5 Angular momentum, mass and spin correlations

In a simulation we have access to the position and the velocity of the particles. Hence, it is easy to determine the angular momenta of our cluster sized halos. In this section we use the direction of the angular momenta of galaxy clusters as vector marks and compare their correlation properties with the alignment seen in the orientation of the mass distribution (Sect. 4). With scalar MCFs, using the absolute value of the angular momentum, the cluster mass and the spin parameter as scalar mark, we will supplement the foregoing result.

It is well known that there exists a correlation between the axes of the mass ellipsoid and the direction of the angular momentum in gravitationally bound N-body systems (Binney & Tremaine 1987). Figure 6 shows that the major axis of the mass ellipsoid tends to be perpendicular to the angular momentum. Compared to the expectation from a purely random distribution, the minor axis favors smaller angles with the angular momentum.

Figure 6: Cumulative plot of the number of clusters and the cosine between the direction of the angular momentum and the major, medium and minor cluster axis, respectively.

Hence, using the direction of the angular momentum as vector mark, we are expecting to see effects comparable to the shape orientations. The vector correlations of the directions of the angular momentum are shown in Fig. 7. There is no clear signal for a direct alignment of the angular momenta as traced by ${\cal A}(r)$. However, the correlation ${\cal F}(r)<0.5$ is smaller than for random directions (shaded area), this indicates that the angular momenta are preferably oriented perpendicular to the connecting vector $\hat{\vec{r}}$on scales up to 30 h^-1 Mpc. The amplitude of the deviation from the random orientation reaches approximately 5% and is clearly outside the fluctuations. With Fig. 6 in mind, the latter result was expected from the correlations of the orientations shown in Fig. 3: For a pair of clusters residing in a filamentary structure the connecting vector is oriented along this filament. The major axes of the orientations tend to align with the filaments, and therefore the angular momenta tend to stand perpendicular to the filaments. A random orientation of the angular momenta in the planes perpendicular to the filaments provides a simple explanation for the absence of any signal in ${\cal A}(r)$, still compatible with the strong signal in ${\cal F}(r)$.

Figure 7: Mark correlation functions using the normalized angular momentum ${\vec{l}}$ with $\vert{\vec{l}}\vert=1$ as vector mark. The shaded area is obtained by randomizing the orientation among the clusters.

Besides the correlations in the directions of the angular momentum vectors, we are also interested in the correlations of their magnitudes. We investigate these correlations with the MCFs discussed in Sect. 3.1, using the magnitude of the angular momentum as a scalar mark. In Fig. 8 the increased $k_{\rm m}(r)$ shows that pairs of clusters with separations $\lesssim$50 h^-1 Mpc tend to have higher mean angular momentum compared to the overall mean angular momentum. The positive covariance up to $\lesssim$15 h^-1 Mpc indicates that both members of close pairs tend to have similar angular momentum. Hence, inspecting both $k_{\rm m}$ and the covariance, we see that close clusters tend to carry a similar amount of angular momentum, larger than the overall mean angular momentum.

Figure 8: Mark correlation functions with the absolute value of the angular momentum of the cluster as scalar mark. The shaded area is obtained by randomizing the mark among the clusters.

There are two possibilities to explain this behavior. On the one hand the mean mass of a close cluster pair could be enhanced, and so the absolute value of the angular momentum would grow simply due to the fact that bigger clusters are under consideration. Indeed, Gottlöber et al. (2002) found such an increase in the mean mass in close pairs (in their case traced by the maximum circular velocity of the halo). Consistently Beisbart & Kerscher (2000) report an enhanced luminosity for close galaxy pairs. On the other hand, the rotational support of close cluster pairs could be higher, meaning that the spin parameter $\lambda$ is enhanced, cp. Eq. (1). To illustrate this further, we investigate the MCFs with the mass and the spin parameter as marks.
Mass: In Fig. 9 we show the MCFs of the cluster distribution with the total mass as scalar mark. The increased $k_{\rm m}(r)$ indicates that close pairs of clusters tend to have higher mean masses than the overall mean mass $\overline{m}$ on scales out to 50 h^-1 Mpc. The signal shows a deviation up to 10% from a purely random distribution. The conditional covariance ${\rm cov}(r)$ shows only a weak positive signal, confined to scales below 15 h^-1 Mpc, indicating that only close pairs tend to have similar masses.

Figure 9: Mark correlation functions with the mass of the cluster halo as scalar mark. The shaded area is obtained by randomizing the mark among the cluster halos.

Spin: In Fig. 10 we show the MCFs of the cluster distribution with the spin parameter $\lambda$ as scalar mark. The increased $k_{\rm m}(r)$ indicates that neighboring pairs of clusters tend to have higher spin parameter $\lambda$ compared to the overall mean spin $\overline{\lambda}$. The signal seen in Fig. 10 shows an enlarged spin parameter of clusters on scales out to 50 h^-1 Mpc, the same range as for the mass. The deviation from random distribution is $\sim$$15\%$. Hence, not only an increased mass accretion but also tidal interactions out to fairly large scales seem to characterize the regions around cluster halos. The conditional covariance ${\rm cov}(r)$shows no signal. This can be explained by looking at the mark variance ${\rm var}(r)$. The signal at small distances deviates about $50\%$ from random, indicating large scattering of the spin parameter. This large dispersion also might be the reason why previous studies did not find an environmental dependence of the amount of the spin parameter. A sufficiently large simulation and also a large number of clusters are needed to get a clear signal in the MCFs for the spin parameter.

Figure 10: Mark correlation function with the spin parameter $\lambda$ of the cluster halo as scalar mark. The shaded area is obtained by randomizing the mark among the clusters.

Putting together all the results shown in Figs. 8-10, it turns out that the increased mean angular momentum is not only caused by an enhanced mass, but also by an enlarged spin parameter of neighboring pairs of clusters. From the present analysis we can not draw any firm conclusions, whether this phenomenon is dominated by tidal interaction or merging processes. There are compelling observational hints (see e.g. Plionis & Basilakos 2002; Schuecker et al. 2001) that clusters in high-density environments show indications of dynamical disruption. Thus it seems to be likely that the enhancement of the angular momentum of close neighbors is caused by a substantial mass accretion (i.e. merging). The concordant behavior of the mass and the spin MCF support this interpretation, at least on scales below $\sim$15 h^-1 Mpc. However, tidal interactions could be the cause for the correlations seen on large scales.