Their dynamics are controlled by Jupiter through frequent close encounters. The dynamical evolution of JF comets has been considered to be chaotic on a ...
Chaotic dynamics of planet-encountering bodies G. Tancredi
Dpto. Astronoma, Fac. Ciencias, Igua s/n esq. Mataojo, 11400 Montevideo, Uruguay. (gonzalo@ sica.edu.uy)
Abstract. The dynamics of two families of minor inner solar system bodies that
su er frequent close encounters with the planets is analyzed. These families are: Jupiter family comets (JF comets) and Near Earth Asteroids (NEAs). The motion of these objects has been considered to be chaotic in a short time scale, and the close encounters are supposed to be the cause of the fast chaos. For a better understanding of the chaotic behavior we have computed Lyapunov Characteristic Exponents (LCEs) for all the observed members of both populations. Lyapunov Characteristic Exponents are a quantitative measure of the exponential divergence of initially close orbits. We have observed that most members of the two families show a concentration of Lyapunov times (inverse of LCE) around 50-100 yr. The concentration is more pronounced for JF comets than for NEAs, among which a lesser spread is observed for those that actually cross the Earth's orbit (mean perihelion distance q < 1:05 AU). It is also observed a general correspondence between Lyapunov times and the time between consecutive encounters. A simple model is introduced to describe the basic characteristics of the dynamical evolution. This model considers an impulsive approach, where the particles evolve unperturbedly between encounters and su er \kicks" in semimajor axis at the encounters. It also reproduce successfully the short Lyapunov times observed in the numerical integrations and is able to estimate the dynamical lifetimes of comets during a stay in the Jupiter family in correspondence with previous estimates. It has been demonstrated with the model that the encounters with the largest e ect on the exponential growth of the distance between initially nearby orbits are not the infrequent deep encounters, neither the frequent and far ones; instead, the intermediate approaches have the most relevant contribution to the error growth. Such encounters are at a distance a few times the radius of the Hill's sphere of the planet (e.g. 3). An even simpler model allows us to get analytical estimates of the Lyapunov times in good agreement with the values coming from the model above and the numerical integrations. The predictability of the medium-term evolution and the hazard posed to the Earth by those objects are analysed in the Discussion section. Key words: chaos, dynamics, comets, NEAs, Lyapunov exponents
1. Introduction At the risking of being crude, we classify the small bodies orbiting between the planets in two groups: those which encounter a planet and those which do not. Let's just consider bodies in the inner Solar System; that is to say objects with perihelion distance within the orbit of Jupiter and low eccentricity. The non-encountering objects are the main belt asteroids and librating asteroids (e.g. Hildas and Trojans),
2 G. Tancredi while the bodies that encounter any planet are the comets and the Near Earth Asteroids (NEAs). Among comets we de ne those belonging to the Jupiter family (JF) as the ones with P < 20 yr, perihelion distance inside Jupiter's orbit (q < 5:2 AU) and Tisserand's parameter between 2 and 3 (approx.) with respect to the planet. Their dynamics are controlled by Jupiter through frequent close encounters. The dynamical evolution of JF comets has been considered to be chaotic on a short-time scale, but until recently there has not been estimates of the degree of stochasticity for a large sample of objects (Tancredi, 1995) (hereafter Paper I). The stochasticity is supposed to derive mainly from close encounters with the planets (especially with Jupiter) and during the interval between such encounters the cometary motion is quasi-regular and predictable. The word encounter or approach has been loosely used, since it can be considered as an encounter the very rare events when the comet enters the sphere of in uence of the planet or the more frequent situation when the object is at conjunction with the planet and the relative distance reaches a minimum (though it could be quite far). Which encounters have a greater contribution to the exponential divergence of initially close orbits, is still not known. A NEA is an object that approaches the Earth's orbit and generally crosses it; more precisely it is an object with asteroidal appearance (inactive) with aphelia Q > 1AU and perihelia q < 1:5 AU. Within this population we found some objects that resemble JF cometary orbits (as de ned before), but most NEAs have aphelion distances far too small to approach Jupiter. Some NEAs in cometary type orbits are expected to be defunct JF comets, i.e. devolatilized comets or objects with a refractory crust that quenches the outgassing. The dynamical link between these two populations is still questionable and the fractional contribution of JF comets to the NEA population is poorly known. As JF comets, a large proportion of NEAs are known to be chaotic on a short-time scale (Whipple, 1995). Frequent close encounters are supposed to be the cause of the chaos, but a proper model accounting for the short-time scale chaos is still to be done. We do not know if the encounters with the low-mass Earth are enough to account for the chaos. Furthermore, everybody agrees on the unreliability of "longterm" integrations to predict the evolution of a particular object but the de nition of "long-term" is still vague. The later point has important implications when we analyze the combined physical and dynamical evolution of JF comets, the origin of the NEAs or the hazard posed to the Earth by these objects. No conclusion can be drawn about the consequences of the dynamics into the physical characteristics of a particular comet or about the source (fate) of a particular NEA, unless
3 we know the past, or future, orbital evolution with certain con dence. Since the starting conditions are generally known with a rather low precision, we may not be able to track the evolution reliably for a long time; i.e. comets and asteroids might remember little from their past, they might have short dynamical memories (Paper I). For comets we also have non-gravitational forces that are poorly known and dicult to model, and they could be considered as uncertainties that reduce the predictability. Lyapunov Characteristic Exponents (LCE) are a quantitative measure of the exponential rate of divergence of nearby trajectories (see (Benettin et al., 1980) and references therein). By computing LCEs for all the already observed JF comets and NEAs we analyse the characteristics of the chaotic behavior of these populations. A LCE is de ned as the following limit ln ( j j=j0 j) (1) LCE = lim lim Chaotic dynamics of planet-encountering bodies
!0;t!1
!0;t!1
t
where j0 j and j j are the deviations of two nearby orbits at times 0 and t respectively. The number of di erent LCEs equals the dimension of the phase space. Let's consider a dynamical system composed by the Sun, the planets and the massless particle. Starting from an elliptical heliocentric orbit, we let the particle evolve due to the gravitational e ects of the Sun and the planets. After repeated encounters, the particle would nally be ejected out of the system into an hyperbolic orbit. According to the de nition of LCE, at t ! 1, the LCEs would be zero for any such particle. Nevertheless, we could calculate nite time estimates of the Lyapunov exponents, so-called Lyapunov Characteristic Indicators (Froeschle et al., 1993).
2. Results A detailed description of the method to compute the LCEs is described elsewhere (Paper I). We implement the renormalization methods presented by (Benettin et al., 1976), (Benettin et al., 1980). In particular the second method is used to nd the directions of the eigenvectors associated with the three positive eigenvalues (three positive LCEs). The system of variables to be used should be canonical. Although the numerical integrations are actually performed in a Cartesian rectangular system (for which position and momentum constitute a canonical set) we prefer to use a Delaunay set of variables to calculate the 6 dimensional phase space distance . Di erent tests show us that for low
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Table I.
Lyapunov times Fraction of JF comets all NEAs NEAs with cum. number q < 1:05 AU 25% 50 66 50 50% 60 117 81 75% 77 294 121 90% 100 516 209
LCE values the Delaunay set gives better results than the Cartesian one, although it does not make any di erence for the large LCE values found in this section (Tancredi, Roig and Sanchez in preparation). On one side we numerically integrate the whole sample of observed JF comets with the starting conditions corresponding to the last observed apparition taken from the 7th edition of the Catalogue of Cometary Orbits (Marsden and Williams, 1992). On the other side we integrate the whole sample of NEAs listed by D. Steel (personal communication) with the starting conditions corresponding to the updated version distributed on May, 1994. The integrations are performed with the RADAU 15th integrator (Everhart, 1985). Ful lling the de nitions of section 1, 145 comets and 307 NEAs were found. We perform integrations of a system composed by the Sun, 7 planets (from Venus to Neptune, with the mass of Mercury added to the Sun), one object and a test particle always close to it. We refer to Paper I for further details on the computing method. Each object was backward integrated for 20000 years and plots of log t vs. log for every object were visually inspected to decide if the horizontality is reached. For a few comets, we continued the integrations up to 200000 yr when log was still decreasing at the end of the integration period. Results for the 145 comets have been presented elsewhere (Paper I). In short, we found 5 objects with a hyperbolic escape; from the remaining 140 objects, 137 (98 %) reach the horizontal behavior. A clear concentration of values between 50 and 100 yr was observed, with very few cases over 150 yr. The median value is 60 yr, and the lower and upper quartiles are 50 and 77 yr, respectively (see Table I). In Paper I it was shown that the stochasticity of JF comets derives mainly from close encounters with the planets and in between two encounters the motion is quasi-regular.
Chaotic dynamics of planet-encountering bodies
5
Figure 1. Lyapunov times vs. the median time between two consecutive close encounters at a distance less than 3 Hill's radii for: a) JF comets, b) NEAs with q < 1:05 AU.
As the object crosses the orbits of many planets, we compare distances in terms of the ratio between the distance to the planet and the radius of the corresponding Hill's sphere, de ned as the Hill's radius. In the three body problem, the Hill's sphere of a planet corresponds to the volume centered on the planet for which its gravitational attraction is larger than the Sun's tidal attraction; and therefore, the perturbations to the orbital elements are more signi cant. The Hill's radius is given by 1 MP 3 1 (2) rH = a P 3 MP + M S where aP is the planet's semimajor axis, MP its mass, and MS the mass of the Sun. In Paper I we found a general correspondence between the Lyapunov times for JF comets and the mean time between two consecutive close encounters (Fig. 1a) (a distance of 3 Hill's radii is assumed as a criterion for close encounters). We do not claim a precise correlation between the two sets of values, we just point out the fact that both sets are concentrated between 50 and 100 yr. For JF comets 90 % of the close encounters are referred to Jupiter. The same procedure described above was used to study 307 NEAs and we did not nd any hyperbolic escape in the 20000 yr integration. Horizontality is reached by 218 objects (71 %) with a clear concentration of Lyapunov times (t ) between 50 and 150 yr. While the dynamical evolution of JF comets show drastic changes in the orbital elements, the evolution of NEAs is pretty smooth with only slight variations of the elements. It is thus meaningful to plot the Lyapunov times against
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Figure 2. Lyapunov times vs. mean perihelion distances over the time span of the integrations for NEAs reaching horizontality. The horizontal error bars represent the excursion in perihelion distance attained by the object in the 20000 yr integration.
an orbital parameter that characterizes the group, i.e. the mean perihelion distance q (Fig. 2, similar to Fig. 2 of (Whipple, 1995)). NEAs that cross the Earth's orbit (i.e. q < 1 AU) show less scatter in t compared to those that only have far approaches to the planet (i.e. q > 1 AU). Putting the limit at q = 1:05 AU (corresponding to objects that at least make an approach to the Earth's orbit within 3 Hill's radii), we nd 173 NEAs ful lling this criterion, with 141 (82 %) of them reaching the horizontality. By comparing the cumulative distribution of Lyapunov times of the whole and restricted samples (see Table I), we conclude that the sample of NEAs with q < 1:05 AU has a more concentrated distribution with a median of 81 yr and 75% of the objects with t < 121 yr. Our data shows two clear behaviors of the Lyapunov times as a function of q. For q < 1 AU we see a concentration around 50 and 100 yr, and for q > 1 AU a very scattered distribution. We have not tried to t any function relating t and q, as (Whipple, 1995) did, since we feel that the data sets show two distinct regimes, with a border at q 1 AU. Furthermore, from the model presented below, we do not nd any reason for the exponential law claimed by (Whipple, 1995). As for JF comets, a general correspondence between the mean time between encounters and the Lyapunov times is observed, in particular for NEAs with q < 1:05 AU (see Fig. 1b). The considerations made for JF comets about the signi cance of the correspondence are also valid in
7 this case. Here again, close encounters are de ned in terms of approaches to a certain number of Hill's radii of the di erent planets. Though most of the encounters are with the Earth (56 %); the contribution of Venus (20%), Jupiter (16%) and Mars (8%) are not negligible. We note a qualitative di erence between the dynamical evolution of the two populations of objects. Whereas for most JF comets low Lyapunov times are associated with an erratic behavior of all the orbital elements, for NEAs the macroscopic variation is smooth and constrained. The reason is quite obvious: the perturbations su ered by the JF comets during encounters with the massive Jupiter are much stronger than the corresponding perturbations of the NEAs during their approach to the inner planets. Nevertheless, as shown in Fig. 1, the "kicks" produced by the planets seem to be enough to induce the observed chaoticity. In spite of the previous dynamical di erence, we obtain Lyapunov times in the same range of values for the two families. The reason for this similarity is better understood if we consider that the direction of the fastest divergence in the Delaunay phase space is, for most of the time and all the objects, along the mean anomaly axis (l element), i.e. in all cases the maximum eigenvalues corresponding to the principal eigenvectors (the inverse of the Lyapunov times) are mainly associated with chaoticity along the orbit; the chaos in other directions could be observable on the other eigenvalues (this point should deserve further studies). The projection of the vector of maximum divergence on the mean anomaly axis has two components: one linearly increasing with time due to di erent mean motions (di erent a) of the initially nearby orbits and a second contribution due to frequent close encounters that produce new variations in a. For chaotic orbits as t increases the second contribution takes over the linear part, and the length of the vector grows exponentially. In Fig. 3, we show the projections of the maximum divergence versor (unit vector) on the 3-angle elements subspace (l; g; h) (l = mean anomaly, g = argument of perihelion, h = longitude of node). The projections along the 3-action elements are negligible and the versor is generally aligned with the l axis (a component with a value 1). The sporadic contributions of the angle elements g and h are more frequent and important for JF comets that for NEAs. Nevertheless, either for JF comets and NEAs the short Lyapunov times seem to be associated with maximum divergences along the orbit. The projection of the other two vectors associated with the positive LCEs are mainly along the angle elements g and h; but the contribution of the action elements, though erratic, are more frequent and important for JF comets than for NEAs. Chaotic dynamics of planet-encountering bodies
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1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 1 0.5 0 -0.5 -1 -5000
0
Time (yr AD)
Figure 3. Projections of the maximum divergence versor (unit vector) on the 3 angle elements subspace (l; g; h) (l = mean anomaly, g = argument of perihelion, h = longitude of node). The data is computed from a backward integration of the evolution of comet 9P/Tempel 1 for 20.000 yr starting from 1989 AD.
The particular case of P/Encke deserves further attention. P/Encke can be considered as a border case between JF comets and NEAs, it has only very distant encounters with Jupiter, just on the edge of our limiting distance of 3 Hill's radii. The orbit decoupled from Jupiter, resembles a NEA orbit. The evolution for a few hundred thousand
9 years shows no erratic variation of the orbital elements like most of the NEAs. Though, its Lyapunov time is among the highest ones for a JF comet (t = 177 yr), is still generally correlated with the time between encounters at distances less than 3 Hill's radii. It seems that the very far encounters with Jupiter are enough to induce a fast chaotic behavior. Chaotic dynamics of planet-encountering bodies
3. A simple impulse model In the previous section we found a general correspondence of the Lyapunov times with the time between two consecutive encounters. The critical approaching distance which leads to an exponential growth of the errors (i.e. a positive Lyapunov exponent) seems to be on the order of 3 times the Hill's radius of a planet. A possible explanation to be tested could be: encounters at closer distances produce larger errors, but they are rare; on the other hand far distance encounters are more frequent, but their contributions are not so relevant for the error growth. Furthermore, in Paper I we found that the largest contribution to the Lyapunov values corresponds to the time for which the object is close to the planet, reinforcing the conclusion that the chaos is induced by the encounters, and between two of them the object moves in a quasi-regular orbit. These hypotheses lead us to develop a simple model to test the previous explanations and to further understand the reasons for this fast chaos. Let's consider a 3D model with the Sun, a planet in a circular orbit and a particle moving in a planet-crossing orbit. We characterize the position of the particle by an action variable (the semimajor axis { a) and a position angle (the true longitude { ). Between the encounters the action variable is constant and the position angle growths linearly with time, at an instantaneous encounter, the action su ers a jump and the angle stays constant. Let's consider a test particle moving before the encounter on an orbit with the same a but the mean anomaly shifted by . At the next encounter the di erence in would become 0 , given by 0 = + e (3) where e is the variation introduced by the di erent values of a after the encounter. The outcoming orbits have a di erence in semimajor axis of ae , which depends on the pre-encounter con guration. The mean motion of the particles will thus di er by ne = (1=a3e=2 ). We can reasonably assume that the variation of the longitude of the node and the argument of perihelion are negligible compared to the variation
10 G. Tancredi on the mean anomaly in the short time between two encounters. At the time of the next encounter , the di erence in true longitude introduced by this encounter will be 3 1 (4) e = ne = 2 a1e=2 ae (For the planet we use the following standard units: semimajor axis ap = 1, mean motion np = 1, planet velocity vp = 1, period Tp = 2). The encounter of particles with a planet in the frame of the Restrict ed Three Body Problem (RTBP) has been analyzed by (Opik, 1976). Many of the equations presented below have been taken from his work. The Tisserand parameter (T ), an approximation of the Jacobi's constant of motion (J ), is related with the relative velocity at the point of intersection of the orbits or encounter velocity by U 2 = 3 ? T . The heliocentric velocity vector of the planet (~vP ) makes an angle with the encounter velocity vector U~ (the apical angle) (see Fig. 4). The semimajor axis is then related with U and by 1 = 1 ? U 2 ? 2 U cos (5) a
For a de ection angle , the pre- and post-encounter apical angles(i and e , respectively) are related by cos e = cos i cos + sin i sin cos (6) where is the angle between the direction in which the velocity is rotated and the plane containing the velocity of the planet (~vP ) and the relative velocity of the particle (U~ ). Note that in the planar case the angle can take the values (0; ), and eq. 6 reduces to e = i . Since the two particles have almost the same pre-encounter apical angle and , the di erence in the post-encounter angles are due to di erent impact parameters. The de ection angle is a function of the impact parameter b, the encounter velocity U and the planet mass , (7) tan 21 = b U 2 Considering that the planet is moving with an heliocentric velocity vector v~p along the y axis, the object heliocentric velocity vector is given by ~v = v~p + U~ . Suppose that at time t = 0 the planet is at the crossing point of the orbits, and the particle has been there at an earlier time t0 . At time t the particle is at (x; y; z ) and the planet at (0; yp ; 0). The mutual distance is q (8) d = x2 + (y ? yp)2 + z 2
Chaotic dynamics of planet-encountering bodies
11
ε ψ θe
θi
Ue Ui
P
vP
Figure 4. Frame centered on the planet (P ) showing the de ection of the relative velocity vector (from U~i ) to U~e ) (see text for description of the symbols).
Approximating with straight lines the motion of the two particles close to the crossing point of the orbits, the distance can be computed in terms of the relative velocity U and its y component (Uy ) by q
d = U 2 (t ? t0 )2 ? 2Uy vpt0 + vp2 t20
(9) Di erentiating the previous equation and equating to 0, we get an expression for the time of minimum distance Uvt tmin = t0 + y 2p 0 U
(10)
The minimum distance (equivalent to the impact parameter b under the assumption of rectilinear paths) is then given by "
b = v p t0 1 ?
#1=2 Uy 2 U
(11)
where Uy =U = cos i. The minimum distance in the planetocentric trajectory (d) would be just a minor correction to the value of the impact parameter for the range of U values we are considering (d2 = b2 ? 2=U 2 ). For a di erence in pre-encounter true longitude , there would be a di erence in the time of closest approach t0 = =ni (ni - preencounter mean motion).
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We di erentiate eqs. 5,6,7,11 , and substitute into eq. 4. Under the assumption of small de ection angles, i.e. e i and ae ai , generally called and a respectively, we get e = f (12) with 6 a sin g(; ; ) (13) f= 2 b U 1 + 2 b U2 where the function g(; ; ) corresponds to @e =@, obtained from the di erentiation of eq. 6. Note that g takes values in the range [?1; 1], and in the planar case it can only be 1. The di erence in true longitude at the time of the next encounter will then be 0 = (1 + f ) (14) The system evolves from one encounter to another following the previous evolution equation (eq. 14). The LCE could then be easily computed by P 1 X ln j0 =j = lim N ln j1 + fij (15)
= lim t!1 t
N !1
PN
i where fi is the fraction f at the ith encounter, and i is the time to
the next encounter.
3.1. Parameters of the model The parameters involved in eq. 13 depend on the planet and the characteristic of the population of encountering objects. Looking at the distribution of orbital elements, a typical semimajor axis a and encounter velocity U for each family can be computed (see Table II). The time between encounters depends on the minimum approaching distance. For a given distance it has a distribution with a long tail. Fig. 5a and 5b show the distribution of encounters with minimum distance less than 3 Hill's radii for JF comets and NEAs. From similar distributions at di erent distance levels, we nd the dependence of the mean time between two encounters on the minimum distance (d). The mean time between two encounters on minimum distance less than d vs. d is presented in Fig. 6a and 6b for JF comets and NEAs respectively ( lled dots). A power law function of the form d =k d0
(16)
13 with ?1:4 (dotted line) is a good t for both data sets in a wide range of distances. For d0 = 3 rH , we get k = 112 yr and k = 225 yr for JF comets and NEAs respectively. Note that these values are computed considering encounters with all the planets, but in the model we assume that all the encounters are due to a single planet: Jupiter for JF comets and the Earth for NEAs. Nevertheless, in the case of JF comets such a distinction is insigni cant since 98 % of the encounters are with Jupiter. For encounters of NEAs only with the Earth, we get k = 218 yr, because although the number of encounters is 60 % of the total, the number of objects encountering the Earth is also lower. In a planar problem there is a relation between the minimum distance (d) and the time between encounters with an exponent -1, while in an isotropic 3D problem the exponent should be -2. The transition between the two regimes should occur at d ' a sin I , where I is the inclination of the particle's orbit respect to the planet's orbit. (Milani et al., 1990), using an improvement of Opik's method (Opik, 1976), were able to develop a semi-analytical estimate of the dependence of with d for the case of low inclination orbits. The two regimes at small and large approaching distance are seen in their Fig. 3, with a transi tion at the range of 2-4 Hill's radius. An extension of Opik's method to the case of JF comets would be questionable, since the method is based on the assumption that the longitude of the node and argument of perihelion are uniformly precessing. This is not the case of JF comets, since generally one of the nodes of the comet's orbit is always xed at the distance of Jupiter, and the time between encounters that are strong enough to keep the node at Jupiter's distance are much shorter than the time for circulation of the node. A proper theoretical model for the dynamics of encounters of JF comets that accounts for the time and strength of the encounters would be desirable but it is out of the scope of this paper. Table II presents values of the parameters involved in eqs. 13 and 16 for each family. Chaotic dynamics of planet-encountering bodies
3.2. Results of the model Given some initial conditions we let the true longitude error evolve via eq. 14, which is evaluated at every encounter. The time between encounters is randomly taken from the distribution of Fig. 5 and the minimum approaching distance is randomly chosen from the power law function presented in eq. 16. Since the time between encounters would reduce to zero if we let the approaching distance increase up to the maximum distance between the orbits, we set a maximum approaching distance for an encounter to be considered, further encounters are disregarded.
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Figure 5. Relative distribution of the number of encounters with minimum distance less than 3 Hill's radii for: a) JF comets, b) NEAs. Similar plots are obtained for other distance levels.
Figure 6. Mean time between encounters as a function of the minimum distance of encounter (measure in Hill's radius) for: a) JF comets, b) NEAs. The data is presented as lled dots. The dotted line corresponds to a power law function, with index = ?1:4, adjusted to the data point at 3 Hill's radii.
Table II. Planet{Object Jupiter{Comet Earth{NEA
Model parameters
9:5 10?4 3 10?6
ay
U
y
0.85 1.2
0.35 1.1
rH (AU) z
kd?o y
0.35 0.01
6.26 1.66
in Standard units; the relevant planet has: semimajor axis ap = 1, mean motion np = 1, period Tp = 2). z rH { Hill's radius
y
Chaotic dynamics of planet-encountering bodies
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Figure 7. Evolution of the LCI vs. time for a model of a Jupiter family comet.
We vary the cut-o distance to analyse the e ects on the Lyapunov times. The angle (eq. 6) is randomly chosen between [0; 2]. After each encounter a new semimajor axis is obtained and the process is iterated. Figure 7 shows an example of the evolution of (computed by eq. 15) with time for a model of a Jupiter family comet with a cut-o distance of 3 Hill's radii (similar plots are obtained for NEAs). A di erent initial seed for the random number generator gives just minor variations on the Lyapunov times for a sucient long integration time. In Fig. 8 we present the dependence of the Lyapunov times on the cuto distance. If we only consider the very close encounters, we do not obtain Lyapunov times as low as if medium encounters were included. On the other hand the very distant encounters make little contribution to the Lyapunov times. An asymptotic behavior is obtained, with a knee around a distance 3-4 Hill's radius. We should note that at large model out of the limits distances we are pushing the application of Opik where it is usually used (just 1-2 Hill's radii). The Lyapunov times for the Jupiter family model is t 30 yr and for the NEA model is t 60 yr, a bit shorter than the times coming from the numerical integrations of Section 3 (compare with Table I), but good enough to consider the model successful. We conclude that medium distance encounters ( 3 Hill's radii) are responsible for the low values of the Lyapunov times and the fast rate of growth of the errors. The Lyapunov times are correlated with the frequency of such encounters.
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Figure 8. The variation of the Lyapunov times with the cut-o encountering distances imposed in the model. The distances are measured in Hill's radius. The lower curve corresponds to JF model particles and the upper curve to NEA ones.
For the planar case (i.e. = 0; ) and with a simpli ed version of the previous model, we are able to analitically obtain similar results (see Appendix I). With a simple model, where the only dynamical e ects considered are the close encounters, we are able to reproduce the results of the full integrations. The chaos is due to the kicks imparted to the orbital elements (in particular the semimajor axis) and its e ect on the position angle (the true longitude) via the variation of the mean motion. Though the true longitude growths linearly with time between the encounters, the slope of this variation is changing with time due to the kicks, and the e ect after multiplying the linear variations is an exponential growth. A plot of the evolution of 0 = vs. time would show a polygonal made of linear segments that at t ! 1 approaches an exponential function. On view of the model, we do not nd any argument for an exponential relation between the Lyapunov times and the perihelion distances of the NEAs, as (Whipple, 1995) claimed. The behaviour characterized by two regimes at di erent perihelion distances (see Section 3) is clearly understood, since NEAs with q < 1:05 AU have medium distance encounters with the Earth with a frequency of 100 ? 200 yrs, while NEAs with q > 1:05 AU have less frequent encounters with the less massive Mars (smaller Hill's radius in AU). Regarding the success of the model to reproduce the chaotic evolution of the planet-crossing bodies, we investigate other applications. The dynamical lifetime of a comet in the Jupiter family is de ned as
17 the time span between the comet's entrance and exit from the family. (Note that we are not talking about the total dynamical lifetime, i.e. from capture from the sources to ejection from the Solar System). A comet belongs to the family if the period is less than 20 yr (a < 7:2 AU) and the Tisserand parameter T less than 3. In our model T is constant and a is changing due to the kicks. Starting with a particle with a value of a given in Table II, we let it evolve for a time t until it reaches the edge of the family. Due to the reversibility of the process, the dynamical lifetime for this particle will then be two times t. Modelling a few thousands of particles we compute a median dynamical lifetime for the Jupiter family 4000 yr, in excellent agreement with previous estimates based on full integrations of ctitious comets in a two-planet system (Lindgren, 1992). Chaotic dynamics of planet-encountering bodies
4. Conclussion and Discussion We have found two families of objects, moving in the 6D phase space of the orbital elements, that have low and similar values of the maximum LCE; those are the Jupiter Family comets and the Near Earth Asteroids. The Lyapunov times are the lowest ever found for a particle moving around the Sun and perturbed by the planets. Low Lyapunov times are due to frequent kicks induced by close encounters with the planets; in particular those encounters that are not unrare and not very far, and at a distance a few times (e.g. 3) the radius of the Hill's sphere of the planet. A similar conclusion was obtained in studies of N-body system. (Goodman et al., 1993) showed that \the encounters which dominate the growth of errors in an N-body system are those on the dividing line between those causing large magni cations and those giving values close to unity. ... Wider enounters occur more often but their e ects tend to cancel. Closer encounters have a large e ect, but occur too rarely." In spite of the similar Lyapunov times for both populations, we recall a di erence in the macroscopic evolution of the orbital elements; while JF comets show an erratic behavior of the elements with frequent large jumps, NEAs' evolution is generally smooth and the variations look quasi periodic and constrained. In both families is possible to nd border cases: e.g the evolution of P/Encke resembles a NEA and some cometary candidates among the NEAs show erratic behavior (reinforcing the hypothesis of a dynamical link between members of the populations). Furthermore, there are important di erences in the dynamical evolution among the members of each family. From integrations of a large sample of NEAs for 200.000 yr, (Milani et al., 1989) made a
18 G. Tancredi classi cation of several groups of dynamical behavior. A few members experience transitions between the groups. The maximum Lyapunov exponent (the inverse of the Lyapunov times) do not seem to be enough to explain the important qualitative di erence stated above, since it is associated with a fast divergence along the mean anomaly axis. We still have to nd other parameters related to the chaotic behavior of the particles that express the macroscopic di erences. The other two positive Lyapunov exponents, and their associated eigenvectors, or the distribution of Lyapunov Local Indicators (Froeschle et al., 1993) or Lyapunov spectra (Voglis and Contopoulos, 1994) may be some alternatives. It is still to be shown whether the objects in both families belong to a common ergodic sea fully connected, i.e. they could go back and for between between the families; or in spite of the similar Lyapunov exponents the families and classes within them are separated. Furthermore, since the dynamical systems under study have singularities, it is not yet demonstrated that for those systems two regions with the same LCE have to be connected. From our data set and from the simple model we could not nd any support for the exponential relation between t and q for NEAs, as claimed by (Whipple, 1995). Nevertheless, we observed a relation between the Lyapunov times and the time between encounters that can be explained if we consider the evolution in phase space as a di usion problem and the results that relate these two quantities via the di usion coecient (Varvoglis and Anastasiadis, 1996). As mentioned in Paper I, a typical relative error in the orbital element determination for a JF comet on the order of 10?7 would lead to 100 % of discrepancy after 1000 yr. Non-gravitational forces are generally 10?7 times smaller than gravitational forces. Due to their erratic characteristic and our poor knowledge, they add to the prediction uncertainty of the past or future evolution of the objects. Even reducing the orbital determination error by many orders of magnitude, we still have to face the non-gravitational e ects that make the estimate of a dynamical memory of 1000 yr still valid. We start losing information about the position of the particle along the orbit, but due to the strength of encounters with Jupiter, this would lead to large errors in the other elements after the encounter. For NEAs the situation does not seem to be the same, the encounters with the Earth are not strong enough to produce large orbital variations, they just induce "microscopic" changes. Nevertheless, regarding the hazard associated with a collision of a NEA with the Earth, the precise knowledge of the mean anomaly is of capital importance. An e-folding time for the error growth in mean anomaly of 81 yr in connection with the typical poor orbital determination would imply large uncertainties in
19 the medium-term fate of these objects. In the simple model described above, the error, and consequently the covariance matrix, grow linearly with time with a certain slope until an encounter where the slope changes generally to a larger value. The covariance matrix is then valid for a given epoch and linear extrapolation could only be done between encounters. The covariance approach would completely fail when the so called "error ellipsoid" centered on the object's orbit (Muinonen and Bowell, 1993) contains at a certain orbital position, part of the planet's orbit. Though we start losing information along the orbit, an analysis of the covariance matrix shows a correlation with the other directions. (Bowell and Muinonen, 1994) computed the Minimum Orbital Intersection Distance (MOID ) for many NEAs. Taken the case of Toutatis, they found = 0:0058 AU. (Muinonen and Bowell, 1993), based on a set of 188 optical observations, made an orbital determination with an error in mean anomaly M = 5:410?6 deg, quite small due to the large observation set. The position error is then d a M = 2:4 10?7 AU. The time taken by the error to grow from d to is given by =d = exp(t=t ) or t = t log(=d). Toutatis's Lyapunov time of is among the shortest ones, t = 55 yr. Substituing these values, we conclude that the covariance approach would fail after 550 yr, but no reliable conclusion about the hazard posed by this asteroid can be made afterwards. Note that this estimate is made under the assumption of a correlation between the error along the orbit and along the directions normal to it. If such a correlation is weak, the prediction using the covariance matrix could be longer. This exempli es the short-term predictability of the hazard problem. Note that Toutatis is a very well observed asteroid, but is not an exceptional case regarding the low Lyapunov times, since small MOIDs are expected to be associated with low t . A continuous follow-up of the most hazardous objects would then be needed, even after the whole population of NEAs was catalogued. Observations just after an encounter will be critical for reducing the uncertainties. We nally note that the chaotic behavior of objects encountering planets is of di erent nature than the better studied chaos related to mean motion or secular resonances. We have already mentioned that in the former problem the phase space (manifold) is no longer compact, nor its implications on the validity of some fundamental theorems. Perturbation mappings are suitable for the resonance problem, while impulse mappings (like the one described in Section 3) are more appropriated for the planet-encountering problem. Several approximations have been developed to model the evolution of planet-encountering bodies (see e.g. (Froeschle and Rickman, 1988) and references tehrein) Chaotic dynamics of planet-encountering bodies
20 G. Tancredi but many of the tools developed to study chaotic systems are still to be applied to this kind of problem.
Appendix I - An analytical computation of the Lyapunov times The Lyapunov exponents ( ) are e ectively estimated by the renormalization procedure where you integrate for a short time step ( ) a particle and a nearby shadow and you calculate the log. of the ratio between the nal and initial distance at time step i(ln (ji j=j0 j) = si ). We then get N P
si (17) N !1 N
can then be understood as the mean of s (s ) divided by . In fact, it has been recently shown that not only the mean (i.e. ) is an invariant for a given orbit, but the whole distribution of s is invariant (Voglis
= lim
and Contopoulos, 1994) (they call the distribution: Lyapunov spectra; and the parameter s: stretching number). For a given probability distribution p(s), the mean is computed as s =
Z +1
?1
s p(s) ds
(18)
If the stretching number s is a function of a parameter b (s = s(b)), with a probability distribution p(b), the fundamental transformation law of probability states that p(s) ds = p(b) db (19) If by considerations of the physics of the problem, we know p(b) and s(b), we then will be able to analytically compute . Although these considerations are possible to be applied to any problem, they are better suited for the case where the stretching number is independent of the position on the phase space (at least this should be valid for the region of the phase space reached by the particle). Otherwise, one has a joint probability distribution p(b; I ), where I is the position in the phase space and b a parameter that characterize the perturbation to the orbit. In this case we have to know the frequency of occurrences of a given in nitesimal box of the phase space centered on I (and derive p(I )); and for each of these boxes; estimate p(b). We will apply the previous method to a simpli ed version of the impulse model described in Section 3. In the planar case we have shown that si = ln j1 + gi fij (20)
21 where g can only takes the value 1, and f depends on some model parameters (i.e. ; k; p0 ; U ). It is a weak function of the semimajor axis and strongly varies with the impact parameter b. We can then take a mean value of the semimajor axis. The stretching number can be expressed as: s = s(b; g). We are under the previous hypothesis of independency of the stretching number s on the position in phase space. Assuming that the variables b and g are independent we have Chaotic dynamics of planet-encountering bodies
Z +1 Z +1 1 s p(s) ds = 2 0 s(b; +1)p(b)db + 0 s(b; ?1)p(b)db ?1 (21) The integrations are e ectively computed up to a maximum value (bmax ) of the impact parameter due to the considerations given in Section 3.2. In section 3.1 we mention that by a semi-analytical model, (Milani et al., 1990) observed a transition situation between two regimes at the range of distance we are considering. The dependence of the time between encounters with b can be tted to a power law (eq. 16) with an index between -1 and -2 in these range of distances, though the exponent depends on the particular problem. This relation leads to the following expression for the probability distribution Z +1
p(b) =
b
?1
bmax bmax
(22)
(Note that we have disregarded again the small di erence between the impact parameter b and the minimum approaching distance d). Inputing the model parameters given in Table II, assuming = ?1:4 and for di erent values of the maximum impact parameter, we numerically compute the integrals of eq. 21 and substitute in eq. 17 to obtain the Lyapunov exponents. We obtain curves of Lyapunov times vs. maximum impact parameter of similar shape to those presented in Fig. 3.2. The Lyapunov times for bmax = 3rH are: 25 yr for JF comets and 80 yr for NEAs; a good enough estimate for such a simple model.
Acknowledgements The suggestions given by G. Valsecchi and A. Milani are very much appreciated. This work was partially supported by Consejo Nacional de Investigaciones Cient cas y Tecnicas (CONICYT) and Programa de Desarrollo de las Ciencias Basicas (PEDECIBA), Uruguay.
22
G. Tancredi
References
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