In this work we re-examine the influence of the microwave background radiation on the propagation of extremely high energy cosmic rays. We present a recalculation of the rate of fractional energy loss in the continuous energy loss approximation including the effects of resonances other than the Δ. We also discuss the distortion of the cosmic ray spectrum due to interactions with the relic photons.Facultad de Ciencias Exacta

Anchordoqui, Luis Alfredo

Dova, María Teresa

Epele, Luis Nicolás

Swain, John D.

Nuclear Physics B

English

El Servicio de Difusión de la Creación Intelectual

ELSEVIER 
UCLEAR PHYSIC, ~ 
Nuclear Physics B (Proc. Suppl.) 52B (1997) 249-252 
PROCEEDINGS 
SUPPLEMENTS 
Opacity of the Microwave Background Radiation to Ultrahigh Energy 
Cosmic Rays 
L.A. Anchordoqui a, M.T. Dova a*, L.N. Epele a, and J.D. Swain b 
aDepartamento de Fisica, Universidad Nacionai de La Plata C.C. 67, (1900) La Plata, Argentina 
bDepartment of Physics, Northeastern University, Boston, Massachusetts, 02115, USA 
In this work we re-examine the influence of the microwave background radiation on the propagation ofextremely 
high energy cosmic rays. We present a recalculation of the rate of fractional energy loss in the continuous energy 
loss approximation i cluding the effects of resonances other than the A. We also discuss the distortion of the 
cosmic ray spectrum due to interactions with the relic photons. 
As soon as the microwave background radia- 
tion (MBR) was discovered [1], it became un- 
avoidable to evaluate the interaction of ultrahigh 
energy cosmic rays (UHECR's) with the relic pho- 
tons. This interaction gives rise to a cutoff in the 
cosmic ray spectrum at energies > 5 x 1019 eV, 
as first shown in 1966 by Greisen, Zatsepin and 
Kuz'min, and then called the "GZK cutoff" [2,3]. 
Recent observations with several air shower ar- 
rays have clearly detected a few UHECR's with 
energies above 1020 eV [4-7]. Since the GZK 
cutoff implies relative proximity of such UHECR 
sources, the origin of such energetic particles has 
became one of the most pressing questions of cos- 
mic ray astrophysics. 
The study of the modification of the spectrum 
due to interactions with the MBR en route to 
Earth, is essential if one is to obtain a deeper 
understanding of the origin of the highest energy 
cosmic rays. There exist in the literature various 
works on the propagation of nncleons in the in- 
tergaiactic medium. [8-17]. In the present work 
we re-analyze the effects of meson photoproduc- 
tion, the dominant energy loss mechanism, using 
the continuous energy loss approximation which 
is expected to be reasonable for the problem at 
hand. This allows us to discuss, in a general way, 
the modification of the spectrum of cosmic rays 
from nearby sources. 
The propagation of cosmic rays through the 
*E-mail: dova@venus.fisica.unlp.edu.ar 
0920-5632(97)/$17.00 © 1997 Elsevier Science B.V. All rights reserved. 
PII: S0920-5632(96)00898-5 
intergalactic medium is described by a kinetic 
transport equation which takes into account hree 
mechanisms for energy losses: redshift, pair pro- 
duction, and pion photoproduction. For nearby 
sources corresponding to propagation times T 
3 × l0 s years, redshift effects can be neglected. 
If one assumes that the highest energy cosmic 
rays are indeed nucleons, the fractional energy 
loss due to interactions with the cosmic back- 
ground radiation at temperature T and redshift 
z = 0, is determined, following the formalism of 
[18], by 
1 dE ckT 
- x (1) E dt 27r2F2 (ch) 3 
/ /  dwr a(wr) K(w~) ln(1 - e -w~/2rkT) Wr 
o 
where w~ is the photon energy in the rest frame 
of the nucleon, w0 is the threshold energy for the 
reaction, K(w~) is the average fraction of the en- 
ergy lost by the photon to the nucleon in the rest 
frame, a(w~) is the total cross section of inter- 
action and F, as usual, is the nucleon Lorentz 
factor. 
The characteristic time for the energy loss due 
to pair production at E > 1019 eV is t ~ 5 x 109 yr 
[19] and therefore it does not affect the spectrum 
of nucleons arriving from nearby sources. Conse- 
quently, for nucleons with E _ 3 x 1019eV, meson 
photoproduction is the dominant mechanism for 
energy loss. 
In order to determine the effect of meson pho- 
250 L.A. Anchordoqui et al./Nuclear Physics B (Proc. Suppl.) 52B (1997) 249-252 
toproduction on the spectrum of cosmic rays it 
is necessary to study the kinematics of photon- 
nucleon interactions. Assuming that reactions 
mediated by baryon resonances have spherically 
symmetric decay angular distributions [13,16], 
the average nergy loss of the nucleon after n res- 
onant collisions is given by 
mR . -- m M 
<K> = 1 - 1 + me- - -  (2) 
i----1 R{-t  
where mR, denotes the mass of the resonant sys- 
tem of the chain, mM the mass of the associated 
meson, mn ° = v ~ is the total energy of the re- 
action in the centre of mass and mR, the mass 
of the nucleon. It is well established from ex- 
periments.that very high energies (v ~ above 
,-~ 3 GeV), the incident nucleons lose half their 
energy via pion photoproduction i dependent of
the number of producted pions. 
In the region dominated by baryon resonances, 
the cross section is obtained as a sum of Breit- 
Wigner distributions constructed from the ex- 
perimental data in the Table of Particle Prop- 
erties [20] considering the main resonances pro- 
duced in N'~ collisions with 7rN, rTrN, and ~rKN 
final states. For the cross section at high ener- 
gies we used the fits to the high-energy cross sec- 
tion Crtotal(P"f) made by the CERN-HERA and 
COMPAS Groups [20]. In this energy range, the 
O'tota I (n')') is to a good approximation identical to 
atota, (PT). 
We perform the numerical integration of (1), 
taking into account he aforementioned resonance 
decays and the production of multipion final 
states at higher centre of mass energies (v ~ ~ 3 
GeV). The )/2 fit of the numerical result of (1) to 
an exponential A exp( -B /E ) ,  similar to that pro- 
posed by Berezinsky and Grigor'eva [10] for the 
region of resonances, gives A = (3.66 -4- 0.08) × 
10-Syr -1, B = (2.87+ 0.03) x 1020 eV. At high 
energies the fractional energy loss was fitted with 
a constant, C = (2.42 + 0.03) x 10 -s yr -1. These 
results are different from those obtained in [10], 
due to a refined expression for the total cross sec- 
tion. From the values determined for the frac- 
tional energy loss, it is straightforward to com- 
pute the energy degradation ofUHECR's in terms 
of their flight distance. The results shown in Fig. 
1, are consistent with those previously obtained 
by Cronin [21], in which only single pion produc- 
tion is considered. Our results therefore support 
the assumption, that the energy loss is dominated 
by single pion photoproduction. 
~10 31 
10 2 ]
r I I I I I I  
lO d' d 1 1 2 
E leVI 
Figure 1. Energy attenuation length of nucleons 
in the intergalactic medium for initial energies of 
102°, 1021 and 1022 eV 
Let us now turn to the modification that the 
MBR produces in UHECR spectrum. We as- 
sume that nucleons propagate in a straight line 
through the intergalactic medium, the evolution 
of the spectrum N(E ,  t) is governed by the fol- 
lowing balance quation 
cON _ O(b(E)N)  + D V2N + Q (3) 
Ot CO E 
that takes into account he conservation of the 
total number of nucleons in the spectrum. In the 
first term on the right, b(E) is the mean rate at 
which particles lose energy. The second term, the 
diffusion in the MBR, may be neglected ue to the 
low density of the relic photons. The third term 
corresponds to the particle injection rate into the 
intergalactic medium (the sources). 
Let us focus our attention on the evolution of a 
single source spectrum. In this case, the solution 
L.A. Anchordoqui etal./Nuclear Physics B (Proc. Suppl.) 52B (1997) 249-252 251 
of eq. (3) reads [11], 
Y(E , t ) -  1 /~  b(E) Q(E~, t') dEg (4) 
with 
fig d/~ t' = t - (5 )  
and Eg -= energy of the nucleon when emitted 
by the source. As there exists evidence that the 
source spectrum of cosmic rays has a power-law 
dependence Q(E) = ~ E -~ (see for example [14]), 
the injection spectrum of a single source can be 
written as Q(E,t) = ~E -'r 5(t - to). At very 
high energies, i.e. where b(E) = C E, the total 
number of particles at a given distance from the 
source is given by 
t~ 
N(E,t) ~ b- -~× (6) 
1 
fE  Eg~ 6 ( t -  - c ln -~)  dEg 
or equivalently 
N(E, t) .~ tc E-'~e - (-~-1)ct (7) 
This means that the spectrum is uniformly 
damped by a factor depending on the proximity 
of the source. 
At low energies, in the region dominated by 
baryon resonances, the parametrisation of b(E) 
does not allows a complete analytical solution. 
However using the change of variables 
fE ~ dP~ = (s)  
with Eg = ~(E, t) and dt = dEJb(Eg), we easily 
obtain, 
t~ 
N(E,t) = b(E) × (9) 
fo ~ ~(E, - t)b[~(E, t)]dt ~)-~5(~ 
and then, the compact form, 
t~ 
N(E, t) = b -~ E-~'~ b(Eg) (10) 
In our case, where we have assumed an expo- 
nential behaviour of the fractional oss energy, 
Eg is fixed by the constraint: A t - Ei (B/E) + 
Ei (B/Eg) = 0, Ei being the exponential integral 
[22].  
Studies are underway of the case of a nearby 
source for which Eg ~ E + AE. In this case it 
is possible to obtain an analytical solution. An 
expansion of b(E + AE) in powers of AE  in eq. 
(5) allows one to obtain an expression for AE: 
AE~{exp[ tb(E)(E+B)]-I}~ J (E E2+B)(II) 
To describe the modification of the spectrum, 
it is convenient to introduce the factor ~, the ratio 
of the modified spectrum to the unmodified one. 
At low energy, the value is 
[" E + AE'~ -~ b(E + AE) 
(12) 
rl = ~, -E- - b(E) 
Equations (11) and (12) describe the spectral 
modification factor up to energies of ~ 95 EeV 
(~ 85 EeV) with a precision of 6% (20%) for a 
source situated at 50 Mpc (100 Mpc). 
4 
¢J  
th i s  work  _ _  
Berezlnsky et a l  . . . . . .  
/i 
~ ,----750 Mpc 
, , 
ld ° 
E[eV] 
Figure 2. Modification factor of single-source en- 
ergy espectra for different values of propagation 
distance and power law index 7 = 2.5 . 
In Fig. 2 we plot the modification factors for 
the case of nearby sources with power law injec- 
tion ('y = 2.5) to compare with the corresponding 
results of ref.[10]. The most significant features 
252 L.A. Anchordoqui etal. /Nuclear Physics B (Proc. Suppl.) 52B (1997) 249-252 
of ~ are the bump and the cutoff. It has been 
noted in the literature that the continuous energy 
loss approximation tends to overestimate he size 
of the bump [12,17]. Fig. 2 shows the bump is 
less pronounced in our treatment. As the bump 
is a consequence of the sharp dependence of the 
nucleon free path on energy, we attribute these 
differences to the replacement of a cross section 
approximated by the values at threshold energy 
[10], by a more detailed expression taking into 
account he main baryon resonances. 
These results allow us to recover the continuous 
energy loss approximation which is the only way 
in which analytical solutions can be obtained for 
a wide range of situations. 
Acknowledgments 
We are grateful to Huner Fanchiotti and Carlos 
Garcia Canal for illuminating discussions, and to 
James Cronin for stimulating our initial interest 
in this subject. One of us (MTD) would also like 
to thank the organizing committee for a most en- 
joyable and interesting conference. 
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15. S. Yoshida and M. Teshima, Prog. Theor. 
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Opacity of the microwave background radiation to ultrahigh energy cosmic rays

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Opacity of the microwave background radiation to ultrahigh energy cosmic rays

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