R.W. Clay , B.R.Dawson , L. Kewley , M. JohnstonHollitt, PASA, 17 (3), 207.
Next Section: Application to the SUGAR
Title/Abstract Page: The Two Point Angular
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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 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 nonuniformity 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 ontime 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  
05  8.4  14.2  278 
510  5.3  8.0  670 
1015  8.0  6.5  161 
1520  7.3  5.2  830 
2025  6.8  5.0  148 
2530  3.5  4. 5  255 
3035  0.7  4.1  541 
3540  1.5  3.9  600 
4045  5.9  3.9  122 
4550  3.6  3.6  747 
5055  1.9  3.6  654 
5560  3.0  3.6  259 
6065  8.5  3.3  957 
6570  3.0  3.5  732 
7075  2.5  3.6  286 
7580  1.4  3.5  590 
8085  5.3  3.7  131 
8590  7.2  3.6  937 
9095  4.6  3.7  812 
95100  1.6  3.9  369 
100105  2.9  4.0  297 
Angular 

 number 
interval  
05  47.9  14.2  766 
510  6.0  22.6  357 
1015  25.8  18.2  109 
1520  9.9  14.7  253 
2025  5.2  12.2  569 
2530  5.4  11.9  333 
3035  17.9  11.8  105 
3540  20.3  9.3  908 
4045  4.9  10.2  339 
4550  1.1  9.9  432 
5055  14.1  9.0  837 
5560  22.3  8.4  952 
6065  14.2  8.9  851 
6570  27.0  8.1  978 
7075  7.1  9.9  263 
7580  20.7  10.6  43 
8085  11.0  10.0  183 
8590  14.5  10.5  118 
9095  5.7  9.6  635 
95100  10.7  10.7  200 
100105  3.2  10.3  553 
Angular 

 number 
interval  
05  100  undef.  585 
510  86.4  59.0  47 
1015  22.3  35.3  212 
1520  24.6  31.1  182 
2025  39.7  29.8  100 
2530  24.6  25.4  165 
3035  10.8  22.6  180 
3540  40.1  16.0  910 
4045  2.9  20.2  371 
4550  37.3  15.7  896 
5055  3.2  15.7  884 
5560  29.3  15.8  851 
6065  27.3  16.2  816 
6570  22.0  16.6  757 
7075  14.0  17.5  642 
7580  22.4  21.0  160 
8085  18.1  20.9  209 
8590  23.8  22.6  151 
9095  2.6  20.9  387 
95100  78.4  28.2  1 
100105  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
© Copyright Astronomical Society of Australia 1997