2 Cluster density distribution

Looking at an optical image of the LMC, it is apparent that the stellar and cluster distribution is not uniform across this galaxy. One of the most striking features besides the prominent star-forming H II regions is the LMC bar, a region of increased stellar density.

In Fig. 1 we plotted the angular distribution of all clusters listed in the BSDO catalogue. Again, the bar structure can easily be recognized. However, the clusters are not evenly distributed in the outer LMC. To make the structures of the cluster density distribution more apparent, Fig. 1 was smoothed with a Gaussian filter with a blur radius of 50 image pixels. The intensity scale of the resulting figure (Fig. 2) was reduced to 15 bins, the darkest shade indicating the highest cluster density. In this way, it is possible to get qualitative information about the density distribution, however, quantitative values of the cluster densities cannot be derived from the image's greyscales alone. All steps were performed with the aid of common image processing tools (GIMP).

Regions of different cluster densities and their spatial extent can be seen in Fig. 2: apart from the prominent bar structure, also the area surrounding the bar is densely populated with star clusters. To the north-east another region of enhanced cluster density is clearly visible, which coincides with LH 77 (Lucke & Hodge 1970) in the supergiant shell LMC 4 (see e.g. Braun et al. 1997) and the constellation Shapley III (McKibben Nail & Shapley 1953). The outermost LMC areas show a considerable drop-off in the cluster density, which is already recognizable in Fig. 1.

Figure 1: Angular distribution of all cluster-like objects derived from the BSDO catalogue. The bar structure is clearly visible.

Figure 2: Density plot of all LMC clusters. The highest cluster density (black) coincides with the bar region. Please note that this picture provides qualitative information about the density distribution. No quantitative values of the cluster densities can be derived from the image's greyscales alone. The location of the supergiant shells LMC 1-LMC 9 are sketched (see Fig. 6, p. 6, in Braun 2001). The position of 30 Doradus is marked with a white cross.

Figure 3: Cluster distribution plotted in cartesian coordinates. The regions that are selected on the basis of the cluster density plot are sketched.

Due to the non-uniform distribution of the star clusters in the LMC, we decided to subdivide the BSDO catalogue into regions of equal cluster densities. It is a difficult task to find a suitable partition: it has to be small enough so that a (nearly) uniform distribution of clusters can be assumed but general density differences between larger areas must still be recognizable and may not be smeared out by a too small partition, and the regions have to be larger than the detection limit for cluster pairs and groups. Based in Fig. 2, we decided to split the input catalogue into five lists: the inner part of the LMC shows in general a higher concentration of star clusters, thus we first distinguish between an inner ellipse (which we call $E_{\rm inner}$) and an outer ring, the remaining outer LMC area which shows an overall low cluster density ( $E_{\rm outer}$). Inside the inner ellipse, there are still regions of varying cluster frequencies. Thus, we further define an ellipse surrounding the bar (which we call $E_{\rm bar}$), a rectangular area that coincides with the bar itself (called "bar''), and an ellipse north-east of the bar corresponding to the location of LMC 4 ( $E_{\rm north}$). All areas are disjunctive, i.e., if a selected region contains one or more selected ellipses or the bar, these "inner'' areas are not considered in the following calculations and simulations, i.e., $E_{\rm bar}$ does not contain the bar, $E_{\rm inner}$ does not contain $E_{\rm bar}$ and $E_{\rm north}$ and so on.

For the following selections and Monte Carlo simulations it is suitable to transform the spherical coordinates of the catalogue entries ($\alpha$, $\delta$ coordinates) into cartesian coordinates (x, y) (see, e.g., Geffert et al. 1997 and Sanner et al. 1999). The gnomonic projection was done via:

  

$$\frac{-\cos\delta \cdot \sin(15 \cdot (\alpha-\alpha_{0}))}{\sin\... ...a_{0}+\cos\delta \cdot \cos\delta_{0} \cdot \cos(15 \cdot (\alpha-\alpha_{0}))}$$ (1)

$$\frac{\sin\delta \cdot \cos\delta_{0}-\cos\delta \cdot \sin\delta... ...a_{0}+\cos\delta \cdot \cos\delta_{0} \cdot \cos(15 \cdot (\alpha-\alpha_{0}))}$$ (2)

(van de Kamp 1967). We adopt $\alpha_{2000} = -69^{\circ}45\hbox{$^\prime$ }$ and $\delta_{2000} = 5^{\rm h}23\farcm6$ as the central coordinates of the LMC (CDS data archive). Figure 3 shows the cluster distribution in cartesian coordinates.

The selection criterion for clusters situated inside an ellipse with the semi major axes a and b, central coordinates x₀ and y₀, and rotation angle $\phi$ is:

 

$$\left(\frac{(x-x_{0})\cdot\cos\phi+(y-y_{0})\cdot\sin\phi)}{a}\ri... ...\left(\frac{(x-x_{0})\cdot\cos\phi-(y-y_{0})\cdot\sin\phi}{b}\right)^{2}\leq 1.$$ (3)

The semi axes, central coordinates and the rotation angle for the selected ellipses are listed in Table 1. A helpful tool for selecting the bar region is the map of the LMC provided by Smith et al. (1987, their Fig. 4) in which prominent features are sketched. The vertices that we choose to cut out the bar are listed in Table 2.

All selected areas are sketched in Fig. 3.

 

 

Table 1: Semi axes a and b (in units of the cartesian system), central coordinates x₀ and y₀ and the rotation angle $\phi$ for the selected ellipses for which uniform cluster densities are assumed (see Eq. (3)).

Region	a	b	x0	y0	$\phi$
$E_{\rm bar}$	0.045	0.020	0.000	0.005	$20^{\circ}$
$E_{\rm north}$	0.030	0.013	0.025	0.040	$20^{\circ}$
$E_{\rm inner}$	0.065	0.055	0.002	0.018	$25^{\circ}$
$E_{\rm outer}$	0.100	0.135	-0.005	0.015	$0^{\circ}$

 

 

Table 2: Vertices defining the selected bar region.

x	y
0.0265	0.0212
0.0337	0.0119
-0.0199	-0.0067