The Two Point Angular Autocorrelation Function and the Origin of the Highest Energy Cosmic Rays

R.W. Clay , B.R.Dawson , L. Kewley , M. Johnston-Hollitt, PASA, 17 (3), 207.

Next Section: Application to the SUGAR
Title/Abstract Page: The Two Point Angular
Previous Section: Cosmic Ray Directions and
Contents Page: Volume 17, Number 3

The Two Point Angular Autocorrelation Function

The two point angular autocorrelation function has been described by Peebles (1980). It involves the analysis of an ensemble of data points with known angular locations. The two point angular autocorrelation function is then the joint probability of finding an object in both of two elements of solid angle placed at an angular separation of $\theta$ as a function of $\theta$. In practice, the probability can be found by comparing the number of ensemble pairs having a particular separation (in a given separation interval) with the number derived for a random ensemble which contains the same number of directions. We can express the function (w) and its uncertainty (dw derived from a consideration of Poisson statistical uncertainties) through the ratio of the number found in the experimental data to the number in the random data (at a particular angular separation and within an angular range) minus one. A positive value at small angular separations indicates a clustering of the data points.

A significant observational difficulty in applying this procedure is in determining a suitable random dataset for the comparison data. The experimental dataset may contain clustering information but it will also contain the effects of non-uniformity of sky coverage, which might have changed over the period of the data acquisition.

In order to determine the function for our complete catalogue of SUGAR data, we created a randomised SUGAR dataset through a procedure which shuffled components of the real SUGAR data. The SUGAR dataset available to us did not contain the altitude and azimuth of the events. We determined these from the event times and the celestial coordinates of the event arrival directions. A new dataset was then generated using randomly chosen Julian dates for the events from the ensemble of dates within the real catalogue. The catalogue was then used to assign horizontal coordinates for each of these events and corresponding celestial coordinates derived as though events from those directions had been observed at the randomly selected Julian dates. In this way, the new ensemble contained the real on-time properties of the array and it also contained the altitude and azimuthal properties of the array. In each case for our analysis of a subset of SUGAR events, we generated 1000 shuffled datasets so that it was possible to determine the number of real events with a particlar angular separation together with the chance probability that such a number would have been exceeded (the number of times the random sets gave a larger value divided by 1000, the number of random trials).


Table 1: The two point autocorrelation function (w) for SUGAR data as a function of angular spacing for showers with energies above

$2\times10^{19}$eV. The column 'number' indicates the number of times that value of w is exceded in 1000 randomised data sets.

Angular

$w\times10^{-2}$

$dw\times10^{-2}$

number
interval      
0-5 8.4 14.2 278
5-10 -5.3 8.0 670
10-15 8.0 6.5 161
15-20 -7.3 5.2 830
20-25 6.8 5.0 148
25-30 3.5 4. 5 255
30-35 -0.7 4.1 541
35-40 -1.5 3.9 600
40-45 5.9 3.9 122
45-50 -3.6 3.6 747
50-55 -1.9 3.6 654
55-60 3.0 3.6 259
60-65 -8.5 3.3 957
65-70 -3.0 3.5 732
70-75 2.5 3.6 286
75-80 -1.4 3.5 590
80-85 5.3 3.7 131
85-90 -7.2 3.6 937
90-95 -4.6 3.7 812
95-100 1.6 3.9 369
100-105 2.9 4.0 297


Table 2: The two point autocorrelation function (w) for SUGAR data as a function of angular spacing for showers with energies above

$4\times10^{19}$eV. The column 'number' indicates the number of times that value of w is exceded in 1000 randomised data sets.

Angular

$w\times10^{-2}$

$dw\times10^{-2}$

number
interval      
0-5 -47.9 14.2 766
5-10 6.0 22.6 357
10-15 25.8 18.2 109
15-20 9.9 14.7 253
20-25 -5.2 12.2 569
25-30 5.4 11.9 333
30-35 17.9 11.8 105
35-40 -20.3 9.3 908
40-45 4.9 10.2 339
45-50 1.1 9.9 432
50-55 -14.1 9.0 837
55-60 -22.3 8.4 952
60-65 -14.2 8.9 851
65-70 -27.0 8.1 978
70-75 7.1 9.9 263
75-80 20.7 10.6 43
80-85 11.0 10.0 183
85-90 14.5 10.5 118
90-95 -5.7 9.6 635
95-100 10.7 10.7 200
100-105 -3.2 10.3 553


Table 3: The two point autocorrelation function (w) for SUGAR data as a function of angular spacing for showers with energies above

$6\times10^{19}$eV. The column 'number' indicates the number of times that value of w is exceded in 1000 randomised data sets.

Angular

$w\times10^{-2}$

$dw\times10^{-2}$

number
interval      
0-5 -100 undef. 585
5-10 86.4 59.0 47
10-15 22.3 35.3 212
15-20 24.6 31.1 182
20-25 39.7 29.8 100
25-30 24.6 25.4 165
30-35 10.8 22.6 180
35-40 -40.1 16.0 910
40-45 2.9 20.2 371
45-50 -37.3 15.7 896
50-55 -3.2 15.7 884
55-60 -29.3 15.8 851
60-65 -27.3 16.2 816
65-70 -22.0 16.6 757
70-75 -14.0 17.5 642
75-80 22.4 21.0 160
80-85 18.1 20.9 209
85-90 23.8 22.6 151
90-95 2.6 20.9 387
95-100 78.4 28.2 1
100-105 -1.8 22.0 433


Next Section: Application to the SUGAR
Title/Abstract Page: The Two Point Angular
Previous Section: Cosmic Ray Directions and
Contents Page: Volume 17, Number 3

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