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]
€
S = F(t)dt0
T
∫
€
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
€
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)
€
= 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
€
E f =43γ 2Ei
Finalphotonenergy Comptonpower: Comptoncoolingtime
€
ν f =43γ 2ν i
Finalphotonenergy Comptonpower: Comptoncoolingtime
Compton:
Synchrotron:
B
SynchrotronselfCompton:
Populationofrelativisticelectronsinamagnetizedregion.Theyproducesynchrotronradiationandfilltheregionwithphotons.Thesephotonsarethenup-scatteredbythesamepopulationofelectrons