The Durham air shower array - III: Evaluation of the array's response to showers

NUCLEAR INSTRUMENTS

AND METHODS

148 ( 1 9 7 8 )

5 2 1 - 5 2 9 ; (~) N O R T H - H O L L A N D

P U B L I S H I N G CO.

T H E D U R H A M AIR S H O W E R A R R A Y - III:

EVALUATION OF THE ARRAY'S RESPONSE TO SHOWERS ALAN C. SMITH

Department of Physics, Universityof Durham, Durham, England Received 4 August 1977 A Monte Carlo investigation of the Durham air shower array is described in which the systematic biases introduced into air shower data recorded by the experiment are evaluated.

1. Introduction

2. The array acceptance

This paper, which is the third in a series of three papers L2) describing the Durham array and how it operates, is primarily concerned with a study of the systematic effects and the accuracy obtainable in calculating extensive air shower parameters using the data obtained from the array. From the reported calculations one can identify and subsequently eliminate those biases which are present in this experiment. Calculations of this nature are a necessary feature in the running of an air shower experiment: they are such an integral part of the study that the results would be rendered incomplete without them. The Monte Carlo technique of experiment simulation, which has been used here, is widely employed to determine the response of experiments to physical phenomena. The success of this approach lies within the random sampling procedures for establishing values of selected parameters and this is particularly valuable when the number of free parameters is large or when analytical functions are difficult to solve. Another great virtue of this technique is that one can counterfeit processes that occur in Nature quite simply and readily in a computer. Modelling algorithms based on this technique are simple to construct and are used for a variety of reasons: 1) To assist the experimenter in selecting data with minimal bias to produce results that truly represent the phenomena observed. 2) To calculate any biases that the data acquisition and analysis may introduce and then to remove them from the analysed data. 3) To determine the accuracy of the data and of the experiment. 4) To assist the experimenter in designing better experiments.

One of the first quantities to be calculated that describes an array's response to air showers is its acceptance. This quantity describes the probability of detecting an air shower event with, for example, a particular size and arrival direction and its magnitude depends upon the array triggering requirement and array geometry. By calculating the acceptance for a variety of situations the most appropriate shower trigger may be deduced which will enhance the shower property that one wishes to study. Knowing also the response of the experiment to the feature of interest it then becomes a simple task to remove the geometrical factor from the data to retrieve the physics. To calculate the array acceptance one needs to know those parameters which will influence the detection of an event. In particular these are the detector coordinates, their areas, their efficiencies and the triggering requirement. This can be varied to find the most suitable condition that gives the response that is required. The acceptance represents a region of square metre steradians over which the array may be considered as being totally efficient in detecting showers and is naturally shower size dependent. The particle densities at each of the triggering detectors that would have occurred for showers of a particular size, core location and zenith angle are calculated according to some structure function, from which the Poissonian detection probabilities are found. These probabilities are then multiplied together to give the array triggering probability for recording a shower with the same size, core location and arrival direction. By integrating over a pre-defined core location area and arrival directions and averaging the detection probability in this region the mean shower detection probability is thus found.

522

A.C. SMITH

Proceeding with this method over all of the array a contour map of the detection probability is constructed. This map is of great use for it is this that shows the core locations of the showers which are most likely to be detected. The contours are dependent upon the size of the shower used to produce them and consequently to gain a full picture of an array's response many such calculations need to be done.

2.1. CALCULATION OF THE AVERAGE TRIGGERING PROBABILITY

Calculation of the average triggering probability is largely based upon the assumption that the number of cosmic ray particles that are detected in an instrument follow Poissonian statistics and if this is so then the detection probability, Pi, may be given by

pi(/liSi, ~>m) = e -Alsl

S(Ai i)~ tl!

'

(1)

dent on the detector and m is the particle threshold, then eq. (1) becomes

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> ~ m ) = e -'ltsi ~, (zl~li)n/~i(n,/'r/) .

(2)

n=O

The function ei is a function of the detecting elements and has to be determined empirically. It can be measured easily for one particle going through the detector. The function for more particles needs to be deduced from that for the single particle. A useful modification to eq. (2) is to rewrite it such that it is not necessary to sum over an infinite number of terms but only k terms where k is the number of particles at which ~i becomes unity for the value of m in question and is:

pi(Aisi, >~m)= 1--e -a's' ~. (A~y n! [1-.,(., m)] . n=O

(3)

For a 100% efficient detector e i is a step function with e i = 0 for n < m and ~ = 1 for n>~m. where A~ is the particle density on a detector of In calculating A~ a lateral structure function has area s~ whose particle threshold for detection is m. to be assumed and the most appropriate one is (m can be set by an electronic device in the laborthat due to Greisen3). This function has held its atory.) ground for many years despite repeated attempts This is true only for detectors with no sampling to better it and it is widely used for shower work errors, but in general the detecting elements tend whose charged particle sizes lie between 103 and to have efficiencies that depend both on the num108 . The use of this function also allows one to ber of particles incident on them and on their parinclude shower age effects which could have a ticle threshold. If the efficiency is defined as e~(n, m), where n is the number of particles inci- bearing on some experiments. For specific case of the Durham array (see fig. 1) the normalised acceptance has been calculated using eq. (3) from which the average p~ for many showers of equal size, that fall randomly over a specified area, whose arrival directions have been modulated by an appropriate zenith angle distribution and whose structure function is that given in table 1, was found. The normalised acceptance is shown in fig. 2 for showers being accepted within a 50 m radius and a 20 m radius of the central detector. Here the zenith angle is less than or equal to 30° and the triggering requirements are eY m those given in the caption. Curves for perfectly efficient detectors and the same acceptance and trigger criteria are also included in fig. 2. This figure Scale 0 2Sm 53 E~ shows which showers are detected with a large probability and thus whose intensities need smallFig. 1. The Durham extensive air shower array. [] Muon specer correction factors to obtain the shower size intrograph; [] hadron chamber; [] density and fast-timing detecput spectrum from the observed shower size spector of area 0.75 m 2 ; [] density and fast-timing detector of area trum. For applications where the array's accept2.0 m 2 ; [] density detector of area 1.6 m 2 ; [] density detector ance need not be known a size cut at 2× 105 is of area ].0 m 2 ; C) density detector of area 0.26 m 2 (liquid scintillator), imposed, essentially because of inaccuracies in

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3. Individual shower simulation Systematic effects introducing biases or inaccuracies into the eventually analysed data can be detected by means of simulating the experiment in a computer. For the simulation studies of the air shower and experiment the theoretical model must counterfeit the real situation as closely as possible and using this model to generate faked events the usual analysis programme must be invoked to analyse them. This gives one the input and output parameters Which can then be compared to determine any shifts in them due either to the experimental technique or to the analysis procedures. The simulation programme used in the present work comprise three distinct parts: the shower simulation, the array simulation and the analysis programme and by examining each one in turn their fez_tures will be made apparent.

////

////

ARRAY

106

Fig. 2. T h e normalised array acceptance (detection probability) for showers falling within 30 ° of the zenith and R m m of the central detector for real (R) and perfect (P) detectors and for two trigger criteria. Perfect detectors have a step-like efficiency function (see text). (a) P, R,. = 20 m ; (b) P, R,. = 50 m ; (c) R, Rm=20m; (d) R, R m = 5 0 m ; (e) P, R m = 2 0 m ; (f) R, R m = 20 m ; (g) P, R m = 50 m ; (h) R, R m = 50 m.

shower parameter values for smaller showers. Thus, knowing the size range over which the array can be successfully used one can study the finer details of the experiment's response by simulating individual showers and observing how accurately they are interpreted.

3.1. THE SHOWER There are many random processes at work in the shower as it develops through the atmosphere but the feature that is most important in this kind of simulation [as opposed to, for example, the shower simulation work of Dixon et al.4)] are the statistical fluctuations of particle numbers at each observation station. Other, perhaps more intuitive, variables are the shower size, core position, arrival direction (modulated in 8) and the shower age. (The zenith angle and shower age distributions of air showers at sea level used in these calculations have been estimated using, as a guide, the results of the observations of the authors of refs. 5-7.) These quantities represent the major parameters of the observed air shower and they are usually the parameters that are of interest to other physicists when they need to have data relevant to a particular shower event. However, they are by no means the only parameters that need to be introduced to the simulation programmes. There are the energy spectra of the shower particles as a function of radius and the effects of several particle species in the shower. Each species will have different response characteristics in different detectors. Detectors with a non-zero energy threshold give erroneous particle density measurements and this eventually results in the observation of a steeper lateral distribution and an incorrect estimation of the shower size. 3.2. THE EXPERIMENT

By "the experiment" is meant the method of

A.C. SMITH

524

detection, data storage and data calibration. In this section of the simulation programme there is the need to take into account the detector response to particles, linearity of the laboratory electronics, pulse shapes, detector saturation effects, coincidence device resolving time and the propagation delays in cables and electronic devices. Although this is a long list it is by no means complete; there are other effects peculiar to each experiment. Some effects will make no significant contribution to the overall response but they should, nevertheless, be included in any simulation programme. It

should be remarked, however, that simulations generally take large amounts of computer time. This is because many variables have to be chosen from random distributions. Consequently, the programmes must be efficient or else insufficient statistics will be accumulated. 3.3. THE ANALYSIS For the simulations to have any real meaning the faked data must be analysed with the analysis programmes that would be used to analyse the real data; this is to see if any biases exist in the

TABLE 1 Features of the shower simulation programme. (a) The shower Size distribution

Range of core impact Zenith angle dependence Azimuth angle dependence Shower age distribution Radius of curvature of the shower front Lateral distribution of shower particles Arrival Particle Shower Energy

time fluctuations number fluctuations particle type distribution of shower particles

Random shower size between 3 x 104 and 108 modulated by a differential input spectrum with 7D = - 2 . 5 for N < 7 x l 0 5 and ~'D = - 3 . 0 for N > 7 × I 0 s Random over a 200 m × 2 0 0 m area with the array (radius 50 m) at the centre. Random zenith angle modulated by a cos l° 0 m -2 s -1 sr -1 distribution. Random. Random age modulated by the distribution 16 ( S - 0 . 7 2 ) 2 ( 1 . 7 2 - S ) 2, (see text) Infinite. 0.4 N r~ r'------L--I 1+ A(N'r'S)=--S-\r/rl \r+rl/ \ 11.4rl (r I = 79 m). Gaussian fluctuations with a = _ + 5 ns Poissonian. Single species ("vertical equivalent calibration particle"). Energies > threshold of detector ( - 10 MeV).

(~S-0.5(

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Y)

(b) The experiment Operational detectors Trigger criteria Detector saturation Analogue laboratory electronics Detector sampling error

Data digitisation

Only the 14 plastic scintillator detectors have been included. Detectors C(A~>4m - 2 ) plus 13, 33 and 53 (all A~>2 m -2) whose density signals exceed the coincidence unit discriminator thresholds. Density measurements up to 80 m -2 are used. Their non-linearity (where applicable) is included. Fluctuations included, generated randomly and modulated by a distribution of the same mean ( = particle number) and standard deviation as experimentally measured. The same resolution of the data (in terms of channel number) as in the experiment.

(c) The Analysis Programme

The same programme that is used to analyse real shower data (see ref. 2) and the same acceptance criteria as those applied to real data are implemented.

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Fig. 3. Scatter plots showing the shower arrival direction accuracies for fast-timing measurement standard deviation of _+ 5 nS (a) and (b) and _+ 10 ns (c) and (d). • represents a shower with a zenith angle less than 10 °. The number to its side is the point's ordinate value.

analysis procedures. Consequently, the shower and experiment simulation programmes must be written with this in mind. The Durham array's data analysis has been described by Smith and Thompson 2) and will only be outlined here. Briefly, after the observed densities and times have been calibrated into useful quantities a least squares minimisation is made on the timing information (a plane is fitted to the data) and then, using the arrival direction thus calculated a weighted least squares minimisation is made on the density data by fitting N, Xcore and Ycore simultaneously. After the analysis about 52% of the simulated showers that triggered the array are found to be outside the array's acceptance (radial distance cut of ~<50 m, zenith angle cut of ~<30°) and thus it

becomes necessary to impose the same restrictions on the faked data as are on the real data. If this is done then meaningful comparisons between the data can be made. For the purpose of the simulations there are included in these calculations as many parameters as are thought justified for the purpose. Table 1 shows those features and assumptions of the shower and experiment which are included amongst the simulation routines. Taking these together with the analysis programme the array biases, resolution of parameters and so on may be investigated. Going one stage further, simulations of this kind can help considerably in the design of new experiments. To simulate 50 showers, according to the prescription of table 1 that satisfy the array coinci-

526

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dence discriminators~), about 5000 showers must be generated in the region of the array. This small success rate is because of the many smaller showers (3 × 104-105) that do not trigger the experiment but which must be generated from an input spectrum if the real situation is to be counterfeited as closely as possible. By simulating many showers like this, conclusions about the quality of the array and its analysis may be drawn.

4. Systematic effects and the accuracy of analysis 4.1. THE ARRIVALDIRECTIONEVALUATION For the measurement of the arrival direction accuracy only three timing detectors have been used (detectors C, 13, 33 in fig. 1 since these detectors, and detector number 53, are used, at present, to trigger the array) and also because three time marks, which give the equivalent of two relative times, represents the least amount of timing information necessary to calculate a shower's arrival direction. Consequently, the data to be quoted will thus represent the worst possible accuracy that is obtainable with this array. The angular accuracy will be considerably improved, therefore, when more timing detectors are included in the angular calculation. Fig. 3a--d show the results from a number of simulated showers in which the timing measurements have standard deviations of + 5 ns and _ 10 ns. Table 2 condenses this information into a more readable form. The errors quoted represent one standard deviation.

Only showers whose zenith angle are analysed to be less than or equal to 30° are used in evaluating shower properties. This is chosen because the proximity of a building near to some of the detectors would automatically impose some anisotropy onto the shower arrival directions and render those showers that were detected with large zenith angles having a reduced number of particles in those detectors directly behind the building, resulting in erroneous shower sizes and core locations. Summarising table 2 for showers whose zenith angle is less than 30° it is evident that the angular resolution is good and entirely adequate for shower work. 4.2. CORE LOCATIONEVALUATION Core location accuracy may be conveniently determined by measuring the difference of distances between the simulated input and output core locations and some fixed point in the array, for example the central detector, for many showers. The mean of the distribution thus calculated is the error in radial distance measurement from this fixed point and is shown in fig. 4 as a function of shower size. Here the ordinate is the difference between the radial distance between the central detector and the input core location (before analysis) and the radial distance between the central detector and the output core location (after analysis); both quantities being measured along the shower front.

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Fig. 4. This figure shows the difference between the simulated input and output core locations and the central detector for showers simulated according to table 1 (also, see text) and for which whose showers were analysed as being within 50 m of the central detector and with a zenith angle of less than, or equal to, 30°. The number beneath each point refers to the number of simulated showers that have been used in calculating that point.

527

THE DURHAM AIR SHOWER ARRAY - II1 TABLE 2

Shower arrival direction zenith and azimuthal angle accuracies for fast-timing measurements with standard deviations of +_5 ns and -4-10 ns.

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The number beneath each point refers to the number of simulated showers that have gone into producing that point. The error bars represent standard errors. The data in fig. 4 have been produced in two parts and hence the discontinuity, in the number of showers simulated, at N = 5 × 105. The showers were simulated according to table 1 but with lower size cuts of 3×104 and 3× 105 . Both data sets were cut at N = 5× 105 and displayed as shown. Core location accuracy can also be measured by calculating the mean length of the vector between the input and output core locations. It can then be calculated what fraction of the data lie within a specified region of the input core location and this is a useful measure of the absolute displacement.

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4.3. SHOWERSlZE EVALUATION By selecting those showers that have been analysed to be within a certain size range and then examining the spectrum of the input showers that, when analysed, give the specified size output, a measure of the bias in shower size introduced by the experiment and analysis procedures can b e calculated. Again simulating showers in the same way as before, that is according to table 1, and then selecting for example, those showers for three size bins and whose zenith angles are less than, or equal to, 30° and whose cores a r e no further than 50 m from the central detector fig. 5 has been produced. It is shown that the average input a n d output shower sizes are not the same and for shower work where the size is important,

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528

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where n represents the number of showers of size N and the errors are one standard deviation. This effect is thought to be mainly due to the minimising package MINUIT 2,9) in present use in the analysis programme and because the analysis of the simulated and genuine data is performed with a different structure function to that used simulating the showers2'8). Another useful relationship that can be obtained from results of this kind is the acceptance function (see fig. 2) deduced by calculating the ratio of the simulated output spectrum (after analysis) to the simulated input spectrum before the shower acceptance test (that is, the shower trigger condition) is applied.

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this correction must be applied. Fig. 6 shows how the mean input shower size varies with the output shower size for showers inside the array acceptance criteria. The equation that best describes this curve

is:

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5. Summary The air shower experiment has undergone several modifications since it began collecting data in October 1975, as the Shower Group's understanding of the experiment and shower analyses have developed. The simulations discussed in this paper represent the present status of the air shower experiment and have shown how accurately the data from it may be considered. Providing that sufficient detailed simulations are undertaken of an experiment of this kind then more of the analysed data can be made use of which would otherwise have been discarded. Here only the data from the plastic scintillator detectors have been used in these calculations. The liquid scintillators ~) will be included, and the simulations repeated at a later date. The new calculations will then give better shower parameter accuracy than is the case at present. The author would like to thank the Northumbrian Universities Multiple Access Computer Unit for the computing facilities provided for this work and the Science Research Council for the award of a Fellowship and for the funds to build the Durham array. Dr. M.G. Thompson and Mr. T.R. Stewart are thanked for many helpful and stimulating discussions. Mr. A. Jones is also thanked for his assistance with the manuscript. References )) W. S. Rada, E. A. M. Shaat, A. C. Smith, T. R. Stewart, M. G. Thompson and M. W. Treasure, Nucl. Instr. and Meth. 145 (1977) 283. 2) A. C. Smith and M. G. Thompson, Nucl. Instr. and Meth.

THE DURHAM AIR SHOWER ARRAY - II1 145 (1977) 289. K. Greisen, Ann. Rev. Nucl. Sci. 10 (1960) 63. 4) H. E. Dixon, J. C. Earnshaw, J. R. Hook, J. H. Hough, G. J. Smith, W. Stephenson and K. E. Turver, Proc. Roy. Soc. Lond. A339 (1974) 133. 5) F. Ashton, A. Parvaresh and A. J. Saleh, Proc. Int. Conf. on Cosmic rays, vol. 8 (1975) 2831. 6) S. N. Vernov, G. B. Khristiansen, A. T. Abrosimov, V. B.

3)

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Atrashkevitch, I. F. Beljaeva, G. V. Kulikov, K. V. Mandritskaya, V. I. Solovjeva and B. A. Khrenov, Can. J. Phys. 46 (1968) 197. 7) S. Hayakawa, Cosmic ray physics (Wiley-lnterscience, New York, 1969) p. 477. 8) W. S. Rada, A. C. Smith and M. G. Thompson, Proc. Int. Conf. on Cosmic rays 8 (1977) 13. 9) F. James and M. Roos, Comp. Phys. Comm. 10 (1975) 343.