Appendix A: Details on modified clusters

Figure A.1: A modified distribution nt(a) of satellites in the sample. Columns (i)-(iv) show the modified distributions which are developed following the interaction with the corresponding disc models. (The models are listed in Sect. 3.) The last two columns reflect the structure of composite models consisting of different regions, as indicated on top. The evolution of the composite models can be compared with the case of individual simple models from which they are constructed. In each graph, line types of the curves reflect different views of nt(a) profiles (see the text). First, snapshots are shown with varying time (top row; $M_{\ast}=1~M_{\odot}$); second, the fraction of satellites aligned with the disc is extracted from the whole sample (middle row); third, the quasi-stationary states are plotted for different $M_{\ast }$ (bottom row). See the Appendix for exact interpretation of different rows of graphs.

We can expand the results of Sect. 3.2 in slightly more detail by examining how the modified cluster distribution depends on the adopted accretion regime and how the quasi-stationary state is eventually established. There is a qualitative difference from the simpler case discussed in the main text, because varying surface density profiles of the disc models (i)-(v) introduce potentially observable concentrations of satellites at certain radii. Also, it is instructive to recover in the graphs different influences resulting from orbital decay, discussed for individual satellites in previous paragraphs (Sect. 2).

In Fig. A.1, the first four columns correspond to simple models adopted from our list. In the remaining two columns, the disc consists of a conjunction of two different models, joined at a given radius: The model denoted (ii)+(iii) is formed by the inner (ii) and the outer (iii) part of the gas pressure dominated disc with the transition at  $100r_{\rm {}g}$. Similarly, the model denoted (i)+(iv) is a conjunction of the radiation pressure dominated disc spreading up to  $10^3r_{\rm {}g}$ with the outer, self-gravitating disc (Collin & Huré 1999). The model parameters determine the final form of plotted curves in their mutual interplay together with the mass $M_{\ast }$ of the cluster members, which remains the characteristic influencing the efficiency of individual contributions to the cluster evolution.

Given a, the values of n_t(a) are distinguished from each other if a substantial fraction of the satellites are inclined in the plane of the disc without opening a gap in it. In such a situation the migration is faster for more massive stars whose number density is then lowered with respect to low-mass stars. Radial distribution of the satellites is influenced by the mode of their migration in the disc, and as a consequence the satellites are transported at different rates towards the centre. As a result, a wiggle occurs in the n_t(a) curve. The particular value of a where the satellites are accumulated at an increased rate, forming a ring-type structure, depends on the details of the model. However, the trend to accumulation at some distance is seen in various other situations and it leads to a bias in the mass function of the cluster.

For example, when the model (ii) is considered, the stars proceed under the regime of density wave excitation in the disc plane. The critical point (below which the gap would have been opened) lies at too small distances, , even for the largest considered mass $M_\ast=10~M_{\odot}$. Another situation develops in the case (iii) where the gap is opened below a=5, 150 and $7\times10^3$ for our three adopted values of satellite masses. This implies different shapes of n_t(a) for $M_\ast=M_{\odot}$ satellites with respect to more massive ones. Furthermore, in the marginally unstable disc, i.e. the case (iv), the point above which the gap could be opened is beyond the outer boundary of the considered region and, therefore, the distributions n_t(a) are sensitive to the mass of the satellites (most of them are inclined in the disc at this stage).

In order to maintain clarity of the graphs we omit the curves corresponding to the model (v). This case is similar to (ii) except for the fact that the gap opens for all three considered stellar masses, which makes the case (v) distributions very similar to each other, quite independent of $M_{\ast }$.

We could see that simple power-law profiles  $\Sigma_{{\rm d}}(r)$ lead to different distributions with respect to more realistic models, which are governed by different processes dominating at the corresponding radius. In discs that are a compound of several parts, the surface density profile contains several transitions between different regions, and hence it is more complicated than a simple power-law. The resulting effects on the boundaries resemble the transitions which we observed previously with simple discs when the mode of radial migration is changed. A local peak of n_t(a) occurs at radii where the satellites enter the disc region with slower radial motion, which is the situation of the composite model (ii)+(iii), or quite on the contrary a dip develops in n_t(a) as it can be seen in the case (i)+(iv).

We observed that a quasi-stationary distribution n_t(a), different from the initial one, was established below  $10^3r_{\rm {}g}$ for rather extended periods of time. However, a decline of n eventually arrives from large a at final stages when the supply of satellites is exhausted. The sample is eventually depleted. This can be also seen in Fig. A.1, the first (top) row, where time evolution is presented for one solar-mass satellites. Dimensionless times are t=10⁶ (solid line), 10⁸ (long-dashed), 10¹⁰ (short-dashed), 10¹²(dotted), and 10¹⁴ (dash-dotted). See footnote 3 for time units. The last curve is not visible in all plots because of a very large t; it can be noticed e.g. in the case (iv) where the evolution is slow and a fraction of satellites persists above .

Complementary to the above-described panels is the second row of graphs where only a part of n_t(a) distribution is plotted, corresponding to the satellites aligned with the disc. Line styles denote time in the same manner as in the first row. Naturally, the population of aligned bodies is less frequent, and hence some curves are missing in the second row with respect to their matching curves in the first row, if no satellites are brought to zero inclination.

Finally, the third (bottom) row shows n_t(a) profiles taken at t=10¹². At this moment a quasi-stationary state is already established in the form depending on the mode of satellite-disc interaction. The profiles are plotted for three different values of the satellite mass, corresponding to $M_{\ast}=1~M_{\odot}$ (thick solid lines), $M_\ast=3~M_{\odot}$ (long-dashed lines), and $M_\ast=10~M_{\odot}$(short-dashed lines). For example, one can check in column (ii) that the structure of the plot is ruled by the density wave regime and acquires the corresponding slope. (The initial slope 1/4 is also indicated with a light dotted line.) The curves clearly reflect the fact that the orbital decay (9) depends on $M_{\ast }$.