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.

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# 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 as a function of . 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).

 Angular 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

 Angular 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

 Angular 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