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3 Probability of tidal capture

The probability of close encounters between star clusters leading to a tidal capture is considered to be relatively small or even very unlikely (Bhatia et al. 1991). However, van den Bergh (1996) suggested that it becomes more probable in dwarf galaxies like the Magellanic Clouds due to the small velocity dispersion of the cluster systems. Furthermore, Vallenari et al. (1998) proposed that interactions between the LMC and SMC might have increased the formation of star clusters in large groups in which the encounter rate and thus the formation of bound binary clusters is higher. This scenario is capable of explaining large age differences between cluster pairs which Leon et al. (1999) refer to as the "overmerging problem''. In this section, we will determine the cluster encounter rates in our selected areas.

Inside the chosen regions we find 491 clusters in the bar, 863 in the remaining parts of $E_{\rm bar}$, 372 objects in $E_{\rm north}$, 1439 clusters in $E_{\rm inner}$ (without $E_{\rm bar}$ and $E_{\rm north}$), and 924 remaining entries in the outer region of the LMC. Assuming a distance modulus of 18.5 mag to the LMC, 1 pc corresponds to $1.999\times10^{-5}$ units in our cartesian system. This leads to a length of 2710 pc and a width of 591 pc for the bar, and thus to a cluster density of $3\times10^{-4}$ clusters ${\rm ~pc}^{-2}$. For $E_{\rm bar}$, the semi major axis corresponds to 2250 pc and its semi minor axis to 1000 pc, resulting in $863~{\rm clusters}/(A_{E_{\rm bar}}-A_{\rm bar}) =
1.6\times10^{-4}$ clusters ${\rm ~pc}^{-2}$. Cluster densities for the other selected areas follow in an analogous manner: $1.2\times10^{-4}$ clusters ${\rm ~pc}^{-2}$ for $E_{\rm north}$, $8\times10^{-5}$ clusters ${\rm ~pc}^{-2}$for $E_{\rm inner}$ (without the northern ellipse and the region surrounding the bar), and $1.2\times10^{-5}$ clusters ${\rm ~pc}^{-2}$ for the outer ring $E_{\rm
outer}$. Please note that, assuming an outer ring with limited boundaries and semi axes corresponding to 5000 pc and 6750 pc, the number of objects in $E_{\rm
outer}$ amounts to 911. However, this does not alter the value of the outer cluster density.

The cluster density is highest in the innermost part of the LMC, the bar region, and it drops off by an order of magnitude towards the outer region. According to Vallenari et al. (1998), cluster pairs can be formed by close encounters which result in the tidal capture of two clusters. The higher the cluster density, the higher the probability for close encounters, and thus the probability for the formation of cluster pairs or groups.

The cluster encounter rate can be determined following Lee et al. (1995):

$\displaystyle \frac{{\rm d}N}{{\rm d}t} = \frac{1}{2} \cdot \frac{N-1}{V} \cdot \sigma \cdot v$     (4)

where N is the number of clusters, V denotes the volume of the galaxy or, respectively, the part of the galaxy under investigation, $\sigma = \pi R^{2}$is the geometric cross section of a cluster with radius R, and v is the velocity dispersion of the cluster system of that galaxy. Typical cluster radii are about 10 pc. For the velocity dispersion of the cluster system we adopt 15 km  $\mbox{s}^{-1}$ as quoted in Vallenari et al. (1998). For the depth of the LMC, and thus for each selected area, we adopt 400 pc (Hughes et al. 1991). This leads to a cluster encounter rate of $20\times 10^{-10}~{\rm yr}^{-1}$ inside the bar. In the ellipse surrounding the bar, the northern region and the inner LMC region, the probability of close encounters is much lower by a factor of 2 to 4, namely $9\times 10^{-10}~{\rm yr}^{-1}$ in $E_{\rm bar}$, $7\times
10^{-10}~{\rm yr}^{-1}$ in $E_{\rm north}$, and $5\times 10^{-10}~{\rm
yr}^{-1}$ in $E_{\rm inner}$. The lowest encounter rate is, as expected, in the outer ring with a value of $0.7\times 10^{-10}~{\rm
yr}^{-1}$. This means that the probability of a close encounter between star clusters is $\approx$30 times higher in the bar than in the outskirts of the LMC.

All results are summarized in Table 3.

The probabilities for cluster encounters are already very low. In addition, the probability of tidal capture depends on further conditions which will not be fulfilled during every encounter. Whether a tidal capture takes place or not depends strongly on the velocities of the two clusters with respect to each other, on the angle of incidence, whether sufficient angular momentum can be transferred, and whether the clusters are sufficiently massive to survive the encounter. Since only very few of these rare encounters would result in tidal capture, it seems unlikely that a significant number of young pairs may have formed in such a scenario.


Table 3: Semi axes a, b in pc and number of clusters found in the selected regions, the resulting cluster densities and the encounter rate ${\rm d}N/{\rm d}t$. For the bar region, a and b do not denote semi axes, but the lengths of a rectangular area.
Region a [pc] b [pc] $N_{\rm clusters}$ $\frac{{\rm clusters}}{{\rm ~pc}^{2}}$ ${\rm d}N/{\rm d}t$

(2710) (591) 491 $3\times10^{-4}$ $20\times 10^{-10}~{\rm yr}^{-1}$
$E_{\rm bar}$ 2250 1000 863 $1.6\times10^{-4}$ $9\times 10^{-10}~{\rm yr}^{-1}$
$E_{\rm north}$ 1500 $\mbox{ }\mbox{~}650$ 372 $1.2\times10^{-4}$ $7\times
10^{-10}~{\rm yr}^{-1}$
$E_{\rm inner}$ 3250 2750 1439 $8\times10^{-5}$ $5\times 10^{-10}~{\rm
outer}$ 5000 6750 911 $1.2\times10^{-5}$ $0.7\times 10^{-10}~{\rm


Table 4: Statistics about the cluster groups found in the selected regions. $N_{\rm tot}$ denotes all clusters found in the corresponding region, $N_{\rm cl}$ is the number of clusters involved in $N_{\rm pairs}$ pairs. The numbers of isolated pairs $N_{\rm 2}$, triple systems $N_{\rm 3}$, and so on can be found in the subsequent columns. Groups consisting of more than eight members do not occur.
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 491 228 166 59 22 5 - 4 - -
$E_{\rm bar}$ 863 306 207 97 20 5 2 1 - 2
$E_{\rm north}$ 372 117 88 36 5 3 2 - - 1
$E_{\rm inner}$ 1439 371 247 131 19 5 4 2 - -
outer}$ 924 93 55 40 3 1 - - - -
Sum 4089 1115 763 363 69 19 8 7 - 3
LMC total 4089 1126 770 366 69 19 9 7 - 3

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