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Chaotic diffusion of small bodies in the Solar System - Marie Curie ...

Chaotic diffusion of small bodies in the Solar System - Marie Curie ...

"§Figure 4: Surface

"§Figure 4: Surface of section for the 3/1 resonance. Thehomoclinic orbit crosses the separatrix of the pendulum andthe chaotic zone extends to Mars-crossing values of .ticularly interesting that, for resonances with , thereexist level curves which are closed (see Fig. 2). Even ifsuch a level curve intersects the §¡separatrix, chaotic motionwill be restricted to small eccentricities, due to energy conservation.This means that#low-eccentricity chaotic orbitscannot access the high-eccentricity regions and are therefore confined at low values of . In the non-averaged problem,chaotic orbits cannot be eternally confined. However,the jumps between different energy levels are controlled bythe magnitude of the remainder that is disregarded whenconstructing the Hamiltonian (6). Apart from resonanceswhich are close to other, lower-order, resonances, the effectsof the remainder should be miniscule for times comparableto the age of the solar system.The equations of motion resulting from Eq. (6) (fordifferent resonances) were numerically integrated, using a2nd-order symplectic implicit , scheme[17].@KBEF©'Surfaces ofsection for (¢the 8/3£ , FandD2'3/1§resonances' ¢,H) , ) are shown in Figs. (3)-£(4). Regular orbits appear as smooth curves onata surfaceof section, while chaotic orbits appear as a cloud of points.We have also superimposed the trace of theseparatrix of thependulum, as calculated in the adiabatic approximation, fordifferent values of ( ). The 8/3 and 3/1 resonances arerepresentative of the two types of mean motion resonancethat exist in the main belt. In the 8/3 ' ¢case, a narrow chaotic ¨ zone at moderate values of is found. Chaotic orbitsare confined in this narrow zone and low-eccentricityorbits cannot reach the high-eccentricity region. The bordersof the chaotic zone are almost tangent to the analyticallycalculated separatrix.In the 3/1 case, however, the topology of the surface ofsection is very different. Low-order resonances can forcethe frequency of perihelion to become'zero.This¢leads toa corotation resonance, when , which appearsas a set of fixed points on the surface of section. In Fig. (4)the fixed ¢ "points are located at¢ ¢¢ ¨ ¡ ? , £ ¢ ( ' ¢ ,composed of two knots which join smoothly at the unstablepoint. Each knot encircles one of the two stable islands(like the ones shown in Fig. 4). Chaotic motion is generatedin the vicinity of the homoclinic orbit. If the homoclinicorbit intersects the separatrix of the pendulum, thetwo chaotic regions merge and large-scale chaos sets in. Wenote though that the adiabatic approximation cannot wellreproduce the inner borders of the chaotic zone. Note thatnow almost circular orbits can random-walk to the higheccentricityregion. In [16] it was shown that asteroids inthe low-order mean motion resonances, associated with theKirkwood gaps, can escape from the main belt because ofthis mechanism.It is of course necessary to check now (i) the behaviorof orbits in the non-averaged 2-D elliptic problem and (ii)how the above results may change by including additionalperturbations in our model, such as the third spatial dimensionand the secular perturbations of Jupiter’s orbit due toits interaction with Saturn.3. ADDITIONAL PERTURBATIONSWe numerically integrated the orbits of a carefully selectedsample of fictitious asteroids, within all four models describedabove. All integrations were performed with a mixedvariable symplectic integrator [18], as it is implemented inthe SWIFT package [19].3.1. Numerical experimentsThe initial conditions were selected, by considering particlesinitially placed in the vicinity of several mean motionresonances of the 2DE model. We studied resonances upto order in the central belt ¨ ¢ ¨ ¦ ( )and ¥¤ § for the outer belt ¨ ¦ ¨ ); 22resonances are studied in total. For each resonance we integratedthe orbits of 60 particles, 30 with free eccentricity(# § #¢#¢ ¢ and 30 ¨ ¡#¢¡ ) with , ¨ sothat the resonance’s width is at maximum. The mean lon-" ". We set " £"gitude was selected so that the angle £ " critical correspondedto the stable equilibrium. Finally, the initial value of was varied, with respect to the nominal resonance location,within a range that agrees with our calculations forthe width of the resonance. A short term ( ¡£¢§¦yrs) integrationwas performed, in order to select 10 chaotic orbitsin each resonance for a 1 Gyrs integration. The selectionwas based on the behavior of the critical angle. For highorderinner-belt resonances it was not always easy to findchaotic orbits; the width of the chaotic domain can be verysmall. In such cases, slowly circulating or librating (butwith large libration amplitude) orbits were selected. Afterselecting our sample of 220 fictitious asteroids we verified,by means of Lyapunov exponent 2 estimates, that more than¢§¨of our selected particles indeed follow chaotic orbits.The orbits of these 220 fictitious asteroids were firstintegrated, for a time corresponding to years, in the¡#¢¥framework of the 2DE model. The orbital¡elementsof Jupiter%¨ ¡ ¢were ¨ ¢ ¦E? set to AU, (present epoch),and ). The same particles were#¢(at I stable) and ¢ " £¨ ¢ ¡, ( ¢ ' ) , unstable). In the £integrable approximation there exists a homoclinic curve,2 The maximal Lyapunov exponent is the mean rate of exponential divergenceof near-by orbits, in a chaotic domain.¡ ¢¡ ¢

T L(yrs)10 610 510 410 32-D E2-D P3-D E10 23-D P2.0 2.5 3.0 3.5 4.0a (AU)Figure 5: The Lyapunov time of the test-particles’ orbitsin all models. For most particles, the value

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