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

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

3. 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 Quasi-satellite orbits Real asteroids in QS-orbits: 2002VE68 – Venus QS; 2003YN107, 2004GU9, 2006FV35 – Earth QS; 2001QQ199, 2004AE9 – Jupiter’s QS ……………………

4. Secular effects: examples Nonplanar circular restricted three-body problem “Sun-Planet-Asteroid” 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. Time scales at the resonance T 1 - orbital motions periods T 2 - timescale of rotations/oscillations of the resonant argument (some combination of asteroid and planet mean longitudes) T 3 - secular evolution of asteroid’s eccentricity e, inclination i, argument of prihelion ω and ascending node longitude Ω. T 1 << T 2 << T 3 Strategy: double averaging of the motion equations Nonplanar circular restricted three-body problem “Sun-Planet-Asteroid”

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

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 or HS-orbit?

11. Asteroid 164207 (2004GU9)

15. Asteroid 2006FV35

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