Chin.Astron.Astrophys.(l990)14/3,248-255 0 Pergamon Press plc A translation of Printed in Great Britain ActaAstrophys.Sin.(1990110/2,106-113 0275-1062/90$10.00+.00

DIFFUSE COSMIC GAMMA RAYS IN DOUBLE

LEAKY BOX MODEL

ZHOU Da-ehuang Department of Earth and Space Science, University of Science and Technology of China

Received 1987 June 8

ABSTRACT Diffuse cosmic gamma rays are produced in the propagation of primary cosmic rays. Using the "double leaky box" model for the propagation a spectrum of the diffuse gamma rays is calculated in close agreement with the observed spectrum.

Key words: Cosmic ray astrophysics-diffuse gamma rays

1. INTRODUCTION

Cosmic gamma ray is an important topic in cosmic ray astrophysics, it is very useful for understanding the origin and propagation of cosmic rays and the composition of the interstellar gas.

Present theories of the propagation of cosmic ray can be roughly divided into two kinds: the "single leaky box" model and the "double (or nested) leaky box" model. In the single leaky box model, the Milky way is regarded as one big box inside which cosmic rays are generated, accelerated, propagated and partially leaked out. In the double leaky box model, sources of cosmic rays (ener- getic active bodies such as supernovae and pulsars) are individual small boxes (Box I), scattered inside the large box (Box II) that is the Milky Way. Cosmic rays are generated in the sources, leaked from them into the interstellar space of the Milky Way; they then propagate further and partially leak out of the Milky Way. In comp- arison, the double leaky box model takes a more comprehensive view of the actual physical processes and hence is better than the single leaky box model in dealing with certain problems. For example, in order to fit the observed spectrum the single leaky box model always overestimates the galactic hydrogen content, much higher than the generally accepted values [1,21. In this paper I shall use the double leaky box model to calculate the spectrum of the diffuse galactic gamma ray and compare the result to the observations.

2. OBSERVED SPECTRUM OF COSMIC GAMMA RAY

The spectra of diffuse cosmic gamma ray observed by the COS-B and SAS-2 experiments are marked with various symbols in Fig.1. The

Double Leaky Box 249

continuous curves are calculated results based on the single leaky box model [3]. The galactic hydrogen content used in [3] is a little lower than the estimate by Gordon and Burton [4]. We see that for medium to low energies, the calculated values are close to the observed values, but at high energies (& 70MeV), the calcul- ations fall below the observations.

Fig. 1 The observed spectra of the diffuse gamma ray and the calculated spectra according to the single leaky box model

3. THE GENERATING MECHANISM

High energy diffuse cosmic ray comes from the decay of the nuclear reaction product no meson and bremsstrahlung and inverse Compton scattering of electrons. As cosmic ray propagates inside the Galaxy, its nuclear components will interact with the interstellar diffuse matter (interstellar H, He, etc.), these nuclear reactions will produce so oesons, which will then decay into gamma photons. We may write this process briefly as

nuclear reaction + w” + 2y

and this is the main source of the higher energy (2 mnoc2/2 91 67.5MeV) gamma rays.

250 ZHOU Da-ehuang

At the same time, electrons in the cosmic ray will react electro- magnetically with interstellar diffuse matter (hydrogen, starlight, infrared light, the 2.7R background photons etc.) and through bremsstrahlung and inverse Compton scattering, produce gamma photons:

e- t nuclear Coulomb field + bremsstrahlung y photons, e- t low energy photon + inverse Compton y photons.

these two processes are the main source of low energy (s 67.5 MeV) gamma rays.

4. CALCULATION OF THE DIFFUSE COSMIC GAMMA RAY

We use the natural unit system, c = h = 1.

1. The nuclear reaction + so + 2y Process 151

Nuclear reactions between cosmic ray and interstellar matter are mainly proton-proton collisions. From studies of high energy nuclear reactions we know that the differential cross section of the production of no mesons in such a process is

where PI = P,, sine is the transverse momentum of the meson, 0 is the angle of emergence, emax = sin-l(P*nn&Pn), P*WIIZJX bein% the maximum momentum in the barycentric system, and En(d3u/dPn ) is the covariant cross section of the so meson generating momentum Pn,

E. $:‘- - A1(E,)(I - ?)~~xp[--EPL/(I + 4mi/S)] I

where

J

___-

Q - (C, + C,Pl + C&)/J1 + 4&/s ; F - x:l’ + $ (P: + m:) ;

I’S is total reaction energy in the centre of mass system, x*11 = P*XX/PY*XIllX, PxIIx is the longitudinal momentum of the meson in the centre of mass system, and

f(E,) - (1 + 23.!5’.‘) (I - %.‘)I;

the constants are A=140, 8~5.43, C1~6.1, Cz=-3.3, C3=0.6, mp=0.938GeV, mn=0.135GeV.

From the differential cross section we obtain the generated spectrum of so mesons,

Q(L) - 4mJzp_,_ i,(E,)nr,[da(E.,E,)/dE.ldE,

where ns is the number density of diffuse hydrogen in the box,

Double Leaky Box 251

jp(Ep) is the cosmic ray proton spectrum, Epnrns 1.22 GeV/nucleon, and n = 1.6 is the factor that converts the contribution from the proton-proton reaction to that from all reactions including those involving the heavier elements.

The spectrum of the gamma ray generated is then

The corresponding differential gamma flux is

where L is the linear size of the box along the line of sight.

2. The Bremsstrahlung Process [61

The differential cross section of an electron with initial and final energies Et and Et and a photon with energy El is

where a=1/137.0.36 is the fine structure constant, re = 2.82 x 10-13cm is the classical radius of the electron, El = Et-Er, and, for unscreened nuclear charge ZO,

PI - ‘pa - Z’Y’. , 1p. - 4 ih (2) :$I, m - 0511MeV,

while for atoms in general,

‘pi - 4 I fi(q,Ei,E,.E,)~,~,(q)dq

where q = KI -Kr-k, Kg, Kr being the initial and final momenta of the electron and k is the momentum of the photon. The lower limit of integration is km/PEfEf. Further expressions of the screen factor i& and ft can be found in [61.

Using do = o(El)dEl, and writing Et = E, we have

f, - 2E’ - 2EE, + E: E,E’

and o(E7) = u(E,)JE, - ur:(f,lp, - f&dEv

For the light elements (II, He), the generating spectrum of gamma rays is

Q(E,) - 4s ,ze ni [J, L(E)u(E,)dE ] ,

and the differential flux is

252 ZHOU Da-zhuang

For the heavier elements, these are

Q(E,) - 4~ - 4lx c Zi(Zi + 1)&i #>I

ICE,) - 4ur:L c z,(z; + l)fli i.(EXfw, - /GddE .>1 J : I In the above expressions for the differential flux, je(E) is the cosmic ray electron spectrum in the box, L is the linear scale of the box along the line of sight and nt is the number density of atom i. The relative abundances of the main elements in interstellar medium are shown in TABLE 1.

TABLE 1 Relative Abundances of Interstellar Elements

ELEMENT H tie c N 1 o _.~~_ __ ___ _~~.. _..__. ~_---.~-

ABUNDANCE I .” 0.12 2.5x 1” ’ 1.3x10 ’ 7.9x 1” ’ .~ ~~-~~~ _ .--- -.

ELEMENT NC M8 hi s PC

ABUNDANCE- 6.)x7- - 4.5x IO-’ 2.5x 10.’ Z.BXIO-’ 2.5X,” ’

3. The Inverse Compton Process 161

In this process, the differential cross section for a photon with energy Er is

Here E is the initial energy of the electron, E = &sTu 3 eV (energy of starlight photon) or (infrared photon).

~<z??r<4E 0 4’ m

m is its rest mass, ers III 0.516 eV

The generating spectrum of the inverse Compton gamma ray is then

Q(Er) - 4% J;, i.(E)dE j: cr(~,,e,~)n,*(e)de - Inn,*8

5 = L o(E,,E)j,(E)lfE .q E

where nph . c = &T, the energy density of interstellar photons, which is about the same as that of infrared photons, DsT W DIR = 0.5 eV/cm3, and je(/?) is the SpSCtrUpl Of COSIIIiC ray electrons in the box. The differential flux is

KE,) - L

Double Leaky Box 253

We add together the cosmic gamma rays produced by the three processes. Obviously, apart from the cross sections, the following factors enter into the calculation: a. The cosmic ray proton and electron spectra in the two boxes. b. The densities of diffuse matter in the two boxes, obviously, the density in Box I should behigher than that in Box II. c. The siees of the boxes, obvious- ly, the siee of Box I is much smaller than that of Box II.

4. Cosmic Bay Spectrum in the Two-Box Model

For simplicity we assume the cosmic nay to be uniform, that is, its intensity is independent of position and direction. Further we neglect any actual differences in the cosmic ray sources and assume them to have more or less the same cosmic ray intensities.

Box II is represented by the interstellar proton and electron spectra as measured near the Earth. The high energy end is not subject to solar modulation and so is close to the interstellar value. The low energy end is affected by solar modulation, it departs greatly from the interstellar spectrum and must be corrected.

Fig.2 Proton and electron spectra in the double

leaky box model

The proton and electron spectra of Box I have to be found by simulation calculation on the double leaky box model. The simulation must reproduce the various cosmic ray observations, including the observed diffuse gamma ray spectrum. The spectra used in our simulation are shown in Fig. 2; they were based on the calculations by K. K. Tang of Chicago University.

In the simple leaky box model, in order to explain the observed gamma ray spectrum, it had to use an excessively high interstellar density as well as a very steep gradient at the low energy end of the electron spectrum 131. Moreover, different authors used greatly different gradients, ranging from E-‘.l’ to EW2*‘, which differed greatly from the observed value. Compared to the single leaky box model, the spectra in the double leaky box model are much lower at the low energy end.

Interstellar hydrogen abundance was obtained by Cordon and Burton in 1976, but according to Refs. [1,2,7] their estimate was too

254 ZHOU Da-zhuang

Fig. 3 The calculated diffuse gamma ray spectra in Box I

10-t

; 10-S i ,0-d

a 10-s

P 10-6

2 y 10-7

f 10-E

Q 10-g

lo"0

; 0.01

F “O.Wl $.

e ‘O“ n IO-’

lo- -i f to- P

z lo- IO-

lo-'

10-1

IO“

10-j 00

Fig. 4 The calculated gamma ray spectra in Box II Fig. 5 The observed and calculated (double leak) gamB8 ray spectra

Double Leaky Box 255

high, and if we use their value then the calculated diffuse gamma ray will be higher than the observed. Accordingly, the value we used in our calculation was about one-half of their value.

Box I is made of various cosmic ray sources, their sizes and concentrations of diffuse matter vary. Also we do not know just how many sources there are along a given line of sight. Hence it is difficult to quantify the diffuse matter in Box I. In our calcul- ation we took the sum of all the diffuse matter so that, after combining with the proton and electron spectra of Box I and the other quantities of the model, it will reproduce the observed diffuse gamma ray. This choice meant that the amount of hydrogen between the Earth and the galactic centre in Box I is about one order of magnitude lower than in Box II. The precise amount of diffuse photons in Box I may not be important because the contr- ibution of Box I by inverse Compton to the gamma ray is far less than that by bremsstrahlung.

5. RESULTS OF THE DOUBLE LEAKY BOX MODEL

The calculated spectra of the diffuse cosmic gamma ray for the double leaky box model are graphed in Figs. 3-5.

Figs. 3 and 4 show that when E, 5 lOOMeV, the contribution from Box I is dominant and bremsstrahlung far exceeds inverse Compton and that when EvLlOOMeV, the gamma ray arising from the decay of x0 mesons is the dominant one.

In Fig. 5, the calculated spectrum (sum of all three processes in the double leaky box, continuous curve) is compared with the observed data points. The agreement is obviously good. Especially in the range El170MeV, the double leaky box model gives much better fit to the observations than does the single leaky box model (cf. Fig. 1).

We conclude that the double leaky box model can reproduce the observed diffuse cosmic gamma ray spectrum. Incidentally, this wor confirms that the abundance of galactic hydrogen should be lower than the Gordon estimate and that low energy end of the interstel- lar electron spectrum should be lower than the usual theoretical estimate.

ACKNOWLEDGEMENT I thank Colleague LI Ti-pei and Dr K. K. Tang of Chicago University for their sustained support and help.

REFERENCES

[ I] Li, Tipei et al., 1. Plzyr. G: Nucl. I’hyl., 7(lYSl), L157.

[ 2 ] Li, Tipei et al.. J. I’Lyr. G‘: Nucl. f’hys., S(lYS2), 1141.

( 3 ] Sacher. W. CL al., Spocc Sci. Rmienv, 36( 1983), 249.

( 4 ] Cordon, M. A. et al., Arrrophyr. J, 208( 1976). 348.

[ 5 ] Stephens, S. A. et al., Arrrophyr. ~purc Sci, 76(1981), 213.

( 6 ] Blumenthal, G. B. ct al.. Rruwwr of Modern Phyr., 42(1970). 237.

[ 7 ] Blitz, L. ct al., Arrruphyr. I., 238(1980), 148.