- Text
- Chaotic,
- Resonances,
- Solar,
- Orbits,
- Asteroids,
- Resonance,
- Neas,
- Planets,
- Particles,
- Population,
- Diffusion,
- Curie

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

"§Figure 4: Surface **of** section for **the** 3/1 resonance. Thehomocl**in**ic orbit crosses **the** separatrix **of** **the** pendulum and**the** chaotic zone extends to Mars-cross**in**g values **of** .ticularly **in**terest**in**g that, for resonances with , **the**reexist level curves which are closed (see Fig. 2). Even ifsuch a level curve **in**tersects **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 **the**refore conf**in**ed at low values **of** . In **the** non-averaged problem,chaotic orbits cannot be eternally conf**in**ed. However,**the** jumps between different energy levels are controlled by**the** magnitude **of** **the** rema**in**der that is disregarded whenconstruct**in**g **the** Hamiltonian (6). Apart from resonanceswhich are close to o**the**r, lower-order, resonances, **the** effects**of** **the** rema**in**der should be m**in**iscule for times comparableto **the** age **of** **the** solar system.The equations **of** motion result**in**g from Eq. (6) (fordifferent resonances) were numerically **in**tegrated, us**in**g a2nd-order symplectic implicit , scheme[17].@KBEF©'Surfaces **of**section for (¢**the** 8/3£ , FandD2'3/1§resonances' ¢,H) , ) are shown **in** Figs. (3)-£(4). Regular orbits appear as smooth curves onata surface**of** section, while chaotic orbits appear as a cloud **of** po**in**ts.We have also superimposed **the** trace **of** **the**separatrix **of** **the**pendulum, 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** ma**in** belt. In **the** 8/3 ' ¢case, a narrow chaotic ¨ zone at moderate values **of** is found. **Chaotic** orbitsare conf**in**ed **in** this narrow zone and low-eccentricityorbits cannot reach **the** high-eccentricity region. The borders**of** **the** chaotic zone are almost tangent to **the** analyticallycalculated separatrix.In **the** 3/1 case, however, **the** topology **of** **the** surface **of**section is very different. Low-order resonances can force**the** frequency **of** perihelion to become'zero.This¢leads toa corotation resonance, when , which appearsas a set **of** fixed po**in**ts on **the** surface **of** section. In Fig. (4)**the** fixed ¢ "po**in**ts are located at¢ ¢¢ ¨ ¡ ? , £ ¢ ( ' ¢ ,composed **of** two knots which jo**in** smoothly at **the** unstablepo**in**t. Each knot encircles one **of** **the** two stable islands(like **the** ones shown **in** Fig. 4). **Chaotic** motion is generated**in** **the** vic**in**ity **of** **the** homocl**in**ic orbit. If **the** homocl**in**icorbit **in**tersects **the** separatrix **of** **the** pendulum, **the**two chaotic regions merge and large-scale chaos sets **in**. Wenote though that **the** adiabatic approximation cannot wellreproduce **the** **in**ner borders **of** **the** chaotic zone. Note thatnow almost circular orbits can random-walk to **the** higheccentricityregion. In [16] it was shown that asteroids **in****the** low-order mean motion resonances, associated with **the**Kirkwood gaps, can escape from **the** ma**in** belt because **of**this mechanism.It is **of** course necessary to check now (i) **the** behavior**of** orbits **in** **the** non-averaged 2-D elliptic problem and (ii)how **the** above results may change by **in**clud**in**g additionalperturbations **in** our model, such as **the** third spatial dimensionand **the** secular perturbations **of** Jupiter’s orbit due toits **in**teraction with Saturn.3. ADDITIONAL PERTURBATIONSWe numerically **in**tegrated **the** orbits **of** a carefully selectedsample **of** fictitious asteroids, with**in** all four models describedabove. All **in**tegrations were performed with a mixedvariable symplectic **in**tegrator [18], as it is implemented **in****the** SWIFT package [19].3.1. Numerical experimentsThe **in**itial conditions were selected, by consider**in**g particles**in**itially placed **in** **the** vic**in**ity **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 **in**tegrated**the** 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. F**in**ally, **the** **in**itial value **of** was varied, with respect to **the** nom**in**al resonance location,with**in** a range that agrees with our calculations for**the** width **of** **the** resonance. A short term ( ¡£¢§¦yrs) **in**tegrationwas performed, **in** order to select 10 chaotic orbits**in** each resonance for a 1 Gyrs **in**tegration. The selectionwas based on **the** behavior **of** **the** critical angle. For highorder**in**ner-belt resonances it was not always easy to f**in**dchaotic orbits; **the** width **of** **the** chaotic doma**in** can be very**small**. In such cases, slowly circulat**in**g or librat**in**g (butwith large libration amplitude) orbits were selected. Afterselect**in**g our sample **of** 220 fictitious asteroids we verified,by means **of** Lyapunov exponent 2 estimates, that more than¢§¨**of** our selected particles **in**deed follow chaotic orbits.The orbits **of** **the**se 220 fictitious asteroids were first**in**tegrated, for a time correspond**in**g to years, **in** **the**¡#¢¥framework **of** **the** 2DE model. The orbital¡elements**of** Jupiter%¨ ¡ ¢were ¨ ¢ ¦E? set to AU, (present epoch),and ). The same particles were#¢(at I stable) and ¢ " £¨ ¢ ¡, ( ¢ ' ) , unstable). In **the** £**in**tegrable approximation **the**re exists a homocl**in**ic curve,2 The maximal Lyapunov exponent is **the** mean rate **of** exponential divergence**of** near-by orbits, **in** a chaotic doma**in**.¡ ¢¡ ¢

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’ orbits**in** all models. For most particles, **the** value

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