Interpretation of High-Energy Stars Observed in Cosmic Rays

High-energy stars appearing in photographic plates exposed to cosmic rays have been named “strue,” because they consist chiefly of relativistic particles concentrated in a cone with a very small angle at the vertex. If one assumes that such a star has arisen as the result of a nucleon–nucleon collision, then from the mean angle of emission of the particles \(\Phi\) one can determine the energy of the primary particle \((E)\) according to the formula: \(E = \dfrac{2Mc^2}{\Phi^2}\) \((M\)—the mass of the nucleon). However

there are almost no data on the interaction of energetic nucleons \((E > 10^{11}\ \mathrm{eV})\) with hydrogen; in most experimental works collisions with heavy nuclei were observed: Ag, Br, or Cu\(^{1,2}\). The interpretation of the stars obtained in this case depends essentially on what path the particle takes in the nucleus, i.e., on the impact parameter of the nucleon on the nucleus. In the case of a grazing impact with a parameter close to the radius of the nucleus, one may interpret the star as a nucleon-nucleon collision and determine its energy from the mean angle of emission (in the general case, however, this estimate of the energy is correct only as to order of magnitude). The criterion that a peripheral collision has occurred was for a long time considered to be the almost complete absence of traces of heavy particles in the star. However, by the present time experimental material has already accumulated which testifies to the limited character of such a criterion. Such, for example, is Teicher’s “star”\(^{3}\). In this star no heavy particles were found, but interpreting it as a nucleon-nucleon collision leads to a number of difficulties, which disappear if it is assumed that this star occurred as the result of a collision of a nucleon with a nucleus. The question therefore arises: when is such an interpretation possible? For this reason Heitler’s work\(^{4}\), devoted to the question of how many heavy particles can be formed in the collision of a nucleon of very high energy \((E > 10^{11}\ \mathrm{eV})\) with a heavy nucleus, is very timely.

As one of the initial assumptions in the work it is taken that the incoming nucleon does not react with the whole nucleus, but only with those particles of the nucleus which lie in its path, i.e., the nucleon cuts out a tube in the nucleus. This assumption seems quite natural and has already been developed both in the domestic (see review\(^{5}\)) and in the foreign\(^{6}\) literature. It is based on the fact that at sufficiently high energy the primary particle and all secondary particles after collision move within a cone whose angular aperture in the rest system of the nucleus is very small. In the case where the primary particle has very high energy, the tube cut out may be regarded as cylindrical. In the general case, upon entry into the nucleus the cross section of the tube is equal to \(\pi r^2\) (where \(r\) is the mean distance between nucleons in the nucleus), and upon exit from the nucleus it is somewhat larger, so that the whole tube has the form of a horn. The criterion of applicability of the tube model consists in the requirement that the angular divergence of the individual particles upon leaving the nucleus, even in a central collision, should be less than \(r\) (in other words, the form of the tube should differ little from cylindrical). This gives

\[ \overline{\Phi^2} \ll \left(\frac{r}{d}\right)^2, \]

where \(d\) is the diameter of the nucleus.

For such nuclei as Ag or Cu,

\[ \left(\frac{r}{d}\right)^2 \simeq 3 \cdot 10^2. \]

If the energy is estimated by the formula

\[ E \sim \frac{2Mc^2}{\overline{\Phi^2}}, \]

then the energy criterion for applicability of the tube model will be

\[ E \gg 2 \frac{Mc^2 d^2}{r^2}. \]

Thus, in the energy region of interest to us this model is applicable.

All particles, both newly produced ones and recoil nucleons making up the tube, after the interaction acquire extremely relativistic energies. Thus slow particles (giving gray and black tracks) can fly out only from the remaining part of the nucleus (i.e., they cannot be direct participants in the collision). The transfer of energy to the nuclear remnant can occur in two ways:

1) Each of the nucleons directly adjacent to the tube (the number of such nucleons is \(\sim 4\dfrac{d}{r}\)), owing to the fact that its neighbor suddenly disappears, receives a momentum of order \(\dfrac{V}{c}\), where \(V\) is the mean interaction energy of nucleons in the nucleus, equal to \(30\) MeV. The total energy transferred to the nucleus owing to this kind of “friction” will be of order \(U_1 \simeq 4\dfrac{d}{r}\dfrac{V^2}{2Mc^2}\sim 14\) MeV, i.e., negligibly small.

2) A nucleus in the middle of which a tunnel has been cut out possesses excess surface energy in comparison with a spherical nucleus having the same number of nucleons. This surface energy gives the main contribution to the excitation energy; in the case of a central collision with a silver nucleus it is equal to \(U_2=105\) MeV. The total energy may be taken, in order of magnitude, as \(U\sim 80\div 150\) MeV. With an increase of the impact parameter this quantity decreases. The number of charged particles “evaporating” from the nucleus at such an excitation, according to (7), is

\[ N_h = 2.5\cdot 10^{-2}(U_{\text{MeV}}-0.7)\sim 2\div 3 \text{ particles}. \]

It is natural to expect that, with such a small mean number of evaporating particles, the fluctuations may be very considerable.

The principal cause producing the fluctuations, according to the authors’ estimates, is the fluctuations in the distribution of charge between the evaporating particles and the particles remaining in the nucleus. The root-mean-square deviation of \(N_h\) from the value 3 is \(\Delta N_h\simeq 1.7\).

Thus, the small number of tracks of heavy particles by no means indicates that a grazing collision has occurred. Therefore, to determine the impact parameter of a nucleon with a nucleus it is necessary to bring in other experimental and theoretical data (for example, as was done in \(^{5}\)).

In addition, Heitler’s paper contains calculations of the number of shower particles on the basis of the concept of multiple and multiple-multiple production. The calculations are illustrative in character, since the author does not specify any concrete theory, but uses a very general scheme with arbitrary coefficients. It is impossible to obtain definite indications of the advantage of one or another theory on the basis of these calculations.

D. Ch.

References

1. Daniel, Davis, Mulwey and Perkins, Phil. Mag. 43, 753 (1953).
2. Kaplon and Ritson, Phys. Rev. 85, 900 (1952); 88, 386 (1952).
3. Teucher, Naturwiss. 39, 68 (1952).
4. Heitler and Terraux, Proc. Phys. Soc. A 66, 929 (1953).
5. I. A. Rozental and D. S. Chernavskii, UFN 52, 185 (1954).
6. Roesler and McCusker, Nuovo Cim. 10, 127 (1953).
7. Le Conteux, Proc. Phys. Soc. A 63, 259 (1950); A 65, 718 (1952).