The model described was integrated for several values of masses, starting from , supposing the KBOs move on circular orbits. We studied the motion of KBOs both on inclined orbits, in order to study the evolution of the inclination with time, and on orbits on the plane of the disc, in order to study the drift of the KBOs from their initial position. The masses of the KBOs, M, were considered constant during the whole integration in order to reduce the number of differential equations to solve. Moreover we use and , N and m being the total number and the mass of the swarm of field particles in which the KBO moves. The assumption that field particles have all equal masses, m, does not affect the results, since dynamical friction does not depend on the individual masses of these particles but on their overall density. The results of our calculations are shown in Figs. 1-4.
In Fig. 1 we plot the values of versus time for planetesimals having masses and . The z coordinate is orthogonal to the plane of the disc and while . The brackets are mean values obtained averaging over a suitable number of orbital oscillations. As shown in the figure the decay of the inclination of the planetesimal of (dotted line) has a timescale larger than the age of the solar system and on the order of . The inclination of the more massive planetesimal (full line) decays in a time . This is due to the fact that dynamical friction effects increase with the mass M of the KBO. In fact (see Fig. 2) when the mass of the KBO is the decay time reduces to . Dynamical friction makes the orbits of KBOs undergo some collimation along the z direction, characterized by a low value of the dispersion velocity. This was expected because it is known that the larger the velocity dispersion along the direction of motion, the lesser the effect of dynamical friction (Pesce et al. 1992). Binney (1977) has found an efficient collimation of orbits along the main axis of the velocity dispersion tensor in the case of an anisotropic axisymmetric system, in which the principal velocity dispersions have constant values.
The result obtained is in agreement with previous studies of the damping of the inclinations of very massive objects by Ida (1990), Ida & Makino (1992) and Melita & Woolfson (1996). In fact in the semi-analytical theory by Ida (1990) the timescale for inclinations damping due to dynamical friction is almost equal to the two-body relaxation time
where is the surface density of small objects and is the Keplerian frequency. The timescales given by Ida's theory for our planetesimals of , , are respectively , , , in agreement with our results.
The effect of dynamical friction on the semi-major axis is plotted in Fig. 3 and Fig. 4. In Fig. 3 we plot versus time for a planetesimal of mass . Here r is the in-plane radial heliocentric distance of the planetesimal while and . As shown, the time required to a planetesimal of the quoted mass to reach is , which is an order of magnitude larger than the damping timescale. This is due, in agreement with what was previously told, to the fact that in the plane the dispersion velocity is larger than that in the z direction. Increasing the mass to the time needed for a planetesimal to reach decreases to (here ) (see Fig. 4). The threshold planetesimal mass, , that starts orbital migration is g. This mass scales with the disk density as:
We recall that we do not take into account the effects of the planets because we considered planetesimals initially at distances , but when the planetesimal moves towards the region of influence of planets the role of these must be taken into account.
We have some difficulty to compare this result with previous studies because the problem of the decay of the semi-major axis has not been particularly studied. So far, many people have assumed a priori that radial migration due to dynamical friction is much slower than damping of velocity dispersion due to dynamical friction. Therefore most studies of dynamical friction were concerned only with damping of velocity dispersion (damping of the eccentricity, e, and inclination, i), adopting local coordinates. Analytical work by Stewart & Wetherill (1988) and by Ida (1990) adopted local coordinates. N-body simulation by Ida & Makino (1992) adopted non-local coordinates, but did not investigate radial migration. Only the density wave approach by Goldreich & Tremaine (1979, 1980) and Ward (1986) considered radial migration. However, the relation of this approach to the particle orbit approach is not clear. Furthermore, a few numerical simulation has been devoted to investigate radial migration.
In any case we shall compare our result with that by Ward (1986) supposing that this density approach describes correctly the radial migration. Following Ward (1986) the characteristic orbital decay time of a disc perturber is given by:
where , and the nondimensional factor C is , depending on the disc's surface density gradient, k, and the adiabatic index, s, for a disc with , c is the gas sound speed (in a planetesimal system this must be replaced by the velocity dispersion). The timescale for the perturber to drift out of a region of radius r is given by (Ward 1986):
Using , calculating supposing that the disc mass is uniformly spread in the region and we find for a planetesimal having , a little larger than the value previously found. We have to remember that Ward's (1986) model supposes that the solar nebula is two-dimensional and that in a finite thickness disc other damping mechanisms may come into play perhaps invalidating Ward's result.
This last calculation shows that in a timescale less than the age of the solar system, objects of the mass of Pluto may move from the region towards the position actually occupied by the planet. This opens a third possibility to the standard scenarios for Pluto formation (in situ formation, formation at and transport outwards) namely that Pluto was created beyond , and then transported inwards. One of the problems of Pluto formation in the first two scenarios is that the growth of Neptune caused accretion to be inhibited and then it is necessary that the time for accreting an object of Pluto mass was shorter than the timescale of Neptune formation. This problem is not present in the third scenario because at heliocentric distances larger than Neptune never induced significant eccentricities on most orbits in the region. Hence the dynamical conditions necessary for growth may have persisted for the whole age of the solar system and consequently Pluto could have formed later than in the other two quoted scenarios. The mechanism responsible for the transport of Pluto from the quoted region to that nowadays occupied might have been dynamical friction.
Three possible objections to this last model are:
1) Moving to greater and greater distances both the disc density and the velocity dispersions decrease and consequently the accretion times increase. Can a Pluto-scale body form at distances of 70 AU?
2) A possible explanation for Pluto's orbital parameters is connected to outwards migration of Neptune (Malhotra 1993). If this works for Neptune, could it also work for Pluto-like objects in the KB? How much might Pluto have moved out? Could that compensate for the effect of the frictional motion?
3) How can one explain the odd orbital parameters of Pluto's orbit (, degrees to the ecliptic)?
Surely, as made clear by the first objection, formation of large bodies is more and more difficult moving away from the inner parts of Solar system. Although growth times at 70 AU are about 4-5 times longer than at 40 AU, Pluto-mass bodies can indeed be grown at this distance, from 1 to 10 km building blocks, in , if the mean disc eccentricity, , and if the KB mass interior to 50 AU was, as previously stated in Sect. 4, 30-50 and continued outwards with (see Stern & Colwell 1997b).
The answer to the second objection is the following.
Orbital migration of a planet can be accomplished through two mechanisms. In the first mechanism a planet and the circumstellar disc interact tidally which results in angular momentum transfer between the disc and the planet (e.g. Goldreich & Tremaine 1980; Ward 1997). The planet's motion in the disc excites density waves both interior and exterior to the planet. If the planet is large enough (at least several Earth masses), it is able to open and sustain a gap. It establishes a barrier to any radial disc flow due to viscous diffusion and it becomes locked to the disc and must ultimately share its fate (this is known as type II drift). In this case both inwards and outwards planet migration are allowed. In fact in a viscous disc, gas inside the radius of maximum viscous stress, , drifts inwards as it loses angular momentum while gas outside expands outwards as it receives angular momentum (Lynden-Bell & Pringle 1974). Neptune's outwards migration is due to the fact that the gas in the Neptune forming region has a tendency to migrate outwards (Ruden & Lin 1986).
If the planet is not able to sustain a gap, the net torque from the disc is still not zero and it migrates inwards in a shorter timescale (type I drift).
Pluto being a low mass planet can migrate only by means of type I drift, this means that it can only migrate inwards.
In the second mechanism a planet can undergo orbital migration as a consequence of gravitational scattering between itself and residuals planetesimals. If a planetesimal in a near-circular orbit similar to that of the planet is ejected into a Solar system escape orbit, the planet suffers a loss of orbital angular momentum and a corresponding change of orbital radius. Conversely, planetesimals scattered inwards would cause an increase of orbital radius and angular momentum of the planet. A single massive planet scattering a population of planetesimals in near-circular orbits in the vicinity of its own orbit would suffer no net change of orbital radius as it scatters approximately equal numbers of planetesimals inwards and outwards. However in some peculiar situations, such as that encountered in the region of Jovian planets, things go differently from this picture (Fernandez & Ip 1984). In particular, as Jupiter preferentially removes the inwards scattered Neptune planetesimals, the planetesimal population encountering Neptune at later times is increasingly biased towards objects with specific angular momentum larger than Neptune's. Encounters with this planetesimal population produce a net gain of angular momentum, hence an increase in its orbital radius. Evidently this situation is not the one present in the outer Solar system region, occupied by Pluto in our model. In other words, there is no reason to suppose that Pluto moved outwards like Neptune.
For what concerns the third objection, a possible answer is that Pluto gained high eccentricity and inclination in a similar way to that described by Malhotra (1993, 1995b). There has, of course, been much speculation as to the origin of the extraordinary orbit of Pluto (Lyttleton 1936; Farinella et al. 1979; Olsson-Steel 1988; Malhotra 1993). All but one (Malhotra (1993, 1995b)) of these speculations require one or more low-probability "catastrophic" events. In Malhotra's (1993, 1995b) model, Neptune's orbit may have expanded considerably, and its exterior orbital resonances would have swept through a large region of trans-Neptunian space. During these resonances sweeping, Pluto could have been captured into the 3:2 orbital period resonance with Neptune and its eccentricity and inclination would have been pumped up during the subsequent evolution.
The phenomenon of capture into resonance as result of some dissipative forces is common in nature. Weidenschilling & Davies (1985) studied resonance trapping of planetesimals by a protoplanet in association with gas drag. Many of the characteristics of this effect have been studied. Patterson (1987) and Beauge et al. (1994) have investigated its cosmogonic implications. The stability of the orbits and capture probabilities have been studied by Beauge & Ferraz-Mello (1993), Gomes (1995). Melita & Woolfson (1996) showed that a three-body system (Sun and two planets) under the influence of both accretion and dynamical friction forces, evolve into planetary resonance when the inner body is more massive. In general capture into a stable orbit-orbit resonance is possible when the orbits of two bodies approach each other as a result of the action of some dissipative process. The transition from a non-resonant to a resonant orbit depends sensitively upon initial conditions and the rate of orbital evolution due to the dissipative effects. Borderies & Goldreich (1984) showed that for single resonance and in the limit of slow "adiabatic" orbit evolution, the probability of capture for the 3:2 Neptune resonance is 100% for initial eccentricity less than and reduces to less than 10% for initial eccentricities exceeding 0.15 (see Malhotra 1995b).
In our model Pluto reaches its actual position in a time ( yr) larger than Neptune's orbital migration timescale ( yr; see Ida et al. 1999). Then Neptune was in its actual position when Pluto reached its own. Before the resonance encounter, Pluto has an initial low eccentricity , because of eccentricity and inclination damping due to dynamical friction, and its migration is very slow ( yr). The capture probability is then very high. The increase in eccentricity and inclination is naturally explained by Malhotra's theory.
© European Southern Observatory (ESO) 1999
Online publication: October 4, 1999