5 Monte Carlo simulations - How many pairs can be expected statistically?

In order to check how many of all the found pairs and multiple clusters can be expected statistically due to chance line-up, we performed statistical experiments:

For this purpose we developed C and C++ software which performs Monte Carlo simulations and analyses of the resulting random distribution.

The simulations are carried out in the cartesian system and we used the same selected areas as mentioned in Sect. 2. The same number of objects which were found based on the BSDO catalogue are now distributed randomly in each region, i.e., 372 objects are randomly spread in an ellipse with semi axes of 0.030 and 0.013 (or 1500 and 650 pc) for the inner, northern region $E_{\rm north}$; 491 objects are stochastically distributed in a rectangular area with lengths of $1.1822\times10^{-2}$ and $5.4189\times10^{-2}$ in units of the cartesian system (corresponding to 591 and 2710 pc) which denotes the LMC bar; 863 objects are arbitrarily placed in the space between the bar and the boundaries of the ellipse described with $E_{\rm bar}$ and so on. In this way, artificial cluster distributions are produced that can be compared with the true distribution in the LMC. To improve statistics this procedure was repeated 100 times for each region.

An example of an artificial cluster distribution is plotted in Fig. 4. The inner part of the LMC, including the bar and the northern region corresponding to LMC 4, is well represented. However, there is a sharp drop-off in the cluster density in the outskirts. The cluster density in the LMC is low in its outer regions, however, the decrease between the inner and outer region ( $E_{\rm inner}$ and $E_{\rm outer}$) is smoother (see Fig. 3). Figure 5 displays the density distribution of the artificial cluster system. Compared with Fig. 2 all prominent features are well represented.

Figure 4: Example of one of the artificial cluster distributions that was created using Monte Carlo simulations. The distribution is plotted in cartesian coordinates and can be compared with Fig. 3, which shows the real LMC cluster distribution.

Figure 5: Density distribution of the artificial cluster system represented in Fig. 4.

The number of chance pairs and groups in our simulated cluster distributions as well as the number of single objects involved in these groups was counted and compared with our findings based on BSDO. As pointed out in the previous sections, two objects are considered as a pair and thus will be included in our list of random pairs or groups if their separation is 20 pc ( $3.9979\times10^{-4}$ in units of the cartesian system) at maximum.

Out of 491 random objects in the bar region, ${\approx} 156 \pm 14$ can be found in ${\approx} 94 \pm 10$ pairs. In reality, the LMC bar comprises 166 pairs which is nearly twice the amount of what we can expect statistically. However, a closer look at the group statistics reveals that of the 59 binary systems found in the real LMC bar cluster distribution, $55 \pm 6$ isolated pairs can be explained as chance pairs based on our simulations, i.e., these numbers agree within the uncertainties. A discrepancy between found and expected figures is more apparent when groups with three or more members are considered. Out of 22 triple systems, only ${\approx} 11 \pm 4$(i.e., $\approx$50%) can be explained with chance superpositions. The number of actually found groups containing four members is twice as much as expected (5 found, 2.49 expected). 4 groups with six members are found, however, their random formation is very unlikely (0.21).

The statistically expected number of chance pairs and groups are summarized in Table 5 for all regions. A graphical display of our results can be found in Fig. 6. The numbers of the cluster groups actually found in the LMC regions are also indicated in Fig. 6 (see also Sect. 4) so that the results of both the real and the simulated cluster distribution can easily be compared.

In the remaining space of $E_{\rm bar}$, 207 pairs are actually found, but only $84 \pm 9$ ($\approx$41%) of them can be explained statistically. Out of 97 isolated pairs, $\approx$66% can be ascribed to chance superpositions. 20 triple systems are found, but only $7 \pm 2$ can be expected. Again, for larger groups the discrepancy increases.

In the northern region, $E_{\rm north}$, $33\%$ ( ${\approx} 29 \pm 6$ pairs) of the 88 ones found can be expected due to chance line up.

For the remaining part of the inner LMC ( $E_{\rm inner}$), $\approx$29% of all pairs are explainable on a statistical basis.

 

 

Table 5: Statistics about the cluster groups found in the simulated regions. The standard deviations are given in brackets. Column labels are as in Table 4.

Region	$N_{\rm tot}$	$N_{\rm cl}$	$N_{\rm pairs}$	$N_{\rm 2}$	$N_{\rm 3}$	$N_{\rm 4}$	$N_{\rm 5}$	$N_{\rm 6}$	$N_{\rm 7}$	$N_{\rm 8}$
Bar	491	156.39	94.18	54.64	11.13	2.49	0.47	0.21	0.01	0.01
		(14.28)	(9.83)	(6.11)	(3.67)	(1.72)	(0.67)	(0.46)	(0.10)	(0.10)
$E_{\rm bar}$	863	152.37	84.05	63.59	7.00	0.89	0.10	0.01	0.01	-
		(15.57)	(9.23)	(7.20)	(2.40)	(0.84)	(0.30)	(0.01)	(0.01)	-
$E_{\rm north}$	372	52.72	28.83	22.44	2.18	0.30	0.02	-	-	-
		(10.24)	(5.70)	(4.41)	(1.34)	(0.56)	(0.14)	-	-	-
$E_{\rm inner}$	1439	136.51	71.50	62.53	3.53	0.19	0.02	-	-	-
		(16.20)	(8.61)	(7.59)	(1.77)	(0.44)	(0.14)	-	-	-
$E_{\rm outer}$	911	13.12	6.58	6.53	0.02	-	-	-	-	-
		(5.16)	(2.55)	(2.54)	(0.14)	-	-	-	-	-
Sum	4076	511.11	285.14	209.73	23.86	3.87	0.61	0.22	0.02	0.01
		(28.82)	(14.92)	(12.12)	(5.10)	(2.16)	(0.85)	(0.48)	(0.14)	(0.20)
Total	4076	515.58	288.01	211.32	24.19	3.97	0.63	0.22	0.02	0.02
		(26.02)	(15.35)	(12.2)	(4.96)	(2.49)	(0.80)	(0.46)	(0.14)	(0.14)

The outer LMC is approximated by a ring (see Sect. 3) in which 911 objects are randomly distributed. These are 13 objects less than are actually found outside of $E_{\rm inner}$. However, these 13 clusters are located so far outside that a region which includes all these objects cannot be assumed to have an overall constant cluster density. No cluster pairs are among these outer objects so that they do not need to be included in our statistics. 55 cluster pairs are found in $E_{\rm outer}$, but only $\approx$12% of them ( ${\approx} 7 \pm 3$ pairs) can be explained due to chance line-up. Though there is a low probability for a triple system (0.02), no larger group occurred in our simulations.

Figure 6: Histogram of the number of cluster groups found in the selected regions of different cluster densities. The solid line denotes the number of groups detected in the real cluster distribution, while the dashed line indicates the number of cluster systems which can be expected statistically. The percentage of the clusters involved in the groups of different member size is also given. See Sect. 5 for details.

Table 5 also lists the sum of all cluster pairs and groups that can be expected if the figures for all regions are added up. The last line of Table 5 gives the group statistics for an entire artificial LMC, i.e., the experiments for the selected regions are put together. Again, as was already noticed for Table 4, the statistics show slight differences since some chance pairs and groups are located across the rims of the selected regions. However, the differences are much smaller than for the "real'' LMC (Table 4). Comparing the results with Table 4 it can be seen that approximately $37\%$( $288 \pm 15$) of the found 770 cluster pairs can be expected due to spatial superposition. Approximately $58\%$ ( $211 \pm 12$) of all 366 binary cluster candidates can be expected statistically. The discrepancy between found and expected cluster groups increases for larger groups.

For each region, the number of pairs that can be expected due to random superposition is much lower than the number of pairs that are actually found: between $56\%$ (in the bar region) and $12\%$ (in the outer LMC ring) of all detected pairs can be explained statistically. It seems that the discrepancy between found and expected pairs also depends on the cluster density of the region, i.e., in the densest bar region the percentage of pairs that can be explained statistically is also the highest, while it is the lowest in the region with the lowest cluster density (the outer ring). It is striking that especially large cluster groups with more than four members scarcely occur in any of the artificial cluster distributions.