Energy spectra of gamma-rays, electrons and neutrinos produced at interactions of relativistic protons with low energy radiation

0.61 MB·English

by  S. R. Kelner

Energy spectra of gamma-rays, electrons and neutrinos produced at interactions of
relativistic protons with low energy radiation
S.R. Kelner∗ and F.A. Aharonian†
Wederivedsimpleanalyticalparametrizations for energydistributionsofphotons,electrons, and
neutrinosproducedininteractionsofrelativisticprotonswithanisotropicmonochromaticradiation
field. The results on photomeson processes are obtained using numerical simulations of proton-
photon interactions based on the public available Monte-Carlo code SOPHIA. For calculations of
energyspectraofelectronsandpositronsfromthepairproduction(Bethe-Heitler)processwesuggest
asimpleformalismbasedonthewell-knowndifferentialcross-sectionoftheprocessintherestframe
of the proton. The analytical presentations of energy distributions of photons and leptons provide
a simple but accurate approach for calculations of broad-band energy spectra of gamma-rays and
9 neutrinosin cosmic proton accelerators located in radiation dominated environments.
0
0 PACSnumbers: 12.20.Ds,13.20.Cz,13.60.-r,13.85.Qk
2

n
a I. INTRODUCTION stantial (10 percent or more) fraction of the proton en-
J ergyistransferredtothesecondaryproduct. Asaresult,
6 whenthe protonenergyexceeds the π-mesonproduction
In astrophysical environments the density of low-
1 threshold,the hadronicinteractions ofprotonsdominate
energy radiation often exceeds the density of the gas
over the pair-production.
component. Under such conditions, the interactions of
]
h ultrarelativistic protons and nuclei with radiation can The cross-sectionofpair-productionis calculatedwith
p dominate over interactions with the ambient gas. These averyhighaccuracyusingthestandardroutinesofquan-
o- interactions proceed through three channels: (i) inverse tum electrodynamics. The cross-sections of photomeson
r Comptonscattering,pγ →pγ′,(ii)electron-positronpair processesare providedfrom acceleratorexperiments and
st production, p+γ → pe+e−, and (iii) photomeson pro- phenomenological studies. Generally, for astrophysical
a duction,pγ →N+kπ. While the inverseComptonscat- applications the data obtained in fixed target experi-
[
tering does nothavea kinematic threshold,the electron- ments with gamma-raybeams of energiesfrom 150 MeV
3 positron pair production and the photomeson produc- to 10 GeV are sufficient, especially in the case of broad-
v tion processes take place when the energy of the pho- bandspectraoftargetphotons,whenthehadron-photon
8 ton in the rest frame of the projectile proton exceeds interactions are contributed mainly from the region not
8 2m c2 ≃ 1MeV and m c2(1+m /2m )≃ 145MeV, re- far from the energy threshold of the process.
6 e π π p
0 spectively. The energy losses of protons in the photon fields, in
. TheprocessofinverseComptonscatteringofprotonsis particularinthecontextofinteractionsofhighestenergy
3
identical to the inverse Compton scattering of electrons, cosmicrayswith2.7Kcosmicmicrowavebackgroundra-
0
8 buttheenergyloss-rateofprotonsissuppressedbyafac- diation (CMBR), have been comprehensively studied by
0 tor of (me/mp)4 ≈10−13. At energies above the thresh- manyauthors(seee.g. Refs.[1,2,3,4,5]). Lessattention
: old of production of electron-positron pairs this process hasbeengiventocalculationsoftheproductsenergydis-
v
is four orders of magnitude slower compare to the losses tributions. Partly this can be explained by the fact that
i
X caused by pair-production. Therefore, generally the in- thecross-sectionsofsecondaryelectronsandgamma-rays
r verse Compton scattering does not play significant role withtheambientphotonssignificantlyexceedsthecross-
a
even in extremely dense radiation fields. sections of interactions of protons with the same target
The cross-section of (e+,e−) pair production (often photons. Thereforetheelectronsandgamma-rayscannot
leave the active regions of their production, but rather
called as Bethe-Heitler cross-section) is quite large, but
trigger electromagnetic cascades in the surrounding ra-
onlyasmall(≤2m /m )fractionoftheprotonenergyis
e p
diation and magnetic fields. The spectra of gamma-rays
converted to the secondary electrons. The cross-section
formedduringthe cascadedevelopmentare notsensitive
of photomeson production is smaller, but instead a sub-
to the initial energy distributions, and therefore simple
approximate approaches (see e.g. Refs. [6]) can provide
adequate accuracies for calculations of the characteris-
∗Moscow Institute of Engineering Physics, Kashirskoe sh. 31, tics of optically thick sources. This does not concern,
Moscow, 115409 Russia; Max-Planck-Institut fu¨r Kernphysik, however, neutrinos which freely escape the source and
Saupfercheckweg 1, D-6917 Heidelberg, Germany ; Electronic ad- thushaveaundistortedimprintofparentprotons. More-
dress: [email protected] over,atsomespecificconditions,thesecondaryelectrons
†Dublin Institute for Advanced Studies, 31 Fitzwilliam Place,
fromthepair-andphotomesonproductionprocessescan
Dublin 2, Ireland; Max-Planck-Institut fu¨r Kernphysik,
coolmainly throughsynchrotronradiationfor which the
Saupfercheckweg 1, D-6917 Heidelberg, Germany; Electronic
address: [email protected] source can be optically thin (see e.g. [7, 8, 9, 10]). Thus
2
the exact calculations of spectra of secondary electrons
are quite important, since the synchrotron radiation of
theseelectronscarrydirectinformationabouttheenergy
spectra of accelerated protons.
The interactions of hadrons with radiation fields can
be effectively modeled by Monte Carlo simulations of
characteristics of secondary products. In particular, the
SOPHIAcode[11]providesanadequatetoolforthecom-
prehensive study of high energy properties of hadronic
sources in which interactions of ultrarelativistic protons
with ambient low-energy photons dominate over other
processes. At the same time, it is useful to have a com-
plementary tool for study of radiation characteristics of
hadronic sources, especially when one deals with sim-
ple scenarios, e.g. interactions of protons with a homo- Figure 1: The total cross-section of the inelastic γ-p inter-
geneous and isotropic source of radiation in which the actions as a function of energy of gamma-ray in the proton
hadronic cascades (i.e. the next generation particles) do rest frame. The calculations have been performed using the
not play an important role. Motivated by this objective, routineof thecode SOPHIA[11].
in this paper we developa simple approachbased onthe
description of the energy distributions of final products
from the photomesonand electron-positronpair produc- E is simply the energy of the photon in the proton rest
tionprocessesinanalyticalforms whichcanbe easilyin- frγame. Thetotalcross-sectionσ(E ),calculatedusingthe
tegrated in any complex model of hadronic interactions γ
routines ofthe code SOPHIA[11], is presentedinFig. 1.
inhighenergyastrophysicalsources. Toacertainextent,
Although the production of pions dominate in the γ-p
thispapercanbeconsideredasacontinuationofourfirst
interactions, some other channels, in particular the ones
paper[12]whereweobtainedanalyticalpresentationsfor
leading to production of K- and η-mesons, contribute
proton-protoninteractions.
noticeably (up to 10 to 20 %) to the overall production
of photons and leptons. These channels are taken into
account in our calculations presented below. They are
II. PHOTOMESON PROCESSES
based on the SOPHIA code which allows simulations of
all important processes belongs to γ-p interactions.
The formation of high energy gamma-rays, elec-
tronsandneutrinosinphotomesoninteractionsproceeds
through production and decay of non-stable secondary
A. Production of gamma-rays
products, mainly π0 and π±-mesons:
γ+p→n π0+n π++n π−+··· (1) The inclusivecross-sectionofproductionofπ0-mesons
0 + −
wheren0,n+ andn− arethenumbersofproducedpions. dσπ0 =S(k,p,pπ0)d3pπ0 (3)
Hereafterwewillassumethatthe density ofthe ambient
medium is sufficiently low, thus the pions decay before dependsonthemomentak, p, pπ0 ofthephoton,proton
they interact with the surrounding gas, radiation and and π0-meson, respectively. Let’s denote by
magnetic fields. We will also assume that (i) both the
relativisticprotonsandthetargetlow-energyphotonsare dw =W(pπ0,pγ)d3pγ (4)
isotropically distributed, and (ii) the energy of colliding
particles the probability of decay of a π0-meson with momentum
p into a gamma-ray photon of momentum p in the
π0 γ
ǫ≪m c2, E ≫m c2, (2) volume d3p of the momentum space. Then
π p p γ
where ǫ is the energy of the target photon, m is the
π dσ =2d3p S(k,p,p )W(p ,p )d3p (5)
mass of π-meson (we will assume that m0 = m+), and γ γ π0 π0 γ π0
π π
Z
E andm aretheenergyandthemassofproton. Those
p p
conditions,whicharealwayssatisfiedinastronomicalen- can be treated as the inclusive cross-section of γ-rays
vironments,allowsusto obtainsimpleanalyticalpresen- production through the chain γ + p → n π0 + ··· →
0
tations for energy distributions of the final products of 2n γ+···.
0
π- decays - photons and leptons. For isotropic distributions of initial particles, the final
The total cross-section of inelastic γ-p interactions is products will be isotropically distributed as well. There-
a function of the scalar E ≡ (k·p)/m , where k and p foreforthedeterminationoftheenergyspectraofγ-rays
γ p
are four-momenta of the photon and proton. The scalar wecanusetheinclusivecross-sectiongivenbyEq.(5)but
3
integrated over the gamma-rays emission angles: the relation given by Eq.(13) is readily fulfilled.
Some features of the energy spectra of γ-rays can be
dE
dσ ≡G (E ,ǫ,cosθ,E ) γ (6) understoodfromtheanalysisofthekinematicsofthepro-
γ γ p γ
Ep cess. In particular, for production of a single π0-meson
the following condition should be satisfied
The function G depends on E , ǫ, E , and the angle
γ p γ
θ between the momenta of colliding proton and photon. 2ǫE (1−β cosθ)>(2m m +m2)c4, (15)
The corresponding differential interaction rate is p p π p π
where β – is the proton speed (in the units of c).
dE p
dw =c(1−cosθ)G (E ,ǫ,cosθ,E ) γ . (7) Since the protons are ultrarelativistic, we will assume
γ γ p γ
E
p β = 1. If the condition of Eq.(15) is not satisfied,
p
thentheinteractionrategivenbyEq.(7)isequaltozero.
Let f (E ) and f (ǫ) be functions characterizing the
p p ph
Therefore, for the case of 4ǫE ≤ (2m m +m2)c4, the
energy distributions of initial protons and photons, i.e. p π p π
function Φ = 0. The integration of Eq.(11) should be
f (E )dE andf (ǫ)dǫ arethe numbersofprotonsand γ
p p p ph
photons per 1cm3 in the energy intervals dE and dǫ. performed over the region1
p
respectively. Since it is assumed that the target photons
m m2
areisotropicallydistributed, theirangulardistributionis η ≥η ≡2 π + π ≈0.313, (16)
0 m m2
described as dΩ/4π. Then the production rate of γ-rays p p
(i.e. number of γ-rays in the energy interval (E ,E +
γ γ As it follows from kinematics of production of a single
dE )persec,per1cm3)canbeobtainedafterintegration
γ pion, the energy of the latter appears within
ofEq.(7)overenergiesofprotonsandtargetphotons,as
well as over the solid angle dΩ: E ≤E ≤E , (17)
πmin π πmax
where
dE dΩ
p
dN (E )=dE dǫf (E )f (ǫ)×
γ γ γ E 4π p p ph E =E x , E =E x (18)
Z p πmax p + πmin p −
c(1−cosθ)Gγ(Ep,ǫ,cosθ,Eγ). (8) are the maximum and minimum energies respectively,
with
Let’s introduce the function Φ defined as
γ
1
Φ (η,x)≡ c(1−cosθ)G (E ,ǫ,cosθ,E )dΩ, (9) x± = 2(1+η) η+r2± (η−r2−2r)(η−r2+2r) ,
γ γ p γ 4π h p (1i9)
Z
where r =m /m ≈0.146.
where π p
Let’s consider now the general case when the total
4ǫEp Eγ massofparticlesproducedinproton-photoninteractions
η = , x= . (10)
m2pc4 Ep is M +mπ. For example, the single-pion production im-
plies M = m , while in the case of two-pion production
p
Then Eq.(8) can be written in the following form M =m +m , etc. In this case the maximum and min-
p π
imum energies of pions are
dN dE
γ p
= f (E )f (ǫ)Φ (η, x) dǫ. (11)
p p ph γ
dEγ Z Ep E˜πmax =Epx˜+, E˜πmin =Epx˜−. (20)
NotethatΦ canbetreatedasafunctionoftwo(butnot
γ with
three) variables. This is connected with the fact that, as
1
we assume, at Ep → ∞ and ǫ → 0, Eγ → ∞, and for x˜ = η+r2+1−R2±
fixedxandη,thefunctionΦ shouldhaveacertainlimit. ± 2(1+η)
γ
h
In other words, at the rescaling
± (η+1−(R+r)2)(η+1−(R−r)2) , (21)
E →λE E →λE , ǫ→ǫ/λ, (12)
p p γ γ p i
where R = M/m . In particular, for R = 1 Eqs. (19)
where λ is an arbitrary number (λ > 1), the function p
and (21) coincide. For R > 1, we have the following
Φ is not changed, therefore the following relation takes
γ inequalities
place.
E <E˜ , E >E˜ . (22)
(1−cosθ)G (λE ,ǫ/λ,cosθ,λE )dΩ= πmin πmin πmax πmax
γ p γ
R = (1−cosθ)G (E ,ǫ,cosθ,E )dΩ. (13)
γ p γ
Numerical cRalculations based on the code SOPHIA [11] 1 For determination of the region allowed by kinematics, we will
show that already at assume mπ± =mπ0 =0.137GeV. Since on the boarder of this
region the function Φγ = 0, this approximation does not affect
ǫ<10−3mπc2, Ep >103mpc2 (14) theaccuracyofnumericalcalculations.
4
The decay of ultrarelativistic π0-mesons with energy
distribution J (E ) within the limits given by Eq.(17), TableI: Numericalvaluesofparameters Bγ,sγ andδγ char-
π π acterizing theγ-ray spectra given by Eq.(11).
results in the energy spectrum of γ-rays
η/η0 sγ δγ Bγ, cm3/s
dN EπmaxdE 1.1 0.0768 0.544 2.86·10−19
γ π
=2 J (E ), (23)
dE E π π 1.2 0.106 0.540 2.24·10−18
γ Z π
E1 1.3 0.182 0.750 5.61·10−18
where 1.4 0.201 0.791 1.02·10−17
m2c4 1.5 0.219 0.788 1.60·10−17
E =max E , π ,E , E <E . (24)
1 γ πmin γ πmax
(cid:18) 4Eγ (cid:19) 1.6 0.216 0.831 2.23·10−17
Below we will be interested in γ-rays with energy E > 1.7 0.233 0.839 3.10·10−17
γ
mπ/2. While in the energy range Eγ <Eπmin the diffe- 1.8 0.233 0.825 4.07·10−17
rential spectrum of γ-rays is flat,
1.9 0.248 0.805 5.30·10−17
Eπmax 2.0 0.244 0.779 6.74·10−17
dN dE
γ π
=2 J (E ), (25)
dE E π π 3.0 0.188 1.23 1.51·10−16
γ Z π
Eπmin 4.0 0.131 1.82 1.24·10−16
within the interval Eπmin < Eγ < Eπmax the spectrum 5.0 0.120 2.05 1.37·10−16
decreases with energy,
6.0 0.107 2.19 1.62·10−16
Eπmax 7.0 0.102 2.23 1.71·10−16
dN dE
γ π
=2 J (E ). (26)
dE E π π 8.0 0.0932 2.29 1.78·10−16
γ Z π
Eγ 9.0 0.0838 2.37 1.84·10−16
The above quantitative features of the spectrum of γ- 10.0 0.0761 2.43 1.93·10−16
rays appear quite useful for the choice of approximate
20.0 0.107 2.27 4.74·10−16
analytical presentations. The results of numerical cal-
culations of the function Φ based on simulations using 30.0 0.0928 2.33 7.70·10−16
γ
the code SOPHIA[11]canbe approximated,with anac- 40.0 0.0772 2.42 1.06·10−15
curacy better than 10% by simple analytical formulae.
Namely, in the range x <x<x 100.0 0.0479 2.59 2.73·10−15
− +
x δγ
Φ (η,x)=B exp −s ln ×
γ γ γ
( (cid:20) (cid:18)x−(cid:19)(cid:21) ) SOPHIA (histograms) and the analytical presentations
given by Eqs. (27) and (29) for two values of η.
2 2.5+0.4ln(η/η0) Eq.(11) provides a simple approach for calculations of
× ln , (27) γ-ray spectra for arbitrary energy distributions of ultra-
1+y2
(cid:20) (cid:18) (cid:19)(cid:21) relativistic protons and ambient photons. The param-
where eters Bγ, sγ and δγ are quite smooth functions of η,
thus for calculations of these parameters at intermedi-
x−x
y = − . (28) ate values of η one can use linear interpolations of the
x −x numerical results presented in Table I. Note that for in-
+ −
teractions of protons with 2.7 K CMBR, the results pre-
At low energies, x < x−, the spectrum does not depend sented in Table I allow calculations of γ-ray spectra up
on x, to ∼1021eV.
2.5+0.4ln(η/η )
Φ (η,x)=B ln2 0 . (29)
γ γ
Finally in the range x>x(cid:0) th(cid:1)e function Φ =0.
+ γ B. Production of electrons and neutrinos
AllthreeparametersB ,s andδ usedinthispresen-
γ γ γ
tation are functions of η. The numerical values of these
parameters are shown in Table I. At η/η = 1, i.e. at The production of leptons in proton-photon interac-
0
the threshold of π0-meson production, B =0. In Fig. 2 tions is dominated by the decay of secondary charged
γ
we show the functions xΦ (η,x) obtained with the code pions. In analogy with Eq.(11), the spectrum of each
γ
5
Figure2: Gamma-rayspectraproducedinphotomesoninteractionscalculatedfortwovaluesofη=4ǫEp/m2pc4. Solidlinesare
calculated using theanalytical presentations given byEq.(11), thehistograms are from simulations using theSOPHIA code.
type of leptons can be presented in the form 1. Energy spectra of positrons, muon antineutrinos and
electrons neutrinos
N (E )dE =
l l l
Fore+,ν¯ andν , the parameterψ ispresentedinthe
dE µ e
=dE f (E )f (ǫ)Φ (η, x) p dǫ, (30) form
l p p ph l
E
Z p η
ψ =2.5+1.4ln , (34)
η
where η is determined in Eq.(10), x = El/Ep and l im- (cid:18) 0(cid:19)
plies one of the following symbols: e+,e−,ν ,ν¯ ,ν ,ν¯ . with
µ µ e e
As in the case of γ-rays, the energy range of leptons, x′ = x− and x′ =x , (35)
forthefixedvaluesofE andǫisdeterminedbykinemat- − 4 + +
p
ics. We obtained analytical forms for the energy spectra where x and x are determined from Eq.(19); η is
+ − 0
for all lepton types, Φl, using numerical results of simu- defined in Eq.(16). Note that here ψ is different than
lations the SOPHIA code. In the range of x′ <x<x′ the relevantfunctionin Eq.(27). InFig.3 andFig.4 the
− +
analyticalpresentationsofdistributionsxΦ andxΦ
e+ ν¯µ
x δl arecomparedwithMonte-Carlosimulationsbasedonthe
Φl(η,x)=Bl exp(−sl(cid:20)ln(cid:18)x′−(cid:19)(cid:21) )× SOIPnHthIAercaondgee.η <4mπ +4 mπ 2 =2.14η ,onlyasin-
mp mp 0
gleπ+-mesoncanbeproduce(cid:16)d. I(cid:17)tdecaystoπ+ →µ+ν .
ψ µ
2 Thepositronsandmuonantineutrinosareproducedfrom
ln , (31)
1+y′2 the decay µ+ →e+ν¯ ν . Since the spectra of e+ and ν¯
(cid:20) (cid:18) (cid:19)(cid:21) µ e µ
fromthe decay of µ+ coincide (see e.g. [13]), the param-
where eters in the Tables II calculated for e+ and ν¯ for small
µ
η should be identical. The slight difference at η < 2η
x−x′ 0
y′ = − . (32) is explained by fluctuations related to the the statistical
x′ −x′ character of simulations.
+ −
At η > 2.14η , a new channel is opened for produc-
0
In range of x<x′ , the function Φ does not depend on tionofν¯ ,becauseofproductionofπ−-mesonsandtheir
− l µ
x: decay: π− → µ−ν¯ . Therefore for large values of η the
µ
parameters characterizing e+ and ν¯ differ significantly
µ
Φ (η,x)=B (ln2)ψ . (33) (see Table II). This can be seen from comparison of re-
l l
sultspresentedinrightpanelsofFig.3and4;forη =30,
For x ≥ x′+ the function Φl = 0. The analytical presen- xΦν¯µ significantly exceeds xΦe+.
tation in the form of Eq.(31) contains three parameters
s , δ and B which themselves are functions of η. The
l l l 2. Muon neutrinos
numerical values of these parameters obtained with the
methodofleastsquaresaretabulatedinTableII(fore+,
ν¯ , ν and ν ) and Table III (for e− and ν¯ ). The values Thedistributionforν isdescribedbyEq.(31)withthe
µ µ e e µ
of x′ and ψ are given below for each type of leptons. same function ψ as for e+, ν¯ and ν , given by Eq.(34),
± µ e
6
Table II: Numerical valuesof parameters s, δ, and B for e+, ν¯ , ν , ν .
l l l µ µ e
η/η0 se+ δe+ Be+, cm3/s sν¯µ δν¯µ Bν¯µ,cm3/s sνµ δνµ Bνµ, cm3/s sνe δνe Bνe, cm3/s
1.1 0.367 3.12 8.09·10−19 0.365 3.09 8.09·10−19 0.0 0.0 1.08·10−18 0.768 2.49 9.43·10−19
1.2 0.282 2.96 7.70·10−18 0.287 2.96 7.70·10−18 0.0778 0.306 9.91·10−18 0.569 2.35 9.22·10−18
1.3 0.260 2.83 2.05·10−17 0.250 2.89 1.99·10−17 0.242 0.792 2.47·10−17 0.491 2.41 2.35·10−17
1.4 0.239 2.76 3.66·10−17 0.238 2.76 3.62·10−17 0.377 1.09 4.43·10−17 0.395 2.45 4.20·10−17
1.5 0.224 2.69 5.48·10−17 0.220 2.71 5.39·10−17 0.440 1.06 6.70·10−17 0.31 2.45 6.26·10−17
1.6 0.207 2.66 7.39·10−17 0.206 2.67 7.39·10−17 0.450 0.953 9.04·10−17 0.323 2.43 8.57·10−17
1.7 0.198 2.62 9.52·10−17 0.197 2.62 9.48·10−17 0.461 0.956 1.18·10−16 0.305 2.40 1.13·10−16
1.8 0.193 2.56 1.20·10−16 0.193 2.56 1.20·10−16 0.451 0.922 1.32·10−16 0.285 2.39 1.39·10−16
1.9 0.187 2.52 1.47·10−16 0.187 2.52 1.47·10−16 0.464 0.912 1.77·10−16 0.270 2.37 1.70·10−16
2.0 0.181 2.49 1.75·10−16 0.178 2.51 1.74·10−16 0.446 0.940 2.11·10−16 0.259 2.35 2.05·10−16
3.0 0.122 2.48 3.31·10−16 0.123 2.48 3.38·10−16 0.366 1.49 3.83·10−16 0.158 2.42 3.81·10−16
4.0 0.106 2.50 4.16·10−16 0.106 2.56 5.17·10−16 0.249 2.03 5.09·10−16 0.129 2.46 4.74·10−16
5.0 0.0983 2.46 5.57·10−16 0.0944 2.57 7.61·10−16 0.204 2.18 7.26·10−16 0.113 2.45 6.30·10−16
6.0 0.0875 2.46 6.78·10−16 0.0829 2.58 9.57·10−16 0.174 2.24 9.26·10−16 0.0996 2.46 7.65·10−16
7.0 0.0830 2.44 7.65·10−16 0.0801 2.54 1.11·10−15 0.156 2.28 1.07·10−15 0.0921 2.46 8.61·10−16
8.0 0.0783 2.44 8.52·10−16 0.0752 2.53 1.25·10−15 0.140 2.32 1.19·10−15 0.0861 2.45 9.61·10−16
9.0 0.0735 2.45 9.17·10−16 0.0680 2.56 1.36·10−15 0.121 2.39 1.29·10−15 0.0800 2.47 1.03·10−15
10.0 0.0644 2.50 9.57·10−16 0.0615 2.60 1.46·10−15 0.107 2.46 1.40·10−15 0.0723 2.51 1.10·10−15
30.0 0.0333 2.77 3.07·10−15 0.0361 2.78 5.87·10−15 0.0705 2.53 5.65·10−15 0.0411 2.70 3.55·10−15
100.0 0.0224 2.86 1.58·10−14 0.0228 2.88 3.10·10−14 0.0463 2.62 3.01·10−14 0.0283 2.77 1.86·10−14
Figure 3: Energy spectra of positrons produced in photomeson interactions calculated for two values of η=4ǫEp/m2pc4. Solid
lines are calculated using the analytical presentations, the histograms are from Monte-Carlo simulations using the SOPHIA
code.
but with different parameters x′ : where ρ=η/η , and
± 0
0.427x , ρ<2.14, x′ =0.427x . (37)
+ − −
x′+ = (0.427+0.0729(ρ−2.14))x+, 2.14<ρ<10, The difference of values of x for ν¯ and ν appears for
 x , ρ>10, thefollowingreason. Atthed±ecayπµ+ →µ+µν¯ themaxi-
+ µ
mumenergyofν¯ isequalto(1−m2/m2)E ≈0.427E ,
 (36) µ µ π π π
7
Figure 4: The same as in Fig. 3 but for muon antineutrinos.
Figure 5: The same as in Fig. 3 but for muon neutrinos.
where m is the mass of muon. On the other hand, the The maximum and minimum energies of the pion corre-
µ
maximum energy of ν¯ , produced at decay of the muon spondingly are
µ
isequaltoE . Withanincreaseofthe parameterη,π−-
π
mesons start to be produced, the decay of which leads
E =x E , E =x E ,, (39)
toν withmaximumenergycomparabletotheenergyof π max max p πmin min p
µ
the pion (a detailed discussion of these questions can be
found in [12]). where
In Fig. 5 the function xΦ (η,x) is shown. The
νµ
histograms are from Monte-Carlo simulations using the
1
SOPHIA code, and the solid lines correspond to the an- xmax = η−2r± η(η−4r(1+r)) . (40)
min 2(1+η)
alytical approximations.
(cid:16) p (cid:17)
Then x′ = x and x′ = x /2. Eq.(40) is obtained
+ max − min
from Eq.(21) if one sets R = 1 + r. These functions
3. Electrons and electron antineutrinos
together with
The electronsand electronantineutrinos are produced
η
through the decay µ−, which in its turn is a product ψ =6 1−e1.5(4−ρ) Θ(ρ−4), ρ= , (41)
η
of the decay of π−-meson. Therefore, for production of 0
(cid:16) (cid:17)
e− and ν¯ , at least two pions should be produced. The
e
production of two pions is energetically allowed if determine the distributions fore− andν¯ givenina gen-
e
eralformby Eqs.(31)- (33); Θ(ρ)is the Heaviside func-
η >η′′ =4r(1+r)≈2.14η . (38) tion (Θ(ρ)=0 if ρ<0 and Θ(ρ)=1 if ρ≥0).
min 0
8
(multiplicity)producedinoneinelasticinteractionofpro-
Table III: Numerical values of parameters s, δ and B for
l l l tons with 2.7 K CMBR as a function of proton energy.
electrons and electron antineutrinos
The results of numerical calculations are obtained using
η/η0 se− δe− Be−, cm3/s sν¯e δν¯e Bν¯e, cm3/s the energy spectra of secondary photons, electrons and
3.0 0.658 3.09 6.43·10−19 0.985 2.63 6.61·10−19 neutrinos, and the total cross-section shown in Fig 1.
4.0 0.348 2.81 9.91·10−18 0.378 2.98 9.74·10−18 Notethatbelowthethresholdofproductionoftwopions
one should have the following relations
5.0 0.286 2.39 1.24·10−16 0.31 2.31 1.34·10−16
1
6.0 0.256 2.27 2.67·10−16 0.327 2.11 2.91·10−16 hn i+hn i=1, hn i=hn i=hn i. (43)
2 γ e+ e+ ν¯µ νµ
7.0 0.258 2.13 3.50·10−16 0.308 2.03 3.81·10−16
The results of calculations based on approximate analy-
8.0 0.220 2.20 4.03·10−16 0.292 1.98 4.48·10−16
tical presentations of functions Φ satisfy these relations
l
9.0 0.217 2.13 4.48·10−16 0.260 2.02 4.83·10−16 with an accuracy of better than 5%.
10.0 0.192 2.19 4.78·10−16 0.233 2.07 5.13·10−16 Note that at very low energies the average number
of γ-rays hn i appears smaller than average number of
30.0 0.125 2.27 1.64·10−15 0.135 2.24 1.75·10−15 γ
positrons hn i. This, at first glance unexpected result
e+
100.0 0.0507 2.63 4.52·10−15 0.0770 2.40 5.48·10−15 is actually a direct consequence of the experimental fact
thatnear the thresholdthe totalcross-sectionσ of π+
π+
production significantly exceeds σ (see Fig. 9).
π0
III. PHOTONS AND LEPTONS PRODUCED AT
INTERACTIONS OF PROTONS WITH 2.7K
CMBR IV. PRODUCTION OF ELECTRON-POSITRON
PAIRS
In this section we compare the energy spectra of
gamma-rays, neutrinos, and electrons produced at pho- At energies below the photomeson production, the
tomeson interactions. For monoenergetic protons inter- main channel of inelastic interactions for protons with
acting with a radiation field with energy distribution ambientphotons proceedsthroughthe directproduction
f (ǫ), the energyspectra ofphotons and leptons canbe ofelectron-positronpairs. Intherestframeoftheproton,
ph
reduced to the calculation of a one-dimensional integral this process is described by the so-called Bethe-Heitler
cross-section. Inastrophysicalenvironments,the process
∞
is more often realized when ultrarelativistic protons col-
dN
= fph(ǫ)Φ(η, x)dǫ, (42) lides with low-energy photons,
dx
Z
ǫ
0 p+γ →e++e−+p (44)
where ǫ =η m2c4/(4E ), and x=E/E is the fraction
0 0 p p p The process is energetically allowed when
of energy of the protons transferred to the given type of
secondaryparticle; Φ is oneofthe functions describedin
γ ǫ>m c2, (45)
the previous section. p e
In Fig. 6 and 7 we show the energy spectra of γ- whereγ =E /m c2istheprotonLorentzfactor,ǫisthe
p p p
rays and electrons (left panels) and all neutrino types
soft photon energy, and m is the mass of electron. The
(rightpanels)producedbyprotonsofenergy1020eVand e
maximumenergyoftheelectron(positron)isdetermined
1021 eV interacting with blackbody radiation of temper-
by the kinematics of the process
ature T = 2.7 K. The results depend only on the prod-
uct Ep ×T, therefore they can be easily rescaled to a γ 2
blackbody radiation of an arbitrary temperature. The Eemax = 1+4γ ǫp/(m c2) γpǫ+ γpǫ−mec2 .
chosen radiation field and proton energies are of great p p (cid:18)q q (cid:19)
(46)
practical interest in the context of origin and intergalac-
This equation is valid when γ ≫1 and ǫ≪m γ c2. In
ticpropagationofultrahighenergycosmicrays. Because p p p
the interval
of interactions with the intergalactic radiation fields, ul-
trahighenergyγ-raysachievetheobserverfromdistances m c2 ≪γ ǫ≪m c2 (47)
e p p
lessthan1Mpc(seee.g. Ref.[14]). Theelectronsrapidly
coolviasynchrotronradiationor,inthecaseofverysmall the maximum electron energy is
intergalacticmagnetic field, initiate electromagnetic cas-
cadessupportedbyinteractionsofelectronsandgamma- E =4γ2ǫ, (48)
emax p
rays with the 2.7 K CMBR. Only neutrinos freely pene-
trate through intergalactic radiationand magnetic fields ThisappliesforE ≪E . Inthelimitofγ ǫ≫m c2
emax p p p
and thus carry a clear imprint of parent protons.
In Fig. 8 we show the average number of secondaries E =m c2γ =E , (49)
emax p p p
9
Figure6: Theenergyspectraofstableproductsofphotomeson interactionsofaprotonofenergy Ep=1020eV with the2.7K
CMBR. Left panel - gamma-rays, electrons, and positrons, right panel- electron and muon neutrinosand antineutrinos.
Figure 7: The same as in Fig. 6, but for a proton of energy Ep =1021eV
i.e. the whole energy of the proton is transferred to one this distribution can be found introducing an additional
of the electrons. δ-function under the integral in Eq.(51):
Let’s denote by dσ the differential cross-sectionof the
process. The interaction rate is dN =c2 dǫdΩf (ǫ)(k·up) δ(ξ−ϕ)dσ. (52)
ph
dξ 4π ǫγ
(k·p) (k·u ) Z p Z
dw =c3 dσ =c2 p dσ, (50)
ǫE ǫγ In particular, the energy distribution of electrons in the
p p
laboratory frame can be calculated using the following
where k and p are four-momenta of the photon and formula
proton, u = p/m c is the four-velocity of the proton,
p p
(k·p)=ǫEp/c2−kpisthescalarproductoffour-vectors. dN =c2 dǫdΩf (ǫ)(k·up) δ(E −c(u ·p ))dσ,
Letassumethatinaunitvolumewehavef (ǫ)dǫdΩ/4π dE 4π ph ǫγ e lf e
ph e Z p Z
photonsbetweenthe energyinterval(ǫ, ǫ+dǫ)andmov- (53)
ing withinthe solidangledΩ. Thenthe number ofinter- where u is the four-velocity of the laboratory frame,
lf
actions per unit of time is and p is four-momentum of electron since the scalar
e
c(u ·p )is equalto the energyofelectroninthe labora-
lf e
N =c2 dǫdΩf (ǫ)(k·up) dσ, (51) torysystem. TheprotonLorentz-factorinthelaboratory
ph
4π ǫγ system also can be considered as a relativistic invariant:
Z p Z
γ =(u ·u ).
p lf p
where the integration is performed over all variables. Note that the integral
Below we perform calculations based on the following
approach. If we are interested in a distribution of some
S ≡ δ(E −c(u ·p ))dσ (54)
variable ξ, which is a function ϕ of particle momenta, e lf e
Z
10
θ , the result can be written in the form
−
S = W(ω,E ,cosθ )×
− −
Z
δ(E −γ (E −V p cosθ ))dE d(cosθ ), (57)
e p − p − − − −
where
d2σ
W(ω,E ,cosθ )= (58)
− −
dE d(cosθ )
− −
isthedouble-differentialcross-sectionasafunctionofen-
ergy and emission angle of the electron in the rest frame
of the proton; ω = (u ·k)/(m c) is the energy of the
p e
photontherestframeofprotoninunitsm c2. Thefunc-
e
tion W has been derived in the Born approximation in
Refs. [15, 16]. The approach used in these papers de-
Figure 8: The multiplicity of photons and leptons produced scribes the production of an electron-positron pair by a
in oneinteraction of a relativistic proton with 2.7 CMBR
photon in the Coulomb field which formally corresponds
to the limit m → ∞. However, we warn the reader
p
that there is a misprint in the cross-section published in
these papers,thereforeweadviseto useEq.(10)ofpaper
by Blumenthal [2], where the typo is fixed. Note that
in Refs. [2, 15, 16] the system of units is used in which
c = ~ = m = 1. Since here we cite to certain equa-
e
tions of these papers, in this section, in order to avoid a
confusion, we use the same system of units.
The presence of the δ-function in the integrand allows
the integration over the variable d(cosθ ), which gives
−
1 dE
−
S = W(ω,E ,ξ), (59)
−
γ V p
p p Z −
where
γ E −E
p − e
ξ ≡cosθ = . (60)
−
γ V p
p p −
Figure 9: The total cross-sections of production of π+ and
π0-mesons as a function of energy of incident gamma-ray in After substituting Eq.(59) into Eq.(53), and using the
the rest frame of proton. The experimental points are taken relation
from the site http://wwwppds.ihep.su:8001.
ω =(u ·k)=ǫγ (1−cosθ), (61)
p p
itisconvenienttoperformtheintegrationoverω instead
is a relativistic invariant, so it can be calculated in any of integration over the angle. Then, for ultrarelativistic
frame of coordinates. The differential cross-section dσ protons (γ ≫1), we obtain
p
can be written in the simplest form in the rest-frame of
theproton,thereforeforcalculationsofS wewillusethis ∞ 2γpǫ
dN 1 f (ǫ)
system of coordinate where = dǫ ph dωω×
dE 2γ3 ǫ2
e p Z Z
c(u ·p )=γ (E −V p cosθ ). (55) (γp+Ee)2 (γp+Ee)2
lf e p − p − − 4γp2Ee 2γpEe
Here E− is the energy and p− = E−2/c2−m2ec2 is the ω−1 dE−
W(ω,E ,ξ). (62)
momentum modulus of electron inqthe rest frame of the p −
−
proton, θ− is the angle between the momenta of photon γp2+ZEe2
and electron. Therefore 2γpEe
When substituting Eq.(60) into (62) we set V = 1, and
p
S = dσδ(E −γ (E −V p cosθ )). (56)
e p − p − − correspondingly ξ = (γpE− −Ee)/(γpp−). The integra-
Z tionlimitsinEq.(62)arefoundfromtheanalysisofkine-
After integration over all variables, except for E and matics.
−