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V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA)

Quasi-satellite orbits in the context of coorbital dynamics. V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA) A.V.Artemyev, A.I.Neishtadt, L.M.Zelenyi (Space Research Institute, Moscow, RUSSIA). Moscow , 2012. Quasi-satellite orbits.

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V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA)

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  1. Quasi-satellite orbits in the context of coorbital dynamics V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA) A.V.Artemyev, A.I.Neishtadt, L.M.Zelenyi (Space Research Institute, Moscow, RUSSIA) Moscow, 2012

  2. Quasi-satellite orbits (based on pictures from http://www.astro.uwo.ca/~wiegert/quasi/quasi.html) 1:1 mean motion resonance! Resonance phase j=l-l’librates around 0 (land l’are the mean longitudes of the asteroid and of the planet)

  3. Quasi-satellite orbits Historical background: • J. Jackson (1913) – the first(?) discussion of QS-orbits; • A.Yu.Kogan (1988), M.L.Lidov, M.A.Vashkovyak (1994)– the consideration of the QS-orbits in connection with the russian space project “Phobos” • Namouni(1999) , Namouni et. al (1999), S.Mikkola, K.Innanen (2004),… - the investigations of the secular evolution in the case of the motion in QS-orbit Real asteroids in QS-orbits: 2002VE68 – Venus QS; 2003YN107, 2004GU9, 2006FV35 – Earth QS; 2001QQ199, 2004AE9 – Jupiter’s QS ……………………

  4. Nonplanar circular restricted three-body problem “Sun-Planet-Asteroid” Secular effects: examples Parameters:

  5. Phase portrait of the slow motion: mathching of the trajectories on the uncertainty curve

  6. Scaling A – the motion in QS-orbit is perpetual

  7. Nonplanar circular restricted three-body problem “Sun-Planet-Asteroid” Timescalesattheresonance T1 - orbitalmotionsperiods T2 - timescaleofrotations/oscillationsoftheresonant argument (somecombinationofasteroidandplanet mean longitudes) T3 - secularevolutionofasteroid’seccentricitye, inclinationi, argumentofprihelionω and ascendingnodelongitudeΩ . T1<< T2 << T3 Strategy: double averaging of the motion equations

  8. Fast variables Symplectic structure Resonant approximation Scale transformation Slowvariables Slow-fast system • approximate • integral of • the problem SF-Hamiltonian

  9. Regular variables Relationship with the Keplerian elements:

  10. Averaging over the fast subsystem solutions on the level Н =ξ Problem: what solution of the fast subsystem should be used for averaging ? QS-orbit orHS-orbit?

  11. Asteroid 164207 (2004GU9)

  12. Asteroid 164207 (2004GU9)

  13. Asteroid 164207 (2004GU9 )

  14. Asteroid 164207 (2004GU9)

  15. Asteroid2006FV35

  16. Asteroid2006FV35

  17. Conclusions: • Row classification of slow evolution scenarios is presented; • The criterion to distinguish between the perpertual and temporarily motion in QS-orbit is established

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