Theory of the Jitter radiation in a magnetized plasma accompanying temperature gradient

0.31 MB·

by  Makoto Hattori

Publ.Astron.Soc.Japan(2016)00(0),1–14 1
doi:10.1093/pasj/xxx000
Theory of the Jitter radiation in a magnetized
plasma accompanying temperature gradient
M.Hattori and K.Fujiki
6
1
AstronomicalInstitute,Graduateschoolofscience,TohokuUniversity,Sendai980-8578,
0
2 Japan

n E-mail:[email protected]
∗
a
J Received2015June3;Accepted2016January28

9
2 Abstract

] Thelinear stability of amagnetized plasmaaccompanying temperaturegradient wasreexam-
E
ined by using plasmakinetic theory. The anisotropic velocity distribution function wasdecom-
H
posedintotwocomponents. Oneisproportionaltothetemperaturegradientparalleltoandthe
.
h other is proportional to the temperature gradient perpendicular to the back ground magnetic
p field. Since the amplitude of theanisotropic velocity distribution function is proportional to the
-
o heat conductivity and the heat conductivities perpendicular to the magnetic field is strongly
r reduced, the first component of the anisotropic velocity distribution function is predominant.
t
s
The anisotropic velocity distribution function induced by the temperature gradient along the
a
[ backgroundmagneticfielddrivesplasmakineticinstabilityandthecircularpolarizedmagnetic

plasma waves are excited. The instability is almost identical to Weibel instability in weakly
1
v magnetizedplasma. However,dependingonwhetherwavevectorsofmodesareparalleltoor
4
antiparallel tothebackgroundmagnetic field,thegrowthrateissuppressedorenhanceddue
0
tobackgroundmagnetic field. Inthestronglymagnetizedplasma,onemodeisstabilizedand
0
8 onlyoneofthemodesremainsunstable.
0 TheJitterradiationspectrumformulaeemittedbyrelativisticelectronswhentheytravelthrough
.
1 themagnetizedplasmawiththeplasmawavesdrivenbytheinstability,arededucedatthefirst
0
time. The synchrotron emission and the Jitter radiation are simultaneously emitted from the
6
1 same relativistic electron. The Jitter radiation is expected to be circularly polarized but with a
: verysmallpolarization degree sincealmostthesameamount ofleftandrighthanded circular
v
i polarizedmagneticwavesareexcitedbytheinstability.
X
Keywords:Plasmas—instabilities—relativisticprocesses
r
a
1 Introduction (Weibel 1959, Ramani and Laval 1978, Okabe and Hattori
2003). By following Okabe and Hattori 2003, we refer this
AstronomicalrolesoftheWeibeltypesinstabilitieshaveexam-
instability as the RL instability since this instability was first
inedsinceitsdiscovery(Weibel 1959). TheWeibelinstability
foundbyRamaniandLaval 1978. TheRLinstabilityhasbeen
hasbeenproposedasagenerationmechanismofthemagnetic
studiedasamechanismofthereductionoftheheatconductivi-
fieldsinastronomicalshockwaves(MedvedevandLoeb 1999,
tiesintheintraclustermedium(Pistinner,LevinsonandEichler
Huntington et al 2015), of the cosmological magnetic from
1996,HattoriandUmetsu 2000)andasamechanismtomain-
zero seed field (Schlickeiser and Shukla 2003), and of origin
tainthesharpinterfaceofthecoldfrontsfoundintheintraclus-
oftheinterstellar turbulent field(TautzandTriptow 2013). It
termedium(OkabeandHattori2003).However,ithasbeenyet
hasbeenalsoknown thatthetemperaturegradientofthether-
unclearwhethertheRLinstabilityplayssignificantroleinsome
malelectrons drivestheWeibeltypeplasmakinetic instability
(cid:13)c 2016.AstronomicalSocietyofJapan.
2 PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0
astronomical situation. Therefore, the studies on the observa- andstronglymagnetized plasma. Okabe andHattori 2003has
tional tests to judge whether the RL instability really plays a shownthatthephysicalessenceoftheRLinstabilityinunmag-
centralroleinthegenerationofthemagneticfieldornotarede- netized plasma is the same as the Weibel instability. Studies
sired.Spectrumdistortionofthecomicmicrowavebackground ontheWeibelinstabilityinamagnetizedplasmaweredoneby
radiationcausedbytheinverseComptonscatteringbythermal Lazar, Schlickeiser and Poedts 2009. The comparison of the
electronswhichbelongstotheanisotropicelectronvelocitydis- results of the RL instability in a magnetized plasma with the
tribution function generated by the temperature gradient pro- results obtained by Lazar, Schlickeiser and Poedts 2009 may
posedbyHattoriandOkabe 2005couldbeoneofsuchtests. helptodeepenourunderstandingsofthephysicsoftheRLin-
stability in a magnetized plasma. Because the physical simi-
The Jitter radiation emitted from the relativistic electrons
larities between the RL and the Weibel instabilities were not
when they travel through the magnetic fields generated by the
clearlyrecognizedbeforeOkabeandHattori2003,thecompar-
Weibelinstabilityhasbeenstudiedasanalternativeinterpreta-
ison was not done by Levinson and Eichler 1992. In this pa-
tion of the gamma ray burst after glow (Medvedev 2000). In
per,wesummarizethecharacteristicsoftheRLinstabilityina
the case of the RL instability, the Jitter radiation is also ex-
magnetizedplasmabycomparingtheWeibelinstability. More
pectedtobeobserved. Thetypicallywavelengthoftheexcited
over,LevinsonandEichler1992didnotshowthenatureofthe
wavesareorderoftheplasmapenetrationlength,thatisc/ω ,
pe
eigen modes. However, to analyze radiative processes when
wherecisthespeedoflightandω istheplasmafrequencyde-
pe
relativistic electrons travel through the excited waves, the na-
finedbyω = 4πe2n /m where eisachargeofanelec-
pe e e
− tureoftheeigenmodesareimportant. Therefore,weshowthe
tron,m isaneplectronmassandn isanelectronnumberden-
e e
eigenmodesoftheunstablemodes.Medvedev2000studiedthe
sity.Whenthesaturationleveloftheamplitudeofthemagnetic
Jitterradiationwhentherelativisticelectronstravelthroughthe
fieldsintheexcited waves aredefinedbythethermalelectron
Weibelturbulence. However,theradiativeprocessesinthesit-
trappingbythemagneticfieldoftheexcitedwaves,theampli-
uationwhentheorderedmagneticfieldandthewavemagnetic
tudeofthemagneticfieldintheexcitedwavesisdescribedby
field generated by the RL instability in a magnetized plasma
ω (v /c)ω wherev istheelectronthermalvelocityde-
ce1 th pe th
∼ coexist,havenotyetbeendone. Therefore,westudytheradia-
finedby 2k T /m ,k istheBoltzmannconstant,T isthe
B e e B e
tiveprocessesinthesituation whentheorderedmagneticfield
electronpthermaltemperatureandω =eB /m cistheelec-
ce1 1 e
andthewavemagneticfieldgeneratedbytheRLinstabilityina
troncyclotronfrequencydefinedbythemagneticfieldstrength
magnetizedplasmacoexist,anddeducethespectrumformulae
oftheexcitedwaves,thatisdenotedbyB . Orbitsofrelativis-
1
oftheJitterradiationinthissituation.
ticelectronsareperturbedbythemagneticfieldoftheexcited
waves. Thedeflection angle, θ ,whentherelativistic electron
d TheJitterradiationhasbeenstudiedforyears(e.g. Landau
withtheLorentzfactorofγ travelsthroughtheexcitedwaves,
andLifshitz 1975,ToptyginandFleishman 1987,Revilleand
isabouttheratiobetweenthewavelengthoftheexcitedplasma
Kirk 2010, Teraki and Takahara 2011, Kelner, Aharonian
waves and the Larmor radius for the relativistic electron, that
and Khangulyan 2013). The emission of electromagnetetic
isθ (1/γ)(ω /ω ) (1/γ)(v /c). Itshowsthat,asfar
d∼ ce1 pe ∼ th wavesduetotheinteractionofrelativisticelectronswithregu-
asnonrelativistic thermalplasmaisconsidered, thedeflection
larandturbulentmagneticfieldswerefirststudiedbyToptygin
angle is much less than the angle spanned by the relativistic
andFleishman 1987. Theemissionarisedbytheregularfield
beamingcone,thatis1/γ. Oncethelineofsightisincludedin
ischaracterized bythesynchrotron emissionandtheemission
the relativistic beaming cone, the line of sight is continuously
raisedbythesmallscaleturbulent fieldischaracterizedbythe
includedinitduringtherelativisticelectrontravelthroughthe
Jitter radiation. They showed that the frequency of the Jitter
excited waves. The emission caused by this situation is the
radiationischaracterizedbyγ2(c/ℓ )andmuchhigherthan
min
Jitter radiation (Medvedev 2000). The observed frequency of
the characteristic frequencyof the synchrotron emissionemit-
the Jitter radiation in this situation is defined by the Doppler
tedbythesamerelativisticelectronwhentheminimumlength
shiftedplasmafrequency,thatisγ2ω .
pe of the coherent scale of the turbulence, ℓ , is much shorter
min
Theextensionoftheabovestudiestothemagnetizedplasma than the non relativistic Larmor radius R = c/ω where
L ce0
isrequiredwhenweapplytheresultstoastrophysicalsituation ω istheelectroncyclotronfrequencydefinedbytheregular
ce0
sincealmostalltheastrophysicalplasmaaremagnetized. The magneticfield. Kelner,AharonianandKhangulyan 2013pre-
plasma kinetic instability driven by electron temperature gra- sented the general formulae of the radiation within the frame
dient in a magnetized plasma was examined by Levinson and workofperturbationtheoryandrestressedthedistinctspectral
Eichler 1992. They found that the unstable modes driven by features of the Jitter radiation in which the emitted radiation
the temperaturegradient exist even in the magnetized plasma. shifted toward higher energy compared to synchrotron emis-
They also showed that there are significant differences of the sion. Toptygin and Fleishman 1987 showed that both syn-
behaviorsoftheinstabilitybetweenweaklymagnetizedplasma chrotron and Jitter radiations appeared in the frequency range
PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0 3
lessthanγ2ω . However,theirturbulentmagneticfieldmod- totheheatcurrentdensityas
ce0
elsdidnothavecertainphysicalgrounds.Inthispaper,wehave m v2
κ ~T = d3~v e ~vf(1), (2)
linkedtheRLinstabilityinamagnetizedplasmatosmallscale − Sp∇ e Z 2
turbulentmagneticfieldsandperformedselfconsistentstudies
whereκ istheSpitzerheatconductivity(HattoriandOkabe
Sp
oftheradiationemittedbyrelativisticelectronsatthefirsttime.
2005) and we neglect a factor of order one deviation. The
Numerical studies of theJitter radiation basedon theparticle-
Spitzerheatconductivityisgivenbyκ λ n k v (Spitzer
Sp e e B th
in-celltechnique(RevilleandKirk 2010,TerakiandTakahara ∼
1956,Sarazin 1988).Therefore,theanisotropicelectronveloc-
2011) has been progressed. Although this method has agreat
itydistributionfunctionisrewrittenbyusingκ as
Sp
potential to deal with complex situations, their studied cases
1 ~v ~T 5 v2
awreeroeulitmsiditeedofinatrhaensgietuwatiitohnosuwrhceurrereℓnmtiinnt>erResLtsu.ptonowwhich f(1)∼κSpkBne v·t2h∇Tee(cid:18)2−vt2h(cid:19)fm. (3)
Sincetheheatconductivities inthedirectionsofperpendicular
The structure of the paper is as follows. In section 2, we
to the magnetic field, κ , are dramatically reduced (Spitzer
revisitplasmakineticinstabilitydrivenbyelectrontemperature
⊥
1956) from the Spitzer value, we propose that the anisotropic
gradientinamagnetizedplasma. Insection3,wepresentthe-
electron velocity distribution function in amagnetized plasma
oryoftheJitterradiationinamagnetizedplasmawhentherel-
isdecomposedintofollowingtwopartsas
ativisticelectronstravelthroughthemagneticfieldsassociated
wdeidthictahteedptloastmheadwisacvuesssieoxncsi.ted by the instability. Section 4 is f(1)∼κSpkB1nevvk∇t2hkTTee(cid:18)25−vvt22h(cid:19)fm
1 v T 5 v2
2 Plasma kinetic instability driven by +κ⊥kBne ⊥v∇t2h⊥Tee(cid:18)2−vt2h(cid:19)fm, (4)
where v and v are electron velocity components parallel to
electron temperaturegradientin a k ⊥
andperpendiculartomagneticfield,andthetemperaturegradi-
magnetized plasma
entsparallelandperpendiculartothemagneticfieldaredenoted
Inthissection,plasmakineticinstabilitydrivenbyelectrontem- by T and T ,respectively. Sinceκ κ ,weadopt
e e Sp
peraturegradientinamagnetizedplasmaaresummarized. The the∇fokllowinga∇pp⊥roximatedformasforthe⊥an≪isotropicelectron
wavelengthsofthemodeswhichweareinterestedin,aremuch velocitydistributionfunctioninamagnetizedplasmaas
smallerthanthemeanfreepathofthermalelectron. Therefore,
v 5 v2
the plasma kinetic theory is used to analyze the linear stabil- f(1)=δf1vtkhcosθB(cid:18)2−vt2h(cid:19)fm, (5)
ity (Ichimaru 1973). The driving force of the instability is
whereδf ǫδ andθ isaninclinationangleofthetempera-
anisotropy of thermal electron velocity distribution function 1∼ T B
turegradientrelativetothebackgroundmagneticfield. Inthe
induced by temperature gradient (Ramani and Laval 1978).
following discussion, we drop a factor of cosθ appeared in
The amplitude of the deviation of the electron velocity distri- B
Eq.(5) since cosθ stays order of one except θ has a value
bution function from the Maxwell Boltzmann distribution is B B
veryclosetoπ/2.
characterizedbyrelativevariation oftheelectrontemperature,
The linear stability of the RL instability in the magnetized
δ =δT /T ,andratiobetweenelectronmeanfreepathanda
T e e
plasmawasfirststudiedbyLevinsonandEichler1992. Inthe
scale of electron temperature variation, ǫ=λ /δx, where we
e
restofthissection,wesummarizethecharacteristicsoftheRL
assumethattheelectrontemperaturevariesδT acrossthescale
e
instabilityinamagnetizedplasmabycomparingwiththechar-
δxalongthetemperaturegradient.
acteristicsoftheWeibelinstabilityinamagnetizedplasmaob-
Theanisotropicelectronvelocitydistributionfunctionwhich
tained byLazar,Schlickeiser andPoedts 2009aimingtohelp
isthedrivingforceoftheplasmakineticinstability,isobtained
physical understandings of the RL instability in a magnetized
by expanding electron velocity distribution function perturba-
plasma. As shown by Ramani and Laval 1978, the modes
tivelyinǫδ followingtheproceduresoftheChapman-Enskog
T
which have wave vectors nearly perpendicular to the temper-
expansion(ChapmanandCowling 1960). Inanunmagnetized
aturegradientarestableevenintheabsenceofthebackground
plasma,theanisotropicpartoftheelectrondistributionfunction
magneticfield. Further,theamplitude oftheanisotropy ofthe
uptothefirstorderinǫδ isdescribedby
T
electron velocity distribution function due to the temperature
f(1)=ǫδTvkv∇thTe (cid:18)52−vvt22h(cid:19)fm, (1) gnreatidciefinetldwhisicnhegislingeibalrylyspmearlpleansddiciusclaursstoedthienbaabcokveg.roTuhnedremfoarge-,
wheref istheMaxwell-Boltzmannelectronvelocitydistribu- wemaysafelyconcludethatthemodeswithwavevectorswhich
m
tion function, v isthevelocity component alongthetem- areparallelorantiparalleltothebackgroundmagneticfieldare
k∇Te
peraturegradient(RamaniandLaval 1978,OkabeandHattori themostrelevanttotheRLinstability inamagnetizedplasma
2003). Theanisotropic velocity distribution is directly related (Levinson and Eichler 1992). Inthe following discussion, we
4 PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0
takek 0limitwherek isthecomponentofthewavevec-
torspe⊥rp→endiculartothem⊥agneticfield. Inthislimit,onlytwo 100
fundamental modes with n=+1,−1arepredominant against 10-5
the higher order harmonics (Ichimaru 1973). The dispersion e0
c
relation is described by the function of ξ± ≡(ω±ωce)/kvth w /i 10-10
where the double sign same order and each sign corresponds w 0 10-15
ton= 1,respectively. Characteristicsoftheunstablemodes e
showth∓edifferentbehaviorsforsmallandlarge|ξ|limits. w| /rc 10-20
First the results for |ξ|≪1 and δf1 ≪1 are summarized. w| 10-25
Imaginarypartsoftheangularfrequencyaregivenby
10-30
ω(n) 1 3δf2kv c2 k3v δf1nω , (6) 1 2 3 4 5 6 7 8 9 10
i ∼ √π(cid:18)8 1 th−ωp2e th∓ 2 ce(cid:19) w ce0/kvth
wherek= ~k,thelasttermtakesuppersign,thatis( ),when~k Fig.1.Thedispersion relations obtained bylinear plasma kinetic stability
| | −
isparalleltothebackgroundmagneticfield,B~ ,andlowersign, analysisforamagnetisedplasma.Theequipartitionmagneticfieldstrength,
0
thatis(+),when~kisantiparalleltothebackgroundmagnetic thatisβ=1,isassumed. Thehorizontal axisisωce0/kvth wherekis
absolutevalueofthewavevectorofthemode.Thedottedlineisanabsolute
field. Ittakesthemaximumvalueatk=km= 2√12δf1ωpe/c. valueofrealpartoffrequencyofunstablemodesnormalized bykvth for
Realpartsoftheangularfrequencyaregivenby δf1=0.3.Thesolidlinesshowimaginarypartsoffrequencyoftheunstable
1 2 modesnormalizedbyωce0fromlefttorightforδf1=0.3,0.03,0.01,0.006
ωr(n)∼±4δf1kvth∓√πδf1ωi, and0.003,respectively.
1 1
δf kv + δf2nω , (7)
∼±4 1 th π 1 ce inamagnetizedplasmaandfirstfoundbyLevinsonandEichler
wherethefirsttermofthelastequationtakesupperorlowersign 1992althoughthephysicaloriginoftheappearanceofthisterm
when~kisparalleltoorantiparalleltothebackgroundmagnetic hasnotbeenclarifiedyet.When~kisparalleltoB~0andn=+1,
field,respectively. InEq.(7),termhigherthanthirdorderinδf theRLinstabilityisstabilizedbytheexistenceofthemagnetic
1
are neglected and k ∼km is assumed. As far as the absolute fields and β>δf1−4 is required for the modes to beunstable.
valuesofthewavevectorsstayaroundk δf1ωpe/c, Ontheotherhand,when~kisantiparalleltoB~0andn=+1,the
∼
existenceofthemagneticfieldsmakethemodesmoreunstable.
|ξr|∼δf1+β−1/2δf1, Forthemodeswithn= 1,situationsbecomeviceversa. The
|ξi|∼β−1/2+δf12, mostunstablemodesapp−earatk=kmwhen~kisantiparallelto
where the plasma β is defined by the ratio of thermal pres- B~0forn=+1and~kisparalleltoB~0forn=−1. Thegrowth
ratesofthesemaximumgrowthratemodesaregivenby
sure to the back ground magnetic field pressure as β =
2nekBTe/(B02/8π) where the mean molecular weight is set ω = δf1 δf12 vthω +1ω , (8)
to be 0.5 which corresponds to the fully ionized pure hydro- i,m √π(cid:18)8√2 c pe 2 ce(cid:19)
gen plasma. The plasma β is expressed by a combination
andrealpartsoftheangularfrequencyofthesemodesaregiven
of the electron thermal velocity, v , the electron cyclotron
th by
frequency for the back ground magnetic field, ω , and the
ce0
plasma frequency as ωce0c/(ωpevth)=(2/β)1/2. Therefore, ωr(±,m)=δf12(cid:18)±8√12vcthωpe±π1ωce(cid:19), (9)
small ξ limit requires that high β, that is β 1, and small
| | ≫
amplitude of the anisotropy, that is δf 1. The first term where the double sign same order. Set the back ground mag-
1
≪
of Eq.(7) exactly coincides with the real part of the RL in- neticfielddirectionaspositivedirectionofzaxis,thex,ycom-
stability for unmagnetized plasma (Okabe and Hattori 2003). ponentsofthemagneticfieldoftheeigenmodessatisfyfollow-
The second term of Eq.(7) is almost equivalent to the equa- ingrelations
tion (18) in Lazar, Schlickeiser and Poedts 2009. This result
B = iB , (10)
kx ky
is natural consequence since the RL instability for unmagne- ∓
tized plasma is essentially equivalent to the Weibel instability wheretheuppersignforn=+1andthelowersignforn= 1.
−
withrelativedirectionaltemperaturedifference,parameterAin Whenthebackgroundmagneticfieldisantiparalleltothetem-
Lazar,SchlickeiserandPoedts 2009,ofδf2(OkabeandHattori peraturegradient,resultsforn=+1correspondstotheabove
1
2003).Thefirstandsecondtermsoftheimaginarypartsexactly resultsobtainedforn= 1andresultsforn= 1corresponds
− −
coincide withtheimaginarypartsoftheRLinstability forun- totheaboveresultsobtainedforn=+1.
magnetized plasma (Okabe and Hattori 2003). The last term Nexttheresultsfor ξ >1aresummarized. Thissituation
| |
inEq.(6)isuniquelyappearedinthecaseoftheRLinstability corresponds to β 1 or δf 1. We solved numerically the
1
∼ ∼
PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0 5
plasmadispersionfunctionusingthecontinuedfractionexpan- ferred to thermal electron trapping condition. By taking into
sions(Kaji 1966,Mccabe 1984)toobtainthedispersionrela- accountthepossibilitythatthegrowthofthewavestopsbefore
tion in this casesincethe asymptotic expansion of theplasma itgetsvthωc−e11km∼1,thesaturationlevelofthemagneticfield
dispersionfunctioncannotbeused. Theplasmaβ issetequal strengthoftheunstablemodesisexpressedas
to1inthefollowing analysis. Whenthedirection ofthetem-
ω =bk v , (12)
peraturegradientisparalleltothebackgroundmagneticfield, ce1 m th
unstablemodesemergewhenk = kforn= 1modesand where b is a non dimensional number satisfying b 1.
k ∓ ± ≤
therearenounstablemodeforn= 1whenk = kwherethe Second argument is that the saturation level is determined
± k ±
doublesignsameorder.Theseresultsareabletobeunderstood by balancing between growth and nonlinear damping of the
becausethemodeswithk = kforn= 1appearedinhigh waves (RamaniandLaval 1978, Levinson and Eichler 1992).
k ± ±
βplasmaarestabilizedbythebackgroundmagneticfieldasβ Levinson and Eichler 1992 estimated the nonlinear damping
approachingto1(seeEq.(6))andonlyk = kforn= 1are rate due to wave-wave interaction for magnetized plasma as
k ∓ ±
remainedunstable. Whenthedirectionofthetemperaturegra- γnl v /(λ β(1+ω /k v )2) v δ /(ǫ˜δ δxβ(1+
th eff ce0 m th th T T
∼ ∼
dientisantiparalleltothedirectionofthebackgroundmagnetic ω /k v )2) where λ is an effective electron mean free
ce0 m th eff
field,unstablemodesappearwhenk = kforn= 1where path reduced from the Coulomb collision mean free path due
k ± ±
thedoublesignsameorder.Eigenmodesoftheunstablemodes tothescatteringbytheexcitedmagneticwavesand˜ǫappeared
arecircularpolarizedmagneticwaves.Forn=+1modewhen in γ is defined by the ratio between the effective mean free
nl
B~0 ~Teandn= 1whenB~0 ~Te,theeigenmodessat- path to the scale of the temperature gradient, δx. The ǫ˜may
k∇ − k−∇
isfyBkx= iBky.Forn=+1whenB~0 ~Teandn= 1 decreasefromtheinitialvalueofǫbecausetheeffectivemean
− k−∇ −
whenB~0 ~Te,theeigenmodessatisfyBkx=iBky. Theab- freepathbecomesshorterasafractionoftrappedthermalelec-
k∇
solutevaluesoftherealpartofωforδf1=0.3isshowninFig.1 trons which are trapped by the magnetic field of the excited
andisorderofδf1kvth. Signoftherealpartispositivewhen waves, is increased as the amplitude of the magnetic field of
n=+1andisnegativewhenn= 1.Therefore,thedirections the excited wave is growing. The decrease may stop at the
−
of thephase velocity of theunstable modes areparallel to the value at where the growth rate balances with the decay rate.
directionoftheheatflow,regardlessofthedirectionofthetem- Sincethequantitative argumentshowthenonlinearsaturation
perature gradients and signs of n. This propagation direction level is defined by this condition depend on detail models of
ofthewaveisoppositetothepropagationdirectionofthewave the temperature gradient of the system, we skip further quan-
excited in a high β plasma. The growth rates of the unstable titative arguments based on this argument. Third argument is
modeshaveanidenticalshapeforn=+1andn= 1.Theun- whetherinitialenergycontentisenoughtoexcitethemagnetic
−
stablemodesexistonlyinthelimitedrangeofthewavevector plasmawaves. Kato 2005showedbyperformingplasmapar-
aroundωce0/vth 9 10−6(B0/3µG)/(Te/106K)1/2 cm−1. ticlesimulationsaimingtostudythenonlinearevolutionofthe
∼ ×
Thegrowthrateshavesharppeakatthewavevectorof Weibelinstabilitythattheinitialdifferenceofthekineticenergy
1/2 caused by the anisotropy of the temperature of the electrons,
1ω 1 2 ω
km= a vcteh0 = a(cid:18)β(cid:19) cpe (11) thatis∆W ∼kBT1−kBT2whereT1andT2aretemperatures
in two different directions, is converted into the magnetic en-
where a runs from 2 to 5as δf runs from 0.3 to 0.002. It is
1 ergyoftheexcitedwaveswhentheamplitudeofthewavesget
naturaltoassumethatonlythemodeswithk=k areexcited
m maximum value. As pointed out by Okabe and Hattori 2003,
bytheinstability. Amplitudeofthemaximumgrowthratede-
thephysicalessenceoftheinstabilitydrivenbythetemperature
creasesrapidlyasdecreasingδf . Whenδf becomessmaller
1 1 gradientisequivalenttotheWeibelinstabilitywithadirectional
than0.002,thegrowthtimescalestartstoexceedtheageofthe
temperaturedifferenceof∆T (δf )2T .Therefore,themaxi-
universe for typical interstellar plasma . So we conclude that ∼ 1 e
mumavailableenergystoredinthemagneticfieldoftheexcited
theinstability sets inonly whenδf >0.002. Thedecreaseof
1 wavesisaboutk (δf )2T .Thissetsupperlimitontheexcited
thegrowthratesasdecreasingδf aremuchmoredramaticthan B 1 e
1 magneticfieldstrengthas
thedecreasesexpectedfromEq.(8).
The nonlinear saturation levels of the excited waves could 2n k δf2T > B12. (13)
bedefined byone ofthe following threemechanisms depend- e B 1 e 8π
ing on the situation. First argument is that the growth of the Forhighβplasma,thisconditionsetsupperlimitonbappeared
unstable modes stop when the Larmor radius of thermal elec- inEq.(12)as
tron gets shorter than the wavelength of the growing mode as
b β1/2aδf . (14)
1
discussed by Okabe and Hattori 2003. This is expressed as ≤
vthωc−e11km∼1whereωce1 iselectrongyrofrequencywiththe Inthecaseofhighβplasma,aslongasb<1thecondition(13)
excitedwavemagneticfieldstrength,B . Hereafter,thisisre- issatisfied.
1
6 PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0
3 Theoryof the Jitter radiation in a B =B sin(k z+ω t),
y1 1 m r
magnetized plasma
whereω isarealpartoftheangularfrequencyoftheunstable
r
3.1 Fundamentals mode. Velocities and orbits generated by the wave fields are
The emission mechanisms when relativistic electrons travel describedby~v1and~x1. Inthefirstorderofv1/v0andB1/B0,
through the magnetized plasma which is filled with the circu- theequationsofmotionfor~v1aregivenby
larpolarizedmagnetic waves excited bytheRLinstability are dv e e
γm x1 = v B + v B ,
examined. When thecoherent length ofthemagnetic fields is e dt −c y1 0 c z0 y1
dv e e
much longer than the gyration radius of the relativistic elec- γm y1 = v B v B ,
e dt c x1 0−c z0 x1
trons,hereafterwereferthisfieldorderedfield,theemittedra-
dv e e
diation is known as synchrotron radiation. Inthis section, we γme dzt1 =−cvx0By1+cvy0Bx1.
drive the radiation spectrum when relativistic electrons travel
Byinserting thezeroth orderorbit asfortheorbit ofelectron,
through the plasma which is occupied by both ordered mag-
thewavefieldsareapproximatelydescribedas
netic fields and magnetic waves generated by the RL instabil-
itydiscussedinSection2. Therearesimilaritiesinprocedures Bx1=B1cos(kv 0+ωr)t,
k
with the derivation of the weak undulator radiation spectrum B =B sin(kv +ω )t.
y1 1 0 r
(Hofmann2004).Setthedirectionoftheorderedmagneticfield k
to z-axis as B~ =(0,0,B ). Assume that the gradient of the Theperturbedvelocitiesinthefirstorderareobtainedas
0 0
erelecctitoronn. Cteomnpseidraetruurensitsapbaleramlleoldteowthiethonrd=ere+d1miangtnheetifcofilleolwdidnig- vx1+ivy1= kvk0−ω+ωsωerv1v−k0ωse0ei(kvk0+ωr)t+Ceiωse0t, (15)
athnealwysaivse.sAwsidthisckuzs<sed0iwnhSeerce.2k,zthiesuanzstcaobmlepmoondenetisolfimthietewdafvoer vz1= kvk0+seω1r⊥−0ωse0cos(kvk0+ωr−ωse0)t. (16)
vectorofthewavesinthecaseofthelowβ plasma. Sincethe SincethelastterminEq.(15)isdescribingthegyrationmotion
growthratetakesthemaximumvalueatk=k ,weassumethat aroundtheorderedfield,itisabletobeabsorbedinthezeroth
m
theexcitedwavesduetotheinstability ismonochromaticwith ordermotionandwesetC=0.Forhighβplasma
thewavevectorof~k=(0,0,−km).Hereafter,wereferkm=k. ωr vth ωr 1,
Thezerothordermotionofrelativisticelectronsaregivenby kv 0 ∼ c kvth ≪
k
followingequationsofmotion andforlowβplasma
dv e ω v
γme dxt0 =−cvy0B0, kvr ∼aδf1 cth ≪1.
0
dv e k
γme dyt0 = cvx0B0, Further,
dvz0 =0, ωse0 avth 1,
dt kv ∼ γ c ≪
th
whereγ isaLorentzfactoroftherelativisticelectron. Theze- is satisfied. Therefore, amplitudes in Eqs. (15) and (16) are
roth order orbits for relativistic electron with a velocity of v0 approximately ωse1/k. As discussed in below, the contribu-
(anequivalent Lorentz factor of γ) and apitch angle of α are tion of the perturbed motion of the relativistic electron to the
givenbysolvingtheseequationsas radiation spectrum happens within a only limited time inter-
vvxy00==vv⊥⊥00csoinsωωssee00tt,,xy00==−ωv⊥sωve0⊥s0e0s0incoωssωe0set0,t, a(v1na6ld)oωwfriTTth′′in∼∼thδ1if/s1ωtic<me0e1..inCTthehraevnraeglfeoasrreeo,fkthvtheke0pTph′hasa≫esef1sa,ciωntosrEes0qTisn.′E∼(q1s5γ1.)≪(a1n5d1)
vz0=vk0,z0=vk0t, and(16)areapproximatedaskvk0tinthefollowingdiscussion.
Undertheseapproximations,thevelocitiesandtheorbitsofthe
wherev v sinα, v v cosαandω ω /γ. The
⊥0≡ 0 k0≡ 0 se0≡ ce0 perturbedmotionaredescribedby
zerothorderorbitishelicalmotionaroundtheorderedmagnetic
ω
fieldwithagyrationperiodofT =2π/ωse0. v1x=− ske1coskv0kt, (17)
Orbits of the relativistic electrons are perturbed due to the v = ωse1sinkv t, (18)
magnetic waves excited bythe instability. Thesimilar studies 1y − k 0k
ω
in the context of scattering of relativistic charged particles by v1z= ske1tanαcoskv0kt, (19)
magneticirregularitieswasstudiedbyParker 1964. Themag- x = ωse1 1 sinkv t, (20)
neticfieldoftheunstablemodeforn=+1isexpressedas 1 − k kv 0 k0
k
ω 1
y = se1 coskv t, (21)
Bx1=B1cos(kmz+ωrt), 1 k kv 0 k0
k
PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0 7
ω 1
z = se1 tanαsinkv t. (22)
1 k kv 0 k0
k
Relativeamplitudesofthevelocityandthepositionoftheper-
B
turbedmotiontothezerothorderorbitare z
v v abv
⊥1 z1 th 1, (23)
(cid:12)v 0(cid:12)∼(cid:12)vz0(cid:12)∼ γ c ≪
(cid:12) r⊥ (cid:12) (cid:12) z (cid:12) ab v 2
(cid:12) ⊥1(cid:12) (cid:12) 1(cid:12) th 1, (24)
(cid:12)r 0(cid:12)∼(cid:12)z0(cid:12)∼ γ2(cid:16) c (cid:17) ≪
(cid:12) ⊥ (cid:12) (cid:12) (cid:12)
fo(cid:12)r low(cid:12) β(cid:12)plas(cid:12)ma where we implicitly assume that the pitch
angle does not take a value close to π/2 so as to avoid being TV cosθ
tanα 1,andv = v2 +v2 andr = x2+y2. For
≫ ⊥1 x1 y1 ⊥1 1 1
highβplasma, p p
v
v v 1v 1 T
⊥1 z1 thβ−1/2 , (25)
(cid:12)(cid:12)vr⊥0(cid:12)(cid:12)∼(cid:12)(cid:12)vzz0(cid:12)(cid:12)∼ γ1 cv 2 δf1 zʼ θ
(cid:12)(cid:12)r⊥10(cid:12)(cid:12)∼(cid:12)(cid:12)z10(cid:12)(cid:12)∼ γ2(cid:16) cth(cid:17) β−1δf1−2. (26) v α^
(cid:12) ⊥ (cid:12) (cid:12) (cid:12)
T(cid:12)heref(cid:12)ore,(cid:12)unl(cid:12)esstheamplitudeoftheanisotropicvelocitydis-
tribution is close to or less than vth/(γc)β−1/2, Eqs.(25) and v
(26) aremuchsmaller than 1. Theseresults guarantee the va-
lidityofourperturbativetreatmentfordescribingthemotionof α
the relativistic electron. When the pitch angle closes to π/2, π v
−−θ
theamplitude ofv1z becomes largeandtheperturbative treat- 2
x
mentisbroken.Bycomparing(19)andthezcomponentofthe
zeroth order electron velocity, the rangeof the pitch angle for
whichtheperturbativetreatmentisacceptable,isobtainedas Fig.2.Orbitofarelativisticelectronunderbackgroundmagneticfield.The
electrontravelshelicalorbit. Thelineofsightistakeninx-zplaneforsim-
π 1v
α< th. (27) plicityandisdescribedbydashedlinewitharrow. Inclinationangleofthe
2 −rγ c lineofsightrelativetozaxisisdefinedasθ.Byrotatingthecoordinatewith
As far asthe pitch angle is inthis range, the amplitude of the angleθasanaxisiny-axis,x-zistransformedintox′-z′. Thex′axisco-
incidewiththelineofsight. Thepitchangleoftheelectronisdefinedasα.
perturbed velocities are less than the amplitude of the zeroth
Thedeviationofthepitchanglefromθisdenotedbyαˆ.
ordervelocities. Therefore,exceptalimitedsmallrangeofthe
pitchangle,ourtreatmentisvalid.
The Fourier spectrum of the electric field of the radiation
emittedfromasinglerelativisticelectronisobtainedbyinsert-
ingtheelectronorbitinthefollowingequation
RE~ˆ(ω)= 2ieπωceiΦZ T2′(~n×(~n×β~0))eiω(t′−1c~n·~r(t′))dt′ (cid:54)
T1′ v 2
+ 2ieπωceiΦZ T2′(~n×(~n×β~1))eiω(t′−1c~n·~r(t′))dt′,(28) v2 γ-v1
T′
1
whereβ~ =~v /candβ~ =~v /c,and~r=~r +~r issuperposi- 1 2
0 0 1 1 0 1
tionofazerothorderorbitandanorbitoftheperturbedmotion
oftheelectron.Thefirsttermcontributesthesynchrotronradia-
tion.Thecontributiontotheradiationspectrumoftheperturbed 2vv1
orbit,~r ,appearedintheexponentofthefirsttermisnegligible.
1
sin
ThesecondtermprovidestheJitterradiation. Similaranalysis ω α
ce
wasdonebyGinzburgandSyrovatskii 1965. Theystudiedthe
Fig.3.Partofarelativisticelectronorbitinx′-yplaneforanelectronwith
polarizationdegreeofthesynchrotronemissionwhenarandom
α=θ. Theemissionconesduetotherelativisticbeamingeffectarede-
fieldissuperimposedonahomogeneousfield. scribedbyconeswithasolidangleofπ/γ2assignedtoelectronswhichare
Theradiationspectrumcontributedfromthesecondtermof denotedbyblackdots.
equation (28) is evaluated as follows. The direction cosine of
thelineofsightisdefinedinx-z planeas~n=(sinθ,0,cosθ).
8 PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0
Adenotesthevelocitiesspannedby 1/γ αˆ 1/γ. Thearc
V − ≤ ≤
zʼ Ais apart of acirclein vx′-vz′ plane with aradius of v cen-
teredontheorigin. ThearealabeledbyBismadebyrotating
the arc A around vz′ axis. Therefore, this surfaceis a part of
thespherewith aradius of v centered ontheorigin. Sincean
V Vy electronvelocityiscontainedinvx′-vz′ planeeveryT asillus-
tratedinFigure2,electronswithvelocities whichdirecttothe
areaBat somefixed moment contribute to the Jitter radiation
onceeveryT.
φ
V ^
−α
xʼ B 3.2 EssenceoftheJitterradiation
Firstly,thephysicalcharacteristicsoftheJitterradiationinthe
2
A /γ lowβplasmaaresketched.Byperformingthecoordinatetrans-
Fig.4.Thezerothordervelocitydistributionoftherelativisticelectronswith formationfromvx-vztovx′-vz′,thefirstorderperturbedveloc-
itiesaretransformedas
afixedγ which contribute tothe Jitterradiation are described invelocity
aspnadcvez.′T,hreesxpe′,ctyivaenlyd. zA′ncaormcplaobneelnetdsboyfvAeldoecnitoietessatrheedveenlootceitdiebsysvpxa′n,nveyd vx′ = ωske1coαˆsθcoskv0kt′, (29)
boyfs−ig1h/t.γT≤heαˆa≤rc1A/γiswahpearertαˆofisarceilractlieveinpvitcxh′-avnzg′leplaonfeelewcitthronasratodituhseolifnve vy=−ωske1sinkv0kt′, (30)
cAeanrteoruenddovnz′thaexoisr.igTinh.isTshuerfaarceeaislabaepleadrtboyfBtheissmphaedreebwyitrhotaatrinagdiuthseoafrvc vz′ = ωske1co1sθcoskv0kt′, (31)
centeredontheorigin.Theelectronswithvelocitieswhichpointtothearea
BinvelocityspaceatsomefixedmomentcontributetotheJitterradiation wheretheresultsareexpanded inthelowestorderofαˆ andt′
onceeveryperiodofT. isusedtodescribetimeatthepositionoftheelectron. Thetra-
jectoryoftheelectrontotalvelocityvectorinvelocityspaceis
describedinFigure5. Thetotalvelocityisasumofthezeroth
When the motion of the electron is relativistic, the radiation
ordervelocityandthefirstordervelocity. Thevelocityvectors
fromtheelectronisconcentratedinthesocalledemissioncone
of the electrons precess around the line of sight. Direction of
whichistheconewithsolidangleofπ/γ2duetotherelativis-
the rotation is clock wise rotation from the observer. The ve-
ticbeamingeffect. Therefore,theradiation fromelectrons are
locity vector traces elliptical trajectory with a minor radius of
observedonlywhenthelineofsightisincludedintheemission
ω /k and a major radius of ω /(kcosθ). In other word,
se1 se1
cones. InFigure2, thezerothorderorbitofanelectronisde-
the directions of the motion of electrons precess around the
scribed. Itshowsthatelectrontravelsalongahelicalorbit. The
lineofsightwhilechanging inclination anglefromω /kcto
se1
lineofsightisdescribedbyadashedlinewitharrow.InFigure
ω /(kccosθ)asshowninFigure5.Asfarasthelineofsight
se1
3, apartof theelectron orbitinx′-y plane whenα=θ isde- isnotclosetoperpendiculardirectiontothebackgroundmag-
scribed. Whentheelectronarrivesatthepositionlabeledby1,
neticfieldandαˆ<1/γ,thelineofsightisalwaysincludedin
theemissionconestartstoincludethelineofsight. Whenthe
theemissionconesofprecessingelectronssince
electron reaches at the position labeled by 2, the line of sight
ω b v 1
exitstheemissioncone.Duringthisinterval,thelineofsightis se1 th . (32)
kc ∼ γ c ≪ γ
kepttobeincludedintheemissioncone.Hereafterwereferthis
Theperiodoftheprecessionismuchshorterthantheduration
intervalasemissionorbit. Adurationwhileanelectrontravels
whileelectronstraveltheemissionorbitsince
theemissionorbitisT′=2/(ωcesinθ). Sincethesmalldevi-
ationofthepitchangleoftheelectronfromthelineofsightis 2/ωce0sinθ = 1 c cotθ 1. (33)
importanttodeterminethecharacteristicsoftheJitterradiation, 2π/kccosθ πavth ≫
thepitchangleisexpressedasα=θ+αˆinthefollowingdiscus- Therefore, the electron emits the radiation at the frequency of
sionasshowninFigure2.InFigure4,thezerothordervelocity theinverseoftheobservedperiodoftheprecessionoftheelec-
distribution oftheelectrons withafixedγ whichcontribute to tron. This is the Jitter radiation. This situation is illustrated
theJitterradiationisshown. InthisFigure,vx′ axisistakento in Figure 6. Suppose that an electron emits electromagnetic
thedirectionoftheline-of-sightandvz′ axisistakentothedi- wavewhenitarrivesatthepositionlabeledby1atthetimet′1.
rectionnormaltotheplanespannedbyvx′andvyaxesasshown Thedistancebetweentheposition1andtheobserverisL. The
inFigure2. Therelativepitch angle to theline ofsight, αˆ, is electromagneticwaveemittedattheposition1arrivesattheob-
showninthisfigure.Inthefollowingdiscussion,theamplitude serverat t1=t′1+L/c. During oneperiod ofthe precession,
ofthevelocity,e.g. γ,isfixedforsimplicity. Anarclabeledby theelectronmovesfromposition1toposition2. Thedistance
PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0 9
traveledbytheelectronduringtheperiodis2π/(kcosθ)which V
is aneffectivewave length ofthe excitedplasmawave forthe z
ʼ
observer whose line of sight is inclined by anangle θ relative
tothewavevectoroftheplasmawave. Theelectronreachesat
position2att′2=t′1+2π/(kcosθv0). Thearrivaltimeofthe
electromagnetic wave emitted at position 2 to the observer is
V
bV
t2=t′2+(L−2π/(kcosθ))/c.Duringthisperiod,theelectron th
emitsrighthandedellipticallypolarizedelectromagneticwaves cos
γc θ
towardtheobserverwherethepolarizationpatterntracestheor- V
bitoftheelectronillustratedinFigure5.Sincethelineofsight y
isalways included intheemissionconeduringthistimeinter-
val,alltheemissionemittedinthistimeintervalisobservedby
theobserverexceptthecasethatthedirection ofthemagnetic
bV
fieldisclosetoperpendiculartothelineofsight. Theobserved th
periodoftheelectromagneticwave,∆t=t2−t1isgivenby V > γc
2π v π xʼ
∆t 1 0 . (34)
∼ kccosθ − c ∼ γ2kccosθ
(cid:16) (cid:17) Fig.5.Atrajectoryofvelocityvectorofarelativisticelectroninvelocityspace
Therefore,theobservedfrequencyoftheJitterradiationis while the electron travels emission orbit. The velocity vector precesses
cosθ around vx′ axis. It rotates clock wise direction from the observer along
νJit∼ aπ √2β−1/2γ2ωpe, (35) elliptical trajectory with a minor radius of ωse1/k and a major radius of
ωse1/(kcosθ)denotedbythicksolidline.
where Eq.(11) was used. The emitted frequency by the Jitter
radiation is the Doppler shifted plasma frequency in β 1
∼
L
plasma. On the other hand, the frequency of the synchrotron
emission(RybickiandLightmann1979)isgivenby 2π/kcosθ bVth
1 3 γc
ν 0.29 γ2ω sinθ. (36)
syn∼ 2π2 ce0
1 2
Theratioofthesetwofrequenciesforafixedγiswrittenas 1/γ
Fig.6.Properties of jittering motion of relativistic electron while it travels
νJit 4cotθ vth −1= 4cotθ mec2. (37) emissionorbit. Theelectronapproachestotheobserverwhileitcarriesout
νsyn ∼ a (cid:16) c (cid:17) a r2kBT theprecession describedinFig.5. Thelineofsightisalwaysincludedin
theemissionconewhiletheelectrontravelstheemissionorbit. Duringone
Itshowsthatasfaraselectronthermalvelocityisnonrelativis-
periodoftheprecession,theelectronmovesadistanceof2π/(kcosθ).
tic, an relativistic electron with a fixed γ emits much higher
frequencybytheJitterradiationcomparedwiththesynchrotron
emission. Thecondition(33)mustbesatisfiedfortheJitterra- isemittedonlywhentheelectronstraveltheemissionorbit,we
diationtobeobserved. Asthelineofsightapproaching tothe adopt following approximations. When the electron is travel-
perpendicular directionofthemagneticfield,theperiodofthe ingtheemissionorbit,thezerothorderorbitisapproximatedas
precessionapproachestothedurationwhileelectronstravelthe straight line. Then, apart of the phase appeared in Eq.(28) is
emissionorbit.Whenthelineofsightisintherangeof evaluatedas
π π v
θ aπ th, (38) 1 ω
2 ≥ ≥ 2 − c ω t′− c~n·~r0 ∼ κ(αˆ)t′, (39)
(cid:16) (cid:17)
the precession period becomes longer than the duration while
2γ2
electrons travel the emission orbit. It shows that except the κ(αˆ) , (40)
≡ 1+γ2αˆ2
smalllimitedrangeofθtheJitterradiationisobservableinany
lineofsightrelativetothemagneticfielddirection. when T′/2 t′ T′/2.Whentheelectronisoutoftheemis-
− ≤ ≤
sionorbit,thatiswhen T/2 t′< T′/2orT′/2<t′ T/2,
− ≤ − ≤
weassumethatthereisnocontributionfromsecondterm.Inthe
3.3 Analyticalformulationofthespectrumofthe
followingcalculation,weusex′-y-z′coordinateinsteadofx-y-
Jitterradiation
z. Inx′-y-z′,thelineofsightis~n=(1,0,0). ThentheFourier
In this subsection, analytical formulae of the spectrum of the spectrumofeachcomponentoftheelectricfieldoftheemitted
Jitter radiation are provided. First the spectrum emitted by a radiationarededucedas
single electron is evaluated. Basic equations are the second
term of right hand side of Eq.(28). Since the Jitter radiation REˆx′(ω)=0, (41)
10 PublicationsoftheAstronomicalSocietyofJapan,(2016),Vol.00,No.0
REˆy(ω)= 2ieπωceiΦZ TT2′′ dt′ωksce1sinkv0kt′ ItshowsthattheemittedJitterradiationfiosrrigπ2ht<haθnd≤edπelli(p5t0i-)
− 2
eiω(cid:16)κ(tα′ˆ)−kω2csve01kcoαˆsθsinkv0kt′(cid:17), (42) cinalyly,apnodlatrhizeerdatliioghotfwthheicmhahjaosratomthaejomrainxoisrirnadzi′uasnidsm1/incoorsaθx.is
×
T′
REˆz′(ω)=−2ieπωceiΦZ T2′ dt′kcωcsoe1sθcoskv0kt′ intThehediermecittitoendopfow~neirsoofbtthaienJeidttearsradiationperunitsolidangle
×eiω(cid:16)κ(tα′ˆ)−−kω22csve01kcoαˆsθsinkv0kt′(cid:17). (43) dpeJditΩ(ω1) =cZ0∞dω|REˆx′|2+|RETˆy|2+|REˆz′|2,
ByusingfollowingformulafortheBesselfunctions = T′e2ωs2e1ω12(2+tan2θ)κ(αˆ), (51)
T 8πk2c3
ω
e−iλsinkv0kt′= ∞ Jn(λ)e−inkv0kt′, (44) ω1≡κ(αˆ)kv0k= 1+γJ2αˆ2, (52)
nX= ωJ 2γ2ω0, (53)
−∞ ≡
equations(42)and(43)areevaluatedas ω kv . (54)
0≡| 0k|
eωω
REˆy(ω)= 4πksce21eiΦ (Jn+1(λ)−Jn−1(λ))T′ Asthelineofsightapproachingtothedirectionofthemagnetic
Xn field,anintervalofthearrivalofpulsestotheobserveremitted
sinc ω nkv T′, (45) by asingle electron while it is in the emission orbit, becomes
× (cid:18)κ(αˆ)− 0k(cid:19) 2 shorter.Whenθ<1/γ,thelineofsightiscontinuouslyincluded
REˆz′(ω)=−4πiekωc2ωcsoe1sθeiΦXn (Jn+1(λ)+Jn−1(λ))T′ minutshtebeemseists1ioinnEcoqn.(e51o)fwthheensinθg<le1e/lγec.tSroinnc.eTthheerleefnogrteh,oTf′t/hTe
sinc ω nkv T′, (46) emissionorbitbecomesshorterasincreasing|αˆ|byafactorof
× (cid:18)κ(αˆ)− 0k(cid:19) 2 1 γ2αˆ2,thedurationwhileasingleelectronistravelingthe
−
whereλisdimensionlessparameterdefinedas epmissionorbit,T′,isreducedtoT′=(2/ωcesinθ) 1−γ2αˆ2.
The spectrum of the Jitter radiation is monochrompatic and al-
ωω
λ se1 αˆ. (47) mostdeltafunction centeredonω=ω . Supposethattherel-
≡ k2cv cosθ 1
0
k ativistic electron has an isotropic velocity distribution and its
Sincethesincfunctionappearedintheseequationsplaysarole
energy distribution is given by the following power law spec-
suchasadeltafunction,theangularfrequencyωisreplacedby
trum
ω =nkv κ(αˆ). Then the order of magnitude of λ for each
n 0
ωnisestimaktedwithahelpofEq.(12)as Ne(γ)=Cpγ−p, for γ1≤γ≤γc (55)
λ = ωn ωse1 αˆ 2nb γvthαˆ< 2nb vth n, (48) whereNe(γ)dγ giveselectronnumberdensitywhichhaveen-
n kv0 kc cosθ ∼ cosθ c cosθ c ≪ ergyfromγmec2to(γ+dγ)mec2.Theemittedpowerperunit
k
where αˆ <1/γisusedsincethemaincontributioncomesfrom solidanglecomingfromunitvolumeinthefrequencyrangeof
| |
the electrons with the pitch angle |αˆ|<1/γ. Since J0(0)= ω1∼ω1+dω1iscalculatedby
w1haincdhJconn(λtanin)≪J 1prfoovridλenm≪ainncwonhternibunti6=ons0,inonElyqst.h(4e5t)eramnds dPJeditΩ(ω1)dω1=fwZ Ne(γ)dγdα4ˆπdφdpeJditΩ(ω1), (56)
0
(46). Since we are only interested in positive frequency, we wheref isvolumefillingfactoroftheregionswherethewaves
w
obtain excitedbytheinstabilityoccupy. Theintegrationbysolidangle
REˆy(ω)=−e4ωπωksce21eiΦT′sinc(cid:18)κ(ωαˆ)−kv0k(cid:19)T2′, TinhEeqi.n(t5e6g)raistiopnerbfoyrmazeidmtuothcaolvaenrgalreeaφBisiltlruisvtiraaltesdinicneFthigeurinete4-.
for θ π grandisaxisymmetricagainstvz′ axis. Let’sdefineanewvari-
≤ 2 able y as y 2γ2ω /ω . Since the frequency of the emitted
0 1
= e4ωπωksce21eiΦT′sinc(cid:18)κ(ωαˆ)+kv0k(cid:19)T2′, radiation dep≡ends on γ and αˆ, integration variables (γ,αˆ) are
abletobetransformedto(y,ω ).Itisperformedby
π 1
for <θ π (49)
2 ≤ ∂(γ,αˆ) 1 dy
dγdαˆ= dydω = dω . (57)
REˆz′(ω)=−4πiekωc2ωcsoe1sθeiΦT′sinc(cid:18)κ(ωαˆ)−kv0k(cid:19)T2′, Forθ>1∂/(γy,,Tω1′/)T isre1writ−te4nωa1s√y−1 1
π
for θ≤ 2 T′ = 1 2ω0 1/2√2−y. (58)
=−4πiekωc2ωcsoe1sθeiΦT′sinc(cid:18)κ(ωαˆ)+kv0k(cid:19)T2′. STincetπhesiJnitθte(cid:16)rrωad1ia(cid:17)tionfroym1/2asingleelectronisobservedonly