1 Introduction

There is ample evidence that central regions of galaxies consist of three main constituent parts: a supermassive dark object (presumably a black hole), a dense cluster of stars, and an accretion flow with disc-type geometry (for a review, see Rees 1998; Kato et al. 1998). Related to these three ingredients are typical length-scales: (i)  $r_{\rm {g}}=GM/c^2\approx1.5\times10^{13}\,M_8\,\rm {cm}$(gravitational radius of the central black hole expressed in terms of its mass ). (ii)  $r_{\rm {c}}=GM/\sigma^2\approx0.5M_8\sigma_{1000}^2\,$pc, the size of an inner star-cluster (where motion of its members is governed mainly by the black-hole gravitational field) written in terms of stellar velocity dispersion $\sigma_{1000}\equiv\sigma/(1000$ kms^-1). A typical number of stars in this region can be of the order $N\approx10^5$. Finally, (iii)  $r_{\rm {d}}\approx10^4 r_{\rm {}g}$ (the size of the disc). For the purpose of this paper these stars play a role of stellar-mass satellites orbiting the center. The disc itself will be described in the thin-disc approximation with the total mass $M_{\rm {d}}\ll M$.

In this paper we assume that the three mentioned subsystems coexist and they form an integral structure for a sufficiently long period of time. This is a reasonable assumption for a range of model parameters (discussed below), and the resulting geometry captures interesting effects of realistic systems, although it is simplified in several respects. The three subsystems are of very different nature, so that their symbiosis and mutual interactions pose a complex, ecological problem (e.g., Shlosman & Noguchi 1993; van der Marel & van den Bosch 1998; Zwart et al. 1997). We confine ourselves to a simple analysis in which the gravitational field of the system is dominated by the central compact mass while long-term dynamics of the satellites is influenced by dissipation in the disc medium and by gravitational radiation. The present paper is motivated mainly by indications of substantial concentrations of both stars and interstellar medium in the galactic central parsec. For example, a toroidal system of stars and gas has been proposed as a plausible explanation of brightness peaks observed in the nucleus of M 31 (Tremaine 1995; Bekki 2000). However, as we want to discuss also the regime in which inspiraling satellites are influenced by gravitational radiation in the field of the massive centre, we need to consider sub-parsec scales which are not yet resolved observationally. This appears relevant for the continuation (towards small length-scales) of the velocity dispersion versus mass relation (Ferrarese & Merritt 2000), which is so important in the discussion of the masses of supermassive black holes. Finally, core-collapsed globular clusters could be another class of relevant objects on scales of masses $\approx$ $10^4~M_{\odot}$, i.e. much less than galactic cores.

We proceed along the same course as several recent works in which further references can be found: Rauch (1999) studied stellar dynamics near a massive black hole, including the effects of general relativity on stellar trajectories; Subr & Karas (1999) examined the long-term evolution of orbital parameters of a satellite colliding with a thin Keplerian disc; Narayan (2000) estimated the relative importance of hydrodynamic drag versus gravitational-radiation decay of the satellite orbit (this author was interested primarily in the case of a body embedded in an advection-dominated flow very near the central hole); Collin & Zahn (1999), on the other hand, explored the fate of satellites moving on the periphery of the disc and at intermediate distances ($\approx$1 pc for $M_8\approx1$) where mutual interaction of the disc matter with the satellites triggers the starburst phenomenon and provides a mechanism of metal enrichment; finally, Takeuchi et al. (1996) and Ward (1997) discussed the problem of satellite migration via gap formation and density waves in the disc, and they gave a detailed overview of related topics which have been widely discussed also by others.

The orbital motion of satellites forming the central cluster is dominated by gravity of the central body; however, the long-term evolution of their trajectories is affected also by many other effects such as gravity of the background galaxy (e.g., Sridhar & Touma 1999), dynamical friction and tidal interactions (Colpi et al. 1999), self-gravity of the disc matter (Shlosman & Begelman 1989; Vokrouhlický & Karas 1998), repetitive collisions with an accretion disc (Syer et al. 1991; Rauch 1995), and gravitational radiation losses (Peters & Mathews 1963). The interaction between the members of the cluster and the disc are of interest also because they may gradually change the composition of the disc (Artymowicz et al. 1993), speed up the evolution of the satellite itself (Collin & Zahn 1999), and they can enrich the environment outside the disc plane (Zurek et al. 1994; Armitage et al. 1996).

Using simple estimates we compare the energy and angular momentum losses via gravitational radiation against the hydrodynamical drag acting on the satellites. We assume that their orbits have, initially, arbitrary eccentricities and nonzero inclinations with respect to the disc plane - the situation which is complementary to the case of planetary formation and their migration inside the disc. For the subsequent (low-inclination) stages of the orbital evolution we employ a simple prescription in which gravitational radiation losses still tend to bring the orbiter towards the centre while hydrodynamical effects are approximated in terms of motion through gas. We build our discussion on a model description (Subr & Karas 1999) in which the satellite suffers from dissipation of the orbital energy during repetitionary star-disc interactions but the disc itself does not change with time. We show evolutionary tracks followed by the satellite in the parameter space of its orbital osculating elements. The results are found quite sensitive to actual values of the disc density, the mass of the satellite, and to several other parameters of the model. We discuss the dynamical evolution of a cluster of satellites which gradually departs from its initially spherical shape and evolves into a flattened system. The initial stage of the evolution (when most of the orbits are still inclined with respect to the disc) is similar to the situation discussed by Pineault & Landry (1994) and Rauch (1995), but for the later stages we must adapt the prescription for star-disc interactions. See also Ivanov et al. (1999) for a complementary discussion of the gradual evolution of a circumbinary disc structure.