Evolution of a system forming a single (transient) fragment which is eventually accreted into the host star because of orbital migration. Top left: stellar mass, disc mass and disc radius. Top right: minimum Q_Toomre and α. The stellar accretion rate is shown in black (see text Sect. 5.2). Middle left: time evolution of fragment orbital separation r_c and mass M_c. Middle right: same with a zoom-in on the early phase, two variant calculations are shown additionally in black (see text Sect 5.2). Bottom left: surface density, centred on the fragment (and thus at different absolute orbital distances), for a selection of times. Bottom right: time evolution of the fragment radius, central and effective temperature as well as the disc temperature at the fragment’s location.