2 Satellite motion influenced by the disc

 The satellite body is assumed to cross the disc at supersonic velocity with a high Mach number, ${\cal{}M}\approx10^2$-10³. We recall (Syer et al. 1991; Vokrouhlický & Karas 1993; Zurek et al. 1994) that passages last a small fraction of the orbital period at the corresponding radius where they occur, and they can be considered as instantaneous, repetitive events at which the passing satellite expels from the disc some amount of material that lies along its trajectory. In terms of inclination angle i and the disc thickness h, the typical ratio of the two periods is $\delta{\approx}h/(r\sin{i})$; this quantity is assumed to be less than unity. Hence, for a geometrically thin disc ( $\epsilon\equiv{h/r}\ll1$), its influence upon inclined stellar orbits can be treated as tiny kicks (impulsive changes of their energy and momentum) at the points of intersection with the plane of the disc. This can be translated using the relation for the speed of sound, $c_{\rm {}s}=\epsilon{\Omega}r$, to the claim that the motion across the disc is indeed supersonic, which is also required for consistency. Smallness of $\delta$ means that the satellite remains outside the disc for most of its revolution around the centre; naturally this condition cannot be ensured at final stages when i is very small.

2.1 Collisions with the disc

Repetitive collisions lead to the gradual change of the orbiter's velocity $\vec{v}\rightarrow\vec{v'}$, which can be expressed by momentum conservation (Subr & Karas 1999):

$$(A+1)\,\vec{v'}=v_{\rm r}\,\vec{e_r}+v_{\vartheta}\,\vec{e_\vartheta} +\left(v_\varphi+Av_{_{\rm {K}}}\right)\vec{e_\phi}$$ (1)

in spherical coordinates ( $\vartheta=\pi/2$ is the disc plane). Here, $A(r)\,\equiv\,{\Sigma_{{\rm d}}}v_{\rm {rel}} \Sigma_{\ast}^{-1}v_\vartheta^{-1}$, $v_{\rm {rel}}$ is the relative speed between the orbiter and the disc matter, $\Sigma_{\ast}$ is the column density characterizing the compactness of the orbiter and defined by $\Sigma_{\ast}=M_\ast/\left(\pi{R^2_{\ast}}\right)$ (quantities denoted by an asterisk refer to the orbiter, $M_\ast \ll M$), and $\Sigma_{{\rm d}}(r)$ is the disc surface density. Rotation of the disc is assumed Keplerian, $v_\phi\,\equiv\,v_{_{\rm {}K}}=\sqrt{GM/r}$.

Let us consider a satellite on an orbit with semi-major axis a, eccentricity e, inclination i, and longitude $\omega$ of the ascending node. Equation (1) implies a set of equations which can be solved numerically in terms of the orbiter's osculating elements, while analytical solutions are possible in special cases (Subr & Karas 1999). As a useful example, we assume a power-law surface-density distribution in the form $\Sigma_{{\rm d}}=K\left(r/r_{\rm {}g}\right)^s\Sigma_{\odot}$( $K={\rm {const.}}$) and we adopt a perpendicular orientation of the orbit, $\cos\omega=0$. We find

$$a = C_1\,x_+\left(x_+^3+C_2\,x_-\right)^{-1},\quad y = 1+C_2\,x_-\,x_+^{-3},$$ (2)

where $x_\pm\equiv1\pm{x}$, $x\equiv\cos{i}$, $y\equiv1-e^2$. Strictly speaking, this formula concerns only the case of orbits intersecting the disc at two points with identical radial distances from the center but it can be used also as an approximation for orbits with arbitrary orientation. We remark that the relative accuracy $\Delta{a}/a$ of the determination of the semi-major axis is better than 15% with reasonable density profiles (s of the order of unity); numerical computations are not limited by assumptions about $\omega$ imposed in Eq. (2). Furthermore, one can write

$$A=\frac{K\Sigma_{\odot}}{\Sigma_{\ast}} \left(\frac{ay}{r_{\rm {g}}}\right)^s\;\sqrt{3-y-2x \over 1-x^2}\cdot$$ (3)

Integration constants C_1,2 are to be determined from initial values of a=a₀, x=x₀, and y=y₀. Then, the temporal history is obtained by integrating over the orbital period, ${{\rm d}}t=2{\pi}a^{3/2}/\sqrt{GM}$, in the form

$$t=\frac{2\pi}{K\sqrt{GM}}\frac{\Sigma_{\ast}}{\Sigma_{\odot}}... ...r{x}) - 2\bar{x} \right) \left( 1 - \bar{x}^2 \right) }}\cdot$$ (4)

Here, a factor missing in Eq. (19) of Subr & Karas (1999) is corrected (no other equations and graphs were affected by that omission). The orbital decay manifests itself in the gradual decrease of a, e and i, for which surface density of the disc is the main factor. The time derivative of the semi-major axis is

$$\dot{a}_{\rm {}col}={{\cal B}}y^{-q_4} \left[\frac{\Sigma_{\... ...t}}\right]^{-1} \sqrt{\frac{3-y-2x}{y(1-x^2)}} \; (2-x-y) \,,$$ (5)

where

$${\cal B}=-BcM_8^{q_1}\mu_{_{\rm {}E}}^{q_2} \left[\frac{\alpha}{0.1}\right]^{q_3} \left[\frac{a}{r_{\rm g}}\right]^{-q_4}$$ (6)

and $\mu_{_{\rm {}E}}\equiv\dot{M}/(0.1\dot{M}_{\rm E})$ is the accretion rate in units of Eddington accretion rate (with a 10% efficiency factor introduced). The factor B and power-law indices  $q_{1\ldots4}$ are determined by details of the particular model adopted to quantify the disc properties. Table 1 gives the values relevant for different regions of the standard disc as well as for the gravitationally unstable outer region (Collin & Huré 1999). Notice that the algebraic functional form of radial dependencies remains identical in all these cases (a power-law), and we can use it with convenience also later for different prescriptions of the satellite-disc encounters.

   

Table 1: Parameters in Eqs. (5), (7) and (9) describing the orbital decay in the case of different regimes and for different disc models.

Regime	Disc	B	q1	q2	q3	q4
col	(i)	$2.3\times10^{-9}$	0	-1	-1	-1
col	(ii)	$2.9\times10^{-5}$	$\mbox{$\,^{1}/_{5}$ }$	$\mbox{$\,^{3}/_{5}$ }$	$\mbox{$\,^{{-4}}/_{5}$ }$	$\mbox{$\,^{11}/_{10}$ }$
col	(iii)	$1.1\times10^{-4}$	$\mbox{$\,^{1}/_{5}$ }$	$\mbox{$\,^{7}/_{10}$ }$	$\mbox{$\,^{{-4}}/_{5}$ }$	$\mbox{$\,^{5}/_{4}$ }$
col	(iv)	4.05	$\mbox{$\,^{-9}/_{7}$ }$	$\mbox{$\,^{1}/_{7}$ }$	0	$\mbox{$\,^{37}/_{14}$ }$
col	(v)	$1.1\times10^{-2}$	-1	$\,^{1}/_{9}$	0	$\mbox{$\,^{13}/_{6}$ }$
gap	(i)	$1.3\times10^{-3}$	0	2	1	$\mbox{$\,^{5}/_{2}$ }$
gap	(ii)	$7.4\times10^{-8}$	$\mbox{$\,^{-1}/_{5}$ }$	$\mbox{$\,^{2}/_{5}$ }$	$\mbox{$\,^{4}/_{5}$ }$	$\mbox{$\,^{2}/_{5}$ }$
gap	(iii)	$2.8\times10^{-8}$	$\mbox{$\,^{-1}/_{5}$ }$	$\mbox{$\,^{3}/_{10}$ }$	$\mbox{$\,^{4}/_{5}$ }$	$\mbox{$\,^{1}/_{4}$ }$
gap	(iv)	$6.1\times10^{-13}$	$\mbox{$\,^{9}/_{7}$ }$	$\mbox{$\,^{6}/_{7}$ }$	0	$\mbox{$\,^{-8}/_{7}$ }$
gap	(v)	$2.2\times10^{-10}$	1	$\mbox{$\,^{8}/_{9}$ }$	0	$\mbox{$\,^{-2}/_{3}$ }$
dw	(i)	$1.8\times10^{-18}$	0	-3	-1	-5
dw	(ii)	$1.3\times10^{-10}$	$\mbox{$\,^{2}/_{5}$ }$	$\mbox{$\,^{1}/_{5}$ }$	$\mbox{$\,^{-3}/_{5}$ }$	$\mbox{$\,^{{-4}}/_{5}$ }$
dw	(iii)	$4.8\times10^{-9}$	$\mbox{$\,^{2}/_{5}$ }$	$\mbox{$\,^{2}/_{5}$ }$	$\mbox{$\,^{-3}/_{5}$ }$	$\mbox{$\,^{-1}/_{2}$ }$
dw	(iv)	$8.2\times10^{-7}$	$\mbox{$\,^{-5}/_{7}$ }$	$\mbox{$\,^{-1}/_{7}$ }$	0	$\mbox{$\,^{5}/_{14}$ }$
dw	(v)	$2.9\times10^{-4}$	-1	$\mbox{$\,^{-1}/_{9}$ }$	0	$\mbox{$\,^{5}/_{6}$ }$

Notation used in table.

Disc models: (i) ... Standard disc with $p=p_{\rm rad}$, s=3/2; (ii) ... Standard disc $p=p_{\rm gas}$, s=-3/5, electron scattering opacity; (iii) ... The same as (ii) but with s=-3/4 and free-free opacity; (iv) ... Marginally unstable self-gravitating disc (solar metallicity, optically thick), s=-15/7; (v) ... The same as (iv) but for zero metallicity, optically thin medium, s=-5/3; cf. Sect. 3 for further details.

Regimes of orbital decay: Orbital decay dominated by star-disc collisions (col), by gap formation in the disc (gap), and by density waves (dw), respectively.

2.2 An orbiter embedded in the disc

As drag is exerted on the satellite body, its orbit becomes circular and declined in the disc plane. Within this framework, orbital eccentricity and inclination are expected to reach zero values in the disc, so that quasi-circular trajectories are relevant near the center.

The orbit evolution is thus reduced to the situation which was addressed by several people (e.g., King & Done 1993; Takeuchi et al. 1996; Ward 1997; Ostriker 1999) in connection with formation and subsequent migration of bodies inside the disc. Two basic modes can be distinguished according to the disc properties and the orbiter mass. First, a gap is cleared in the disc if the satellite's Roche radius exceeds the disc thickness, , and simultaneously $M_{\ast}\,{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil $\disp... ...cr >\cr\sim\cr}}}}}\,M_{\rm {}gap}\,{\approx}\,\sqrt{40\alpha}\,\epsilon^{5/2}M$(Lin & Papaloizou 1986). Motion of the satellite is then coupled with the disc inflow, so that

$$\dot{a}_{\rm {}gap}={{\cal B}}.$$ (7)

On the other hand, if the satellite is unable to create the gap, the gas drag is imposed on it through quasi-spherical accretion. The resulting radial drift is weaker by a factor  $\epsilon\ll1$than the drift caused by density-wave excitation (Ward 1986; Artymowicz 1994). Hence, such a satellite migrates inward mainly due to the latter effect on the time-scale

$$t_{\rm {}dw}= \left(CM_{\ast}\Omega\right)^{-1}\epsilon^2 \left(\frac{M^2}{\pi{}r^2 \Sigma_{{\rm d}}}\right),$$ (8)

where C is a dimensionless constant of the order of unity. Using Eq. (8) and ${{\rm d}}a/{{\rm d}}t \approx r/t_{\rm {}dw}$ we obtain

$$\dot{a}_{\rm {}dw}=\frac{M_\ast}{M_{\odot}}\,{{\cal B}}.$$ (9)

Substantial differences in the satellite migration are thus introduced in the model already within this very simplified picture where the process of satellite sinking is driven by the gas medium. For the region of the gas pressure dominated standard disc and for a self-gravitating zero-metalicity model (v), the chance of opening the gap increases with decreasing radius. The situation is opposite in the case (i) (radiation pressure dominated) and for the solar-metallicity disc (iv).

2.3 The orbital decay due to gravitational radiation

Figure 1: Contours of f are plotted as a function of inclination and eccentricity for the case (ii). Function f(i,e) characterizes the relative importance of energy losses in Eq. (16). Values of f are indicated with contour lines. A saddle-type point develops between two neighbouring contours, f=0.150 and f=0.152, which are plotted with dashed lines. Its exact location depends on the disc model but the overall picture remains very similar in other cases, too.

The orbiting companion emits continuous gravitational radiation whose waveforms are of particular relevance for gravitational wave searches from compact binaries in the Milky Way. Possible approaches to their observational exploration have been discussed by several people (e.g., Nakamura et al. 1987; Dhurandhar & Vecchio 2001; Hughes 2001). The average rate of energy loss which the orbiter experiences via gravitational radiation over one revolution can be written in terms of orbital parameters (Peters & Mathews 1963),

$$\dot{E}_{\rm {}gw}=\frac{32}{5}\frac{G^4}{c^5} \frac{M^3M_{\... ...style{\frac{73}{24}}e^2 +\textstyle{\frac{37}{96}}e^4\right).$$ (10)

Corresponding to Eq. (10) are the change of semi-major axis

$$\dot{a}_{\rm {}gw} -1.28\times10^{-7}c M_8^{-1}\frac{M_{\ast}}{M_{\odot}} \left[\fra... ...\textstyle{\frac{73}{24}}e^2+ \textstyle{\frac{37}{96}}e^4}{(1-e^2)^{7/2}}\cdot$$ (11)

and the loss of angular momentum

$$\dot{L}_{\rm {}gw}= \frac{32}{5}\frac{G^{7/2}}{c^5} \frac{M... ...-e^2\right)^{2}} \,\left(1+\textstyle{\frac{7}{8}}e^2\right).$$ (12)

The above formulae (10)-(12) assume that the satellite star follows an eccentric orbit in Schwarzschild geometry of the central massive body. Gravitational radiation losses compete with those caused by star-disc encounters. We are thus interested in the relative importance of these mechanisms, which is characterized by the ratio ${{\cal R}}_{\rm {}col/gw}=\left[{{\rm d}}a/{{\rm d}}t\right]_{\rm {}col}/ \left[{{\rm d}}a/{{\rm d}}t\right]_{\rm {}gw}$:

$${{\cal R}}_{\rm {}col/gw}=\frac{5K}{32\pi}\frac{M}{M_\ast} \... ...ight]^{-1} \left[\frac{a}{r_{\rm {g}}}\right]^{s+5/2} f(x,e),$$ (13)

where, for whichever of the models described by the power-law density profile,

$$f(x,e)= \frac{\left(1+e^2-x\right)\left(1-e^2\right)^{s+5/2... ...}{24}e^2+\frac{37}{96}e^4}\; \sqrt{2+e^2-2x \over 1-x^2}\cdot$$ (14)

For the standard thin disc model (i) we obtain

 

$${{\cal R}}_{\rm {}col/gw}^{\rm {}(i)} \approx 1.8 \times 10^{-2}M_8\mu_{_{\rm {}E}}^{-1} \left[ \frac{\alpha}{0... ..._\ast}{M_{\odot}} \right]^{-1} \left[ \frac{a}{r_{\rm g}} \right]^{4} f(x,e)\,,$$ (15)

while for a gas pressure dominated disc (ii)

 

$${{\cal R}}_{\rm {}col/gw}^{\rm {}(ii)} \approx 1.5 \times 10^{2}M_8^{6/5}\mu_{_{\rm {}E}}^{3/5} \left[ \frac{\al... ...t}{M_{\odot}} \right]^{-1} \left[ \frac{a}{r_{\rm g}} \right]^{29/10} f(x,e)\,.$$ (16)

For the model (iii) we find

 

$${{\cal R}}_{\rm {}col/gw}^{\rm {}(iii)} \approx 8.6 \times 10^{2}M_8^{6/5}\mu_{_{\rm {}E}}^{7/10} \left[\frac{\al... ...M_\ast}{M_{\odot}}\right]^{-1} \left[\frac{a}{r_{\rm g}}\right]^{7/4} f(x,e)\,.$$ (17)

We show the functional form (14) in Fig. 1 where the factor f(x,e) determining the dependence of ${{\cal R}}_{\rm {}col/gw}$ on inclination and eccentricity is plotted for the case (16). In other words, f(x,e) represents that part of the drag ratio that is independent of the disc medium and the satellite physical properties; only the two mentioned orbital parameters play a role here. Typically, for a solar-type star it is only on eccentric orbits that f becomes small enough to bring ${{\cal R}}_{\rm {}col/gw}$below unity. The required eccentricity is rather high, and such a satellite would be trapped or disrupted directly by the central hole. Otherwise, ${{\cal R}}_{\rm {}col/gw}\gg1$ for  and for standard values of the disc parameters ( , $\mu_{_{\rm {}E}}\approx1$). This means that direct hydrodynamical interaction with the disc plays a dominant role in the orbital evolution of satellites crossing the disc, unless the medium is extremely rarefied (e.g., an ADAF; Narayan 2000). Notice that the point f(1,0)=0 (i.e. a fully circularized orbit inclined into the disc plane) is the exception, where the adopted approximation of instantaneous collisions breaks down.

Analogous to ${{\cal R}}_{\rm {}col/gw}$, one could explore the relative ratio of the hydrodynamical versus gravitational radiation losses in other regimes of the satellite-disc interaction. In this way, the relevant formulae (those which apply in the course of orbiter evolution) are

 

$${{\cal R}}_{\rm {}dw/gw} 2.3\times 10^{-12} M_8 \mu_{_{\rm {}E}}^{-3} \left[ \frac{\alpha}{0.1} \right]^{-1} \left[ \frac{a}{r_{\rm {g}}} \right]^8 ,$$ (18)

$${\cal R}_{\rm {}gap/gw} 5.2\times 10^4 M_8 \mu_{_{\rm {}E}}^2 \left[ \frac{\alpha}{0.1} \... ...ac{M_\ast}{M_{\odot}} \right]^{-1} \left[ \frac{a}{r_{\rm {g}}} \right]^{1/2} ,$$ (19)

for the inner, radiation pressure dominated disc (i), and

 

$${{\cal R}}_{\rm {}dw/gw} 1.0\times 10^{-4} M_8^{7/5} \mu_{_{\rm {}E}}^{1/5} \left[ \frac{\alpha}{0.1} \right]^{-2/5} \left[ \frac{a}{r_{\rm {g}}} \right]^{19/5} ,$$ (20)

$${\cal R}_{\rm {}gap/gw} 6.3\, M_8^{4/5} \mu_{_{\rm {}E}}^{2/5} \left[ \frac{\alpha}{0.1} ... ...rac{M_\ast}{M_{\odot}} \right]^{-1} \left[ \frac{a}{r_{\rm {g}}} \right]^{13/5}$$ (21)

for the middle region (ii). Here, the eccentricity-dependent factor was omitted upon the finding that orbits are almost circular when inclined in the plane of the disc. It is evident from Eqs. (13)-(21) that gravitational radiation can have a visible impact only at small a, especially when Eq. (18) applies (cp. also Fig. 5 and related discussion below).

Figure 2: Evolutionary tracks of the satellite are plotted in the plane semimajor axis a (units of  $r_{\rm {g}}$) versus time (years). Two very different cases are shown. a) Left: solar-type satellite ( $\Sigma _{\ast }=\Sigma _{\odot }$). The three curves correspond to different initial eccentricities (from top to bottom), e0=0, 0.4, and 0.8. Each curve is divided into three segments. Solid line indicates the period of motion outside the disc (when collisions occur with large orbital inclination); dashed line corresponds to density-wave driven motion in the disc; dotted line is the late stage with gap formation. b) Right: a compact satellite ( $\Sigma _{\ast }=10^7\Sigma _{\odot }$). Only the first stage is resolved here. In all cases, a0=100, x0=0. Gravitational radiation contributes to orbital changes independent of inclination but its impact is clearly important near the centre, where a decreases rapidly.

2.4 Evolutionary tracks of the satellite

Now we explore the evolutionary tracks of the satellite in the parameter space of osculating elements. We start by considering two effects: gravitational-wave losses in the approximation of Eqs. (10)-(12), and hydrodynamical drag acting on the satellite according to Eq. (1) twice per revolution. Dissipation operates with an efficiency depending on the type of satellite, and it provides a mechanism for the separation of different types of bodies in the phase space of a cluster.

Typical results of orbit integrations are presented in Fig. 2. Notice the big difference in time-scales relevant for non-compact stars (left panel) when compared with compact ones (right panel). In the former case, hydrodynamical drag is more pronounced. It gradually changes the orbital plane, while gravitational radiation can be safely neglected. The satellite sinks in the disc where the impulsive approximation (5) loses its validity and it is substituted by motion in the disc plane. Time-scales are generally longer in the latter, compact satellite case, although gravitational wave emission speeds up the evolution at very late stages ( ). For definiteness, we adopt the disc model (iii) and the condition of  $\tan{i_{\rm {}tr}}=h/r$ when the transition in the disc occurs. We checked by modifying the value of  $\tan{i_{\rm {}tr}}$ by a factor of 10that the qualitative picture of orbital evolution does not depend on its exact choice and also that numerical results remain similar.

2.5 Limitations of the adopted approximations

For simplification, we ignored the effects of gradual change of the mass of both the satellite and the central body. Furthermore, we assumed that the interaction has no effect on the structure of the satellites. This is a plausible assumption for stars with column densities much larger than that of the disc, while it is inadequate for giants which must quickly lose their atmospheres (Armitage et al. 1996). Also, we did not consider various effects acting on the disc structure (e.g. torques imposed on it by the dense cluster of stars; Ostriker 1983). Although all these effects will be important for a complete unified treatment of accreting black holes in active galactic nuclei, they can be neglected without losing the main physical effects influencing the satellite motion in the present simplified scheme.

We note that the significance of direct orbiter's collisions with the disc material is controlled by a characteristic time-scale $\tau\,\propto\,\Sigma_{\ast}/\Sigma_{{\rm d}}(r)$, which we expressed in terms of the orbital decay $\dot{a}$. Let us recall that dimensionless parameter characterizing compactness of the orbiter can be introduced in different ways. While $\Sigma_{\ast}/\Sigma_{{\rm d}}$ stands directly in the description of star-disc collisions, the usual factor $\varepsilon=GM_{\ast}R_{\ast}^{-1}c^{-2}$ determines the importance of general relativity effects near the surface of a compact body: $\varepsilon\approx10^{-6}\left(\Sigma_{\ast}/\Sigma_{\odot}\right) \left(R_{\ast}/R_{\odot}\right)$. We treat motion in the Newtonian regime and we only take the possibility of the satellite capture into account by removing the orbiter from a sample if its trajectory plunges too close to the central mass, below a marginally stable orbit. Another dimensionless quantity has also been designated as the compactness parameter when describing accretion onto compact objects, $\tilde{\varepsilon}=L\sigma_{_{\rm {}T}}r^{-1}m_{\rm {}e}^{-1}c^{-3}$. It considers the effect of radiation luminosity L acting through a cross-section  $\sigma_{_{\rm {}T}}$ in the medium, however, we can safely ignore radiation pressure on macroscopic satellites hereafter.

The gradual and monotonic decrease of eccentricity is overlaid with short-term oscillations if the disc mass is non-negligible (Vokrouhlický & Karas 1998). Also the satellites' inclination converges to a somewhat different distribution (instead of a strictly flattened disc-type system) when two-body gravitational relaxation is taken into account (Subr 2001; Vilkoviskiy 2001 - preprint). On the other hand, complementary to the scenario of the satellites grinding into the disc is the picture of enhanced star formation in the disc plane, in which case the stars are born with zero inclination (Collin & Zahn 1999). But these effects, as well as evaporation processes operating in the cluster, as suggested by various Fokker-Planck simulations (e.g., Kim et al. 1999), remain beyond the scope of the present paper.

We could see that different mechanisms (of which we considered particular examples) affect the orbital motion rather selectively, depending on the orbiter's size and mass. One thus expects separation of different objects in the cluster phase space. In order to verify this expectation we examine in the following paragraph a simple scheme which captures gradual changes in the structure of the cluster. Such a discussion is required: indeed, in the absence of sufficient resolution which would enable tracking of individual stellar paths in nuclei of other galaxies, one needs to inspect the overall influence on the members of the cluster, namely, the change of the radial distribution of the satellites in terms of their number fraction and average inclination.

Let us note that the accretion flow is supposed to remain undisturbed by the presence of the embedded cluster. This assumption gives an upper limit on the total number of stars inside the radius  $r_{\rm {d}}$($\approx$ $10^4r_{\rm {}g}$), and on the fraction of those dragged into the disc plane in this region. A simple smooth disc can be destroyed, especially in the process of gap formation (the case of sufficiently large $M_{\ast }$ and small h); the models (i) and (v) are particularly susceptible to the occurence of multiple gaps. Very roughly, if  $\approx2\%$ of the total number of N=10⁴ satellites get aligned with the disc at late times (a result of our computations for the model (iii)), then their Roche lobes might cover an area of the order $\approx0.01Nr_{\rm {d}}r_{_{\rm {}L}}$. This is just comparable with the total disc surface for $M_{\ast}\approx1~M_{\odot}$, however, recall that there are more conditions for the gap formation depending on the disc model. Also, effects of two-body relaxation and of satellite scattering in the gravitational potential of the disc are expected to decrease the fraction of aligned bodies when these effects are taken into account.