TeV Electron Spectrum for Probing Cosmic-Ray Escape from a Supernova Remnant

0.24 MB·English

by  Norita Kawanaka

Draftversion January14,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
TEV ELECTRON SPECTRUM FOR PROBING COSMIC-RAY ESCAPE FROM A SUPERNOVA REMNANT
Norita Kawanaka1, Kunihito Ioka1, Yutaka Ohira1 & Kazumi Kashiyama2
Draft version January 14, 2011
ABSTRACT
Oneofthemostessentialbutuncertainprocessesforproducingcosmic-rays(CRs)andtheirspectra
is how accelerated particles escape into the interstellar space. We propose that the CR electron
1 spectra at & TeV energy can provide a direct probe of the CR escape complementary to the CR
1
nuclei and gamma-rays. We calculate the electron spectra from a young pulsar embedded in the
0
supernova remnant (SNR), like Vela, taking into account the energy-dependent CR escape. SNRs
2
would accelerate and hence confine particles with energy up to 1015.5eV. Only energetic particles

n can escape first, while the lower energy particles are confined and released later. Then the observed
a electron spectrum should have a low energy cutoff whose position marks the age of the pulsar/SNR.
J
The low energy cutoff is observable in the &TeV energy window, where other contaminating sources

3 are expected to be few due to the fast cooling of electrons. The spectrum looks similar to a dark
1 matter annihilation line if the low energy cutoff is close to the high energy intrinsic or cooling break.
The future experiments such as CALET and CTA are capable of directly detecting the CR escape
] features toward revealing the origin of CRs.
E
Subject headings: accelerationof particles – cosmic rays – pulsars:general– supernova remnants
H
.
h
1. INTRODUCTION accelerated at the shock with a large Mach number is
p
s = 2 and harder than that expected from the observa-
- The origin of cosmic-rays (CRs) is a long standing
o tions. Moreover, the non-linear theories of DSA predict
problem since its discovery. The number spectrum of
r a harder source spectrum s . 2.0, which seems to make
t the nuclear component of CRs can be fitted with a
s broken power-law: N(ε) ∝ ε−s with s ≃ 2.7 below thecontradictionbetweentheoriesandobservationseven
[a ε ≃ 1015.5eV (the ”knee” energy). The charged CR woIrtseh.as been proposed that these observational facts
particles are considered to propagate diffusively with an
canbeinterpretedbythemodeloftheenergy-dependent
3 energy-dependent manner (the higher energy particles
v can diffuse faster), so the intrinsic CR spectrum at the CR escape from the SNR (Ptuskin & Zirakashvili 2005;
2 sourceshouldbeharderthanthatobservedattheEarth. Drury et al. 2009; Reville et al. 2009; Caprioli et al.
4 2009;Gabicietal. 2009;Ohiraetal. 2010a;Fujita etal.
Although there are some ambiguities in the energy de-
1 2010;Casanovaetal. 2010). Intheescapemodelitisas-
pendence on the diffusion coefficient, the source spectral
1 indexisconsideredtobearounds≃2.2−2.4(e.g. Strong sumedthatthemostenergeticparticlesleavetheSNRat
. thebeginningoftheSedovphaseand,astheshockslows
9 & Moskalenko 1998).
down and the magnetic field decays, the lower energy
0 It is often argued that the CRs with energy smaller
particles, which have been confined around the shock
0 than the knee energy (or even up to 1018eV) are
front by the magnetic field, can leave the shock grad-
1 originated in our Galaxy. The most widely accepted
: paradigm for the galactic CR production process is the ually. This process can largely affect the CR spectrum
v detectedatthe Earth(Ohiraetal. 2010a;Capriolietal.
i diffusive shock acceleration (DSA) at supernova rem- 2010). Infact, the spectralindex ofthe CRescapingthe
X nants(SNRs). ThetheoryofDSA(forreviewsseeBland-
SNRs isdifferentfromthatoftheCRinsidetheSNR
r ford&Eichler1987;Malkov&Drury2001)cannaturally esc
s,
a derivethepower-lawspectrumofparticlesacceleratedin
the SNR shock. In fact, H.E.S.S. detected TeV gamma-
β
ray emissions from the shell of young SNRs (Aharonian s =s+ , (1)
esc
α
et al. 2004, 2005), and Fermi and AGILE have detected
GeV gamma-ray emissions from middle-aged SNRs in-
wheretheescapeenergyε andthenormalizationfactor
esc
teracting with nearby molecular clouds, which are likely
oftheCRproductionrateareassumedtobeproportional
to be generated via hadronic process, i.e. inelastic colli- to t−α and tβ, respectively. For example, in the phe-
sions between CR protons accelerated at SNRs and am- nomenologicalmodelbyGabicietal. (2009)α∼2.6(see
bient protons (Abdo et al. 2009b, 2010a,b, c; Tavani et
alsoOhiraetal. 2010a). Thereforetheobservedspectral
al. 2010; Giuliani et al. 2010). However, in the conven- index can get softer than the conventional value (s≃2)
tional DSA theory the predicted spectral index of CRs
if the normalization factor of the spectrum gets larger
with time (i.e. β >0; Ohiraet al. 2010a),so we may in-
[email protected] terprettheobservedCRspectruminthecontextofDSA
1Theory Center, Institute of Particle and Nuclear Studies, withthe escapemodel. Moreover,recentgamma-rayob-
KEK (High Energy Accelerator Research Organization), 1-1
servationsofmiddle-agedSNRsinteractingwithmolecu-
Oho,Tsukuba305-0801, Japan
2Department of Physics, Kyoto University, Kyoto 606-8502, lar clouds by Fermi and AGILE show that those spectra
Japan are fitted by a broken power-law with a break energy of
2 Kawanaka et al.
∼1−10GeV (see the referencesshownabove),andthey andCTA)coulddirectlyprovetheenergy-dependentCR
can be interpreted by the energy-dependent CR escape escape if those electrons/positrons are generated by a
from an SNR (Aharonian & Atoyan 1996; Gabici et al. pulsar embedded in an SNR. This is quite a natural sit-
2009; Ohira et al. 2010b)3. However, these gamma-rays uation to be realized because a pulsar should be gener-
arethesecondaryemissionsofCRsfromanSNR,andso ated in the center of a core-collapse supernova and CR
these observations are only the indirect evidences of the electrons/positronsareconsideredtobegeneratedinthe
CR escape scenario. pulsarwindnebula formedinside theSNR.IftheCRes-
On the other hand, the direct observations of CR capereallyoccursinayoungnearbypulsar/SNRsystem,
electrons/positrons have been also greatly advanced. the electronspectrum fromit will show a unique feature
The PAMELA satellite discovered the excess of the CR as explained in the following and, when it is observed,
positron fraction (Adriani et al. 2009) and, as for the that will be the first direct evidence of the CR escape
flux ofCR electronsplus positronsthe experimentssuch scenario.
as ATIC/PPB-BETS (Chang et al. 2008; Torii et al.
2. ASIMPLEEXAMPLE
2008b), Fermi (Abdo et al. 2009a; Ackermann et al.
2010) and H.E.S.S. (Aharonian et al. 2008, 2009) have To illustrate our main idea, we show a clear example
revealedanexcessfromthe conventionalmodel(see also of the effect of the energy-dependent CR escape in the
Meyer et al. 2010 for the energy calibration between direct electron observations. In the calculations of CR
Fermi and H.E.S.S.). These results suggest that there electrons/positrons,we usually assume that they are in-
are some additional electron/positron sources. Possible jected into the interstellar matter (ISM) from a pulsar
candidates include astrophysical sources such as pulsars with a spectral form of
(Shen 1970; Atoyan et al. 1995; Chi et al. 1996; Zhang Q (ε )=Q ε−α, (2)
& Cheng 2001; Grimani 2007; Kobayashi et al. 2004; e e 0 e
Bu¨esching et al. 2008; Hooper et al. 2009; Yuksel et al. where ε is the energy of electrons/positrons. The ob-
e
2009;Profumo 2008;Malyshev et al. 2009;Grasso et al. served electron spectrum f(ε ,r,t) can be obtained by
e
2009; Kawanaka et al. 2010; Heyl et al. 2010; Blasi & solving the diffusion equation
Amato 2010), supernova remnants (Shaviv et al. 2009;
∂ ∂
Blasi2009;Blasi&Serpico2009;Fujitaetal. 2009;Huet f =D(ε )∇2f + [P(ε )f]+Q (ε ,r,t), (3)
e e e e
al. 2009; Biermann et al. 2009; Mertsch & Sarkar 2009; ∂t ∂εe
Ahlers etal. 2009),gamma-raybursts (GRB;Ioka 2010;
where D(ε ) = D (1+ε /3GeV)δ is the diffusion coef-
Kistler & Yuksel 2009), microquasars(Heinz & Sunyaev e 0 e
ficient and P(ε ) is the energy loss rate. Here we adopt
2002) and white dwarf pulsars (Kashiyama et al. 2010). e
D = 5.8×1028cm2s−1, δ = 1/3 that is consistent with
Dark matter annihilations/decays (e.g. Hooper 2009) 0
the boron to carbon ratio according to the latest GAL-
andthepropagationeffect(Delahayeetal. 2008;Cowsik
PROPcode,andP(ε )=−bε2withb=10−16GeV−1s−1
& Burch 2009; Stawarz et al. 2010) are also the possi- e e
which includes the energy loss via synchrotron emission
bleprocessesformakingtheCRelectron/positronexcess
andinverseComptonscatterings(withThomsonapprox-
(for the comprehensive review, see Fan et al. 2010).
imation). Then, if electrons/positrons are injected from
In the near future within a few years, Alpha Mag-
a point-like source instantaneously (i.e. Q (ε ,r,t) ∝
netic Spectrometer - 02 (AMS-02) experiment will mea- e e
sure the positron fraction up to ∼1TeV (Beischer et al. δ(t)δ(r)), the observed spectrum is simply written as
2009; Pato et al. 2010;Pochon 2010) and CALorimetric
EelleeccttrroonnsTpeelcetsrcuompe4(uCpAtLoE∼T)1e0xTpeeVrimwietnhtawnillenexerpgloyreretshoe- f ∼ πQ3/02εd−e3α (1−btεe)α−2e−(r/ddiff)2, (4)
lution better than a few percent (Torii et al. 2008a). In diff
addition, the future Cherenkov Telescope Array (CTA) where d ∼ 4D(ε )t is the diffusion length of CR
diff e
will be able to measure the CR electron spectrum up to electrons/positrons. This spectrum is roughly propor-
∼ 15TeV (CTA consortium 2010). These experiments tional to ε−α−3pδ/2exp(−r2/(4D(ε )t)) up to the sharp
will open the window to the CR astrophysics with the cutoff at εe ∼ 1/(bt), and exponenetially damps beyond
TeV electron and positron components. Especially, as e
thediffusionlengthd (Atoyanetal. 1995;Ioka2010).
Kobayashi et al. (2004) have pointed out, in the TeV diff
However, if the energy-dependent escape of CR parti-
energybandafewnearbysourcesmayleavespectralsig-
cles from the shock is taken into account, the electron
natures and we will be able to see a spectral shape of
spectrum would have a sharp cutoff in the low energy
CRelectrons/positronsfromasinglesource,whileinthe
side because lower energy CRs cannot escape into the
lower energy band we can see only the superposed spec-
ISM. The energy of particles which are marginallycapa-
trum from multiple sources.
ble of escaping to the ISM ε is generally determined
In this paper, we suggest a possibility that the precise esc
by the confinement condition of CR particles, i.e. the
measurement of the CR electron spectrum in the very
equalitybetweenthediffusionlengthofparticlesandthe
high energy band (&1−10TeV, expected with CALET
characteristic size of the system:
3Recentlysomeauthorshavetriedtoexplainthisspectralbreak D (ε )
by considering the other processes such as the propagation of the l = sh esc ∼R , (5)
diff sh
SNRshockinthemolecularclouds(Uchiyamaetal. 2010), orthe ush
two-step acceleration of cosmic-ray particles by the shocks gener-
atedbytheturbulencebehindtheSNRshock(Inoueetal. 2010). where Dsh, ush and Rsh are the diffusion coefficient
4 Hereafter we express the spectrum of electrons plus positrons around the supernova remnant shock, the shock veloc-
asjust”electronspectrum”. ity, and the size of the system, respectively. Then the
Escaping CR Electron Spectrum 3
1000 where Le,pr(t) is the electron/positron production lumi-
2V] nosityandεe,min issettobe1GeV. Weassumethatthis
Ge luminosity is proportional to the spin-down luminosity
1 of the pulsar:
-sr 100
-2-1m s Le,pr(t)∝ (1+1t/τ0)2, (8)
3εΦ [ee 10 wwi/toh eessHcc.aaEpp.See .eeSAff(fTf0eeI8ccC)tt wishreerleatτe0d∼to102th−e4yseuarrfaiscethmeaspgnine-tdicowfineltdimoefsctahlee,pwuhlsicahr
ux PPB-BFEerTmSi (Shapiro & Teukolsky 1983).
Fl H.E.S.S.(09) In order to evaluate the CR spectrum in the escape
1
101 102 103 104 105 scenario, we should assume the time evolution of ε .
Energy ε [GeV] esc
e In this study we adopt two models for the functional
form of ε . In the first model, we assume for ε (t) a
Fig. 1.—Theelectronspectrumpredictedfromatransientsource esc esc
withtheescape energyofεesc=7TeV (solidline)added withthe power-lawbehavior,anddetermineitsnormalizationand
background model (dotted line), compared with the ATIC/PPB- power-law index according to the hypothesis that SNRs
BETS/H.E.S.S./Fermidata. The spectrum without assumingthe are responsible for the observed CRs with the energy
tehnaertgay-sdoeuprecnedeanttres=ca7p0e0piscaflrsoomshtohwenE(adratshhepdroldinuec)e.sWe±e apsasiursmae from∼1GeV up to the knee energy (∼1015.5eV). Then
time tage = 1.0×104years ago with total energy Ee+ = Ee− = εesc(t) should reach the knee energy at the end of the
0.5×1049ergandspectralindexα=2.4. free expansion phase (i.e. the beginning of the Sedov
phase; t ) and should decrease down to 1GeV at t≃
energy ε is generally a function of time t due to the Sedov
esc
evolution of Dsh, Rsh and ush. 105/2tSedov (i.e. the end of the SNR expansion; Gabici
Fig. 1 shows the electron spectrum assuming that et al. 2009; Ohira et al. 2010a):
only electrons and positrons above the escape energy
εesc are injected instantaneously. The energy of elec- ε (t)=106.5GeV× t −2.6. (9)
trons/positrons which is initially ε becomes ε /(1+ esc t
esc esc (cid:18) Sedov(cid:19)
bε t )asaresultofradiativecooling(wheret isthe
esc age age
time since the emission), and we can clearly see a sharp As the second model for the evolution of εesc we
cutoff of the spectrum at that energy, which is the ef- adopttheonediscussedbyPtuskin&Zirakashvili(2005),
fect of energy-dependent escape of CR particles. If such which takes into account the modification of a shock
a spectrum is confirmed by the future experiments, it structure due to the CR pressure, as well as the non-
would be the strong support for the CR escape scenario linear dissipation of magnetic turbulence. They solve
of the particle acceleration process at the SNR, and we the steady-state equation which determines the energy
may also get the information of ε at the time of the density of the magnetohydrodynamic turbulence W:
esc
source age tage. u∇W(k)=2(Γ (k)−Γ(k)−Γ (k))W(k), (10)
cr l nl
3. MODELOFCRELECTRON/POSITRONESCAPE whereu is the flowvelocity (here itis equalto the shock
Letusconsiderthemoresophisticatedmodelthandis- velocity u ), k is the wave number of the turbulence,
sh
cussed in the previous section. We consider a pulsar andΓ ,Γ andΓ arethewavegrowthrateattheshock
cr l nl
emitting electrons and positrons embedded in the SNR dueto theCRstreaminginstability,the dampingrateof
(i.e. it has not been evacuated from the SNR by the waves in the background plasma due to the ion-neutral
natal kick). A pulsar is considered to be an efficient e± and electron-ion collisions (linear damping), and due to
factory, because its rotating magnetic field would pro- thewave-waveinteractions(non-lineardamping),respec-
duce a strong electric field around the pulsar and then tively. The mathematicalexpressionsfor these functions
a large number of e± pairs would be produced via elec- areshownin Ptuskin& Zirakashvili(2005),andby solv-
tromagnetic cascades. The created pairs would stream ing this equation (while the term of linear damping is
away by a centrifugal force as a pulsar wind that ends neglected), we obtain the threshold particle energy for
with a termination shock where the acceleration of elec- escape as a function of the shock velocity u . Since we
sh
trons/positrons may occur. Hereafter we assume the e± know the time dependence of the shock radius and the
production rate per energy from a pulsar having a spec- shockvelocityintheSedovphase,wecanderiveε (t)as
esc
trum with a cutoff power-law shape: a function of time. When the age of the SNR is younger
than . 105years, the evolution of ε (t) in this model
esc
ε
N˙ (ε ,t)=Q (t)ε−αexp − e , (6) can be approximated as
e,pr e 0 e ε
(cid:18) e,cut(cid:19) t −1.0
where the high energy break is fixed as ε = 10TeV. ε (t)≃105.5GeV× . (11)
e,cut esc
t
According to the gamma-rayobservationsofPWNe, the (cid:18) Sedov(cid:19)
electrons/positrons accelerated up to the energy of 10-
Fig. 2 shows the evolutions of ε (t) in two models
100TeV seem to exist in the nebulae (e.g., Aharonian et esc
described above. Once we fix the time dependence of
al. 2006), so this assumption is reasonable. Here Q (t)
0 ε , we can describe the number luminosity per energy
is given to satisfy esc
of escaping electrons and positrons as
∞
L (t)= dε ε N˙ (ε ,t), (7) N˙ (ε ,t)=N˙ (ε ,t)Θ(ε −ε ), (12)
e,pr e e e,pr e e,esc,1 e e,pr e e esc
Zεe,min
4 Kawanaka et al.
2/5
t
Ptuskin and Zirakashivili (2005) ×Θ ε (t′)−ε . (16)
106 Gabicci oeot lainl.g ( 2c0u0to9f)f esc e(cid:18)t′(cid:19) !
V]105 If the decreasing rate of εesc is faster than the adiabatic
Ge cooling rate, −(2/5)εesc/t, then a part of confined elec-
[e104 trons/positronscanescapethe SNRshock. Byusingthe
ε
gy 103 samelogicinderivingEq. (21)ofPtuskin&Zirakashvili
er (2005) in the case of expanding media, we can evaluate
En102 the spectrum of such particles as
101 N˙ (ε ,t)=−δ(ε −ε (t)) ∂εesc + 2εe
e,esc,2 e e esc
∂t 5t
100 (cid:18) (cid:19)
103 Age [year1]04 105 ×Ne,conf(εe,t). (17)
Fig.2.— The threshold energy for escape from the SNR shock Hereafter we consider the spectrum of escaping elec-
εesc intheSedovphaseasafunctionoftime. Thethicksolidline trons/positronsnumberluminosityperenergyasthesum
and the thin solidline correspond to the model by Ptuskin & Zi-
of above two components:
rakashvili(2005)andGabicietal. (2009),respectively. Thecutoff
which appears in the spectrum due to the energy loss (including
theKNeffect)duringthepropagationisalsoshown(dashedline). N˙e,esc(εe)=N˙e,esc,1(εe)+N˙e,esc,2(εe). (18)
where Θ(x) is the step function5. We neglect the radiative energy loss of elec-
Thereisanotherfluxcomponentwhichshouldbetaken trons/positronsduringtheconfinementforsimplicity. As
intoaccount. Theelectrons/positronswhichhavetheen- wehaveshowninEq. (17),theelectronfluxofthesecond
ergy lower than ε (t) are confined in the SNR and lose component N˙ (ε ,t) is determined by the difference
esc e,esc,2 e
theirenergyadiabatically(Ptuskin&Zirakashvili2005). between the decline rate of the escape energy ε and
esc
As ε (t) decreases with time, some of the confined and the energy loss rate of electrons/positrons. In the cases
esc
adiabatically cooled electrons/positrons can escape the showninthisstudy,thedeclinerateoftheescapeenergy
shock surface when their energy becomes greater than is ∼αε /t∼3×10−9(ε /1TeV) t/104yr −1GeV sec−1
e e
εesc(t). First, the CR electron/positron number per en- (α ≃ 1 − 2.6, depending on the model) , while the
ergy confined in the SNR can be written as (cid:0) (cid:1)
adiabatic cooling rate and the radiative cooling rate is
t dε′ ∼1.2×10−9(ε /1TeV) t/104yr −1GeV sec−1, ∼bε2 ∼
N (ε ,t)= dt′N˙ (ε′,t′) e e e
e,conf e e,pr e dε 10−10(ε /1TeV)2GeV sec−1, respectively, where for the
ZtSedov e e (cid:0) (cid:1)
×Θ(ε (t′)−ε′), (13) latter we take the cooling rate for the interstellar space
esc e (see Sec. 2). Therefore, even if we take into account
where ε′ is the energy of the electrons/positrons at the the radiative cooling the flux of the electrons/positrons
time t′,eand it is ε at the time t. which have once been confined in the SNR does not
e
The adiabatic loss is determined by the expansionlaw change its order from our calculation. Moreover, in the
of the SNR, interstellar space the diffusion timescale for TeV elec-
trons can be estimated as
dε R˙
e =− SNRε , (14) r2
dt R e t ∼
SNR diff
4D(ε )
e
where RSNR and R˙SNR are the radius of the SNR shell ε −1/3 r 2
and its expansion velocity, respectively. In the following ∼1.4×104yr e , (19)
we assume that the SNR is in the Sedov phase, in which 3TeV 300pc
(cid:16) (cid:17) (cid:18) (cid:19)
the time dependence of R is expected to be propor-
SNR andthereforetheenergylossofTeVelectronsduringthe
tional to t2/5. Then the energy of electrons/positrons propagation is at most ∆ε /ε ∼ 1−(1+bt ε )−1 ∼
e e diff e
confined in the SNR would evolve as 10%. Then we can say that in this energy range both
t′ 2/5 of the energy losses in the PWN and in the interstel-
ε (t)=ε′ , (15) lar space are small. However, if the magnetic field in
e e t thePWN/SNRisstronglyamplifiedfromtheinterstellar
(cid:18) (cid:19)
value, the energy loss rate due to the synchrotron emis-
and therefore, the distribution function of confined elec-
sionmaybe fasterthanthatdue totheadiabaticexpan-
trons/positrons can be evaluated as
sion of the SNR, and even than that due to the decline
t t 2/5(1−α) rateoftheescapeenergy. Insuchcase,theconfinedelec-
Ne,conf(εe,t)= dt′Q0(t′)ε−eα t′ trons/positrons cannot escape the SNR later and only
ZtSedov (cid:18) (cid:19) the first component N˙ would be emitted and ob-
×exp −εe(t/t′)2/5 served at the Earth. Aen,eyscw,1ay, as either the strength of
ε the magnetic field in the PWN/SNR nor its time evolu-
(cid:18) e,cut (cid:19) tion is generally uncertain, it is difficult to estimate the
5 Strictlyspeaking, thespectrum of theescape fluxhas afinite
width around the escape energy εesc. However in the usual case radiative cooling rate with a moderate strength of the
thestepfunctionisagoodapproximation(Ptuskin&Zirakashvili magnetic field in a reliable way. In the following, we in-
2005;Capriolietal. 2009). vestigate only the case that the radiative energy loss is
Escaping CR Electron Spectrum 5
not significant. electron/positronflux seemsto havea highenergydrop-
ping arounda few TeV. This dropping is quite naturally
4. OBSERVEDELECTRONSPECTRUMINTHEESCAPE
explainedinthecontextoftheastrophysicaloriginofCR
SCENARIO
electrons/positrons because the number of the sources
Now that we have the time evolution of the escap-
contributing to the TeV energy band is quite small ac-
ing CR particle flux from the source, we can obtain the
cording to the birth rate of SNe/pulsars in the vicinity
observed electron spectrum by solving the propagation
of the Earth (Kobayashi et al. 2004; Kawanaka et al.
of CR electrons/positrons with the diffusion equation
2010). In fact, since the pulsars which contribute to the
shown in Eq.(3). Once we know the Green’s function
electron flux at the energy ε should be younger than
e
of this equation with respect to the time and position, the cooling time of electrons/positrons t ∼ 1/(bε )
cool e
G(t,r,ε ;τ), we can obtain the observed electron spec-
e and should be located closer to the Earth than the dif-
trum as fusion length d ∼ 2 K(ε )t , the number of the
diff e cool
t pulsars contributing to & TeV band should be as small
f(t,r,εe)= G(t,r,εe;τ)dτ, (20) as p
Zti
ε −5/3 R
wsthaertreed,tiwihsicthheistaimsseumwehdentothbee epqaurtailclteoitnjectioinn hthaes NPSR(εe)∼6 TeeV 0.7×10−5yr−1kpc−2 (2,7)
Sedov (cid:16) (cid:17) (cid:18) (cid:19)
following discussions.
whereRisthelocalpulsarbirthrateperunitsurfacearea
The mathematical description of G(t,r,ε ;τ) was de-
e of our Galaxy. If we can separate the contribution of a
rived by Atoyan et al. (1995),
singleyoungsourcefromtheobservedelectronspectrum,
N˙ (ε ,t )P(ε ) r2 we can get the information of the CR injection into the
G(t,r,εe;t0)= π3e/,e2sPc (εe,0)d0 (ε e,)03 exp −d (ε )2(21,) ISMfromthatsource. Forthis reason,we especiallypay
e diff e,0 (cid:18) diff e,0 (cid:19) attention to the TeV spectral features of CR electrons
whereε istheenergyofelectrons/positronsatthetime from a pulsar in the followings.
e,0
t which are cooled downto ε at the time t, and d is In Figs. 3 and 4 we show the time evolutions of CR
0 e diff
the diffusion length given by electronspectrum fromanearbypulsar accordingto the
modelsofthe escapeenergyε (t)adoptedinthe previ-
εe,0 D(x)dx −1/2 oussection(seealsoFig. 2). Ietsicsclearthatthereexistsa
d =2 , (22)
diff P(x) low energy cutoff in each spectrum correspondingto the
(cid:20)Zεe (cid:21) value of ε (t) at that time. The spectral shapes gener-
esc
as shown in Eqs.(10) and (11) in Atoyan et al. (1995). allydependonotherparameterssuchasthe highenergy
In deriving the energy loss rate P(εe), we use the for- break of the intrinsic electron spectrum, the spectral in-
mulation shown by Moderski et al. (2005) dex, the duration of electron/positron injection from a
pulsar, and the total energy of CR electrons/positrons.
4σ ε2 B2 4ε ε
P(ε )= T e + dε u (ε )f e γ (2,3) However, the sharp cutoff feature in the low energy side
e 3m2c3 8π γ tot γ KN m2c4 of the spectrum is almost independent of these proper-
e (cid:20) Z (cid:18) e (cid:19)(cid:21)
ties. In Fig. 3, we can see that the low energy cutoff of
where σ is the Thomson cross section, u (ε )dε is
T tot γ γ eachspectrum is slightly broadened comparedwith that
the energy density of interstellar photons with the en-
in Fig. 4. This is because the model adopted in this
ergy between ε and ε +dε (including CMB, starlight
γ γ γ figure assumes that ε (t) decreases more rapidly than
and dust emission; Porter et al. 2008), and B is the in- esc
in the case of Fig. 4 and so the CR electrons/positrons
terstellar magnetic field which we here set as 1µG. Here
inthebroaderenergyrangecanreachthe observerwhile
the function f (x) is the correction factor to include
KN in the case of Fig. 4 where ε (t) decreases slowly the
the Klein-Nishina effect. According to Moderski et al. esc
low energy cutoff becomes very narrow. In either case
(2005), this function can be expressed as
the dropoff in the low energy side of the spectrum is so
9g(˜b) steepthatoneshouldassumetheintrinsicspectralindex
fKN(˜b)= ˜b3 , (24) as hard as α.0−1 if we neglect the energy-dependent
CR escape effects. As we mentioned in the last section,
where˜b=4ε ε /(m c2)2, if the magnetic field
e γ e
The CR electron spectrum with the age and distance
g(˜b)= 1˜b+6+ 6 ln(1+˜b) similar to the Vela pulsar (tage ≃ 104year, r ≃ 290pc),
2 ˜b whichisthoughttobesurroundedbythesupernovarem-
(cid:18) (cid:19)
nant (Aschenbach et al. 1995), is shown in Fig. 5. Here
11 1
− ˜b3+6˜b2+9˜b+4 we show the spectrum with the escape model of Ptuskin
12 (1+˜b)2 & Zirakashvili (2005) as well as the spectrum without
(cid:18) (cid:19)
−2+2Li (−˜b), (25) the CR confinement in the SNR. In addition, the elec-
2 tron flux which have not been confined in the SNR (i.e.
and the function Li2(z) is the dilogarithm N˙e,esc,1) and that which have once confined and escaped
0 ln(1−t)dt later (N˙e,esc,2) are shown. We can see that the latter
Li2(z)= . (26) component dominates the flux around the low energy
t
Zz cutoff. In this case we can detect the electron flux from
According to the recent experiments, especially Vela pulsar with such a sharp spectral cutoff by near
H.E.S.S. (Aharonian et al. 2008), the background CR future missions suchas CALET (we assume the geomet-
6 Kawanaka et al.
1000 ricalfactorSΩtimes the observationtime T ≃5yearsas
2V] ∼220m2sr days) because the assumed electron/positron
Ge flux is sufficiently large and the low energy cutoff comes
1 beyondthe highenergydropping ofthe backgroundflux
1- sr 100 inferred by H.E.S.S. (Aharonian et al. 2009). If such
-s a sharp cutoff feature is confirmed, that would be the
2
-m strong evidence of the energy-dependent escape of CR
Φ [e 10 particles from the acceleration site. As we mentioned
3 ATIC in the last section the second component N˙ would
e HESS(08) e,esc,2
εFlux PHPEBS-BFSEe(0rTm9S)i nstortonbgeaonbdsetrhveedeleicftrtohnes/mpaogsnitertoincsfioenldceinconthfienePdWinNthies
1 PWN/SNR lose their energy too rapidly to escape the
101 102 103 104 105
Energy εe [GeV] SNR later. However, the first component N˙e,esc,1 also
has a sharp cutoff in the low energy side of its spectrum
Fig.3.— The electron spectra from a single pulsar surrounded
byasupernovaremnantwithdifferentages;3×103 (dotted line), and so even in this case it is still possible to prove the
5×103 (dashed line), 1×104 (solid line) and 2×104years (dot- CR escape from an SNR from the electron spectrum. If
dashed line). As for the model of the escape threshold energy the threshold energy εesc(t) at that time is smaller than
εesc(t) we adopt the results by Gabici et al. (2009; see Fig. 2). afewTeVthenthecutofffeaturewouldbehiddenbythe
Weadoptthebackgroundmodelofexponentiallycutoffpower-law background flux. In fact, assuming the phenomenologi-
with an index of -3.0 and a cutoff at 1.5TeV, which is similar to
cal model of ε (t) adopted by Gabici et al. (2009) and
that shown in Aharonian et al. (2008) and reproduces the data esc
in ∼10GeV-1TeV well, and we show the spectrum including this Ohiraetal. (2010a),thelowenergyendofthe spectrum
background bythicklines. Weassumethat asourceatr=290pc from the Vela pulsar would be buried in the background
fromtheEarthproducese± pairswithtotalenergyEe+ =Ee− = flux because the escape energy at the age of Vela SNR
0th.5e×hi1g0h48eneregrg,ydburreaatkioεne,τc0ut==10140yTeeaVr,.spectral index α=2.0and (t∼he1c0u4tyoeffarf)eabteucroemdeusestmoatlhleeretnheargny∼-deapfeenwdeTnetVe,scaanpdesoof
1000 CR electrons/positrons may not be resolved.
2]
V As we mention in Sec. 1, the CR escape from a source
e
G (i.e. SNR) seems to be an important process for the ob-
-1sr 100 served CR spectrum below the knee (∼ 1015.5eV) and
1 the brokenpower-law gamma-rayspectra observedfrom
-s
2 SNRsinteractingwithmolecularclouds. Asforthespec-
-m trum of CR protons and nuclei we can see only the su-
Φ [e 10 perposed flux from multiple CR sources with different
3e HESSA(T0I8C) ages, and so it is impossible to see the direct evidence
εux PPB-BFEerTmSi ofenergy-dependentCRescapefromasingleCRsource.
Fl HESS(09) On the other hand, in the ∼1−10TeV band of the CR
1
101 102 103 104 105 electron spectrum we may be able to get the spectrum
Energy εe [GeV] from a single source, and then we expect the sharp low
energy cutoff showing the boundary between runaway
Fig.4.—ThesameplotswithFig.3.,butusingtheescapemodel
CR electrons/positrons and confined CRs whose energy
byPtuskin&Zirakashvili(2005; seeFig. 2).
isnothighenoughtoescapefromtheSNRshock. There-
1000 fore, this cutoff can be the first direct evidence that the
2]
V energy-dependent CR escape really occurs at the source
e
G of those CR electrons/positrons.
1
-sr 100
1 5. SUMMARYANDDISCUSSIONS
-s
-2m In this paper we show the possibility of getting the
Φ [e 10 ccoommppoonneetnnottt a12l eSvNidRenshceocokfbtyheobesneerrvginyg-dtehpeenTdeeVntCRCReleecstcraopnespfreocmtrutmhe.
3 ATIC
εe HESS(08) TheescapeofCRparticlesisthefundamentalprocessin
ux PPB-BFEerTmSi emitting the CR particles acceleratedatthe SNR shock,
Fl HESS(09) andthe historyofthe escape energyε (t), whichis the
1 esc
101 102 103 104 105 threshold energy of accelerated particles not to be con-
Energy ε [GeV]
e finedbuttoescapetheshockintotheISM,isessentialin
Fig.5.—Theelectronspectrumfromasinglepulsarsurrounded determining the observedCR spectrum. Although there
by a supernova remnant (thick solid line). As for the model for aresomeindirectsuggestionstothisscenariofromrecent
the escape threshold energy εesc(t) we adopt that of Ptuskin & gamma-rayobservationsofSNRsinteractingwithmolec-
Zirakashvili (2005; see Fig. 2), and the background model (thin
ular clouds, we have not verified the energy-dependent
dotted line) is the same as used in Fig. 3 and Fig. 4. The spec-
trumwithoutassumingtheenergy-dependent escape(thindashed CRescapemodelofSNRsdirectlyfromtheobservations.
line), the flux from N˙e,esc,1 and N˙e,esc,2 (long-dashed line and Inorderto seethehistoryofCRinjections fromSNRs
dot-dashedline,respectively),andtheerrorbarsexpectedfromthe we focus on the lepton component of CRs. In the en-
5-yearsobservationbyCALET(SΩT =220m2 srdays;reddown- ergy band larger than ∼ TeV, it is expected that the
wardtriangles)arealsoshown. electron/positron flux from a few nearby young astro-
Escaping CR Electron Spectrum 7
physicalsources(e.g. Vela pulsar)canbe observed,then and should be determined from the analysis of the par-
we will be able to get the information about the intrin- ticle acceleration processes in the pulsar wind nebula as
sic CR spectrum from a single source at a certain time, well as the radiative cooling of electrons/positrons dur-
whichhasnotbeenwellunderstoodbecausetheobserved ing the acceleration. However, these processes have not
nuclear CR spectrum consists of the contributions from beenfullyunderstoodyetandsoitisdifficulttogivethe
multiple sources and it is impossible to resolvethe spec- maximumelectron/positronenergyfromthe firstprinci-
trum from a single source. Especially if a nearby young ple. For the present purpose, we are interested in the
pulsar embedded in an SNR emits sufficiently large flux existence of the low energy cutoff due to the energy-
ofCRelectrons/positrons,theyareinjectedintotheSNR dependent CR escape and unless the intrinsic high en-
shockandonlytheelectrons/positronswithenergylarger ergybreak ε is much smaller than the escape energy
e,cut
than ε (t) cango through the SNR andpropagateinto ε attheobservationtime,therewouldbeasufficiently
esc esc
the ISM. As a result, the observed electron spectrum large electron/positron flux above ε and so the cutoff
esc
from that pulsar may have the low energy cutoff corre- feature would be detected clearly enough to probe the
sponding to the escape energy ε (t) at the observation CR escape scenario.
esc
time. In order to detect such spectral feature the low We should also mention that the position of the low
energycutoff(i.e. ε (t)attheobservationtime)should energy cutoff depends on the model of the time evolu-
esc
come above & 1−10TeV because otherwise the cutoff tion of the CR escape energy and the highly unknown
would be buried in the background flux and would be magnetic and radiation fields, which produce the energy
hard to resolve. losses of electrons/positrons.
The possibility that there is such a low energy cut- The energy-dependent escape of CR elec-
off in TeV electron spectrum has firstly pointed out by trons/positrons may be confirmed by the observations
this study, and taking into account the variation of the of the radio to γ-ray emissions from an SNR with a
CR escape history from the SNR and/or the age of the pulsar. Theelectrons/positronsescapingtheSNRwould
pulsar, we can predict a variety of spectral shapes that radiate via synchrotron emissions and inverse Compton
have not been considered in the context of astrophysi- scatterings just outside the SNR shock. If we can
cal sources. For example, if the high energy cutoff of observe the radio to γ-ray intensity around the nearby
thespectrum,whichisdeterminedbytheintrinsiccutoff young SNR with a pulsar and can fit that spectrum by
energyatthepulsarandtheradiativeenergylossofelec- the leptonic model with the electrons/positrons having
trons/positronsduringtheirpropagation,isnearlyequal a low energy cutoff in their energy distributions, that
to the low energy cutoff determined by the escape from would be the support of the energy-dependent escape of
the SNR, the resulting spectrum would have a narrow- CRs from the SNR (in preparation).
line like shape and it would be the similar feature to
thatexpectedfromtheannihilationofdarkmatterparti- We thank K. Kohri, A. Mizuta, K. Nakayama and Y.
cles. WhethertheobservedCRelectronspectrumisorig- Suwa for useful discussions. This work is supported in
inated from a single nearby source would be determined part by the World Premier International Center Initia-
by measuring an anisotropy of CR electrons/positrons tive (WPI Program), MEXT, Japan and the Grant-in-
(Mao & Shen 1972; Bu¨esching et al. 2008; Ioka 2010). Aid for Science Research, Japan Society for the Pro-
Notethatthehighenergybreakenergyofthe electron motion of Science (No. 22740131 for NK; No.19047004,
spectrum from a pulsar ε generally depends on time 21684014,22244019,22244030for KI).
e,cut
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