Lecture10–SynchrotronRadia3onII

Date:9thSeptember2019

Bhaswa3Bha?acharyya

ElectrodynamicsandRadia3veProcessesI

August-September2019IUCAA-NCRAGraduateSchool

bhaswa3@ncra.3fr.res.in

Reference:1)RybickiandLightman2)Ghisellini:h?p://www.brera.inaf.it/uten3/gabriele/total.pdf

SynchrotronRadia3on(Recap)SynchrotronRadia3onisradia3onfromachargemovingrela3vis3callythatisacceleratedbyamagne3cfield.

Non-rela3vis3cmo3onofachargeacceleratedbyamagne3cfield:CyclotronRela3vis3cmo3onofachargeacceleratedbyamagne3cfield:Synchrotron

Cyclotronradia3onsummary

Letustakeacharge(sayq)andputitinuniformmagne3cfieldB

ForceF=qvxB=qvB (IfBisorthogonaltov)F

B

v

ForceF=qvxB=qvB=mv2/rL=Centripetalforce

rL=mv/qBLarmorRadius/GyroRadius

Cyclotronfrequency

ForceF=mv2/rL=mωLrL

ωL=qB/m

T=2π/ωL=2πm/qBTimeperiod

νL=ωL//2π=qB/2πm=2.8MHzperGaussforelectronFrequencyisindependentofpathradiusandpar3clevelocity

Powerspectrawillpeakatasinglefrequency

Acceleratedchargedpar3clewillradiateaccordingtotheLarmorformula

Cyclotronradia3onAstrophysicalapplica3onDiscovered~40yearsback

Cyclotronlinesfromtheaccre3ngx-raypulsars

In1977J.Trumperiden3fiedacyclotronemissionlineintheaccre3ngpulsarHerculesX-1Trumperproposed:hotelectronsaroundneutronstarmagne3cpolesarerota3ngaroundastrongBfieldof~5x1012Gauss,givingrisetoanabsorp3onlineat~40keV.

Directlyprobethemagne3cfieldsoftheneutronstarsProbegeometrySeeninmorethan30sourcesSimula3ons+Observa3ons

Referto:h?ps://www.cosmos.esa.int/documents/13611/404108/200808_Schoenherr.pdf/ecff8c8e-f1e7-4f30-b3c8-66d682e20a13

Observa3on Modeling

Rela3vis3ceffects:fromCyclotrontoSynchrotronRadia3on

Assump3onv<<c(nonrela3vis3cpar3cles)forCyclotronNowwedescribewhathappenstotheradia3onofachargeacceleratedinaBfieldwhenthespeedsapproachcforSynchrotron

ReviewRela3vis3ceffectsdiscussedinLecture5

Lorentztransforma3onsof3me: Δt=Δt’γ

Lorentztransforma3onsofFrequency: ν=ν’/γ

Rela3vis3ceffects:fromCyclotrontoSynchrotronRadia3on

LarmorFrequency FrequencyofGyra3on

Cyclotron Synchrotron

νL=ωL/2π=qB/2πm νB=ωB/2π=qB/2πmγ

Larmorradius RadiusofGyra3on

Periodofrota3on

T=2π/ωL=2πm/qBPeriodofrota3on

T=2π/ωB=2πmγ/qBTheperioddependonpar3clevelocity(Lorentzfactorgamma)andasthevelocityapproachesc,theperiodincreases.

SynchrotronRadia3onEmissionpa?ern

Arela3vis3celectronmovingaroundaBfield.

CyclotrontoSynchrotron:-  startwiththeradia3on

pa?ernintheelectronrestframe(whereweknowtheradia3onpa?ern)

-  thenwedoaLorentztransforma3onfromtherestframetothelabframe.

Synchrotronradia3on:Mo3onofultra-rela3vis3cpar3clesaroundthemagne3cfieldlines

SynchrotronRadia3on

Equa4onsofMo4onofapar4clewithrela4vis4cvelocity:

Changeofrela3vis3cmomentumdp/dt

γisconstant|v|constant

Forceonthepar3cleisperpendiculartothemo3on.

Considerapar3cleofmassmandchargeq

SynchrotronRadia3onHelicalMo4on:

Separa3ngthevelocitycomponentsalongthefieldandinaplaneperpendiculartothefield

ParallelcomponentofvisconstantBut|v|=constantSoperpendicularcomponentofvisconstant

Mo3onofthepar3cleiscombina3onofcircularmo3onAnduniformmo3onalongthefield

Helicalmo3onofthepar3cle

SynchrotronRadia3on(Totalpowerradiated)

Totalemi?edradia3on(FromLecture7)

zero

Totalemi?edradia3onfromchargedpar3cleswithvelocityv

SynchrotronRadia3on(Totalpowerradiated)

Wehavemanypar3cleseachhavingapitchangle.Sotheperpendicularvelocityneedstobeaveragedoverallpitchangles(α).

Totalemi?edradia3on

Totalemi?edradia3on

Totalemi?edradia3onforelectrons

SynchrotronRadia3on(Totalpowerradiated)

Validforelectrononly

Validforelectrononly

SynchrotronRadia3onEmissionpa?ern

Beaming:Importanttomakeadis3nc3onbetweenemi?edradia3onandreceivedradia3on.Receivedradia3onwillbesuchthattheobservercanseeitonlywhenthenarrowbeampointstowardstheobserver,àradia3onappearstobeconcentratedonanarrowcone.Observerwillseeradia3onfromapar3cleonlyforasmallfrac3on2/γofitsorbit.Observerwillseepulseofradia3onconfinedtoa3memuchsmallerthanitsgyra3onperiod.SpectrumwillbespreadoverregionmuchbroaderthanωB/2π

Restframeofelectron Laboratoryframeofreference

SynchrotronRadia3on(spectrum)

Thespectrumofsynchrotronradia3onmustberelatedtodetailedvaria3onofelectricfieldseenbyanobserver

Becauseofbeaming,emi?edradia3onappeartobeconcentratedaboutpar3cle’svelocity

Angulardistribu3onofradia3onemi?edbyapar3clewithperpendicularaccelera3onandvelocity

SynchrotronRadia3on(spectrum)

Emissionconesatvariouspointsofanacceleratedpar3clestrajectoryObserverwillseepulsefrompoint1and2alongthepar3clespath,wherethesepointsaresuchthattheconeofemissionofangularwidth1/γincludesthedirec3onofobserva3ona=Δs/ΔθΔθ=2/γ(fromgeometry)Δs=2a/γ

SynchrotronRadia3on(spectrum)

Equa3onofmo3on

Since|Δv|=vΔθandΔs=vΔtwehave

Δs=2a/γ

SynchrotronRadia3on(spectrum)

Timest1andt2atwhichpar3clepassespoints1and2aresuchthatΔs=v(t2-t1)

Timest1Aandt2Abethearrival3mesofradia3onatthepointofobserva3on,t1A–t2Aislessthant1-t2byΔs/c(3mefortheradia3ontomoveΔs)

SynchrotronRadia3on(spectrum)

Sinceγ>>1wehave

Widthofobservedpulsesissmallerthangyra3onfrequencybyafactorofγ3Sothespectrumwillbebroadwithcutofffrequency1/ΔtACri3calfrequency:

SynchrotronRadia3on(spectrum)

Time-dependenceoftheelectricfieldinapulseofsynchrotronradia3on

SynchrotronRadia3on(spectrum)

Electricfieldisfunc3onofγθ,whereθispolarangleaboutthedirec3onofmo3onBeamingeffect

tis3memeasuredinobserver’sframe,zeroof3me(andpathlengths)whenpulseiscenteredonobserver.θ~s/at~(s/v)(1-v/c)Rela3onbetweenθandt

SynchrotronRadia3on(spectrum)

Electricfield

FouriertransformofElectricfield

Changingvariableofintegra3onto

SynchrotronRadia3on(spectrum)

Timeaveragedpowerperunitfrequency

Constantofpropor3onality

Totalpower

SpectrumdW/dωdΩispropor3onaltothesquareofE(ω)

Integra3ngoversolidangleanddividingbyorbitalperiod

SynchrotronRadia3on(spectrum)

Totalpower

Previousresults

Forhighlyrela3vis3ccase,powerperunitfrequencyemi?edbyeachelectronis

SynchrotronRadia3on(singlepar3clespectrum)

CyclotronvsSynchrotronRadia3on(singlepar3clespectrum)

Samephysicaloriginbutdifferentspectra

νL=qB/2πm

Cyclotronspectrasinglelineat Synchrotronspectrum

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)

NofactorofγintheformulaotherthaninωcThespectrumcanbeapproximatedbyapower-lawoveralimitedrangeoffrequency.Forthatrangeletusimagine,

Nega3veslopeinP(ω)-log(ω)plot

Oxenthespectraofastronomicalradia3onhasaspectralindexthatisconstantoverafairlywiderangeoffrequenciesexamples=-2forRayleigh-Jeanslaw

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)Numberdensityofpar3cleswithenergiesbetweenEandE+dE

Numberdensityofpar3cleswithenergiesbetweenγandγ+dγ

TotalpowerradiatedperunitvolumeperunitfrequencyisN(γ)dγ3messinglepar3cleradia3on

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)

Totalpowerradiatedperunitvolumeperunitfrequencyforanelectrondistribu3on

Changevariableofintegra3onx=ω/ωc

consideringtobeconstant

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)

Totalpowerradiatedperunitvolumeperunitfrequencyforanelectrondistribu3on(approximatecalcula3on)

Forpowerlawdistribu3onofelectrons,

Totalpowerradiatedperunitvolumeperunitfrequencyforanelectrondistribu3on(detailedcalcula3on)

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)

SynchrotronRadia3on(spectralindexforpower-lawelectrondistribu3on)

ü  Angulardistribu3onofsingleradia3ngpar3cleisbeamed(1/γ)

ü  Singlepar3clespectrumextendsupto~ωcSpectrumfunc3onofω/ωc

ü  Formul3par3clesystem,powerlawdistribu3onofenergieswithindexp

Spectralindexofradia3ons=(p-1)/2ü  Radia3onishighlypolarized

Synchrotronemissionfromapar3cle.Radia3onconfinedtotheshadedregion

SynchrotronSpectra(transi3onfromcyclotrontosynchrotronemission)

Cyclotronradia4onThechargeismovinginacircle,sotheelectricfieldvaria3onissinusoidal

Followtypicalsynchrotronspectrumastheelectron’senergyisvariedfromnon-rela3vis3cthroughhighlyrela3vis3cregime.

Cyclotron-synchrotronradia4onWhenv/cincreases,higherharmonicsoffundamentalfrequencyωBbegintocontribute

Synchrotronradia4onChargeismovinginacircle,andtheradia3onisseenonlyfora3nyamountof3mewhenthecone1/γpointstowardstheobservers.Superposi3onofintegralmul3pleofωB.

SynchrotronSpectra

Timedependenceofelectricfieldfromarapidlymovingpar3cleinamagne3cfield

Powerspectrum

Porb

SynchrotronSpectraForveryrela3vis3cveloci3esv~c,theoriginallysinusoidalformofE(t)hasnowbecomeaseriesofsharppulseswhicharerepeatedat3meintervals2π/ωB.Thespectruminvolvesalargenumberofharmonics,theenvelopeofwhichapproachesF(x).

Whydoweseecon3nuousspectruma)  Asthefrequencyresolu3onbecomeslargerwithrespecttoωBorotherphysicalbroadeningmechanismsfillsinthespacesbetweenthelines(thereisadistribu3onofpar3clewithdifferentenergiesandthegyra3onfrequencyωBispropor3onalto1/γàthespectraofpar3cleswillnotfallonthesamelines)b)Emissionfromdifferentpartsoftheemiyngregionmayhavedifferentvaluesanddirec3onsofthemagne3cfields,sotheharmonicsfallatdifferentplacesintheobservedspectrum.

Theelectricfieldreceivedbytheobserverfromadistribu3onofpar3clesconsistsofarandomsuperposi3onofmanypulsesoftheabovekind.Netresultissumofspectrafromindividualpulses.

SynchrotronSpectra

Singleelectronspectra

manyelectronspectra

Dis3nc3onbetweenreceivedandemi?edpower

Dopplershixofsynchrotronradia3onemi?edbyapar3clemovingtowardstheobserver

ReceivedpulsesarenotatfrequencyωBbutappropriatelyDopplershixedbecauseofprogressivemo3onofpar3cletowardsobserver.IfT=2π/ωBistheorbitalperiodoftheprojectedmo3on,then3me-delayeffectwillgiveaperiodbetweenthearrivalofpulsesTA

Dis3nc3onbetweenreceivedandemi?edpower

ThefundamentalobservedfrequencyisωB/sin2α

Forusualsitua3onencounteredinastrophysicsoneshoulduseexpressionofemi?edpowertogiveobservedpower.Abovecorrec3onduetohelicalmo3onarenotimportantformostcasesofinterest.

Synchrotroncooling3me

Ifweknowthetotalemi?edpowerwecancalculatethecooling3meofanensembleofelectronsemiyngsynchrotron.

Example:ConsiderasupermassiveblackholeinanAc3veGalac3cNucleus.Themagne3cfieldaroundtheblackholeisoftheorderof1,000GTheLorentzfactorisalsooftheorderof1,000,sotheelectronscooldownona3mescaleofjust0.77seconds.

Ifinsteadyoucalculatethesamecooling3meveryfarawayfromtheblackhole,wheregammaiss3ll1,000buttheBfiledismuchsmaller(e.g.,B~1e-5G),thecooling3meisthenoftheorderof250Myr.

Polariza3onofSynchrotronRadia3on

Radia3onfromsinglechargeisellip3callypolarized.Foradistribu3onofpar3clestheradia3onispar3allylinearlypolarized.Polariza3onforfrequencyintegratedradia3onis75%(Problem6.5inR&L)Forpar3cleswithpower-lawdistribu3onofenergythedegreeofpolariza3on,

=75%forp=3

Largescalemapofthegalac3cmagne3cfieldcanbemeasuredfrompolariza3onofradioemissioncomingfromsynchrotronprocesses.rela3vis3cpar3clesinteractwiththeinterstellarmagne3cfieldandemitpolarizedsynchrotronradia3on

SynchrotroninAstrophysics:LargescalestructureofGalac3cMagne3cField

SynchrotronemissionfromCrabnebula

Refertoh?p://www.jeff-hester.com/wp-content/uploads/2015/11/Crab_Annual_Reviews.pdf

IntheCrabnebula,spiralingelectronsemiyngop3calphotonshavealife3meofonly∼100yr,andthoseemiyngX-raysliveonlyafewyears.Suchelectronscouldnothavebeenacceleratedinthe1054,supernovacollapsethatspawnedtheCrabnebula.Theirenergysourcewasapuzzleun3lthediscoveryoftheCrabpulsarin1968.

TheCrabPulsarPowerstheNebula

SynchrotronemissionfromCrabnebula

ColorcompositeoftheCrabsynchrotronnebulashowingaChandraX-rayimageinblue,avisiblelightmosaictakenwithHSTingreen,andaVLAradioimageinred.Thepulsarisseenasthebrightbluepointsourceatthecenteroftheimage.

Emissionfromhigh-energyelectronsisbrightestnearthecenterofthenebula,closetowheretheyareinjected.Movingoutwardthroughthenebula,thespectrumbecomessoxer.

SynchrotronemissionfromCrabnebula

Refertoh?p://www.jeff-hester.com/wp-content/uploads/2015/11/Crab_Annual_Reviews.pdf

Fig:TheintegratedspectrumoftheCrabsynchrotronnebula,fromAtoyan&Aharonian(1996)

Theelectronenergiesshowncorrespondtopeaksynchrotronemissionassumingamagne3cfieldof300µG.MostoftheemissionfromtheCrabisemi?edbetweentheop3calandX-raybands.Thehighestenergyγ-raysareduetoinverseComptonradia3on.

EndofLecture10

NextLecture:12thSeptember

TopicofnextLecture:Synchrotronselfabsop3on(Chapter6ofRybicki&Lightman)

h?ps://www.cv.nrao.edu/course/astr534/SynchrotronSpectrum.html