Positron annihilation as a cosmic-ray probe

0.26 MB·English

by  Yutaka Ohira

Mon.Not.R.Astron.Soc.000,1–6(2012) Printed15January2013 (MNLATEXstylefilev2.2)
Positron annihilation as a cosmic-ray probe
Yutaka Ohira1⋆, Kazunori Kohri1 and Norita Kawanaka2
1Theory Center, Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba 305-0801, Japan
2Racah Institute of Physics, The HebrewUniversity,Jerusalem 91904, Israel
2
1
0
Accepted 2011December 28.Received2011December27;inoriginalform2011November21
2

n
a ABSTRACT
J Recently, the gamma-ray telescopes AGILE and Fermi observed several middle-aged

4 supernova remnants (SNRs) interacting with molecular clouds. A plausible emission
mechanism of the gamma rays is the decay of neutral pions produced by cosmic ray
]
(CR)nuclei(hadronicprocesses).However,observationsdonotruleoutcontributions
E
from bremsstrahlung emission due to CR electrons. TeV gamma-ray telescopes also
H
observed many SNRs and discovered many unidentified sources. It is still unclear
.
h whether the TeV gamma-ray emission is produced via leptonic processes or hadronic
p processes.InthisLetter,weproposethatannihilationemissionofsecondarypositrons
- produced by CR nuclei is a diagnostic tool of the hadronic processes. We investigate
o
MeV emissions from secondary positrons and electrons produced by CR protons in
r
t molecular clouds. The annihilation emission of the secondary positrons from SNRs
s
can be robustly estimated from the observed gamma-ray flux. The expected flux of
a
[ the annihilation line from SNRs observed by AGILE and Fermi is sufficient for the
future Advanced Compton Telescope to detect. Moreover,synchrotron emission from
2
secondarypositrons and electrons and bremsstrahlungemission from CR protons can
v
be also observed by the future X-ray telescope NuSTAR and ASTRO-H.
0
4 Key words: acceleration of particles – cosmic rays – supernova remnants.
1
4
.
3
0
1 INTRODUCTION TeV gamma-ray emission is produced via inverse Compton
1
scattering of CR electrons (leptonic processes) or hadronic
1 The origin of cosmic rays (CRs) is a longstanding problem
: processes. TherearemanyTeV-unIDsources inourgalaxy,
v in astrophysics. Supernova remnants (SNRs) are thought
and their emission mechanisms are also still unclear. Old
i to be the origin of Galactic CRs. The most popular SNR
X or middle-aged SNRs are the candidates of the TeV-unID
acceleration mechanism is diffusive shock acceleration (e.g.
sources(Yamazaki et al.2006;Ohira et al. 2011b).Thus,it
r Blandford & Eichler1987).Infact,Fermi andAGILE show
a iscrucialtoidentifytheemissionmechanismofgammarays.
that middle-aged SNRs (∼ 104 yrsold) interacting with
Positronsandneutrinosareproducedinhadronicprocesses.
molecular clouds emit GeV gamma rays with broken power
On the other hand, they are not produced in leptonic pro-
lawspectra(e.g.Abdoet al.2009,2010;Tavani et al. 2010;
cesses.1 Therefore,directorindirectdetectionsofthesepar-
Giuliani et al.2011)(Hereafter,thesereferencesarereferred
ticles will enable us to identify the emission mechanism in
to as OBS). Considering a stellar wind before the super-
gamma-ray sources.
nova explosion, the molecular cloud has been swept away
Recently,theCR positron excess hasbeen providedby
bythestellarwind.Theinnerradiusofthemolecularcloud
PAMELA (Adrianiet al. 2008).Although theorigin of this
becomes typically about a few tens of parsecs. Then, the
excess has been investigated in many theoretical studies
SNR collides with the molecular cloud at typically 104 yrs
(Kashiyama et al. 2011; Kawanaka et al. 2010; Ioka 2010,
later and the broken power law spectra of GeV gamma
and references therein), it still remains unclear. SNRs have
rays can be interpreted as the inelastic collision between
been discussed as the candidates of the positron sources
molecular clouds and CR nuclei that have escaped from
(Fujita et al. 2009a; Blasi 2009; Shaviv,Nakar& Piran
the SNR (hadronic processes) (Ohira, Murase & Yamazaki
2011a).However,observationsdonotruleoutcontributions
from bremsstrahlung emission due to CR electrons (lep-
tonic processes). SNRs have been also observed by TeV
gamma-ray telescopes, and it is still unclear whether the 1 Strictlyspeaking,leptonicprocessescanproducepositronsvia
e−+γ→e−+e−+e+butthisreactionrateissuppressedbythe
finestructureconstantcomparedwiththegamma-rayproduction
⋆ E-mail:[email protected] rateofinverseComptonscattering.
(cid:13)c 2012RAS
2 Y. Ohira, K. Kohri and N. Kawanaka
2 RELEVANT TIMESCALES
106
In this section, we briefly summarize relevant timescales of
105 CRs in a molecular cloud. Physical quantities of molecular
cloudsassociatedwithgamma-raysourceshavenotbeenun-
e [ yr ]104 ddeenrsstiotyodisinabdoeutatiln.O=b1s0e2rv−at1io0n3scmsu−g3g,estthethsaiztethiseogfatshneuomrbdeerr
al
sc of10pc(OBS),andthemagneticfieldinmolecularcloudsis
me 103 aboutB=1−100µG(Crutcher et al.2010).InthisLetter,
Ti
for the molecular cloud we adopt values of the gas number
102 densityn=300cm−3,thesizeRc =30pcandthemagnetic
field B = 30µG. We consider ionization and inelastic col-
101 lisions with nuclei as cooling processes of CR protons. On
105 106 107 108 109 1010 1011 1012 1013 theotherhand,weincludeionization, bremsstrahlung,syn-
Kinetic enrgy [ eV ] chrotronemission,andinverseComptonscatteringwiththe
cosmic microwave background (CMB) as cooling processes
Figure1.TimescalesofinteractionforCRsinamolecularcloud ofsecondarypositronsandelectrons.ForSNRsobservedby
withthenumber densityn=300cm−3,the sizeRc =30pcand FermiandAGILE,SNRsinteractwithmolecularcloudsand
the magnetic field B = 30µG. The blue and green lines show the expansion of the SNRs is strongly decelerated. There-
thecoolingtimesofsecondarypositrons(andelectrons)andCR fore, we heredo not consider theadiabatic cooling.
protons, respectively. The red line shows the annihilation time We here define τ = E/E˙ as a cooling time, where
cool
of positrons. The black solid line shows the diffusion time with E and E˙ are the kinetic energy and the energy loss rate,
χ=0.01. respectively.Weadoptan expressionof E˙ forprotonsgiven
in Mannheim & Schlickeiser (1994), and that for positrons
and electrons given in Strong& Moskalenko (1998). The
timescale of direct annihilation of positrons and electrons
is represented by
1
τ (E)= , (1)
a,± n σ (E)v(E)
∓ a
2009;Biermann et al.2009;Mertsch & Sarkar2009).Hence,
where n ,σ and v are the number density of positrons or
itisimportanttoobservethepositron productionatSNRs. ± a
electrons, the cross section of the annihilation (Dirac 1930)
Gamma-ray telescopes showed that the photon flux
andthevelocityofsecondaryelectronsorpositrons,respec-
above 100MeV from SNRs with an age of about
104yr is F = ∞ dE(dF/dE) = 10−7 − tively. The electron density, n−, is dominated by thermal
10−6photon>s1−010McmeV−2 (ORB1S0)0.MIefVthe gamma rays have a electronsandthepositrondensity,n+,isverysmall,sothat
we can neglect theannihilation of secondary electrons.
hadronic origin, about the same numbers of positrons are
The escape time through diffusion is written by
produced. When those positrons lose their energy suffi-
ciently, about 80 percent of them would make positro- τ (E)= Rc2 , (2)
niums with ambient electrons, and one fourth of them d 4D(E)
would decay into two photons with the energy of 511keV
where D(E) is the diffusion coefficient. It is notable that
(e.g. Prantzos et al. 2010). This means that we can ex-
so far thediffusion coefficient around an SNR has not been
pect the 511keV photon flux of the order of 10−8 −
understoodwell.Thusweassumethediffusioncoefficientin
10−7photons−1cm−2. This flux is sufficiently high for five-
molecular clouds to be
years observations of the future Advanced Compton Tele-
scope (ACT) to detect. D(E)=1028 χv(E) E 0.5cm2s−1 , (3)
In this Letter, considering the interaction between c (cid:16)10GeV(cid:17)
CR protons and molecular clouds, we estimate anni- where c is the velocity of light. Here χ = 1 might be rea-
hilation emission of secondary positrons produced by sonableastheGalacticmeanvalue(Berezinskii et al.1990),
the CR protons. We partly refer to analyses of previ- but χ = 0.01 should be preferred in surroundings of SNRs
ous works concerning the 511keV line from the Galac- (Fujita et al. 2009b, 2010; Torres et al. 2010; Li & Chen
tic center (Agaronyan & Atoyan 1981; Beacom & Yu¨ksel 2010), so that we use χ=0.01 in thisLetter.
2006;Sizun, Cass´e & Schanne2006)andtheGalacticplane In Fig. 1 we show those relevant timescales. The blue
(Stecker 1967, 1969), and we apply their treatments to line shows the cooling time of secondary electrons and
middle-aged SNRs interacting with molecular clouds. Re- positrons where the synchrotron cooing dominates above
cent review of positron annihilation can be found in 100GeV,thebremsstrahlungcoolingdominatesfrom1GeV
Prantzos et al. (2010). to100GeV,andtheionization lossdominatesbelow1GeV.
We first provide some timescales for CR protons, sec- Thegreen lineshows thecooling timeof CR protons where
ondary positrons and electrons inside a molecular cloud the pion production cooing dominate above 1GeV and the
(Section 2). We then solve energy spectra of the secondary ionization loss dominates below 1 GeV. CR protons (about
positronsandelectrons(Section3)andcalculatetheannihi- 1GeV) producing most positrons do not cool and escape
lation line flux as well as the continuum spectrum (Section as long as the SNR age is smaller than the cooling time of
4). Section 5 is devoted to thediscussion. 1GeV protons.Cooling timesofionization, bremsstrahlung
(cid:13)c 2012RAS,MNRAS000,1–6
Positron annihilation as a cosmic-ray probe 3
100 negligible, sothatweneglectthediffusionescape.Thenthe
solution for secondary positrons and electrons, f , can be
10-2 ±
expressed by
s ] 10-4
arbitrary unit 111000-1--860 f±(t,E) = |E˙(1E)|ZEE0(t,E)dǫ Q±ǫ(t′,dǫǫ)′
E) ) [ 10-12 ×exp(cid:18)−ZE τa,±|E˙|(cid:19) , (5)
Log( f( 1100--1164 where t′ is time for cooling from E0 to ǫ, and defined by
10-18 t′= E0 dǫ′ =t− ǫ dǫ′ , (6)
10-20 Zǫ |E˙(ǫ′)| ZE |E˙(ǫ′)|
105 106 107 108 109 1010 1011 1012 1013
and E (t,E) is the initial energy of positrons or electrons
Kinetic energy [ eV ] 0
before they cool to E and definedby
Figure 2. Energy spectra of CR protons, secondary positrons E0 dǫ
andelectrons fors=2.8andnt=2.8×1014cm−3s.Thegreen, Z |E˙(ǫ)| =t . (7)
red and blue lines show CR protons, secondary positrons and E
electronsspectra, respectively. Wecanneglecttheannihilation lossforsecondaryelectrons
because thetimescale is muchlonger than the SNRage.
Using the code provided by Kamae et al. (2006);
emission andinelastic collision areinverselyproportionalto
Karlsson & Kamae(2008),wecalculatesecondarypositrons
thegasdensity.Therefore,spectralevolutionsofallCRsbe-
andelectronssourcespectra,Q ,andthegamma-rayspec-
±
low1GeVdependsononlythedensity.Theblacklineshows
trum produced by decaying unstable hadrons such as pions
the escape time due to diffusion with χ=0.01. The escape
and kaons. In addition, for the high-energy electron source
of secondary electrons and positrons is negligible compared
Q , we consider knock-on electrons produced by Coulomb
−
with theenergy loss.
collisions with CR protons (Abraham,Brunstein & Cline
The typical energy of positrons produced by the CR
1966). The knock-on electrons contribute somewhat to the
protons is about E ∼ 100MeV (Murphy 1987). Then its
t continuumemission as bremsstrahlung emission.
cooling timeis about 3×104(n/300cm−3)−1yr.Therefore,
To obtain the source term of secondary positrons
SNRswith an age longer than its cooling time can produce
and electrons Q (t,E), we need to calculate the spectral
±
lowenergypositrons.Oncepositronshavecooledto100eV,
evolution of CR protons. CR protons are injected from
theyformpositroniumsthroughchargeexchange.Thereare
SNRs to molecular clouds in an energy-dependent way
four possible spin configurations of the positroniums. One
(Ohira, Murase & Yamazaki 2010). Moreover, the diffusion
of them has the total spin 0 (singlet) and the three others
escape from molecular clouds should be considered for CR
have the total spin 1 (triplet). The singlet state produces
protons above 10GeV (see Fig. 1). However, we do not
511keV linephotons,andthetripletstateproducescontin-
needtocalculate theprecisespectrumofCRprotonsabove
uum photons below 511keV. Because both the singlet and 1GeV because thanks to AGILE and Fermi, we have al-
the triplet state are produced at the same rate, one fourth
ready known spectra of CR protons above 1GeV. That is,
of positroniums can decay into two 511keV line photons.
we do not need to calculate the spectral evolution due to
It is remarkable that the annihilation time is approxi-
thediffusion escape for CR protons.On theotherhand,we
mately 10 times longer than the cooling time of positrons
have to calculate the evolution of the spectrum of CR pro-
at around 10−100MeV. Then while the positron is being
tonsbelow1GeVbecausegamma-rayspectradonottellus
cooled, approximately 10–20 percent of positrons directly
thespectrumofCRprotonsbelow1GeV.Furthermore,the
annihilatewithelectrons.Theyproducecontinuumphotons
injectionofCRprotonsisthoughttostopwhenSNRshock
above 511keV.
collides with molecular clouds (Ohira et al. 2011a), that is,
the injection of CR protons stopped of the order of 104yrs
ago.Therefore,weassumethatCRprotonsareinjectedasa
3 ENERGY SPECTRA OF SECONDARY deltafunctionintimeandaneffectivesteep-spectrum,Q ,
CR
POSITRONS AND ELECTRONS instead of neglecting thediffusion escape
In this section, to calculate the photon spectrum from sec- QCR(t,E)=qCR(E)δ(t) , (8)
ondarypositronsandelectronsproducedbyCRprotons,we
where q (E) is
calculateenergyspectraofpositronsandelectrons,f (t,E). CR
±
An evolution of CR spectra is described by q (E)∝ E+m c2 E E+2m c2 −1+2s , (9)
CR p p
∂f + ∂ E˙(E)f + f + f =Q(t,E) , (4) where m (cid:0)is the prot(cid:1)on(cid:8)m(cid:0)ass. This ex(cid:1)p(cid:9)ression gives an in-
∂t ∂E τ (E) τ (E) p
(cid:0) (cid:1) a d jection term of p−s, where p is the momentum of the CR
where Q(t,E) is the source term of CRs, the third and the proton.ObservationsshowthatspectraofCRprotonshave
fourth terms of the left hand side are sink terms due to broken power-law forms and the spectral index above the
annihilation and diffusion, respectively.As shown in Fig. 1, breakiss=2.7−2.9exceptforW44(OBS).InthisLetter,
thediffusionescapeforsecondaryelectronsandpositronsis weadoptthesinglepowerlawwiths=2.8(Thisassumption
(cid:13)c 2012RAS,MNRAS000,1–6
4 Y. Ohira, K. Kohri and N. Kawanaka
doesnotaffectthefluxoftheannihilationlinesignificantly). 103
Then,thesolutiontoequation(4)forCRprotons,f ,can
CR
beexpressed by
102
fCR(t,E)= E˙(EE˙0((Et,)E))qCR(E0(t,E)) . (10) -2-1m s ] 101
c
V
As mentioned in previous section, the cooling of CRs be- e
low 1GeV dependson only thedensity.Hence,thesolution E [ 100
d
(equation (10)) dependson only nt. F/
d
2
To calculate source terms of secondary electrons and E 10-1
positrons, Q (t′,E), in equation (5), we approximately use
±
the present spectrum of CR protons, f (t,E), by chang-
CR
ingQ±(t′,E)intoQ±(t,E)inequation(5).Thisisbecause 10-2104 105 106 107
CR protons (about 1GeV) producing most positrons do
Photon energy [ eV ]
not cool and escape for SNRs observed by Fermi and AG-
ILE.Thesolutionforthesecondarypositronsandelectrons
(equation (5)) depends on only nt below 100GeV. Here- Figure 3. Photon spectrum from an SNR with F>100MeV =
10−6photons−1cm−2,nt= 2.8×1014cm−3s and s=2.8. The
after, we use nt as the parameter to describe the system
black dashed line shows the total spectrum. The red, blue and
(for example, nt= 2.8×1014cm−3s for n=300cm−3 and green lines show the annihilation spectrum, the bremsstrahlung
t=3×104yr).Afterthecooling timeof CR protonsabove spectrum from secondary positrons and electrons, and the
1GeV(nt>1.9×1015cm−3s),allCRshavealreadycooled bremsstrahlungspectrumfromCRprotons,respectively.
and we do not expect any emission.
In Fig. 2 we show the energy spectra of the CR pro-
tons, secondary positrons and electrons given in equations 100
(5) and (10), where s = 2.8,nt = 2.8 ×1014cm−3s. For
CR protons (green line), the spectrum below 100MeV is
modified from the initial spectrum qCR(E) by ionization. eV10-1
M
The electron spectrum (blue lines) is dominated by knock- 0
0
1
on elAectsrtoenasdyb-esltoawte10soMluetVio.n is obtained by changing E0 / FeV>10-2
to infinity in equation (5). The steady-state spectrum of 1k
1
5
positrons below the typical energy of the source term Et is F 10-3
approximately obtained by using the following approxima-
tion.
10-4
Q+(E)=q+δ(E−Et) , (11) 1013 1014 1015
nt [ cm-3 s ]
where q is the total production number of the secondary
+
positrons per unit time. Then, the steady-state spectrum
below Et is given by Figure 4. The line shows F511keV/F>100MeV as a function of
ntfors=2.8.
q Et dǫ′
f (E)= + exp − . (12)
+ |E˙(E)| (cid:18) Z τ |E˙|(cid:19)
E a,+ Q (t)=|E˙(100 eV)|f (t,100 eV) . (13)
Ps +
Onefourthofpositroniumsdecayintotwo511keVphotons.
4 RADIATION SPECTRUM Thus the photon flux of the 511keV line, F , is given
511keV
by
In this section, we calculate the radiation spectrum from
secondarypositronsandelectronsatnt=2.8×1014cm−3s. F (t)= QPs(t) . (14)
In this case, primary CR electrons are negligible as 511keV 8πd2
long as the ratio of primary CR electrons to CR pro- where d is the source distance. The line width of the anni-
tons is K < 0.01 (see Fig. 2). We calculate syn- hilation lineof positroniums emitted from molecular clouds
ep
chrotronemission,inverseComptonemissionwithCMBand is 6.4keV (Guessoum, Jean & Gillard 2005). Wehere use a
bremsstrahlungemissionfromsecondarypositronsandelec- Gaussianprofilewiththe6.4keVwidthasthelinestructure.
trons(Strong, Moskalenko & Reimer2000),bremsstrahlung Fig. 3 shows the photon spectrum normalized so as
emission from CR protons (Schuster& Schlickeiser 2003), to make F = 10−6photons−1cm−2, where nt =
>100MeV
and annihilation emission from secondary positrons (e.g. 2.8×1014cm−3sand s=2.8. Weagain notethat thiscon-
Sizun et al. 2006). Energy spectra of secondary positrons ditionistypicalformiddle-aged SNRsobservedbygamma-
and electrons, obtained from equation (5), are shown in ray telescopes. Annihilation emission (red line) dominates
Fig. 2. at aroundthe511keV range. Itis notablethat synchrotron
When positrons cool to ∼ 102eV, they form positron- emission and inverseCompton emission with CMB induced
iums through charge exchange. The production rate of the by secondary positrons and electrons do not contribute in
positroniums, Q , is obtained by the energy range of Fig. 3. Moreover, synchrotron emission
Ps
(cid:13)c 2012RAS,MNRAS000,1–6
Positron annihilation as a cosmic-ray probe 5
by secondary positrons and electrons depends on the mag- studiesforCRcompositionsarealsoremarkable(Ahn et al.
netic field, the maximum energy of CR protons and the 2010; Ohira & Ioka 2011)
spectral index of CR protons (s). Bremsstrahlung emission The spectra of secondary positrons and electrons pro-
of CR protons (green line) also depends on s. Note that duced by CR protons are different from a single power law
bremsstrahlungemission of CR protonscan beobservedby because of their cooling and injection spectra (see Fig. 2).
the future X-ray telescope, NuSTAR (Harrison et al. 2005) The spectra of secondary positrons and electrons below
and ASTRO-H(Takahashi et al. 2010). 100MeV would become harder than the CR proton spec-
For the estimation of the annihilation line, we can ne- trumaround1GeV.Thenumberofsecondarypositronsand
glect theescape loss aslong as τ <τ for E <100MeV. electrons can be larger than that of primary CR electrons
cool d
Moreover,mostpositronsarecooledbytheionizationlossas when the SNR age is larger than about 10 percent of the
longasB <1mG(n/300cm3)1/2,sothatthemagneticfield cooling time of CR protons due to the inelastic collision.
(B),thesizeofmolecularclouds(R )andthenormalization EvenwhenthenumberoftheprimaryCRelectronsislarger
c
of the diffusion coefficient (χ) are not important. Hence, thanthatofsecondarypositronsandelectrons,thespectrum
the ambiguity of the annihilation line is only nt. Fig. 4 is affected by thecooling. These cooling effects of positrons
showsthefluxratio,F /F ,asafunctionofnt, and electrons might be a reason why radio spectra are dif-
511keV >100MeV
where s = 2.8. The ratio does not depend on the number ferent from expected from gamma-ray spectra observed by
ofCRsandthedistance.Thefluxratio,F /F , Fermi (Uchiyamaet al. 2010). However, the radio emission
511keV >100MeV
becomes constant after nt = 2.8×1014cm−3s because al- from SNRs may be originated from inside the SNR, but
mostallpositronshavecooledto100eVbythattime.From observed gamma rays may be originated from outside the
Fig. 4 we expect F ∼ 10−7photoncm−2s−1 from SNR.Thatis,adifferentcomponentmayproducetheradio
511keV
SNRs with nt = 1014 −1015cm−3s and with F ∼ emission. We may have to build more detailed models to
>100MeV
106photoncm−2s−1. The expected flux of the annihilation comparethetheoryandobserveddateinfuture.Itisanin-
line, F , can be sufficient for five-yearsobservations of teresting future work to compare between observations and
511keV
ACTtodetect(Boggs et al.2006).Thereforewecanexpect thetheory for individualSNRs.
to detect the annihilation line of secondary positrons pro-
ducedbytheCR protonsinSNRsobservedbyAGILE and
Fermi.
ACKNOWLEDGMENTS
We thank K. Ioka for useful comments. This work is sup-
ported in part by grant-in-aid from the Ministry of Educa-
5 DISCUSSION AND SUMMARY
tion, Culture, Sports, Science, and Technology (MEXT) of
InthisLetter,wehaveinvestigatedtheMeVemission spec- Japan, No. 21684014 (Y. O.), No. 22740131 (N. K.). K.K.
trumduetoCRprotonsfromSNRsinteractingwithmolec- was partly supported by the Center for the Promotion of
ularclouds.Wefoundthatfortypicalmiddle-agedSNRsob- Integrated Sciences (CPIS) of Sokendai, and Grant-in-Aid
served by AGILE and Fermi, secondary positrons can cool forScientificResearch on Priority AreasNo.18071001, Sci-
toanenergysufficienttomakepositroniums.Wecalculated entificResearch(A)No.22244030 andInnovativeAreasNo.
annihilation emission of the positrons and other emissions. 21111006.
The expected flux of the annihilation line is sufficient for
the future gamma-ray telescope, such as ACT, to detect.
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