4 Conclusions

Energy and angular-momentum losses of a satellite colliding with an accretion disc were examined. We explored how the relative importance of several secular effects depends on the parameters of the model, namely, the osculating elements of the satellite trajectories, surface density of the disc, and the mass and compactness of the orbiter bodies. We verified within the thin-disc analysis that hydrodynamical drag is, typically, more important for the long-term orbital evolution than gravitational radiation losses. Decay of the orbit due to interaction with the gaseous environment brings the orbiter gradually towards the center. This drag acts at different regimes depending on parameters of the orbiters and of the disc. The magnitude of the drag determines the rate of orbital decay and it influences the resulting structure of the cluster, especially the characteristic slope of n_t(a) density distribution.

Gravitational radiation losses dominate over the hydrodynamical dissipation if the disc has low density and/or the orbiter is of very high compactness (a neutron star or a black hole). The two influences are thus complementary and they can operate simultaneously at different regions of a given source, assuming the thin-disc scheme is valid farther away from the center, whereas advection-dominated flow takes over at distances of the order of $\approx$ $10^2r_{\rm {}g}$. One could thus expect the final stages of the orbital decay to be governed by emission of gravitational waves and by tidal interaction, while the initial transport from the outer cluster is ensured by other causes, possibly the hydrodynamical collisions together with dynamical friction. Our results are still only indicative in this part because several effects were ignored which must be taken into account in realistic models, namely, distribution of the satellite orbital parameters may be more complicated if two-body collisions between the orbiters are considered.

Another open question is the impact of star-disc interactions on the mass function of the satellites. We observed that $M_{\ast }$, $\Sigma_{\ast}$ and  $\Sigma_{{\rm d}}$ are the crucial parameters determining the cluster evolution, but the parameter range spans an enormous interval for different types of objects. Compact bodies have $\Sigma_{\ast}\approx10^9\Sigma_{\odot}=1.3\times10^{20}\,{\rm {}g\,cm^{-2}}$and hence they are affected less by collisions with the disc than solar-type stars, which are aligned with the disc plane more rapidly. Consequently, the initial mass function is modified towards a higher abundance of compact stars residing in inclined orbits, and vice versa. However, in the relatively dense environment of the disc, the stars accrete at an enhanced rate, they will soon gain sufficient mass and eventually collapse, producing additional compact bodies. A detailed discussion of their subsequent evolution under the influence of the disc environment remains beyond the scope of the present discussion. We only remark that solar-mass satellites can substantially multiply their masses during 10⁶ years (Collin & Zahn 1999), assuming that their own radiation and the effect of gaps do not halt further accretion. This is almost two orders of magnitude shorter than the expected quasar life-time (Haehnelt & Rees 1993) and hence the effects of star-disc interaction should not be neglected.

As a concluding remark, let us recall de nouveau that the final stages of an orbiter located near the center are relevant for gravitational wave experiments. The radiation losses govern the orbital evolution if the satellite is compact enough; on close orbits ( $e\approx0.9$, $a\approx10^2r_{\rm {g}}$) the influence of gravitational radiation is comparable to the effects of star-disc collisions, even if the medium is relatively dense, $\Sigma_{{\rm d}}\approx 10^5\,{\rm {}g\,cm^{-2}}$.

Acknowledgements

The authors acknowledge discussions with Suzy Collin about gravitationally unstable regions of accretion discs, and useful comments and suggestions by the referee, who helped us to improve clarity of the text. Support from the grants GAUK 188/2001, GACR 205/00/1685, and 202/99/0261 is also gratefully acknowledged.