1411
Progress of Theoretical Physics, Vol. 77, No.6, June 1987
Behaviour of Cosmic Rays in the Atmosphere
at Super High Energy Region
Akinori OHSAW A and Seibun Y AMASHIT A*
Institute for Cosmic Ray Research, University of Tokyo
Tanashi, Tokyo 188
*Science and Engineering Research Laboratory
Waseda University, Shinjuku 162
(Received October 2, 1986)
Propagation of high energy cosmic rays in the atmosphere is calculated analytically relative to
their longitudinal as well as lateral aspects, based on a model using the accelerator results on multiple
meson production.
Main concern is the energy spectrum, I( > E), and the lateral energy spread, <E2r2>, of particles
of various components in each resultant atmospheric cosmic· ray phenomenon, called "family" in
emulsion chamber experiments.
Discussions are made on how each constituent assumption in the model affect the above·
mentioned quantities.
§ 1.
Introduction
Cosmic rays are used for the study of nuclear interactions in the energy region
exceeding those of accelerators, and emulsion chamber is one of the most suitable
detectors for it because of its superior resolution in position and energy measurements.!)
Recently the energy region of its main concern has been shifted to 1016~ 10 18 eV
owing to the increase of accelerator energy. In this energy region the observation of
atmospheric cosmic-ray phenomena is the unique way for the study because of low
cosmic-ray intensity. For this way of study the detailed knowledge on the propagation of cosmic rays in the atmosphere is indispensable.
From the above point of view we try in this article to clarify the behaviour of high
energy cosmic-ray particles in the atmosphere with relation to the emulsion chamber
experiment. There have been several similar attempts. 2) Most of them pertain only
to one-dimensional behaviour - - energy spectrum of particles at various observation levels - - , due to its mathematical simplicity. One-dimensional information,
however, is limited, the limitation being all the more serious in view of the increasing
prominence of large transverse momentum component among the particles produced
in higher energy nuclear interactions. Therefore the three-dimensional treatment,
yielding the energy spectrum together with the spatial distribution of particles at
various levels, is necessary in order to make more satisfactory analysis of the events.
In this article, as the first step toward this line of study, we will make a calculation
by introducing a simplified model for elementary processes (x-distribution obtained
by the accelerators, energy independent transverse momentum distributions for
hadrons pr'oduced in the multiple production, and so on), and will examine in what
1412
A. Ohsawa and S. Yamashita
manner each of these assumptions affects the calculated characteristics of the event.
More refined treatment, taking up scaling-violating energy spectrum for produced
particles, Pr-distribution getting harder with incident energy, collision cross-section
increasing with energy, etc., shall be treated in the subsequent papers as the correction
to the gross behaviours obtained here.
Series of works by Shibata et a1. 3 ) have been along the same line of study and they
have shown that their exponential-type x-distribution, deduced from the emulsion
chamber experiment, can reproduce quite well the multiple meson production phenomena of wide energy range, from the accelerator to the extensive air shower experiments. As their framework is similar to ours, we shall describe here the outline of
their work and clarify the differences between the two.
For the charged pions produced in an atmospheric nuclear interaction, they adopt
the following energy-angular distribution (Fig. l(a»:
(1°1)
with a=N,,/(2/3K) and b=(2/3)K/(N"E*), and in solving the diffusion of cosmic-ray
particles in the atmosphere they use the successive method. That is, they treat the
atmospheric nuclear collisions directly as discrete processes. Their approximation
consists in that, for the discussion of the (e, r) and pion components, they retain only
the pions of the first and up to the second generation, respectively, owing to the large
complexity of mathematical handling.*) Their method has proved to be nice in
treating the emulsion chamber data with the observed energy 1014~ 10 15 eV, where
only relatively small number of successive atmospheric interactions make important
contribution. They have succeeded in giving exact lateral distribution functions
within the limit of their own above-mentioned simplifying assumptions.
Our main interest, on the other hand, lies in clarifying the effect which each
constituent assumption in the model produces on the quantities observed by emulsion
chamber experiment, whose main interest now lies in the phenomena with energy 10 16
18
~ 10 eV, where a considerable number of successive atmospheric nuclear interactions would be involved. In the present paper, we adopt the x-distribution given by
the accelerator experiments and treat the whole atmospheric propagation as continuous. We solve the diffusion equation. In our present model the energy variables
appear always in fractional forms, so that we can get exact solutions, the
electromagnetic cascade shower theory being the good paragon for our method. 4)
The present type of the calculation yields only the average behaviour of the
events and we should always keep the ~uctuation in mind. Effect is especially
important in the nucleon primary case, because the first interaction depth fluctuates
greatly.5) We introduce the first interaction depth of the primary particle, to, instead
of fixing it to be zero, to treat it properly. Fluctuation effects arising from the other
causes would be small, because the events of our concern, having the energies of
1017 eV, will involve particles, the number of which is as large as 103 •6)
*) Shibata has also included the contribution of higher generations in his treatment of extensive air
shower, where low energy particles make significant contribution, by assuming the extreme case that the
energy of concerned particles is very large or very small.
Behaviour of Cosmic Rays in the Atmosphere
§ 2.
(a)
1413
Basic assumptions
Skeleton
The scenario about the cosmic rays in the atmosphere is the following, as far as
the high energy particles are concerned. A high energy primary cosmic-ray particle
(a nucleon or a nucleus), incident upon the earth's atmosphere, collides with an
atmospheric nucleus, inducing multiple pion production. The produced charged pions
Gollide again with atmospheric nuclei during their passage in the atmosphere.
Repeating the above (hadronic cascade process), a number of pions are produced and
propagate in the atmosphere. On the other hand, neutral pions, produced
simultaneously with charged pions, decay into two y's and those y's undergo
electromagnetic cascade process. Thus a number of electromagnetic component
(electrons, positrons and photons, abbreviated as (e, y)'s) traverse the atmosphere
together with hadronic component. Those components are observed as the bundle of
showers of parallel direction in the emulsion chamber. The event is called "family".
Our main interest lies in the energy and spatial distribution of these particles constituting the family.
We here give some comments to make the basic assumption clearer.
1) The primary cosmic-ray particle is assumed to be a nucleon with energy Eo,
and the primary heavy nucleus may be approximated to be the bundle of A nucleons
with energy Eo/A, where A is the mass number. The primary energy region of our
main concern is 1016~ 1018 eV.
2) We assume that only the pions are produced in the multiple particle production, neglecting other kinds of particles comprising only small fractions, such as
kaons, heavy mesons, etc. The case of exotic interactions7) will be discussed elsewhere.
3) We assume recoil transverse momentum PTi (i = Nand 7l-) of the leading
particle, transverse momentum PT of produced pions in the multiple production, Qvalue m" in the Jr°~2y decay, and Coulomb scattering Es of electrons, as the sources
of lateral spread.
4) The detector of our concern is the emulsion chamber, which detects both the
showers of (e, y)'s and of hadronic origin. Its detection energy threshold for showers
is around 1 TeV.
5) Jr~ f1.+ J) decay is neglected because only the high energy particles are our
concern.
(b)
Inelasticity and recoil of leading particle
We assume the incident particle i(i=N and Jr) induces a collision of inelasticity
Ki and receives PTi by recoil. WE; treat the leading particle separately from those
produced through the multiple pion production, because it retains large fraction of
incident energy in average.
The distribution of the leading particle after collision is
¢;(Eo, E, (J)dEd(J=o(E-(l- K)Eo)dElj;;((J)d(J ,
(2·1)
1414
A. Ohsawa and S. Yamashita
where
¢;((J)d(J=
o( 8- Pi)
2Jr8
(2·2)
d(J .
According to the accelerator data, the recoil PTi of surviving nucleon or pion
seems to be comparable to that of pions produced through the multiple production,
and we assume PTN = PT7r= PT here as the first step.8)
We assume that inelasticity distribution is uniform between and 1.9 ) During the
calculation, we will take appropriate averages of inelasticity pertinent to the cases.
°
(c)
Multiple meson production
We assume that production spectra of pions produced by N-air and Jr-air collicollision. Possible difference appears in the
sions have the same form as that of
small x region and it will be discussed later. According to the accelerator data on
PP~ Jr+ X,IO) the production spectrum of charged pions in the multiple particle production can be factorized well into x and PT parts, i.e.,
PP
1 d 2 (J
(J
(2·3)
dxdPT
where x is the Feynman variable, defined to be x=E/Eo.
(see Fig. l(a))
¢(x) is given empirically by
¢(x)dx=A (1-x)4 dx .
x
(2.4)
On the other hand, since the transverse momentum distribution is independent of the
incident energy, we assume, for the sake of simplicity,
(2·5)
Changing the variables from x and PT to energy E and emission angle (J, we
obtain the production spectrum of pions
E)dE
¢(Eo, E, (J)dEd(J=¢ ( Eo
Eo ¢((J)d(J,
(2·3')
where ¢((J) has the same form as ¢i((J) of (2·2), except that PH is replaced by <PT>.
Hereafter we will use PT to express the average value, instead of <PT>.
Constant A in (2·4) is determined by the energy conservation:
2
TKEo=Eo
11°
x¢(x)dx,
giving
A=130 K.
Taking the simple average of the inelasticity, we get A=5/3.
The multiplicity obtained from the distribution (2·4) is given by
(2·6)
1415
Behaviour of Cosmic Rays in the Atmosphere
50
m(Eo) =
(a)
11
c/J(x )dx
<lEo
",(xl
= A[ln Eo
E
40
,.
':
,.
,:
,.
20
10
(d)
oL--L__~~-=~~~~
o
(2,7)
12'
where E is an adjustable parameter.
m(Eo) given by (2·7) with A=5/3 and
E=e-25/12(GeV) is shown in Fig. l(b). It
reproduces the energy dependence of multiplicity up to Eo=5 TeV .. In the higher
energy region multiplicity increases more
rapidly than In Eo and this increase is
attributed to the increase of particles at
small x region. The effect due to
enhancement at small x region will be
discussed in § IV.
,.
30
_liJ
0.1
0.2
x
0.3
0.4
0.5
__
0.6
= E/Eo
~
7[0-"
2y decay
We examine to which extent this process affects the lateral spread of (e, y)
component. y's emitted from 7[0 have the
(b)
30
'0
UJ
E
10
o ~..-L~~~~L-~~~~~__~~~~~~~~~~
!S(GeV)
10
10·
10'
10'
Eo (GeV)
5·10
5'10'
5'10·
5'107
PRIMARY ENERGY
Fig. l. (a) Production spectrum of pions (solid line) through multiple meson production, ¢(x)dx, in
terms of Feynman variable x=E/Eo.
Dotted line is ¢'(x) of (4'6) with xo=O.l, for the purpose to examine the lower limit of x,
effective to the propagation. Dashed line is (1'1) (with a=30 and Nn=lO), which is assumed by
Shibata et al. in their calculation.
(b) Primary energy dependence of the multiplicity (solid line). Dotted line is the best fit to
the accelerator data, <n c h)=O.88+0.44(ins)+O.1l8(ins)2. Dashed lines are the multiplicity·energy
relations corresponding to ¢'(x) of (4 ·6) with various values (attached to the line) of xo.
1416
A. Ohsawa and S. Yamashita
distribution in the rest frame of
20(E* - E*)dE* dQ*
47[
7[0
,
which transforms into the distribution in the laboratory frame
where r is the Lorentz factor of
of energy Eo is given by
¢r(Eo, E, ())dEd()
7[0.
Namely the production spectrum of y's from
1
7[Eow(Eo, E) o(e - w(Eo, E))dEd()
7[0
(2·8)
with w(Eo, E)=JmlNEoE-m//E02.
(e)
Isothermal atmosphere
We adopt the isothermal model of the atmosphere.
sphere at depth t is given by
The density of the atmo-
(Ho: scale height)
and the distance 1 between the depths tl and Mtl < t2 ) is
l=-Ho
ln~>
§ 3.
Formulation for various components
In this section we derive the three-dimensional (energy and spatial) distribution
of various components; nucleon, pion and (e, y). The differential distribution of
particles at depth t is denoted by
Fi(Eo, E, r, (), t)dEdrd() ,
U=N,7[ and (e, y))
(3·1)
where r is the position of the particles from the center on the horizontal plane at the
depth t and () the angle between the particle track and the vertical line (see Fig. 2).
(a) Nucleon component
The diffusion equation for nucleon component is
aFN -----at=
1 () .----ar--x;
aFN
1 F N (E 0, E ,r, ())
pet)
,t
+ }N jdE'd()'¢N(E', E,
()')FN(Eo, E', r, ()- ()', t).
(3·2)
On the right-hand side of the equation the first term represents the effect of lateral
displacement, the second and the third decrease and increase due to the collision.
¢N(Eo, E, ()) is given by (2·1).
1417
Behaviour of Cosmic Rays in the Atmosphere
By a Fourier transformation via
exp{iP· r+ iQ· (J), we obtain
aFN _ Hop. aFN
at
t
aQ
L,~.{
,
1
= -;:;;FN(Eo, E, P, Q, t)
1
+ AN(l- KN)<P N( Q)
L
~1,~\;-j7
~~~/
E
XFN(Eo, 1_ KN' P, Q, t).
(3·2')
In the equation t and Q can be treated as
variables, while E and P are to be left as
parameters. The variable transformation,
Fig. 2. Illustration of two dimensional vectors r
(position) and 0 (angle), which appear in the
·distribution function F(E, r, 0, t).
a~N = -
t = t'
and
Q = Q' - P H o lnt' ,
modifies Eq. (3·2') into
LFN(Eo, E, P, Q', t)
+ AN(l ~ KN/O( PEIQ' - PHolntl)FN( EO'l_EKN , P, Q', t),
(3·2")
where we have dropped the prime in r.
For the lateral distribution, what we need are the solutions of Eq. (3·2") for Q=O
and Q=- PHo In(t/T).5l Thus, it is sufficient to solve Eq. (3·2") putting Q=PHo
X In(t;/t), and then to set t;= t and t;= T, corresponding to the pair of required solutions.
The equation to be solved is
a:; = -
LFN(Eo, E, P, t;, t)
+ AN(l ~ KN/O( PTNfio
Since
e~l,
P
In
~)FN( Eo, 1-EKN' P, t;, t).
(3·2"')
we expand the Bessel function in Eq. (3·2"') into series and we get
We try the following form as the solution of Eq. (3·3):
A. Ohsawa and S. Yamashita
1418
= (
X ~1
P 2J[,2 p 2)n
T4£2
fN(S, n, ~, t) .
(3·4)
Equating the term (-PT 2H 02P2/4E 2)n on both sides of Eq. (3·3), we obtain
afN = -1+(1-KN)s+2n .r (
at
AN
J N S, n,
E
t)
c;,
(3·3')
Taking the average for the inelasticity, we get
at; =!1N(s+2n)fN(S,
n,~,
t)
+( ~; y «1- K;r+ 2n - 2) (In ~ YfN(s, n-1, ~, t),
(3·3")
where !1N(S) is the attenuation coefficient of a nucleon, given by
(3·5)
«l-KN)S) is the average value of (l-KN)S and equals l/(s+l).
condition is
Since the initial
FN(Eo, E, r, (), to)=o(E-Eo)o(r)o«() ,
(3·6)
we obtain
(3·6')
The solutions for Eq. (3·3") under the initial condition (3·6') are given by
(3·7)
and
fN(S, 1,
~, T)=( ~; Y Ao(s)ef.'N(S)(T-t o);l([!1N(S) - !1N(S + 2)]) ,
where Ao(s) and ;l(t,to, ~, a) are defined by
Ao(s)=
«1- KNY>
AN
and
;let,
to,~, a)=e-atLtdz
(In ~ye-az.
We write ;l(T, to, T, a) as ilea) for the sake of simplicity.
1419
Behaviour 01 Cosmic Rays "in the Atmosphere
(b)
Pion component
Diffusion equation for pion component is
+ P,,(E,
Here P,,(E, r,
(J,
(3·8)
r, (J, t) .
t) is the production spectrum of pions at depth t, and is given by
P,,(E, r, (J, t)=
1=
dE' d(J' ¢(E', E, (J')
x[L F,,(Eo, E', r, (J-(J', t)+ }N FN(Eo, E', r, (J-(J', OJ ,(3·9)
where ¢(Eo, E, (J) is given by (2·3'). The first and the second term in the brackets are
the contributions of pion and nucleon collision, respectively. As in Eq. (3·3), we try
the following form as the solution of Eq. (3·8):
(3·10)
after the same treatment as in the case of nucleon component.
Eq. (3·8)
al" _ (s+2n)I" (s,
----at-fl."
n,~,
Then, we get for
+ 2n) IN (s, n,~, t )
t ) + ¢(s AN
(3·11)
where fl." (s) is the attenuation coefficient of pions defined by
-1 +«1- K"Y>+¢(s)
A"
(3'12)
and ¢(s) is the Mellin transform of the production spectrum (2·4), i.e.,
¢(s)=
11
dx ¢(X)X S
•
(3 ·13)
As will be seen in (3·12), the attenuation coefficient of pions "is composed naturally of
the loss term due to collision and of the gain terms due to the leading pion and the
newly produced pions by interaction.
The initial condition for the solution of Eq. (3·8) is
F,,(Eo, E, r, (J, to)=O ,
l420
A. Ohsawa and S. Yamashita
corresponding to
so that the solution for n=O is
(3-14)
with
Bo(s)
This solution shows that the pions, produced by the nucleon at depth t', cfJ(s)/it N
x exp[,uN(S)(t' - to)], attenuates by exp[,u,,(s)( T - t')] down to the observation level T
(see § 4.6).
The solution for n=l, which gives the lateral spread, is divided into three parts:
(3 -15)
each of which corresponds to the lateral spread due to PT, PTN and PTrr, respectively.
Their explicit expression is omitted because the derivation is straightforward.
(c)
(e, y) component
Neutral pions produced at various depths give rise to (e, y) component through 7[0
~2y decay and subsequent electromagnetic cascade process. The y-ray, emitted at
the position (r', t') with the angle (J', transforms, at the observation level T, into an
(e, y) bundle whose center lies at r' + 1(J', 1 being the distance between t' and T; 1
= - Ho In(t'IT). That is,
F(e,r)(Eo, E, r, (J, T)= jdE'dt'dr'd(J'PlE', r', (J', t')
X(7[+y)(E',E, r-r'-l8', (J-(J', T-t'),
(3-16)
where (7[+ y) denotes the number of (e, y)'s at T, which originate from the primary
y-rays at t'.
P/E, r, (J, t) in (3-16) is the production spectrum of y-rays produced at depth t,
which is given by
PlE, r, (J, t)=
1'"
dE'd(J'cfJr(E',E,
(J')~P,,(E"
r, (J-(J', t),
(3-17)
where cfJ/Eo, E, (J) is given by (2-8). The factor 1/2 expresses that the number of
produced neutral pions is a half that of charged pions, given by (3-9).
Applying Fourier transformation to (3-16), (3-17) and (3-9), we get
F(e,r)(Eo, E, P, Q, t)=(27[)2jdE'dt'P/E', P, Q-PHoln
x(7[+y)(E', E, P, Q, T-t'),
P/E, P, Q, t)=7[ jdE'cfJ/E', E, Q)P,,(E', P, Q, t)
~, t')
(3 -16')
(3-17')
1421
Behaviour of Cosmic Rays in the Atmosphere
and
P,,(E, P, Q, t)=27[ fdE' ¢,,(E', E, Q)
x [LF,,(Eo, E', P, Q, t)+ }N FN(Eo, E', P, Q,
t)].
(3·9')
Since we have to put Q=O in (3·16') in obtaining the lateral distribution of (e, y)
component, we need the solution FN(Eo, E, P, Q, t) and F,,(Eo, E, P, Q, t) with Q
= - P H o In(t/T), which can be obtained by putting ;= T in !N(S, n, ;, t) and !,,(s, n,
;, t), respectively. Thus we get from (3·17')
(3·17")
where
_ ¢(s+2n) (
) (
t
gr (S, n, t ) - s+2n+1!h s, n, t + InT
)2 ¢(s+2n-2)
(
)
s+2n+1!h s, n-1, t
+(.!!'!:..!£)2(ln_t)2( ¢(s+2n-2)
PT
T
s+2n
¢(S+2n-2») (
) (
)
s+2n+ 1 !h S, n-1, t 3·18
and
(3 ·18')
"In (3·18) the third term represents the spread due to 7[°-->2y decay while the rest due
to hadronic transverse momenta PTN, PT" and PT.
The explicit formula for (7[+ y) is given in Appendix 1. The quantities we need
can be obtained by putting t = T and to= t' in (AI ·1) in the Appendix. That is
(7[+ y)(Eo, E, P, 0, T- t')
X ~o 00
(
r
1
(27[)227[i)
ds (Eo)S
E E1
Es2H02p2)m
,
4E2
h(s, m,;= T, T- t),
(3·19)
where
h(s, 0,;= T, T- t')=Ni(s)eAi*(S)(T-t')
and
h(s, 1,
;= T,
T - t')
=Ha (s+2)Hp (s)e AP ·(S)(T-t')Jl.(t, t', T, [Ap*(s)-;la*(s+2)]).
(3·19')
With the explicit expressions for p" and (7[+ y) given by (3·17") and (3·19), we obtai~
F(e,r/Eo, E, P, 0, T)
1422
A. Ohsawa and S. Yamashita
r (Eo)S
1
E E~
1
(2J[)22J[i Jds
x (;;
r
co (
loTdt'gr(s, n, t')h(s+2n, m,
~= T,
T- t'),
(3-20)
which can be rewritten as
(3 -20')
The solution for n=O is given by
!r.e,r)(S, 0, ~= T, T)
= loTdt'gr(S, 0, t')h(s, 0,
=
S!l
~=T, T-t')
{¢A~) N;(S)S(/-LN(S), Ai*(S»
+ ¢i:) Bo(s)Ni(s)[S(/-LN(S), A;*(S» - S(/-L7r(s), Ai*(S»]}
(3-21)
with
ea(T-to) _
Sea, b)
eb(T-to)
a-b
In (3 -21) the first term is due to the first generation, produced directly by the nucleon
and the second due to higher generations. l/(s+l) represents the effect of J[0--->2r
decay.
The solution for n=l is
!r.e,r)(S, 1, ~= T, T)
=ITdt'grCs, 1, t')h(s+2, 0,
to '
~= T, T- t')
(3- 22)
= f/l,Ws, 1, T) + !r.~~ri(s, 1, T) +!r.<f."')h, 1, T) .
(3 - 22')
Three terms express the lateral spread due to hadronic transverse momenta, J[°--->2r
decay and Coulomb scattering, and their explicit expressions are omitted because of
their complexity.
§ 4.
Discussion
We derived in the preceding section the formulae for the particle distribution
functions for various cosmic-ray components in the family at the observation level T,
under the initial condition of a single primary nucleon with energy
We shall
discuss in this section the effect which each assumed elementary process exerts on the
Eo.
Behaviour of Cosmic Rays in the Atmosphere
1423
quantities to be observed in emulsion chambers. The quantities of our main concern
in this article are the integral energy spectrum, 1(Eo, E, T), and the mea~ square
lateral spread, <E2r2>, defined by
1(Eo, E, T)=
1= 1=
dE
2JrrdrF(E , r, T)
and
Numerical calculation is made by the saddle point method and the constants necessary for it are tabulated in Table 1.
Strictly speaking, the energy of the hadronic shower, observed by the emulsion
chamber, is only a part of it, i.e., that part of the hadronic energy which has been
converted to (e, r) component: Eh(r)=krE h. However in this article we will always
refer to the whole energy of the hadron, Eh, and the effect of kr will be discussed
elsewhere in detail.
4.1.
Attenuation coefficients
Values of attenuation coefficients, given by (3· 5), (3 ·12) and (AI' 4) for nucleon,
pion and (e, r) components, respectively, are shown in Fig. 3. ,u,,(s)=/h*(s) holds at
s=1.04.
Generally the functions !i(S, n, t)'s, (i=N, Jr, or (e, r)), consist of the terms
proportional to eP-N(s+2m)t, eP-"(s+2m)t and e A1 *(s+2m)t with m=O, "', n, as can be seen in,
e.g., (3·21) and (3·22)'. Their relative magnitudes show that the dominant dependences are as
(4 ·1)
and
e A1 *(S)t
fi.e,r)(S, n, t)(X [
eP-~(S)t
,
(s~l)
(s > 1)
Table 1. Constants.
AN =80
A~
Xo
PTN
Pm
Pr
Es
mrr
=120
=37.1
=0.4
=0.4
=0.4
=0.021
=0.14
=7.0
Ho
T, to
(g/cm2 )
(GeV/c)
(GeV)
(GeV /c 2 )
(km)
mJ.p. of nucleon in the air
mJ.p. of pion in the air
radiation length of the air
Pr of recoil nucleon
Pr of recoil pion
PT of produced pions
scattering constant of electron
mass of pion
scale height
atmospheric depth in units of AN
1424
A. Ohsawa and S. Yamashita
\\
~
z
'""
~
a:
u:
::£
0
z
0
i=
«
:::J
z
UJ
~ ~~
-1
"'-.
lI-
«
-2
o
1l.(S)
}iN (5)
I'-----
2
3
i.l (5)
4
5
AGE PARAMETER 5
Fig. 3. Attenuation coefficients of various components in units of nucleon collision m.f.p. (AN). {.IN(S),
{.lrr(S) and Al*(S) are for nucleon, pion and electromagnetic components, and their explicit expressions are given by (3'5), (3'12) and (AI'4), respectively.
at sufficiently large t, irrespectively of n.
4.2.
Number and total energy of hadrons and (e, y) 's
Figures 4 and 5 show the depth variation of total number of particles I(E) and
total energy IE for hadrons and (e, y)'s, respectively. The total energy is expressed
as
IE
1 f ds (Eo)S f(s, 0, t).
E= 27ft s-l E
(4 ·2)
The relation given by (4'1) is shown in Fig. 4 for (e, y) component with s=1.80 and
E o/E=10 3 at T- to=12, showing that the dependence is correct.
At Eo/E-Hx), (4·2) for hadrons and (e, y)'s have the form
(IE)h =-l-f ds (Eo )SB ( ) I'rr(S)(T-to)
E
27fi s-l E
0 s e
and
(IE)ce,r) =-l-f ds (Eo )Sl[oo']eA1*CS)(T-tO)
E
27fi s-l E
E
'
(4' 2')
respectively (see (3'14) and (3·21) for exact formulae). When Eo/E-+=, the integrand has a simple pole at s=1.0, and the depth dependences reduce to el'rr(1)(T-t o) and
e A1 *Cl)CT-to), respectively. Because ,u,,(1) = -0.111 and Ih*(l)=O, (IE)ce,7)/E is independent of depth, while (IEh/E decreases as e-O.lCT-to).
4.3.
Energy spectrum
Figure 6 shows the energy spectrum of various components in the family, at the
observation level T = 2, 6 and 12, showing that the energy of the nucleon component
is not superior to those of the other components at any observation level (even at T
1425
Behaviour of Cosmic Rays zn the Atmosphere
10'
,
,,
,,
,
10'
,
,
,
,
,
....
,,
.,.-
-- ......
~~"'o.
...... - .......
--
W
A
---:::- 10'
(f)
0
"UJ
UJ
a:
a..
>(!)
a:
u.
z
«
0.2
UJ
o
a:
(e.rJ
W
-'
u
F
UJ
1.0
UJ
10'
10'
10'
-'
h
Eo =10'
E
UJ
0.1
1.0
«
I0
I-
ID
::<
:J
a
>
a:
z
UJ
UJ
(f)
ID
0
0.01
10
, .... - .....
I
10
2
--,
-,
10'
,,
(e.rJ
,
4
6
DEPTH
8
T-to
10
12
14
Fig. 4. Transition curve (depth variation) of
hadron (putting nucleons and pions together)
(solid line) and (e, r) components (dashed line).
Number of particles, l( > E), is the integral one
with particle energy exceeding E. The figures
attached to the curves are the primary energy
Eo/E. The thin solid line, near the curve of (e,
r) component of E o /E=10 3 , is eP.(S)(T-t o ) with s
= 1.80, which reproduces asymptotic dependence of (e, r) component correctly (see § 4.2).
2
6
4
DEPTH
8
10
T-to
Fig. 5. Transition curve (depth dependence) of
total energy, for hadron (putting nucleons and
pions together) and (e, r) component, shown by
the solid and dashed lines, respectively.
=2), due to the longer collision m.f.p. of pions. Also we can see that the difference
between the hadronic and (e, y) energy spectra is similar for different depths.
It is interesting to examine the case where the primary particle is a heavy nucleus,
for example, an iron nucleus (A=56). As we treat an iron nucleus as equivalent to
a bundle of A nucleons with monochromatic energy Eo/A, the curves are obtained by
translating those for a nucleon to the left and to the top by a factor A. The result
is summarized as follows. At T=2 (airplane altitude), most of the observed hadrons
with E/Eo> 10- 2 are nucleons, their spectrum being steep reflecting the monochromatic
primary energy at the top. However, at T=6 (high mountain altitude) we cannot
distinguish the nucleons from pions by their energy spectra.
We have taken into account nucleon-air and pion-air collisions as the sources of
A. Ohsawa and S. Yamashita
1426
. Table II. First generation pions, produced by the nucleon, to the total.
E/Eo
T-to=2
T-to=6
T- to=12
lO-1
100 %
75 %
45%
55 %
25 %
45%
20 %
10- 2
lO-3
pions. It is interesting to see which of
the two is more effective for pion production. Rough tendency will be seen from
Table II. Considering that the event of
our concern has the energy Eo=1016 eV
and that the detection threshold for
pions is E=10 13 eV, the energy region
giving satisfactory statistics of family
member particles is E/Eo~ 10- 3 . Table
II shows that the pions produced by
pions contribute considerably, owing to
their longer collision mJ.p. The situation is similar also for (e, y) component.
10'
U)
10
w
--'
u
1=
0::
-0:
CL
1.1.
o
0::
.lJj
:;:
~
z
4.4. Spectral indices of hadrons and
10-'
E
Eo
Fig. 6. Energy spectrum of various component in
the case of a single nucleon primary. Dotted,
solid and dashed lines are for nucleon, pion and
(e, y) component, respectively. The figures
attached to the curves are the thickness (T
- to) between the starting and observation
point.
(e, y)'s
Spectral index is the fundamental
parameter in discussing the energy
spectrum of particles, and related to the
extent of shower development. It is
given by the position of the saddle point,
i.e.,
Eo 1
a j(s
. 0 T)]
In-=---[ln
E
s as
' ,
(4 ·3)
for the component concerned.
Figure 7 shows the indices of hadrons (putting nucleons and pions together) and
(e, y)'s at various observation levels. S(e,r) > Sh holds because (e, y)'s are at the
downstream of hadrons, the equality holding at Eo/E-'>oo.
The discussion in § 4.1 implies that both the energy spectrum I( > E) and mean
square lateral spread <E2r2> have the same saddle point, defined by
In ~
=
~ - fJ.'(s)( T - to) ,
(4 '4)
when Eo/E-'> 00. In (4'4), fJ.(s) stands for fJ.N(S), fJ.,/s) or ,h*(s), depending on the
component concerned. As the mathematical complication 'reduces significantly at
Eo/E -'> 00, it is interesting to see to what extent the condition of Eo/E -'> 00 realizes in
the actual case Eo/E=102~ 106 • Figure 8 shows the spectral index S plotted against
1427
Behaviour of Cosmic Rays in the Atmosphere
1.5
x
w
0
~
...J
-......
1.0
T-to =
12
12
- ...... 6
«
0::
IU
"''''
w
[L
(/)
-------0.5
6
2
2
10'
10'
10'
ENERGY
10·
10'
Eo/E
Fig. 7. Indices of energy spectrum vs primary energy at T - to=2, 6 and 12, for hadronic (putting
nucleons and pions together) (solid line) and (e, y) (dashed line) component.
1.5
0::
w
IW
~
«
0::
«
[L
lli
«
0.5
10'
10'
10'
10'
10·
Eo
E
Fig. 8. Spectral index vs primary energy at the depth T - to= 2, 6 and 12, for energy spectrum (solid
line) and <E2r2> (dashed line). Indices for <E2r2> are not the function of only the thickness T - to
but of depths T and to independently, and all the curves for the combinations of (T, to) to give the
same T - to lie in the hatched area.
the incident energy defined by (4·3), for I( > E) and <E2r2) in the case of pion
component. There exists small difference
which causes the energy dependence of the form C:X(Eo/E)L1S to <E2r2), as shown in
1428
A. Ohsawa and S. Yamashita
l~
....q
.......................................... .
,,"~~C=-=~
200
'00
~
E
r-.~,~ .................~ ............................................ .
50
:>"
'"
t;
4
N
0
~le.
"to.
"
10- 2
30
"
~
20
NL..
'ttl
V
"
",.0
10-' c-________~--------~~~'-'~.~2--~~~----~ 10
5
3
Eo/E
Fig. 9. Primary energy (or threshold energy) vs mean square lateral spread, <E 2 r 2 ), of the particles
in the family, at the observation level T=6. Dotted, solid and dashed lines are for nucleon, pion
and (e, y) components, respectively. The figures attached to the curves are the starting point to
of the primary particle. Scale on the right-hand side is the one by assuming PT=O.4(GeV!c) and
Ho=7.0(km).
Fig. 9.
4.5.
Mean square lateral spread, <E2r2)
We here discuss the quantity, <E2r2), as the one to express the lateral spread,
expecting rough validity of the relation, E· r=constant.
Figure 9 shows the threshold energy (or primary energy) dependence of <E2r2)
for each component at the observation level T=6. <E2r2)'s vary slowly with the
energy as discussed in the previous subsection 4.4. <E2r2) for the nucleon and the
pion components are quite similar, showing again that we cannot distinguish nucleon
and pion component by the lateral spread.
It is interesting to see to what extent each process giving rise to the lateral
spread, affects <E2r2) of various components. Figures 10(a) and (b) show the relative contribution of eachprocess in (3 ·15) and (3·22'), respectively. In Fig. 10(a), in
the case of pion component, contribution of PT is 90% and those of leading particle's
recoils are 10% at most. We should, however, keep in mind that PTN=Pnr=PT is
assumed in the calculation. If PTi has larger value than PT, it gives rise to a factor
(PT;/PT)2 to the process concerned. From Fig. 10(b) one sees that the contribution of
PT (including PTi) is dominant (90 %) for <E2r2), in the case of (e, r) component,
compared with those of mrr and Es. In the region s< 0.5 the spread due to Es becomes
larger than that due to PT, which is caused by the fact that the e.m. cascade development governs the behaviour dominantly in this region, as discussed in § 4.1 (see !r.e,7)
1429
Behaviour of Cosmic Rays in the Atmosphere
- ........ - - - - - - - . - - - - - - - - - .................. PT
--- - - -- ----- ...
~PT"
10-'
10~
.....
PTN
~
l-
mIt
II
~
ui
~
10--
, .
'-
~
'~""'
":
10"
.
.
10-'
.
:.
:.
10-4 L...l.--'-----'-----'---'---'--'-----'-----'---'---'--'-----'-----'---'---'--'----'
o
0.5
1.0
1.5
AGE PARAMETER 5
(a)
AGE PARAMETER
5
(b)
Fig. 10. Relative weight of each process for lateral spread in terms of <E2r2> at the observation level
of T=6 (see Eqs. (3'15) and (3'22')). Recoil of the leading particle (PTN, PT.), transverse
momentum of produced pions (PT), 7[0->2r decay (m.), and multiple scattering (E 8 ) are taken into
account for the sources of lateral spread. Figures (a) and (b) are for pion and (e, r) component,
respectively. The solid, dashed and dotted lines represent the starting points of to=O, 2 and 4,
respectively.
in (4'1)).
4.6.
Equilibrium between production and attenuation
Generally speaking, the particles produced at high altitude dissipate their
energies during their traverse, while those produced near the chamber cannot have
high energy. Thus we can expect that there exists the optimum height, the particles
produced at which contribute most effectively to the family at the observation level.
We will see the situation in the case of pion component. The production
spectrum of pions at the depth t' is given by (3·9). The produced particles attenuate
by the factor exp[,lL7r(s)(T-t')] and the total number of pions observed at Tis
T
F7r(E, T)=l dt' P7r (E, t')eP.,,(S)(T-t') .
(4·5)
to
As P7r (E, t')cx:. ep."(S)t' at sufficiently high energy, the integra:nd is independent of t', i.e.,
all the values of t' contribute equally to F7r(E, T). In other words the equilibrium is
realized between the production and the attenuation of pions. It may be related to
the efficient production of high energy pions, characteristic of the scaling-type of
production spectrum.
1430
A. Ohsawa and S. Yamashita
4.7. Lower limit of x effective to the propagation
We say that only the high energy part of the production spectrum is effective to
the propagation of cosmic rays in the atmosphere, because of the strong energy
dissipation during their propagation. To see the details of the above statement we
examine here the effect of the lower limit of ¢(x) for the hadrons which significantly
affect the propagation, by introducing another form of x-distribution:
¢'(x) = ¢(x)[2- B(x-xo)] ,
(4 ·6)
where B(x) is the step function. This amounts to retaining the original distribution
for x>xo, and doubling it for x<xo, as shown in Fig.l(a). We examine the increase
of 17/Eo, E, T, xo) with Xo.
We can neglect the breakdown of energy conservation by this modification, if we
restrict Xo< 0.1.
Figure 11 shows the ratio of 1n(xo), calculated by the (3'14) for new production
spectrum ¢'(x), to 1n(xo=0) for the original production spectrum ¢(x).
Defining the "effective xo" to be the point where the ratio increases by' 10%, the
effective values of Xo are shown in Table III for various Eo/E and T - to. It shows
that, since xo(min) = E/Eo, considerably small values of x are still effective to the
propagation. This situation is in contrast to the case when we discuss the
:L
~"0~
.
,:./
_1
_____ - 1 - __' - ' - - - - - - ' - - - - L v .
oII
o
x
"'6
:&
o
1=
«
0::
Fig. 11. Lower limit of pion production spectrum, effective to the propagation, at the thickness of
T-t o=2(solid line), 6(dashed line) and 12(dotted line). New production spectrum ¢'(x) of (4'6),
which is twice ¢(x) (see (2'4» at x<xo, is introduced, and Irr(>E)'s, the number of pions in the
family, are compared for both production spectra.
Table III.
Lower limit of x effective to the propagation.
T-t o
Eo/E=102
Eo/E=104
Eo/E=106
2
6
12
1.5 X 10~2
1 x 1O~"
4 X 10~3
9x 1O~:l
< 10-
3 X 10~2
4 x 10~2
4
5xlO"
2 x 10~"
Behaviour of Cosmic Rays in the Atmosphere
1431
"uncorrelated" pion intensity without imposing the condition that they belong to the
same family. Because in this case s is equal to ~2.0 reflecting the spectral index of
primary nucleon (s < 1 for family phenomena), the Xo effective to the propagation
becomes very large, as can be seen in Mellin transform of (4·6),
4_ 8+1+_
6 _. x 8+2+_
4 _ 8+3+_
1 _ 8+4J
¢'(S)=¢(s)+A[lx 8+_
s 0 s+ 1 x 0
s+2 0
s+3 x 0
s+4 x 0
(4·6')
•
And we can see also from Table III that only the high energy part becomes
important when the observation level lies deep.
Detailed discussion on this subject will be given elsewhere in relation to different
types of production spectrum.
Acknowledgements
We thank the member of Brasil-Japan Emulsion Chamber Collaboration for their
valuable discussions at many occasions and also Dr. T. Shibata for his careful reading
of the munuscript and enlightening comments.
Appendix I
- - Cascade Function for Emulsion Chamber Experiment-By the emulsion chamber experiment we observe showers of an electron, a
positron and a photon as (e, y) component without distinction. Then the cascade
function, (7[+ 1') of the r-primary is given by
(7[+r)(Eo, E, P, Q=O, t)
1
( ( EO)8 1
(27[)227[i Jds If E
(AI ·1)
where
h(s, 0,
~=
t, t)=Ni(s)e,,*(8)(t-t o)
(AI ·2)
and
(AI ·3)
Subscripts a, (3 and i denote to make the summation of 1 and 2.
His) are given in Table IV. And
Ni(s), HaCs+2) and
(AI ·4)
where Xo is the radiation length of the air.
References
1)
c. M. G. Lattes et a!. (Brasil-Japan Emulsion Chamber Collaboration), Prog. Theor. Phys. Supp!.
No. 47 (1971), 1.
C. M. G. Lattes, Y. Fujimoto and S. Hasegawa, Phys. Rep. 65 (1980), 151.
1432
A. Ohsawa and S. Yamashita
Table IV.
Notations in the cascade function.
i, a,!3
1
2
N;(s)
H 2(s) + H4(S)
H,(s)-H.(s)
Ha(s +2)
H,(s+2)+H3(S+2)
H 2 (s +2)- H,(s +2)
HP(S)
His)
-H.(s)
H;(s)
O"n+A,(S)
At{s)-Ms)
Hids)
C(s)
A,(s)-Ms)
(A,(S), A2(S), B(s), C(s) and
0"0
O"o+Ms)
At{s)-Ms)
B(s)
A,(s)-Ms)
are familiar ones in the cascade theory.)
]. A. Chinellato et al. (Brasil-Japan Emulsion Chamber Collaboration), Prog. Theor. Phys. Suppl.
No. 76 (1983), l.
2) S. Hayakawa,]. Nishimura and Y. Yamamoto, Prog. Theor. Phys. Suppl. No. 32 (1964),104.
Y. Pal and B. Peters, Danske Videnskab Selskab, Mat.-Fis. Medd., Vol. 33 (1964).
A. Ohsawa, Prog. Theor. Phys. Suppl. No. 47 (1971), 180.
H. Oda and S. Dake, Prog. Theor. Phys. 53 (1975), 516.
K. Kasahara and Y. Takahashi, Prog. Theor. Phys. 55 (1976), 1896.
K. Kasahara, Nuovo Cim. Serie 11 46A (1978), 333; AlP Conf. Proc. No. 49 (1979), 162.
Y. Takahashi, AlP Conf.· Proc. No. 49 (1979), 166.
3) E. Konishi, T. Shibata, E. H. Shibuya and N. Tateyama, Prog. Theor. Phys. 56 (1976), 1845.
E. Konishi, T. Shibata and N. Tateyama, Prog. Theor. Phys. 57 (1977), 142, 44l.
T. Shibata, Prog. Theor. Phys. 57 (1977), 882, 1605, 1950; CRL-Report-53-77-12, Inst. Cosmic Ray
Res., Univ. of Tokyo (1977).
4) K. Kamata and]. Nishimura, Prog. Theor. Phys. Suppl. No.6 (1958), 93.
]. Nishimura, Handbuch der Physik, Vol. XLVI/2 (1967), l.
5) S. Miyake, Prog. Theor. Phys. 20 (1958), 844.
6) S. Yamashita, A. Ohsawa and J. A. Chinellato, Proc. Intern. Sympo. on Cosmic Rays and Particle
Physics. Tokyo (Inst. Cosmic Ray Res., Univ. of Tokyo, 1984), 30.
S. Yamashita, A. Ohsawa, ]. A. Chinellato and E. H. Shibuya, ibid. 46.
A. S. Borisov et al. (Pamir Collaboration), ibid. 3.
M. Amenomori et al. (Mt. Fuji Collaboration), ibid. 76.
]. R. Ren et al. (China-Japan Collaboration), ibid. 87.
]. R. Ren et al. (China-Japan Collaboration), Proc. Intern. Cosmic Ray Coni (San Diego), VOl. 6
(1985), 317.
N. M. Amato, N. Arata and R. H. C. Maldonado, ibid. 324.
E. H. Shibuya, ibid. ·332.
]. R. Ren et al. (China-Japan Collaboration, Mt. Fuji Collaboration), ibid. 336.
A. S. Borisov et al. (Pamir Collaboration), ibid. 340, 344.
M. Amenomori et al. (Mt. Fuji Collaboration), ibid. 348.
S. Yamashita, A. Ohsawa and]. A. Chinellato, ibid. 364.
S. Yamashita, A. Ohsawa, ]. A. Chinellato and E. H. Shibuya, ibid. 368.
7) C. M. G. Lattes, Y. Fujimoto and S. Hasegawa, Phys. Rep. 65 (1980), 15l.
]. Bellandi F. et al. (Brasil-Japan Emulsion Chamber Collaboration), AlP Coni Proc. No. 85
(1982), 133, 317.
]. A. Chinellato et al. (Brasil-Japan Emulsion Chamber Collaboration), Proc. 18th Intern. Cosmic
Ray Coni (Bangalore), Vol. 11 (1983), 77.
]. A. Chinellato et al. (Brasil-Japan Emulsion Chamber Collaboration), Proc. 19th Intern. Cosmic
Ray Coni (San Diego), Vol. 6 (1985), 250, 356, 360; Vol. 8 (1985), 310.
8) M. G. Albrow, D. D. Barber, A. Bogerts, B. Bosnjakovic, ]. R. Brooks, A. B. Clegg, F. C. Erne, C.
N. P. Gee, A. D. Kanaris, D. H. Locke, F. K. Loebinger, P. G. Murphy, A. Rudge, ]. C. Sens, K.
Terwilliger and F. van der Veen, Nucl. Phys. B51 (1973), 388.
]. Whitmore, S. ]. Barish, D. C. Colley and P. F. Schultz, Phys. Rev. Dll (1975), 3124.
]. Whitmore, B. Y. Oh, M. Pratap, G. Sionakides, G. A. Smith, V. E. Barnes, D. D. Carmony, R. S.
Christian, A. F. Garfinkel, W. M. Morse, L. K. Rangan, L. Voyvodic, R. Walker, E. W. Anderson,
H. B. Crawley, A. Firestone, W. ]. Kernan, D. L. Pari5:er, R. G. Glasser, D. G. Hill, M. Kazuno, G.
McClellan, H. L. Price, B. Sechi-Zorn, G. A. Snow, F. Svreck, A. R. Erwin, E. H. Harvey, R. ].
Behaviour of Cosmic Rays in the Atmosphere
1433
Loveless and M. A. Thompson, Phys. Rev. DI6 (1977), 3137.
9)
J. Whitmore, S. J. Barish, D. C. Colley and P. F. Schultz, Phys. Rev. DB (1975), 3124.
M. G. Albrow, A. Bagchus, D. P. Barber, A. Bogaerts, B. Bosnjakovic, J. R. Brooks, A. B. Clegg,
F. C. Erne, C. N. P. Gee, D. H. Locke, F. K. Loebinger, P. G. Murphy, A. Rudge, J. G. Sens and F.
10)
J. R. Johnson, R. Kammerud, T. Ohsugi, D. J. Ritchie, R. Shafer, D. Theriot, J. K. Walker and F.
van der Veen, Nuc!. Phys. B54 (1973), 6.
E. Taylor, Phys. Rev. D 17 (1978), 1292.