BE/L3.
(6)Itfollowsthat:sin2αc=BP/BS=L3(3(11/L)+1)1/2(7)Usingthisformulatoevaluatesin2αc,henceαc,asafunctionofL,weobtainthefollowingplot:5.
ThemagneticsignaturesofinterchangeobservedbyGalileoinJupiter'smagnetosphereindicatethattheinward-movinguxtubeshavemagneticeldstrengthstypicallyhigherthanthesurroundingplasma.
Ifthetotal(plasmaplusmagnetic)pressureinsidetheuxtubeisequaltothatoftheambientplasmaoutside,showthatthesmallchangeineldstrengthδB(insideminusoutsideeld)isrelatedtoacorrespondingchangeinplasmapressureδpasfollows:δp/po=2(δB/Bo)(1/βo)(8)3wherethesubscript'o'indicatesquantitiesoutsidetheuxtube,andβ,asusual,equalstheratioofplasmapressuretomagneticpressure.
Usingthisformula,calculateδp/poforvalues:(i)Bo=1700nT,δB=10nT,βo=0.
05;and(ii)Bo=1700nT,δB=25nT,βo=0.
05.
SolutionThesumofthemagneticandplasmapressuresoutsidetheuxtubemaybewrittenasB2o/(2o)+po.
Ifthisquantityremainsconstantaswecrossintotheuxtube,wemayexpressthisbytakingazerodifferentialbetweeninsideandoutside,asfollows:d(B2/(2o)+p)=0≈2BoδB/(2o)+δp.
Rearranginganddividingbypo,weobtainδp/po≈Bo(δB/o)(1/po)=2(δB/Bo)(1/βo),since,bydenitionpo=βo(B2o/(2o)).
Usingthisapproximationandthevaluesgiven,weobtainvaluesofδp/poofabout(i)-0.
24and(ii)-0.
59.
6.
ConsiderthetypicalinformationforMercuryandtheEarthinthetablefromthelecturenoteswhichcomparesthemagnetopausestand-offdistancesofvariousplanets.
Assumingthatthedipolemagneticpressureoftheplanetbalancessolarwinddynamicpressureatthemagnetopausestandoffpoint,calculatetheratioofsolarwinddynamicpressuresjustupstreamofMercury'sandtheEarth'smagnetospheres.
SolutionThetableinquestionindicatesthatthedipolemagneticpressureatMercury'sdaysidemagnetopauseisapproxi-matelyproportionalto(ignoringdipoletilteffects)[MM/(1.
4RM)3]2(i.
e.
themagneticpressureisproportionaltothesquareoftheexpectedeldstrength).
HereMMisMercury'smagneticdipolemoment.
FortheEarth,thisquantitywillbe[ME/(10RE)3]2.
Takingtheratio,weobtain(MM/ME)2(106/1.
46)(RE/RM)6.
Usingreasonablevaluesoftheplanetaryradii,thisevaluatesto6.
7.
(N.
B.
IthinkthevalueofthemagneticmomentofMercuryshouldbemorelike4*104ME,basedonMessengerdata-notealsotheusualvariabilityexpectedinsolarwindparameters).
7.
ChapmanandFerraro(1930)developedamodelofaplasmacloudinteractingwiththeEarth'sdipolemagneticeld.
Thismodelmaybeappliedtoinvestigatethebehaviourofthemagneticeldgeneratedbythemagne-topausecurrents.
Inthispicture,theEarth'smagneticdipoleissituatedattheorigin(Earthcentre)andthedipoleaxisisorthogonaltotheupstreamsolarwinddirection.
Themagnetopauseisthenmodelledasaninniteconductingplane,perpendiculartotheupstreamsolarwindvelocity,andsituatedaperpendiculardistanceofRMPfromtheplanet'sdipoleaxis.
MagnetopausecurrentsowonthisplaneandgenerateanadditionaleldwithintheEarth'smagnetospherewhichisequivalenttothatofanidenticalmagneticdipole,knownasthe'im-agedipole',situatedoutsidethemagnetosphereatadistance2RMPfromtheEarth'scentrealongthedirectionanti-paralleltotheupstreamsolarwindvelocity.
WedenethexaxistopassthroughtheEarth'scentre(wherex=0)alongthisdirection.
Usingthismodel,calculateandmakeaplotoftheratioBTOT/BDIPasafunctionofdistancealongthexaxis,fromtheEarth'ssurfacetothemagnetopauseplane.
Here,BTOTisthetotalmagneticeldstrengthduetotheactualandimagedipolescombined,andBDIPistheeldstrengthduetotheplanetarydipolealone.
SolutionFortheplanetarydipolealone,theeldstrengthoutsidetheEarthandinsidethemagnetopause,alongthexaxis,isgivenbythefunctionBD(x)=(BER3E/|x|3)(usingthenomenclatureofQuestion4).
Nowwemayexpresstheeldoftheimagedipolesituatedatx=2RMPasthefunctionBD(x2RMP)=(BER3E/|x2RMP|3).
Addingthetwo,weobtain:BT(x)=BD(x)(1+|x|3/|x2RMP|3).
HenceBT(x)/BD(x)=(1+|x|3/|x2RMP|3),whichisalwaysgreaterthanunity.
Aplotofthisquantityversusx/REisgivenbelow,usingareasonablevalueRMP=10RE.
45ProblemSet1:'PlanetaryMagnetospheres'Section1.
Considertheinductionequationforanideal,collisionlessplasmathreadedbymagneticeldB,andhavingbulkowvelocityu:Bt=*(u*B)Consideracontinuous'patch'ofplasma(seeNotes)whichisdenedbyasurfaceS,boundedinspacebyacurveΓ.
Astheplasmamoves,ΓwillgenerallychangeshapeandtheareaofSwillgenerallychangevalue.
ConsideraninnitesimallysmallelementofthemovingcurveΓwhichisdenedbyavectorincrementdl.
Showthat,duringaninnitesimaltimestepdt,themotionofthiselementchangesthemagneticuxΦBthroughthepatchbyanamount:dΦB=B·((udt)*dl),whereBanduarethelocalvaluesofeldandvelocity.
Hence,showthattheconvective,orco-movingtimederivativeofthemagneticuxthroughthepatchmaybewritten:DΦBDt=t(ΦB)+ΓB·(u*dl),whereΦBisequaltothesurfaceintegralSB·dS.
MakinguseofanappropriateMaxwell'sequationandOhm'slawfortheplasma,demonstratethevalidityofthe'frozen-in'condition,i.
e.
DΦBDt=0.
2.
InSolvedProblem4,youwillndtheformulaforthemagneticeldstrengththeEarth'sdipoleeld,andtheequationdescribingtheshapeofadipolarmagneticeldline.
Thecorrespondingradialandmeridionaldipoleeldcomponentsaregivenby:Br=2BEcosθ/(r/RE)3Bθ=BEsinθ/(r/RE)3Usingthisinformationandappropriatephysicalconstants,calculatethegradientdriftvelocityug=W⊥qB3B*B(seeNotes)ofprotonswiththefollowingproperties,driftingintheEarth'smagnetosphere:(a)W⊥=1keVand10keV,r=8RE,θ=90(i.
e.
equatorial).
(b)W⊥=1keVand10keV,θ=60,choosersothatprotonisonsameeldlineasthoseinitem1above.
(c)W⊥=1keVand10keV,θ=30,choosersothatprotonisonsameeldlineasthoseinitems1and2above.
3.
InSolvedProblem7,ChapmanandFerraro's'inniteconductingplane'carriescurrentswhichamplifythemagneticeldneartheEarth'smagnetopausebyafactoroftwo.
Usethe'pressurebalance'argumentfromthelecturestocalculatethechangeintroducedinthestandoffdistanceRMPofactitiousplanet'smagnetopause,atxedsolarwinddynamicpressure,whentheeldisampliedinthisway(assumetwicethestrengthofapuredipoleeldatthemagnetopause).
Considernowaddinganinteriorplasmapressurenearthemagnetopauseofthisctitiousplanet,suchthattheplasmaβparameterthereattainsavalue5.
WhateffectdoesthishaveonRMP64.
Consideractitiousmagnetospherewhererotationaleffectsarenotimportant,andtheonlyforcesareduetotheplasmapressuregradientandthemagneticJ*Bforce.
Ifthesystemisinperfectforcebalance(i.
e.
thesumofthesetwoforcesatanypointisidenticallyzero),explainwhytheplasmapressurewillbeuniformallthewayalongamagneticeldlineNowconsideranidealizedmagnetospherewhererotationplaysanimportantroleinforcebalance,andthemagneticeldissymmetricabouttherotational/magneticequatorialplane.
Themagneticforce,centrifugalforceandplasmapressuregradientalwaysaddtozeroatanypointinthesystem(weneglectallotherforcesforsimplicity).
Byconsideringforcebalanceinthedirectionparalleltothepoloidalmagneticeld(zeroBφ),explainwhytheadditionofthecentrifugalforceontheplasmacausesplasmapressuretochangealongthemagneticeldline.
Demonstratewhytheproleoftheplasmapressurecanbedescribedbytheequation:P(ρ)=P0exp[(ρ2ρ20)/(2l2)],(9)whereρ=rsinθdenotescylindricalradialdistance,theeldlinecrossestheequatoratρ=ρ0,andthescalelengthl≈(2kT/miω2)1/2.
Assumptions:theplasmatemperatureTandangularvelocityωareconstantalongaeldline;theplasmaisquasi-neutral,behavesasanidealgas,andiscomposedofionsofmassmiandelectronsofmassme.
7Solutions1.
Theelementdlchangespositionbyudtinthetimestep.
Thecorrespondingsurfaceareacoveredbytheelementduringthismotionisthusaparallelogramhavingthesevectorsasedges,andmaythusbewrittendS=(udt)*dl-heretheusualconventionisfollowed,whereasurfaceelementisrepresentedbyavectorlyingorthogonaltoitself.
dΦB,bydenition,isthescalarproductofmagneticeldandsurfacevector,i.
e.
theuxofmagneticeldthroughthesurface.
TheintegralrepresentsthechangeinΦBduetothemotionofalloftheelementsdlwhichmakeupthemovingperimeterΓ.
Ingeneral,however,themagneticelditselfwillhaveanexplicittimedependence-i.
e.
anobserverataxedpointinspacewillseeBchangewithtime.
Duetothiseffect,thechangeinΦBcanbewrittendΦB=dtSBtdS.
Theco-movingderivativeis:dΦBdt+dΦBdt,whichis:ΓB·(u*dl)+SBtdS=Γdl·(B*u)+S*E·dS=Γdl·(B*u)+ΓE·dlwhereEdenotestheelectriceld,andwehaveused*E=BtUsingtheidealizedOhm'sLawE=u*B,weobtain:dΦBdt=Γ(B*u)·dl+Γ(u*B)·dl,whichiszero.
2.
IfIhaven'tmadeanyerrors,theevaluationofB*Bgives(helpfromMathematica!
):3B2ER6Esinθ(1+cos2θ)r7(1+3cos2θ)φToobtainug,wemultiplythisexpressionbyW⊥/(qB3)andobtain:(W⊥/q)3r2sinθ(1+cos2θ)BER3E(3cos2θ+1)2φ=(W⊥/q)3(LREsin2θ)2sinθ(1+cos2θ)BER3E(3cos2θ+1)2φ(10)Herewehaveeliminatedrusingthedipoleeldlineformula(L=8forthisproblem).
IfweuseappropriatevaluesRE=6370km,andBE=3*105T,weobtainthefollowingvaluesfortheenergyW⊥=1keV:|ug|≈1005m/s(θ=90),1.
05m/s(θ=60),0.
524m/s(θ=30).
ForthecaseW⊥=10keV,multiplythesevaluesbyten.
(Thisproblemrequiresmuchalgebra,sopleaseletmeknowifyouspotanymistakes!
)83.
Balancingmagneticpressureofapuredipolewiththesolarwinddynamicpressure:BDIP(RMP)2/(2o)=12oBER3ER3MP2=PSWRMP=12o1/6B2ER6EPSW1/6(11)Lookingatthisequality,weseethatifwereplaceBDIP(RMP)by2BDIP(RMP),thenRMPwillincreasefromthepuredipolevaluebyafactor41/6≈1.
26.
Ifwenowintroducetheplasmaβvalueaswell(ratioofplasmapressuretomagneticpressure),thenthetotalpressure(plasmaplusmagnetic)atthemagnetopausecanbewritten:(1+β)(2BDIP(RMP))2/(2o)Sothepressurebalancebecomes:(1+β)(2BDIP(RMP))2/(2o)=12o(1+β)2BER3ER3MP2=PSWRMP=12o1/6(1+β)1/641/6B2ER6EPSW1/6Hencethenon-zeroplasmapressureincreasesRMPbyanadditionalfactor(1+β)1/6=61/6≈1.
35.
4.
Theequationofforcebalanceparalleltothemagneticeldis:dPds+N2(mi+me)ρω2cosψ=0,whereNistotalparticlenumberdensityandψistheanglebetweentheelddirectionandthecylindricalradialdirection(i.
e.
thelocaldirectionperpendicularlyoutwardsfromtheaxisofsymmetry).
Notethatwedon'tneedtoconsideranyotherforce,sincetheparallelcomponentofJ*Biszero,bydenition.
Sincethecentrifugaltermalwayspointsoutwards(positivedirection),werequiredPdstobenegative,i.
e.
pressurePmustincreaseaswetravelalongaeldlinefrompolarregionstoequator(connementofplasmaintoadisc-likeshape).
Sinceanelementoflengthdsalongtheeldcorrespondstoachangedρ=dscosψ,wehave:dPdρ+P2kT(mi+me)ρω2=0,whereP=NkTfortheplasma.
Integratingbetweenanarbitrarypointontheeldlineandtheequator(denotedbysubscript'0'):dPP=(mi+me)ω22kTρdρ,ln(P0/P)=(mi+me)ω22kT12(ρ20ρ2)P=P0exp[(ρ2ρ20)/(2l2)],wherel2=2kT(mi+me)ω2≈2kTmiω2,sincemi>>me.
9ProblemSet21.
Explainwhythevolumeofaunitmagneticuxtube(i.
e.
thevolumeperunitmagneticux)isgivenbytheintegraldsBalongtheeldline,wheredsislengthelementalongtheeld,andB(s)islocaleldstrength.
Considernowacoldplasma(quasi-neutral,withonespeciesofpositiveion)inarotatingmagnetosphere(asinProblemSet1).
ShowthatthenumberofionsNicontainedperunitmagneticuxcanbeexpressedas:Ni=Po2kTexp[(ρ2ρ2o)/(2l2)]dsB,(12)wheretheintegralisagainalongtheeldline,pressureisdenotedbyP,cylindricalradialdistancebyρ,andquantitiesatthemagneticequatoraresubscriptedwith'o'.
listhelengthscalefromProblemSet1,whichinvolvesthetemperatureTandplasmaangularvelocityω,bothconstantalongtheeldline.
2.
Derivetherst-orderdensityandtemperatureperturbationsgiveninthesectionon'InterchangeMotions':σn(1)=n·uu·nσP(1)=γP(·u)u·PYoumaynditusefultoconsidertheperturbedequationsforconservationofmass,andforadiabaticchangeinplasmapressure(thissecondconditionmaybeexpressedasD(Pnγ)Dt=0-azerocomovingtimederivative).
Forsimplicity,assumethattheunperturbedplasmahaszerovelocity.
3.
ConsiderasphericallysymmetricinwardowofmaterialbeingaccretedontoastarofmassM.
Assumethatthematerialisfreelyfallingundertheinuenceofthestar'sgravity,startingfromrestatinnitedistance.
Theaccretionrate˙Misconstantandequalto4πr2ρMv,whererisradialdistancefromthestar'scentre,ρMisdensityofthematerialandvisthevelocity.
Explainwhythisequalityisvalidinthesteady-stateow.
Assumenowaverysimpliedestimateforthemagneticeldstrengthforthestar,basedonadipole'sradialdependence:B(r)/r3,whereisthestar'smagneticmoment(weignoretheangulardependenceforsimplicity).
Usingthisinformation,showthattheapproximateAlfv`enradiusRAofthesystem,wherethedynamicpressureoftheinow(ρMv2)equalsthemagneticpressureofthestar,satisesthedependence:RA∝4/7˙M2/7M1/710ProblemSet31.
The'propeller'mechanismmayacttoejectinfallingmaterialfromthemagnetosphericboundaryofarapidlyrotating,magnetizedstar.
Inasimplepicture,materialinstantaneously'attaches'totherotatingeldatthemagnetosphericradiusRandstartstorotatewiththestellarangularvelocityS.
Thepropellermechanismiseffectivewhenthevelocityofthe'attached'materialexceedsthelocalescapevelocityfromthestar.
Showthatthisconditionisequivalentto:R>21/3Rc,whereRcisthecorotationradius(i.
e.
theradiusatwhichtheangularvelocityofacircularorbitaboutthestarisequaltoS).
2.
ConsideraPolarbinarystarsystemwherethemagneticdipoleofthewhitedwarfisorthogonaltotheorbitalplaneofthetwostars.
Assumethatthemagneticeldatandinsidethecouplingregion,whereaccretingmaterialstartstoowalongeldlines,isdominatedbythedipolareldofthewhitedwarf.
Calculatetherangeinradialdistance(inunitsofwhitedwarfradii)coveredbythecouplingregioncorrespondingtoan'arc-shaped'accretionshockonthewhitedwarfsurfaceextendingbetweenmagneticcolatitudesof20and28.
BywhatfactordoestheeldstrengthchangeoverthecouplingregionConsideranindividual'blob'intheaccretingmaterialwhichischannelledbythemagneticeldontothewhitedwarfsurface.
δArepresentsthe'cross-section'areaoftheblob,locallyperpendiculartotheeld.
Estimate,usingadipoleeldmodel,thefactorbywhichδAchangesastheblobfallsfromtheinneredgeofthecouplingregiontothewhitedwarfsurface.
(Ifradialdistanceisdenotedbyrandmagneticcolatitudebyθ,theequationofadipoleeldlineisr=LRwdsin2θ,whereLRwdisthedistanceatwhichtheeldlinecrossesthemagneticequator.
Themagneticeldstrengthduetothedipoleisproportionaltothequantityr3(1+3cos2θ)1/2).
3.
ConsideraPolarsystemwithasingleactiveaccretionshockwhichemitselectroncyclotronradiation.
'Peaks'inthecontinuumemissionofthesystemoccuratwavelengthscorrespondingtoharmonicsoftheelectroncyclotronfrequency.
Iftwooftheadjacentharmonicpeaksoccuratopticalwavelengthsof7146Aand6125A,estimatethemagneticeldstrengthatthelocationoftheemissionregiononthewhitedwarfsurface.
4.
ConsiderGhoshandLamb'spictureofthemagnetictorqueactingbetweenamagnetized,accretingstaranditssurroundingaccretiondisc.
Whatwouldhappentothecorotationradiusfollowinganunusualtransientepisodeofstronglyenhancedaccretion,which'spinsup'thestartoahigherangularvelocityIfthemagnetosphericradiusRinstantlyreturnstoits'quiet'valueimmediatelyfollowingthisepisode,butnowthecorotationradiusliesinside21/3R.
Whatwouldhappentotherateofaccretionontothestar'ssurfaceWhatwouldhappentotheareasofthediscwhichareattachedto'forward-swept'and'backswept'eldlinesWhatwouldbetheconsequencesforthespinrateofthestar11