ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR THE INTENSITY OF AIR-SHOWER CORES RELATIVE TO AIR SHOWERS BY W. E. HAZEN Associate Professor of Physics Project M784 OFFICE OF NAVAL RESEARCH, U. S. NAVY DEPARTMENT CONTRACT N8onr-55800, ONR PROJECT NR021-064 September 24, 1951

THE INTENSITY OF AIR-SHOWER CORES RELATIVE TO AIR SHOWERS W. E. Hazen, Randall Laboratory of Physics University of Michigan, Ann Arbor, Michigan The following discussion of the probability that an air-shower event in a detector of finite area be due to the passage of the shower axis through the detector supersedes the discussion in IV (a) and V (a) of the report "Air-Shower Cores" of August, 1951. If we let a be the detector area and R the radius of the circle within which a shower axis must strike in order to display n electrons within the detector area, we have r100 00 P = j af(N)dN i rR2f(N)d1 Nm Nm for the probability that an event with n or more electrons within a is due to a shower axis within a. The quantity f(N)dN is the experimentally determined frequency of occurrence of shower events with a total number of electrons N at the point of observation.l,2,3 It has been found that f(N)dN = KN dN, where V= 2.5 for N<106 and 2.9 for N>106. In setting up the denominator in the expression for P, it was assumed that the detector area is small (:or'circular). Actually, the region within which a given-size shower must strike is not circular for a rectangular detector; but a more exact 1 R. W. Williams, Phys. Rev. 74, 1689 (1948). 2 J. M. Blatt, Phys. Rev. J5, 1584 (1949). 3 J. Ise and W. B. Fretter, Phys. Rev. 76, 933 (1949).

treatment shows that the area of the region is not very different from the area of the circle. Strictly speaking, N and R are related through an integral of the lateral-distribution function over the area of the detector. However a sufficiently good approximation is obtained by assuming that the average particle density within the detector is the same as the particle density at the center of the chamber; whereupon n 0.454 N(1+4R) exp-42/3, a R using the analytical expression of Bethe. 1 For the case of a shower axis within the detector area, which must be considered in establishing the value for Nm, the above approximations are less valid. A graphical solution gives an average value of n/N = 1/200 for the case of axes passing through a rectangular area 25 by 80 cm. In the experiment, n was chosen as 100. The calculated value of P for these particular conditions is found to be 0.1, which agrees with the cloud-chamber observation. The agreement may be taken as additional evidence favoring the assumed shower-size distribution and the assumed lateral distribution. The assumed lateral distribution is probably too steep near the origin for the smaller showers;2 a correction for this would lower the calculated value for P. On the other hand, the shower cores are probably multiple, which would result in a higher calculated value for P.