1 Introduction

The study of orientation effects between galaxy clusters has a long and controversial history in cosmology. In a seminal study Binggeli (1982) claimed that galaxy clusters are highly eccentric and oriented relative to neighboring clusters if lying at separations smaller than 15 h^-1 Mpc. Further he found anisotropies in the cluster distribution on scales up to 50 h^-1 Mpc. Following studies found no or weak statistical significance for orientation effects between neighboring clusters or between cluster orientation and the orientation of the central dominant galaxy, cp. Struble & Peebles (1985), Flin (1987), and Rhee & Katgert (1987). Remarkable was the apparent absence (Ulmer et al. 1989) or weakness (Rhee & Latour 1991) of orientation effects in projected X-ray contours of clusters, but it should be noted that the cluster samples at this time were small. Analyzing a large set of 637 Abell clusters, Plionis (1994) found highly significant alignment effects on scales below 10 h^-1 Mpc that become weaker but extend up to 150 h^-1 Mpc. More objectively selected, but smaller cluster samples seemed to put into question the reality of this signal, cp. Fong et al. (1990) and Martin et al. (1995). However, Chambers et al. (2000) found significant nearest neighbor alignment of cluster X-ray isophotes using data from Einstein and ROSAT. With the advent of new rich cluster catalogues as the optical ENACS survey (Katgert et al. 1996) and the X-ray based REFLEX survey (Böhringer et al. 1998), the question of orientation effects in clusters should attract renewed attention. Sufficiently large and well defined cluster samples showing only weak contamination by projection effects seem to be necessary to clarify this uncertain situation.

Strong stimulus to study orientation effects in clusters came from early ideas that a possible relative orientation between neighboring clusters or of clusters in the same supercluster should reflect the underlying structure formation mechanism. Binney & Silk (1979) proposed that tidal interactions of evolving protocluster systems may lead to the growth of anisotropies of clusters and to relative orientation effects. Later, van Haarlem et al. (1997) used numerical simulations of CDM models to demonstrate that clusters are elongated along the incoming direction of the last major merger. In the same spirit, West (1994) found that clusters grow by accretion and merging of surrounding matter that falls into the deep cluster potential wells along sheet-like and filamentary high density regions. Therefore, the cluster formation is tightly connected with the supercluster network that characterizes the large-scale matter distribution in the universe. High-resolution simulations showing this effect are described by the Virgo collaboration, cp. Colberg et al. (2000). Onuora & Thomas (2000) found a significant alignment signal up to scales of 30 h^-1 Mpc for a $\Lambda$CDM model, whereas in a $\tau$CDM model the signal extended only up to scales of 15 h^-1 Mpc.

To quantify the alignment of the galaxy clusters, we use a large $\Lambda$CDM simulation in a box of 500 h^-1 Mpc side length. We identify a set of 3000 clusters. As statistical tools we employ mark correlation functions (MCF), as introduced to cosmology by Beisbart & Kerscher (2000). In this article we will extend this formalism to allow for vector valued marks. The direction of the major axis of the mass ellipsoid serves as the vector mark. Tightly connected with the elongation of clusters is its internal rotation. According to Doroshkevich (1973) and White (1984), the primary angular momentum of bound objects is due to tidal interaction between the elongated protostructures after decoupling from cosmic expansion and before turn-around. More recent studies find that the angular momentum of dark matter halos is later modified by the merging history of their building blocks, cp. Vitvitska et al. (2002) and Porciani et al. (2002a, b). Therefore, we utilise the angular momentum as an additional mark for the study of the correlation of inner properties of simulated clusters, and we compare it with the orientation effects.

The plan of the paper is as follows. In the next section, we describe our numerical simulation, the selection of a cluster sample and the precision with which we can determine structure parameters from it. Next we discuss the MCFs that are relevant for our studies. In particular, we use special MCFs for vector marks to quantify correlations of orientation. In Sect. 4 we investigate correlations in the spatial orientation of clusters both in 3D and in the projected mass distribution. In Sect. 5 we present a MCF analysis using the angular momentum, mass and spin taken as vector and scalar marks, respectively. We conclude with a summary of the results.