Theoretical Model for Time Evolution of an Electron Population under Synchrotron Loss

0.12 MB·English

by  Siddharth Malik

Bull.Astr.Soc.India(2015)00,1–15
5 Theoretical Model for Time Evolution of an Electron
1
Population under Synchrotron Loss
0
2

n Siddharth Malik1∗
a
J 1DepartmentofEarthandSpaceSciences,IndianInstituteofSpaceScienceandTechnology,Thiruvananthapuram-695547,India

5 2SemiconductorLaboratory,DepartmentofSpace,GovernmentofIndia,Mohali-160071,India
1

] 18thMarch2015
E
H
.
h Abstract.
p
Manyastrophysicalsourcesradiateviasynchrotronemissionfromrelativistic
-
o electrons. The electrons give off their kinetic energy as radiation and this radiative
r lossmodifiestheelectronenergydistribution. Ananalyticaltreatmentofthisproblem
t
s ispossibleinasymptoticlimitsbyemployingthecontinuityequation. Inthisarticle,
a
weareusingaprobabilisticapproachtoobtaintheanalyticalresults. Thebasiclogic
[
behind this approach is that any particle distribution can be viewed as a probability
1
distribution after normalizing it (as is done frequently in statistical mechanics with
v
ensembles containing very large numberof particles). We are able to reproducethe
0
8 established resultsfrom ournovelapproach. Same approachcan be appliedto other
0 physicsproblemsinvolvingspatialortemporalevolutionofdistributionfunctions.
5
0
3. 1. Introduction
0
5
1 Synchrotron radiation is emitted by relativistic charged particles circling magnetic field lines.
: Highlyrelativisticparticlesareabundantinouruniverseandtheradiationdetectedfromarange
v
of astrophysical sources originate due to this process. Supernova remnants, pulsars, Gamma
i
X
Ray Bursts and large-scale jets emanating from certain galaxies are some of the examples. In
r this article, we are describingthe time evolutionof a populationofelectronsemittingsynchro-
a
tron radiation. We used probabilitytheory to arrive at the results conventionallyderived using
the continuityequation. These results are widely used in modelingastrophysicalsources. Our
approachisgenericandcanbeusedforsimilarphysicalproblems.
Total energy of a relativistic electron can be expressed as E = γm c2 where m c2 is the
e e
∗email:[email protected]
2 S.Malik2015
rest energy and γ is the Lorentz factor ( 21−v2/c2 ) of an electron moving with velocity v.
The synchrotron power P depends on tphe energy E (equivalently lorentz factor γ) and the
syn
larmorradiusoftheelectron. Astheelectronradiates,itsenergy(equivalentlyγ)reduces. Time
evolution of the Lorentz factor of a synchrotron radiating electron of mass m , charge e and
e
initial lorentz factor γ in the presence of a magnetic field density B can be derivedas follows
0
(seeappendixforthederivation)
γ σB2
γ(t)= 0 A= t (1)
1+Aγ t 6πm c
0 e
whereσ istheThompson’sCrosssection.
t
Insteadofmonoenergeticelectrons,realisticsystemshaveelectronsofacertainenergydistri-
bution. Mostastrophysicalsystemsemittingsynchrotronradiationdonothavesufficientparticle
density to achieve thermal equilibrium. Hence instead of a relativistic Maxwellian, the energy
distributionisexpectedtobea‘power-law’oftheform:
n(γ)dγ= K γ−pdγwhereγ <γ<γ (2)
e m u
whereγ andγ aretheminimumandmaximumlimitsofthedistributionrespectively,K isan
m m e
arbitraryconstantdependingonthephysicalparametersofagivensystem. Thisisalsocalledthe
‘non-thermal’distribution. Thisdistributiondoesnotremainthesameovertimeduetosynchro-
tron loss sufferedby each electron. However,the time evolutionof the electronpopulationnot
onlydependsontheenergylossbutalsoontheinjectionoffreshparticlesintothesystem. Our
aimistoobtainthedistributionn′(γ(t))atanygiventimet.
We assume a constantmagneticfield B and two typesof injection: (i) One Shotinjection
and(ii)ContinuousInjection. We assumethatthefreshparticlesthatareinjectedinthesystem
alwaysfollowthenon-thermalpowerlawgiveninequation2.
2. One ShotInjection
Oneshotinjectionmeansthatatt=0weinjectelectronsinthesystemwhichfollowthepower-law
distributioninequation2. Ifwedivideequation2bythetotalnumberofparticlesinthesystem
N,itbecomesaprobabilitydistribution:
p(γ )dγ = Kγ−pdγ whereγ <γ <∞ (3)
0 0 0 0 m 0
Wehaveassumedthatγ →∞. K = K /N andγ istheminimumlorentzfactorofthedistribu-
u e m
tion. p(γ )=istheprobabilitythatanelectronhasaninitialLorentzfactorbetweenγ andγ +
0 0 0
dγ
0
If we substitute γ as ∞ in equation 1, then γ(t) will tend to 1. Therefore, even if the
0 At
maximumlorentzfactor in infinite initially, it will to tend to some finite value say γ at a later
u
timet.
TheoreticalModelforTimeEvolutionofanElectronPopulationunderSynchrotronLoss 3
Thesystematalatertimetcanberepresentedas,
p(γ)dγ= Kγ−pdγwhereγ <γ<γ (4)
m u
K = K /N; p(γ) =probabilitythataelectronhaslorentzfactorbetweenγandγ+dγatagiven
e
timet.
Letusconsidertworandomvariables:
1. γ whichrepresentstheLorentzfactorofanelectronattimet=0
0
2. γwhichrepresentstheLorentzfactorofanelectronatsometimet
Inordertofind p(γ),wecanusethetheoremfortransformationofrandomvariablesfromprob-
abilitytheory(MyersYeandWalpole2007).
Theorem 1. Suppose that X a continuous random variable with probability distribution f(x).
Lety=u(x)defineaone-onerelationshipcorrespondencebetweenthevaluesofXandY sothat
the equation y = u(x) can be uniquely solved for x to obtain x = w(y). Then, the probability
distributionofY isgivenis:
g(y)= f(w(y))|J| (5)
where J =dw/dyandiscalledthejacobianofthetransformation
Inourcase x = γ , f(x) = p′(γ ), y = γandg(y) = p(γ),u(x)isgivenbyequation2. So,
0 0
usingTheorem1,wecanwrite,
dγ γ−p
p(γ)= p′(γ (γ)) 0 =K (6)
0 dγ (1−Aγt)−p+2
or,
γ−p
p(γ)dγ= K dγ where γ′ <γ<γ′ (7)
(1−Aγt)−p+2 m u
Atalatertimet,minimumandmaximumvaluesofγhavechangedwhichcanbefoundout
bysubstitutinginitialminimum(γ)andmaximum(∞)valuesinequation2.
γ 1
γ′ = m γ ≡γ′ = (8)
m 1+Aγt c u At
Note,thatevenifatt=0,maximumpossibleLorentzfactorforanelectronwasinfinite,after
a timet, maximumpossibleLorentzfactor= γ . NoelectroncanhaveaLorentzfactorgreater
c
thanγ atagiventimet.
c
4 S.Malik2015
Figure 1. Initial and Time Evolved ParticleDistribution for one shot injection given by equation 2 and
equation9respectively
Equation7givesustheprobabilitydistribution. Forobtainingtheparticledistributionfunc-
tion,wehavetomultiplytheprobabilitydistributionfunctionbyN (totalnumberofparticles).
γ−p
n(γ)dγ= K dγ where γ′ <γ<γ (9)
e(1−Aγt)−p+2 m c
whereK = NK. So,theparticledistributionfunctionattimetforoneshotinjectionisgiven
e
byequation9. Figure1showsbothinitialandtimeevolveddistributionfunctions. Thegraphis
plottedbetweenγ andγ .
m c
3. Continuous Injection
InthecaseofContinuousInjection,freshelectronscontinuouslygetaddedandmakeupforthe
loss in energy space. Now, total number of particles will also vary. We can treat Continuous
Injection as a series of one shot injections. For simplicity, let’s assume that at each instant of
timeweinjectN numberofparticlesandeachinjectionfollowsequation1withsame p,γ and
0 m
γ =∞.
u
n(γ )dγ =K γ−pdγ whereγ <γ <∞ (10)
0 0 e 0 0 m 0
TheoreticalModelforTimeEvolutionofanElectronPopulationunderSynchrotronLoss 5
K isdeterminedfromtheequation
e
∞
n(γ )dγ = N (11)
0 0 0
Z
γm
N = N t (12)
0
Now,againconsidertworandomvariablesγ andγinexactlythesamemannerasdescribed
0
fortheoneshotinjectioncase. Eachelectroninthesystemwasinjectedduringaparticularinjec-
tionandeachinjectionevolvesindependentofotherinjections. Ifwelookatanyoneinjection,
it evolves in a manner similar to what we discussed in previoussection for one shot injection.
Theoverallevolutioncanbeviewedasasummationorintegrationoverallinjections. Ifweare
lookingattimetwecanassigntwoothervariablestoeachelectront andt ,t isthetimewhen
1 2 1
the electronwasinjected andt is the time elapsedsince its injectionorthe time forwhichthe
2
electronhasbeenthereinthesystem. Botht andt variesfrom0totsuchthatt +t =t.
1 2 1 2
Define:
p(γ,t ) ProbabilitythatarandomelectronhasaLorentzfactorγattimetandhasbeentherein
2
thesystemfortimet .
2
p(γ) ProbabilitythatarandomelectronhasLorentzfactorγattimet.
p(t ) Probabilitythatarandomelectronhasbeenthereinthesystemfortimet .
2 2
p(γ/t ) ProbabilitythatarandomelectronhasaLorentzfactorγgiventhatithasbeentherein
2
thesystemfortimet .
2
Fromdefinitionofconditionalprobabilitydistribution,weknowthat:
p(γ,t )= p(t )p(γ/t ) (13)
2 2 2
Astheinjectionisidenticalateveryinstant, p(t )isauniformdistribution:
2
1
if0≤t ≤t
p(t2)= t 2 (14)
0 otherwise

p(γ/t )istheprobabilitythatarandomelectronhasaLorentzfactorγgiventhatithasbeen
2
thereinthesystemfortimet . Ifweknowtheinitialdistributionandtimeelapsed,then,thefinal
2
distributionisgiveninasamewayasforoneshotinjectioncase(Oncewefixt ,wearetalking
2
aboutaparticularinjection,sotheproblemisidenticaltooneshotinjectioncase). Therefore,
γ−p
p(γ/t )= K where 0≤t ≤t (15)
2 (1−Aγt)−p+2 2
6 S.Malik2015
where,K = K /N ,puttingequation14and15inequation13:
e 0
K γ−p
p(γ,t )= 0≤t ≤t (16)
2 t (1−Aγt)−p+2 2
So,farwehavenotimposedanyconditiononγ.Nowfromdefinitionofmarginaldistribution
functionweknowthat:
p(γ)= p(γ,t )dt (17)
2 2
Z
Substitutingequation16inequation17:
t K γ−p
p(γ)= dt (18)
Z t (1−Aγt)−p+2 2
0
K γ−p+1 γ p−1
p(γ)= 1−(1− ) (19)
A(p−1)t" γ #
c
γ andγ′ aredefinedbyequation8.
c m
Dependinguponγ andγ ,wecanhavetwocases: slowcooling(γ <γ )andfastcooling
c m m c
(γ >γ ).
m c
Thelossbecomesconsiderableonlyforthoseelectronsforwhichtheradiativetimescaleis
lessthantheageofthesystem(t < t) becauseonlythoseelectronswilldissipateenergyata
rad
fasterratethantheageofthesystem. Radiativetimescaleisgivenas:
E
t ≡ (20)
rad dE/dt
So,lossbecomesconsiderableforthoseelectronsforwhich
E
t ≡ ≤t (21)
rad dE/dt
Fromequation1,wecaneasilyderive:
dγ Aγ 2
= 0 = Aγ2 (22)
dt 1+Aγ t
0
UsingaboveequationandtherelationE =γm c2,wecaneasilyderivethefollowingrelation:
e
dE d(γm c2) dγ
= e = m c2 = Aγ2m c2 (23)
e e
dt dt dt
TheoreticalModelforTimeEvolutionofanElectronPopulationunderSynchrotronLoss 7
Substitutingaboveequationinequation21weget,
γm c2 1
e <t or, γ> =γ (24)
Aγ2m c2 At c
e
Hence, Equation21 is equivalentto equation24. Both are just differentrepresentationsof
thesameconcept.Bothimpliesthattheenergylossisgreaterforelectronshavingalorentzfactor
greaterthanγ ,weexpectthepowerlawtobesteeperintherangeγ>γ .
c c
Therefore,ifγ < γ ,onlyasmallfractionofelectronsinthesystemwillbecoolingdom-
m c
inated. Hencewereferittoasslowcooling. However,ifγ > γ , alargefractionofelectrons
m c
iscoolingdominatedandhencewereferittoasfastcooling.Theelectrondistributionatagiven
timeisdifferentforbothcases. Hence,wehavediscussedthemseparatelyincomingsubsections.
3.1 SlowCooling
Theexpressioninequation19isvalidonlyforγ < γ < γ . Fromequation8,weknowthatthe
m c
maximumattainablelorentzfactorforaelectronatatimetisgivenbyγ = 1/At. Acontinuous
c
injection can be seen as a superimpositionof various sequentially injected one shot injections.
Hence,wecanapplytheresultsderivedforoneshotinjectiononaparticularsub-injectionofthe
continuousinjection.
Consideracontinuousinjectionstartedatt = 0andcurrenttimeast. Now,takeaoneshot
injectioninjectedattimet intothesystem. Then,anyelectronwhichwasapartofthisinjection
1
has been inside the system for t = t−t seconds. Therefore, the maximumattainable lorentz
2 1
factorforsuchaelectronis1/At .
2
Supposewewanttocalculate p(γ)forsomeγ ≥ γ = 1/At. Onlythoseelectronscanattain
c
thisvaluewhosemaximumattainablelorentzfactorisgreaterthanthevalueofγatwhichweare
calculating p(γ)i.e. electronsforwhich
1
>γ (25)
At
2
or,
1
t < (26)
2 Aγ
Therefore,forcalculating p(γ)forγ > γ ,upperlimitoftheintegralintheequation18will
c
changeto1/Aγinsteadoft.
Hence,forγ>γ
c
A1γ K γ−p
p(γ)= dt (27)
Z t (1−Aγt)−p+2 2
0
8 S.Malik2015
K
p(γ)= γ−(p+1) γ>γ (28)
A(p−1)t c
Equation19and28combinedgives p(γ)forγ ≤ γ < ∞,buttherewillbesomeelectrons
m
whichwillhaveLorentzfactorbelowγ also(electronshavinginitialLorentzfactorjustabove
m
γ willloseenergytogobelowγ . Fortheseelectronsalso,wewillhaveadifferentexpression.
m m
The minimumLorentzfactor attainable at time t is γ′ , which is given by givenby equation8.
m
Butthisistheminimumfortheveryfirstinjectioninjectedatt = 0. Forelectronsbelongingto
any later injection (whichwere injected at time t > 0 and has been there for time t < t), the
1 2
minimumattainableLorentzfactorisγ /(1+Aγ t ).So,ifwearelookingataγ<γ ,onlythose
m m 2 m
electronshaveanon-zeroprobabilityofhavingthisLorentzfactorwhichwereinjectedsuchthat
theminimumattainableLorentzfactoratpresenttimeislessthanγ.
γ
m <γ (29)
1+Aγ t
m 2
γ−γ
t > m (30)
2 γγ
m
So,forcalculating p(γ)forγ<γ ,lowerlimitoftheintegralintheequation18willchange
m
to(γ−γ )/γγ insteadoft. So,forγ<γ
m m m
t K γ−p
p(γ)= dt (31)
Zγ−γm t (1−Aγt)−p+2 2
γγm
Kγ−(p+1) γ p−1 γ p−1
p(γ)= − 1− γ<γ (32)
A(p−1)t γ ! γ !  m
 m c 
3.2 FastCooling
Themainlogicremainsthesameforfastcoolingalso,equation16and17arealwaysvalid.Ifyou
arelookingintheregionwhereγ >γ ,then,upperlimitintheintegralinequation18shouldbe
c
asgivenbyequation24insteadoft. Similarlyifyouarelookingintheregionwhereγ<γ ,the
m
lowerlimitoftheintegralinequation18shouldnotbezero. But,asgivenbyequation30. Now,
in case offastcooling,if youarelookingattheregionγ < γ < γ , onlythoseelectronshave
c m
anon-zeroprobabilityofhavingthisLorentzfactorwhichwereinjectedsuchthattheminimum
attainableLorentzfactoratpresenttimeislessthanγandthemaximumattainableLorentzfactor
atpresenttimeisgreaterthanγi.e. injectionswhichsatisfiesbothequation26and30. So,both
thelimitsoftheintegralinequation18willchangeaccordingequations24and30.
So,forfastcoolingandγ <γ<γ ,
m c
A1γ K γ−p
p(γ)= dt (33)
Zγ−γm t (1−Aγt)−p+2 2
γγm
TheoreticalModelforTimeEvolutionofanElectronPopulationunderSynchrotronLoss 9
K γ−2
p(γ)= γ <γ<γ (34)
A(p−1)tγp−1 m c
m
Forγ>γ ,onlyupperlimitchangesbecauseγ>γ butnot<γ ,sotheexpressionforp(γ)
m c m
isexactlysameasequation26.
Similarly, for γ < γ , only the lower limit changesbecauseγ < γ but notγ > γ , so the
c m c
expressionisexactlysameasequation30. Asummaryofformulasfor p(γ)indifferentregimes
forbothslowandfastcoolingcaseisgivenbelow:
SlowCooling
Kγ−(p+1) γ p−1 γ p−1
p(γ)= − 1− γ<γ (35)
A(p−1)t γ ! γ !  m
 m c 
K γ−p+1 γ p−1
p(γ)= 1−(1− ) γ ≤γ≤γ (36)
A(p−1)t" γ # m c
c
K
p(γ)= γ−(p+1) γ>γ (37)
A(p−1)t c
FastCooling
Kγ−(p+1) γ p−1 γ p−1
p(γ)= − 1− γ<γ (38)
A(p−1)t γ ! γ !  m
 m c 
K γ−2
p(γ)= γ <γ<γ (39)
A(p−1)tγp−1 m c
m
K
p(γ)= γ−(p+1) γ>γ (40)
A(p−1)t c
Theseareprobabilitydistributions,forobtainingparticledistributionfunctions,multiplyall
of these equationsby N = N t (totalnumberof particles). It will simply change K/t in all the
0
aboveequationstoK .
e
K
N t× = N K = K (mentionedearlier) (41)
0 0 e
t
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Theparticledistributionfunctionsaregivenbelow:
SlowCooling
K γ−(p+1) γ p−1 γ p−1
n(γ)= e − 1− γ<γ (42)
A(p−1)  γ ! γ !  m
 m c 
K γ−p+1 γ p−1
n(γ)= e 1−(1− ) γ ≤γ≤γ (43)
A (p−1)" γ # m c
c
K
n(γ)= e γ−(p+1) γ>γ (44)
A(p−1) c
FastCooling
K γ−(p+1) γ p−1 γ p−1
n(γ)= e − 1− γ<γ (45)
A(p−1)  γ ! γ !  m
 m c 
K γ−2
n(γ)= e γ <γ<γ (46)
A(p−1)γp−1 m c
m
K
n(γ)= e γ−(p+1) γ>γ (47)
A(p−1) c
Figure 2 andFigure3 showsthe variationofparticle distributionasa functionof γ forthe
slowcoolingcaseandfastcoolingcaserespectivelyinlogarithmicscale.
4. Standard Result
Thestandardresult(Wallace&1979Chapter3)obtainedbysolvingthecontinuityequationsays
thatelectrondistributionsaremulti-powerlawswithpowerindexesdefinedasfollows:
• pforγ <γ≤γ andp+1forγ <γ
m c c
• -2forγ <γ≤γ andp+1forγ <γ
m c c