Chin.Astron.Astrophys.(l991115/4,416-422 % Pergamon Press plc A translation of Printed in Great Britain Acta Astrophys.Sin.(1991)11/3,220-226 02751062/91$10.00+.00
COSMIC RAY AND HYDROGEN DISTRIBUTIONS AND
COSMIC GAMMA - RAYS1
ZHOU Da-ehuang Department of Earth and Space Sciences, University of Science and Technology of China, Hefei
K. K. Tang Laboratory for Astrophysics and Space Research, Enrico Fermi Institute, University of Chicago
ABSTRACT We calculate the spectrum of the diffuse cosmic gamma ray in the single and double leaky box models for several galactic distribution laws of cosmic rays and hdydrogen. The results show that LI Ti-pei's distribution law for the cosmic rays is the best and that the number of interstellar hydrogen molecules should be less than Gordon's value divided by 1.7. The observed spectrum of gamma rays can be reproduced by a suitable choice of the galactic distributions within certain ranges.
Key words: Cosmic ray-Interstellar hydrogen---Gamma ray
For the diffuse gamma rays generated when high-energy cosmic rays interact with interstellar matter in their passage in the galaxy, both the single leaky box model [l-3] and the double leaky box model  were able to give a calculated spectrum close to the observed spectrum. However, in the references [l], 121 and , the nucleid and electron components of the cosmic ray are both assuoed to be distributed uniformly in the galaxy while in the reference 131, only the electron component is assumed to have a non-uniform distribution. The question whether the cosmic ray distribution is uniform or not has not been decided; researches in several directions seem to indicate that the galactic distribution should not be uniform, rather, it should vary from place to place [5-101. In this paper, we shall use the disribution laws given in these papers to calculate anew the gamma ray spectrum for the single and double leaky box models.
2. GAMMA-RAY SPECTRUM FOR A GIVEN GALACTIC DISTRIBUTION OF COSMIC RAY
As the nucleid and electron components of the cosmic ray propagate in the galactic space, the nucleid component interacts with the hydrogen and helium etc. of the diffuse matter to generate nuclear reactions, the reaction product x0 meson will decay into gamma photons; the electron component interacts electromagnetically with 1 Received 1990 June 8; passed by the Referee: 1990 November 27.
Cosmic Ray Distributions 4!7
hydrogen, interstellar radiation, infrared radiation and the cosmic background radiation and produces, through bremmsstrahlung and inverse Compton scattering, gamma photons.
In the uniform distribution model of the cosmic ray, the rate of generation of gamma ray varies only with the matter concentration at the various localities. In the non-uniform model, the rate depends not only on the concentration, but also on the variation in the cosmic ray intensity. Let the differential gamma ray rates the two components be q(R) and qe(R), R being the galactocentric radius. Then the differential spectrum of the gamma ray is
J(E,) - & JJJ 5 tq.(Er,W + dE,,R)lnH(R,Z - 0)
Here, dQ is the observed solid angle, r is the heliocentric distance of the volume element dv, R is the galactocentric distance in the galactic plane, & = lOkpc, 2 is the distance above the galactic plane, m(R,Z=O) is the number density of interstellar hydrogen in the galactic plane, I(R) is the cosmic ray intensity, and a gaussian distribution with height is assumed for the interstellar hydrogen, with scale height ZH = 50 pc of Hz, and for HI,
ZH = 120 pc (R < 10 kpc) q 120 pc t 50(R- 10) pc (R > 10 kpc).
3. COSMIC RAY DISTRIBUTIONS
The cosmic ray distributions given in Refs. [3,5-71 belong to the same type, namely, for R< 5 kpc, the intensity increases with R, reaching a maximum at R- 5 kpc and for R> 5 kpc, the intensity decreases with R. For R< 10 kpc,  gives the smallest electron amplitude, and  the greatest; for R> 10 kpc,  gives the smallest amplitude. For the SN and PS origin of cosmic ray models the distribution given in  is roughly the same as in , their most remarkable feature being that, for the outer parts of the galaxy, the amplitude is small and varies rapidly. Detail can be seen in Fig. 4 of  relating to the CDS-B data, for n= 1 and n= 2, (the meaning of n will be given below), which we shall denote as Li-I and Li-II. The actual distribution chosen should have higher values of p, a parameter related to the outward diffusion of the cosmic ray from the galaxy.
The distribution laws given in  and  are of the negative exponential type. For [a], it is
I(R)/I(Bg) = exp(1.25 - R/8), (R < 10 kpc) = exp(1 - R/10), (R > 10 kpc);
For , it is
418 ZHOU Da-zhuangand K. K.Tang
I(R)/I(RO) = exp Is(R-&Ill
where for the electron component, -0.21 < s< -0.09 and for the nucleid component, -0.055 < SC tO.015.
For the electron distribution of 191, its lower bound for R< 10 kpc is close to that in [81, while its upper bound exceeds greatly other authors' values: for R> 10 kpc, its upper bound is close to that of [81, and its lower bound is close to that of .
For the nucleid distribution of , the lower bound for the R< 10 kpc region and the upper bound for the R> 10 kpc region are both approximately equal to a uniform distribution and close to the value at R= 10 kpc; the upper bound for R< 10 kpc is lower than all other authors, and the lower bound for R> 10 kpc is close to that of 181 and slowly decreases with R.
The main cosmic ray distributions are graphed in Fig. 1. For the Bloemen distribution, the limiting values are shown.
A comparison between the calculated and observed spectrum of the gamma ray in the directions towards and away from the galactic centre will indicate which of the distributions is the best.
A Sacher W A:Stecker
B L, I 2- B:Li II
c Bloemcn -A C:Bbxmen
-6 5 - 2
0 2 4 6 8 IO 12 I4 16 0 2 4 6 8 0 12 I4 IS
Fig. 1 Galactic distribution of cosmic ray electrons (a) and protons (b)
4. DISTRIBUTION OF INTERSTELLAR HYDROGEN
The galactic distribution of interstellar hydrogen comes from Gordon's work 1111. Studies [3, 6, 12, 131 show Gordon's atomic hydrogen distribution to be correct, but his molecular hydrogen density was overestimated. According to [61, Gordon's value for the molecular hydrogen should be divided by 2(M/&)", where M is the O/H ratio at R and ho, that at R= 10 kpc, and n should be between 1 and 2. We consider separately R= 1 and n= 2. These cases will be referred to as Type I and Type II respectively. For the values of AI/&, used above, see Fig. 1 of [61.
Cosmic Ray Distributions 419
5. OTHER CONSIDERATIONS
The r-rays from bremsstrahlung and x0 are dominant and the contribution by inverse Compton process is small (I 5X), hence the variation caused by the cosmic electron distribution is also small. For simplicity, we neglect the Compton component.
For the rates from bremsstrahlung and x0 decay in the single leaky box model, we we the theoretical values of Sacher [Z] and Stephens (141.
For the double leaky box model, the cosmic ray and diffuse matter in Box I (source region of cosmic ray) are still regarded as uniformly distributed, while in Box II (galactic space outside the source), their distributions are taken to be the same as in the single leaky box model. According to , in this model, in the Earth-galactic centre direction, Bremsstrahlung is dominant in Box I and no decay is dominant in Box II. Hence in this direction we need only calculate the x0 decay component when considering the distribution of cosmic nucleids in Box II, the y-rate being taken from (41. The difference in the y-rate between (41 and (141 comes mainly from the use of different cosmic proton spectra.
Fig. 2 The y spectrum in the single leaky box model in the galactic centre direction (a) and in the anticentre direction (b). Solid line is from bremsstrahlung (LiI
CR-distribution, Type II H-distribution). Dotted line is from YC decay (LiI CR, Type I H)
420 ZHOU Da-rhuang and K. K. Tang
6. RESULTS OF CALCULATION
The main results of our calculation are 8s follows:
In the single leaky box model, in the galactic centre direction, agreement between the calculated y-ray spectrum and the observed spectrum is found 1) for the Sacher distribution of cosmic ray and the hydrogen distribution with n equal to either 1 or 2 and 2) for the cosmic ray distributions of Stecker, Stephens and Bloemen (Fig. 1) and the hydrogen distribution with n= 2. In the anticentre direction, however, the calculated spectra using the above cosmic ray distributions turned out be higher than CM-B data, about twice 8s much; the Sacher bremsstrahlung spectrum was the closest to the observed spectrum. These results show that, for the region R > 10 kpc, the amplitudes of the cosmic nucleid distributions of all the above authors are somewhat too high, thus leading to too high calculated y spectra.
In the single leaky box model, for Lis distribution of cosmic ray (for both n= 1 and Z), the calculated y-spectrum agrees with the observed spectrum in both the centre and anticentre directions. This shows that Lis distribution of cosmic ray is more reasonable than the other authors, particularly in the anticentre direction
f f h Anti-Center L
Fig. 3 The y-spectrum in the double leaky box model in the galactic centre direction (8) and in the anticentre direction (b) . Solid line is from bremsstrehlung
(approximate uniform CR distribution in Box I). Dotted line is from so decay (LiII CR- distribution, Type I H-distribution).
Cosmic Ray Distributions 421
where it does not have the difficulty encountered in the other cases, the resulting differential y-spectrun agrees very well with the COS-B and 050-7 data.
As representative of the single leaky box model, the calculated spectrum using Lis cosmic ray distribution is shown in Fig. 2.
Since Lis distribution proved to be the best in the single leaky box model, we use it again for the double leaky box model. The result is shown in Fig. 3.
Our calculation also showed that, in the anticentre direction, for both the single and double leaky box models, because atomic hydrogen contribution is dominant, the two types of hydrogen distribution gave very close y-spectra, the difference being only * 3%.
7. DISCUSSION AND CONCLUSIONS
One point in our calculated results is worth noting. In the anticentre direction only Lis distribution of cosmic ray succeeds in explaining the CM-B data, while all other distribution5 give too high values. This shows that these other distributions have too high amplitudes for the outer parts of the Galaxy, and also, too slow variations. Among these, Bloemens ha5 the largest amplitude and slowest variation. In [9-101, Bloemen repeatedly discussed the question of the distribution of cosmic nucleids in the outer parts of the Galaxy. In  he rejected models with lower amplitudes and faster variation5 and affirmed those with higher amplitude5 and slower variations; bowever, in [lo], he only gave the calculated spectrum in the galacitc centre direction and not in the anticentre direction. Thus, the distribution in the outer parts of the Galaxy cannot be considered as settled. Our calculations here favour the low-amplitude, fast-varying model of Li, in preference to the high-amplitude, slow-varying model of Bloemen.
From our calculation and analysis of the diffuse y-rays when the distribution of cosmic rays is taken into account, we arrive at the following conclusions:
1) Of the cosmic ray distributions given by the several authors, that by Li is the most reasonable, for both in the directions of galactic centre and anticentre, and for whichever of the two distributions of hydrogen we take, the y-spectrum calculated with Lis cosmic ray distribution always agrees with observed epectrum. Of the other distributions, Sachers distribution of cosmic electrons is basically reasonable; for the region outside the Sun, the distributions of cosmic nucleids of Sacher, Stecker, Stephens and Bloemen are all too high, hence fail to account for the differential y-spectrum in COS-B.
2) In regard to distribution of galactic hydrogen, Gordons distribution of atomic hydrogen is reliable and can be used as is; but his distribution of molecular hydrogen is too high and should be revised downwards. From our calculations, it appears that
422 ZHOU Da-zhumg and K. K. Tang
Cordons value for the interstellar hydrogen should be divided by a factor between %M/& and Z(M/&,), in agreement with the conclusion of 16). In the anticentre direction , atomic hydrogen is dominant and molecular hydrogen is unimportant.
3) By either matching a high cosmic ray distribution with 8 low hydrogen distribution or a low cosmic ray distribution with a high hydrogen distribution, we can aluays arrive at a reasonable spectrum for the diffuse r-ray. Therefore, when the hydrogen distribution is not very certain, the distribution of cosmic ray is not precisely determined and can vary within certain extents. The present work is merely an assessBent of the 8V8ilable cosmic ray distributions with the observed distribution of the diffuse y-ray.
ACKNOWLEDGEMENT We thank Colleague LI Ti-pei for suggestions and support of the present work.
[II [21 [31 141
[51 [61 r71
1111 llBl [I31 1141
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