DYNAMICS OF THE EDGEWORTH{KUIPER OBJECTS Andrea Milani Space Mechanics Group, University of Pisa Dipartimento di Matematica, Via Buonarroti 2, 56127 Pisa, Italia E-mail: \
[email protected]" WWW: \http://adams.dm.unipi.it/"
ABSTRACT The recently discovered Edgeworth{Kuiper Objects (EKO) have orbits in a region of the phase space whose dynamics had previously been investigated, either in connection with Pluto or seeking for stable regions for surviving primordial material. The theories are adequate to explain, and predict, the stability regions, and also to constrain theories on the origin of these bodies. To the contrary, the exact location and mechanism of the stability boundary, the slow di usion phenomena occurring nearby, and a quantitative theory of transport from the E{K belt to the Centaurs and short periodic comets regions, are the challenges to be met.
TRANS NEPTUNIAN PLANETS AND COMETS Before they were discovered, two main arguments had been used to advocate the existence of trans-neptunian objects: they can be labeled with the names trans-neptunian planets and transneptunian comets. All the speci c predictions, based upon residuals in the positions of the other planets (Lowell, Pickering), of trans-neptunian planets (with a mass larger than the mass of the Earth) have not been con rmed: the mass of Pluto is ' 1; 000 times too small. However, the systematic survey set up as a result of Lowell's legacy resulted in the discovery of Pluto (Hoyt 1980). A more convincing version of the argument simply points out to the sudden drop, in the averaged density of material left in the planetary system, immediately beyond Neptune (Stern 1991). Although little is known about the density of the protoplanetary disk as a function of the heliocentric distance, a sudden drop by several orders of magnitude appears unlikely. The observational evidence for protoplanetary disks extending well beyond 100 AU , as in the case of -Pictoris, makes this argument stronger. Thus, any dynamically stable region beyond Neptune, especially at low eccentricities and inclinations (where the accretion processes can be most e ective) is a potential reservoir of \planets", which are expected to be on regular and stable orbits. The size of these bodies depends upon the accretional process and the successive collisional evolution, which are not yet well constrained. This rst argument on trans-neptunian \planets" was also used by Edgeworth and Kuiper. 1
A trans-neptunian disk of comet-sized bodies has been advocated as an alternative source for short periodic comets. It solves several problems left open by the classical Oort cloud theory: as an example it is more stable with respect to stellar and molecular cloud perturbations, and appears to be more suitable to supply short periodic comets with the observed inclination distribution (Duncan et al., 1988). Thus, any dynamically unstable region beyond Neptune can supply comets to the observable region of the phase space (let us say perihelion q < 2:5 AU ). However, too quickly unstable orbits cannot be a long term source of comets; the strongly unstable regions need to be continuously replenished from a more stable region. Whether this transfer occurs because of escape of marginally stable orbits, becoming unstable after several 109 years, because of stellar perturbations, or because of collisions, is not yet clear; most likely all three phenomena contribute to the comet ux, in di erent proportions and with di erent dynamical routes. This second argument on trans-neptunian \future comets" was already proposed by Edgeworth and Kuiper, and more systematically investigated by Fernandez and Ip (1984). To assess the diculty of the present day research on the dynamics of the EKO, it is important to understand that we are trying to have it both ways, namely we would like to prove that there is a stable reservoir of primordially accreted bodies, and at the same time this very reservoir should be the main source of Centaurs and short periodic comets. This forces us to study the frontier region of marginally stable orbits, and to take care of the complex shape (with \fractal" boundary) of the stable and unstable regions. Essentially for this reason the research on the dynamics of the EKO is at the cutting edge of modern celestial mechanics, while the study of the stable orbits and of the unstable orbits by themselves would not require very innovative methods.
PLUTO AND THE 2:3 RESONANT GROUP Pluto can be dynamically stable, over time scales longer than a few 1; 000 years, only because of the 2 : 3 mean motion resonance with Neptune. Only when computers became powerful enough to allow either direct numerical integration of the outer solar system for millions of years (starting from Cohen and Hubbard, 1965) or the computation of semianalytical theories for long term perturbations (starting from Williams and Benson, 1971), the long term behaviour began to be understood: as an example, the libration of the argument of perihelion ! was discovered. Understanding of the long term stability of the orbit of Pluto was achieved in the 80's when the machines called, at the time, \supercomputers", became available (Kinoshita and Nakai 1984, Sussman and Wisdom 1988, Milani et al., 1989). These \super-computations" can now be reproduced with a personal computer, as I have done to produce the data for Figure 1. The best way to describe the long term behaviour of an orbit is to use some kind of proper elements. The technical de nition of the latter does not matter, the essential property is that they are quasi{ integrals, that is their fractional change is small even over a very long time, comparable to the age of the solar system. The methods to compute proper elements for Pluto are di erent from the one used for the major planets and for main belt asteroids; I am using here synthetic methods derived from Project LONGSTOP (Roy et al., 1988), and from the computation of proper elements for Hildas (Schubart, 1982) and Trojans (Schubart and Bien, 1987, Milani, 1993). For Pluto, the critical argument is 3P L ? 2NE ? $P L (with P L the mean longitude of Pluto, NE the mean longitude of Neptune, $P L the longitude of perihelion of Pluto), it librates around 2
0.15
80
0.1
79.95
Proper ampl. (DEG)
delta a (AU)
180 . The proper amplitude of libration is the principal mode of this libration; it can be obtained by Fourier transform of the data points with respect to the polar angle around the centre of Figure 1a, and is 79 in the angle and 0:15 AU in the semimajor axis for Pluto. The proper elements allow to detect very slow di usion processes, e.g. for Pluto a change by 0:15 of the libration amplitude in 100 Myr, as shown in Figure 1b.
0.05 0 −0.05 −0.1 −0.15
Fig.
1:
100 Pluto:
semimajor axis;
79.9 79.85 79.8
79.75 150 200 250 0 the corresponding 50 oscillation (a) the libration of the critical argument and crit. arg. (DEG) Time (Myr) (b) the change in the proper amplitude of libration with time, over 100 Myr.
in 100
An orbit is, by de nition, chaotic if it has a positive Lyapounov Characteristic Exponent (LCE); this means that two in nitesimally nearby orbits diverge by a factor exp(1) over the Lyapounov time, the inverse of the maximum LCE. Estimates of LCE can be derived along with any numerical integration, by solving the variational equations (the linearised equations of relative motion). Pluto has a Lyapounov time TL , of about 10 Myr. If Pluto is chaotic, does this mean it is unstable? Does this mean it can originate from anywhere? Numerical integrations even longer than the ones used here (e.g. Kinoshita and Nakai 1996) show that Pluto is actually stable in relative terms, with time-scale for chaotic di usion of several Gyr, and no macroscopic instability occurring over the entire age of the solar system. How could Pluto have been formed in such an eccentric and inclined orbit? Capture in resonance of an orbit initially nearly circular and coplanar is possible, if some dissipative process pushed Neptune and Pluto together. Capture in the 2 : 3 resonance, and pumping up of the eccentricity, is likely; capture in the ! libration, and pumping up of the inclination, is more dicult. The physics of such dissipative process does not matter, provided it is slow (as a result of the adiabatic invariant theory). A self consistent theory could account for the out-bound motion of Neptune as a result of the net exchange of momentum with the planetesimals scattered by the outer planets. There is order of magnitude agreement between such models (Fernandez and Ip 1984, 1996) and the total amount of dissipation required to raise the eccentricity of Pluto to its present level: it corresponds to an increase of the semimajor axis of Neptune by 5 AU. With the discovery of the Edgeworth{Kuiper Objects (EKO), the 2 : 3 resonance appears as a group; most have low inclination, only Pluto has ! libration. Their distribution, both in eccentricity 3
and in inclination, appears well explained by the dissipative model (Malhotra, these proceedings); thus the current location of the EKO can be used as a constraint on the dissipative processes occurring in the early phases of the outer solar system. Numerical integrations indicate long term stability for a large 2 : 3 libration region. The geometry of the stability boundary is very complex, and can be explained at low to moderate eccentricities by the secular resonances (Morbidelli 1996). The stability boundary at high eccentricities is still not entirely understood; orbits which are chaotic but undergo slow di usion can result from low strength resonances; e.g. for 1993 SB my numerical experiments show an irregular interaction between the libration of the main critical argument of the 2 : 3 resonance with Neptune and the circulation of the 1 : 3 main critical argument with Uranus (which are close to a 5 : 4 ratio).
THE OTHER BELT: RESONANCES AND STABLE REGIONS The long term stability of a trans-neptunian disk was established in the early 90's by analytic theory (Knezevic et al., 1991) and by numerical approximate integrations (Gladman and Duncan, 1990; Holman and Wisdom, 1993). Analytically we can plot the location of the main secular resonances. Together with the mean motion resonances, they de ne the unstable region. Higher order resonances (of both kinds) cause minor instabilities, not resulting in delivery to short periodic comet orbits. Theoretical predictions on the stability boundary have been con rmed by numerical integrations (Duncan et al., 1996), e.g. the destabilising e ect of the secular resonances between the perihelia and the nodes of the EKO and of Neptune. Large scale numerical experiments by approximate symplectic integrators show that an initially low eccentricity and inclination population is very stable for semimajor axes between 42 and 47 AU and beyond 49 AU. Below 42 AU there are almost only the stability islands corresponding to mean motion resonances. The predicted stability regions correspond to the locations of the EKO found so far. A small gap around 38 AU might be explained by the dissipative theory (see below). Proper elements have been computed for EKO with accurate enough orbits (that is observed at more than one opposition; see Marsden, these proceedings) by Knezevic and myself. Superimposed on the web of resonances, they show the dynamical structure of the belt (Figure 2). This representation shows well that the EKO belong to two dynamically distinct populations, with low eccentricity and non resonant, high eccentricity and resonant (2 : 3, but also 3 : 5 and 3 : 4).
THE INNER EDGE AND THE \KIRKWOOD" GAPS The Edgeworth{Kuiper Objects form another belt like the asteroid one. If for every asteroid with mean motion n we de ne a \dual" object with mean motion n0 such that inner resonances with Jupiter are transformed into outer resonances with Neptune: n=nJ = nN =n0 , then the \dual" asteroid has semimajor axis ? a0 = aN nnN0 2=3 = aNaaJ
Figure 3 shows \duals" of main belt asteroids (with absolute magnitude H < 13) and EKO, both with observed arcs not too short. 4
Inner Edgeworth−Kuiper belt, proper elements version 7.3 0.35 N2/3 U1/3 0.3
s−s8
Proper eccentricity
0.25
0.2
g−g8
0.15 N3/5 0.1 g+s−g8−s8 0.05
0 39
belt,43 proper elements version 40 Inner Edgeworth−Kuiper 41 42 44 45 7.3 46 Proper semimajor axis (AU)
47
16
14 U1/3 Proper inclination (DEG)
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10
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8 s−s8 g+s−g8−s8
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2 0 secular and mean motion resonances in the Edgeworth{Kuiper belt; the circles The 40 42 43 46 47 indicate the 39 proper elements of41the objects observed over44 at least 45two oppositions; lines across Proper semimajor axis (AU) circles indicate proper libration amplitudes for EKO locked in some mean motion resonance with Neptune; P indicates Pluto. Above: proper semimajor axis{proper eccentricity plane; below: proper semimajor axis{proper inclination plane. Fig.
2:
5
These gures suggest that the EKO discovered so far might belong to the inner edge of a possibly much larger trans-neptunian belt. Note the analogy of the inner edge of the Edgeworth-Kuiper Belt with the outer asteroid belt (3:3 to 4:3 AU ). The latter has several properties di erent from the ones of the main asteroid belt: there are no families, the size distribution is less steep, all the orbits in the region are strongly chaotic but those of the real objects are quite stable, many orbits are a ected by low order mean motion resonances (Milani and Nobili, 1894, 1985). The outstanding problem in the dynamics of the outer asteroid belt is a complete explanation of the Hecuba gap corresponding to the 2 : 1 resonance with Jupiter. Although it is currently believed that gravitational perturbations are enough to empty the gap, over a very long time span, the possible role of dissipative phenomena in the early solar system remains to be established. There is no symmetry between inner and outer resonances as long as dissipative processes are concerned. Thus the open question for the EKO: is there a gap at the location of the 1 : 2 mean motion resonance with Neptune? The problem remains open, because of two opposite arguments which could be used. The dissipative origin of the Pluto group at the 2 : 3 resonance would imply that a group should have originally formed at the 1 : 2 resonance (see Malhotra, these proceedings). On the other hand it is known since Message (1970) that the periodic orbit at the 1 : 2 resonance bifurcates into an unstable symmetric periodic orbit and a stable asymmetric one (with double period). This implies that a small amplitude of libration is impossible. Morbidelli et al. (1995) and Gallardo and FerrazMello (1996) have shown that alternation can occur between small amplitude libration around the asymmetric orbit and large amplitude libration; this would suggest that the long term stability region, if any, is very small, and that the 1 : 2 resonance might correspond to a gap. This problem could be solved by a better understanding of the dissipative phenomena in the early solar system, by a more complete theory of chaotic di usion in the apparently ordered regions inside the mean motion resonances, and ultimately by observations.
THE INNER EDGE AND THE ORIGIN OF THE COMETS Collisions and chaotic di usion (along the edge of the resonances) both contribute to the supply of unstable EKO, hence to the supply of Centaurs and short periodic comets. The two e ects should not be superimposed linearly, but they have a synergistic e ect: collisions can push an EKO from a slow di usion to a faster di usion region. For an EKO at ' 40 AU , the orbital velocity is ' 4 ? 5 km=s. The escape velocity from an EKO with diameter 200 km is ' 80 m=s. A change in velocity V ' 80 m=s corresponds to a change in semimajor axis by ' 1:3 AU . The relative velocity between EKO being 1 km=s, collisions are much less e ective in disrupting than in displacing; displacement of small fragments can be enormous. Moreover, unstable resonances are always nearby. This has many important implications on the interrelation between the collisional evolution of the EKO and their dynamical history. (1) Although EKO are very likely to be collisionally evolved (Farinella and Davis, 1996), the outcome of the collisions, that is the families, cannot be identi ed dynamically, because they are by far too dispersed. 6
The two belts 0.4
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o = EKO (multi) + = EKO (single) . = aster.
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The 40 45 50 two belts 55 60 65 Osculating semimajor axis (AU for TNO, dual for ast.)
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Edgeworth-Kuiper belt \dual" 65 image of70the asteroid main 35 40 45 superimposed 50 55to the60 75 belt. Circles indicate EKOOsculating observedsemimajor at more axis than(AU oneforopposition, TNO, dual forcrosses ast.) are single-opposition EKO, dots are \dual" asteroids. Above: osculating semimajor axis { osculating eccentricity plane; below: osculating semimajor axis { osculating inclination plane. Fig.
The 30
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(2) Comets can be transported trough resonances, from anywhere in the inner edge region (36 to 50 AU ), because a strongly unstable resonance is always within a distance (in velocity space) comparable to the escape velocity from an EKO of the size range observed so far. On the contrary, few comets can originate from near circular and near coplanar orbits well beyond the 1 : 2 resonance, unless a di erent mechanism plays some role (e.g. passing stars, for much larger semimajor axes). (3) Flux computations can not be performed consistently unless they account for both collisions and variable di usion rates. The di usion rates due to chaos are location dependent on a scale which is smaller than the average velocity of fragments of a large collisional event, thus the linear superposition of di usive and collisional e ects must give a completely wrong ux. (4) The size distribution of EKO will depend upon location, in particular upon semimajor axis and resonant state, and does not need to be the same for the EKO population and for the comets; if the latter are mostly collisional fragments fallen into resonances, their size distribution can be steeper.
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