We study the formation of over 6000 compact groups (CGs) of galaxies identified in mock redshift-space galaxy catalogues built from semi-analytical models of galaxy formation (SAMs) run on the Millennium Simulations. We select CGs of 4 members in our mock SDSS galaxy catalogues and, for each CG, we trace back in time the real-space positions of the most massive progenitors of their 4 galaxies. By analysing the evolution of the distance of the galaxy members to the centre of mass of the group, we identify 4 channels of CG formation. The classification of these assembly channels is performed with an automatic recipe inferred from a preliminary visual inspection and based on the orbit of the galaxy with the fewest number of orbits. Most CGs show late assembly, with the last galaxy arriving on its first or second passage, while only 10-20 per cent form by the gradual contraction of their orbits by dynamical friction, and only a few per cent forming early with little subsequent contraction. However, a SAM from a higher resolution simulation leads to earlier assembly. Assembly histories of CGs also depend on cosmological parameters. At similar resolution, CGs assemble later in SAMs built on parent cosmological simulations of high density parameter. Several observed properties of mock CGs correlate with their assembly history: early-assembling CGs are smaller, with shorter crossing times, and greater magnitude gaps between their brightest two members, and their brightest galaxies have smaller spatial offsets and are more passive.
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