OUTLINESpecialrelativityDopplereffects(beaming)SynchrotronComptonscattering

RadiativeprocessesinHigh-EnergyastrophysicsLaraNava

http://arxiv.org/archive/astro-ph

2012

ghisellini

LuminosityL(bolometric):energyirradiatedpersecond[erg/s]

• monochromaticL(ν)[erg/s/Hz]

• inagivenenergyrangeL[ν1-ν2][erg/s]

LGRB=1050-1054erg/s

SUN Galaxy

Gamma-RayBurst(GRB)

FluxF(bolometric):energypassingasurfaceof1cm2persecond[erg/cm2/s]

• monochromaticF(ν)[erg/cm2/s/Hz]

• inagivenenergyrangeF[ν1-ν2][erg/cm2/s]

FluenceS:energypassingasurfaceof1cm2integratedoverthedurationoftheemission[erg/cm2]

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S = F(t)dt0

T

∫

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S = F ⋅ TIfF(t)=constant

T=durationoftheemission

Exercisesfortomorrow:

1.  estimatethefluxoftheSunontheEarth(pg.7-8)

2.  estimatethefluxofaGRBwithL=1052erg/satz=2(pg.8)

3.  estimatethefluenceoftheGRBinexercise2assumingthatthefluxisconstantandtheemissionlasts20seconds.HowmuchtimedoesittaketocollectthesamefluencefromtheSun?

Specialrelativity

Considerarulerandaclockbothmovingwithvelocityv.Wecandefinetwodifferentreferenceframes:1.  Kthatseestherulerandtheclockmovingatvelocityv2.  K’thatseestherulerandtheclockatrest

Forsimplicity,weconsideramotionalongthex-axis

€

β =vc

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Γ =11− β2

Specialrelativity:lengthcontraction

timedilation

Exercisesfortomorrow:

4.  estimateβandΓ(theLorentzfactor)ofanobjectmovingatv=1010cm/s.Isthisobjectmovingatarelativisticvelocity?(relativisticvelocity=Γisappreciablydifferentthan1)

5.  estimatethevelocityvandβ ofaparcelofmattermovingwithaLorentzfactorΓ=100(typicalLorentzfactorofthefluidinGRBs)

Let’snowtakeapictureoftheruler!Picture(ordetector):collectsphotonsarrivingatthesametime,butnotnecessarilyemittedatthesametime!

Consideranextendedobject(abar)movingwithvelocityβcandreflecting(oremitting)photons.l’=properlengthl=l’/Γ

ThephotonemittedinA1att=tiafteratimeΔtereachesH.Inthemeantime,thebarmovesfromitsinitialpositionA1B1tothefinaloneA2B2.ThephotonemittedinB2reachesthedetectoratthesametimeofthephotonemittedatearliertimesinA1.

Let’snowtakeapictureoftheruler!Picture(ordetector):collectsphotonsarrivingatthesametime,butnotnecessarilyemittedatthesametime!

Consideranextendedobject(abar)movingwithvelocityβcandreflecting(oremitting)photons.l’=properlengthl=l’/Γ

Definition

Definition

Theobservedlengthdependsontheviewingangle:• reachesthemaximum(equaltol’)forcosθ=β• isequaltol’/Γforθ=90°• iszeroforθ=0°Tokeep:• viewingangle(betweendirectionofphotonsreachingtheobserverandthevelocityofthesourceofphotons)isimportant• distinguishbetweenemissiontimeandarrivaltime

Exercisesfortomorrow:

6.  Figure3.1inGhisellini2012:demonstratethattheobservedlengthHB2(seeeq.3.8)reachesamaximumforcosθ=β andthatthismaximumlengthisequaltol’.

Considerthefollowingsituation:relativisticelectronemittingradiation

ElectronstartstoemitwhenitisinAandstopswhenitreachesB.ThedifferencebetweenemissiontimesisΔte.Thefirstphoton(emittedatA)afterΔtereachesD.Theelectroninstead,afterΔtereachesBandemitsthelastphoton.WhatisthedifferenceinthearrivaltimesΔta?

Forθ=0°(electronismovingtowardus)

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= Δ ′ t eΓ(1− β2)1+ β

= Δ ′ t eΓ1

Γ2(1+ β)=

Δ ′ t eΓ(1+ β)

Timecontraction!

Forθ=90°

Timedilation=usualspecialrelativity(Lorentztransformations)

AberrationoflightAnotherveryimportanteffectoccurringwhenasourceismovingatrelativisticvelocitiesisaberrationoflight.

TrajectoryofthephotonappearsinclinedAnglesaredifferentindifferentframes

Forθ’=90°sinθ=1/ΓIfΓ>>1thensinθ≈θIsotropicsourceemitshalfofitsphotonsatθ’<90°Observerseeshalfofphotonsbeamedinaconeofsemiaperture1/Γ

1/Γ

K’ K

Synchrotronemission

Twoingredients:relativisticparticlesandmagneticfieldWhatisresponsibleforthiskindofradiationistheLorentzforce,makingtheparticletogyratearoundmagneticfieldlines:changeinvelocitydirection=acceleration=radiationThevelocitymodulusdoesnotchange,becausetheLorentzforcedoesnotwork.

Totalpoweremittedbythesingleelectron:

Forisotropicdistributionofpitchangles:

Totalpoweremittedbyasingleparticlewithpitchangleθ:

Synchrotroncoolingtime

Exercisesfortomorrow:

7.  Estimatethesynchrotroncoolingtimeofanelectronemittinggamma-raysinaGRB(γ=200andB=106Gauss)andcompareitwiththecoolingtimeofanelectroninthevicinityofasupermassiveAGNblackholeandintheradiolobesofaradioloudquasar(seesection4.2.1)

B

Synchrotronspectrumandtypicalfrequency

Exercisesfortomorrow:

8.  Estimatethetypicalsynchrotronfrequencyoftheelectroninexercise7a)  intheframeatrestwiththeemittingelectronb)  intheframeoftheobserverthatseetheelectronmovingtowardhimwithaLorentzfactorΓ=100.

€

νS = γ 2Be

2πmec

SynchrotronspectrumandtypicalfrequencyEmissionfrom1singleelectron

Electronenergydistribution:

Inhigh-energyastrophysicsisoftenapower-lawdistribution:

Theresultingspectrum:power-lawsegments

Ingredients:photonsandelectrons

DirectComptonscattering:whentheelectronisatresttransferofenergyfromphotontoelectron

InverseComptonscattering:electronhasaenergy(greaterthanthetypicalphotonenergy)transferofenergyfromelectrontophoton

Comptonscattering

DirectComptonscattering

electronatrestandincomingphoton

Whentheenergyoftheincomingphoton(asseenbytheelectron)issmallwithrespecttomec2theprocessiscalledThomsonscattering

Whentheenergyoftheincomingphoton(asseenbytheelectron)iscomparableorlargerthenmec2theprocessinintheKlein-Nishinaregime.

Crosssection:

InverseComptonscattering

electronatrestandincomingphoton

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E f =43γ 2Ei

Finalphotonenergy Comptonpower: Comptoncoolingtime

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ν f =43γ 2ν i

Finalphotonenergy Comptonpower: Comptoncoolingtime

Compton:

Synchrotron:

B

SynchrotronselfCompton:

Populationofrelativisticelectronsinamagnetizedregion.Theyproducesynchrotronradiationandfilltheregionwithphotons.Thesephotonsarethenup-scatteredbythesamepopulationofelectrons