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Appendix A: Equations of motions

The reference frame that was used to describe the equations of motion of a particle in our simulation was centred on the density peak of the bulge. The galaxies themselves moved on an eccentric orbit around the cluster centre. Additionally, we rotated our coordinate system in such a way that the x-axis of the reference frame always pointed away from the centre of the cluster. This means that our coordinate system is not an inertial system, but an accelerated and rotated one.

The equation of motion of a particle is (A.1)The first two terms are the gravitational forces of the galaxy f_gal and the cluster f_cl, which acts on a particle. The gravitational force of the galaxy was calculated by direct summation of the two-body interaction of the particles, and the gravitational force of the cluster was calculated by an analytical force field as described in Sect. 2.1: The third term of Eq. (A.1) is the force of inertia a₀. It has the same strength but inverse direction as the force that accelerates the galaxy on its orbit around the cluster centre. The distance of the cluster centre and the origin of ordinates is r₀. The direction of the force of inertia points away from the cluster centre, which means that in our rotating reference frame along the direction of the x-axis, that (A.4)Since it is a rotating reference frame, we have to take the centrifugal f_ce, Coriolis f_co and Euler force f_eu into account. The orbital plane of the galaxy within the cluster is chosen so that the angular velocity of the motion of the galaxy around the cluster centre is pointing in the z direction. This angular velocity is the same as the angular velocity that is used to rotate the reference frame: After an expansion of the cluster potential in a Taylor series to the first order around the origin of ordinates, the equations of motions are

If one assumes a circular orbit for the galaxy , these equations of motion simplify to the Hill’s approximation.

Appendix B: Properties of encounter galaxies

The typical time between two encounters depends on the local galaxy density n(r) and the local velocity dispersion σ(r) of the cluster, which are functions of the cluster centric distance r. We take all those encounters into account that pass the infalling galaxy within a distance of p_max = 60 kpc: (B.1)where Σ is the geometrical cross section of such a galaxy-galaxy encounter . The local galaxy density was derived by assuming a galaxy distribution that follows a power law: (B.2)The parameters N₀ = 154 and β = 0.6 were determined by a fit to the radial galaxy distribution of those galaxy clusters of the Millennium II simulation (Boylan-Kolchin et al., 2009) that have a mass between 2.4−4 × 10¹⁴M_⊙.

Cluster members with total masses down to 0.1 M_gal were taken into account, where M_gal was the mass of the infalling galaxy. The total masses of the galaxies were taken from the corresponding semi-analytic model of (Guo et al., 2011). The local velocity dispersion was calculated numerically as a solution of the spherical Jeans-equation for the cluster potential. The occurrence of an encounter was simulated by randomly placing the particle that represents the perturber galaxy on a sphere with a radius of 200 kpc around the infalling galaxy. The velocities of the flying-by galaxies followed a Maxwell-Boltzmann distribution. The velocity vector of the flying-by galaxy pointed to the centre of the sphere, but offset by the impact parameter of the encounter. The impact parameter was chosen randomly based on the geometrical cross section, so that the probability of an impact parameter p ∈ [ p_min,p_max ] is given by (B.3)One also has to consider that the flying-by galaxy will by focused on the infalling galaxy by gravitation. We corrected the offset to take this gravitational focusing into account by following Spurzem et al. (2009).

The masses of the perturber galaxies were chosen randomly by following the mass distribution of the aforementioned galaxy clusters of the Millennium II simulation (Boylan-Kolchin et al., 2009). To determine the mass distribution, a power-law ansatz was used: (B.4)The fit yielded α = 2.0.

© ESO, 2015