1. Институт прикладной математики им. М.В.Келдыша РАН Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

2. “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Keldysh Institute of Applied Mathematics, Russian Academy of Sciences Alexey Golikov, Andrey Tuchin

3. Essential objectives “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Orbit measurements: interpretation, information processing, ballistic and navigational mission support, etc. o ground supported trajectory measurements (GSTM): range range rate o measurements by the strup down Orbit determination: determination of all orbital parameters taken into account essential orbit perturbations Maneuver optimization: planning the scheme of maneuvers, error estimation of maneuver realization Landing on the surface of Ganimede: optimal scheme of descent session by using of thrusters

4. Ganymede Lander: Ganymede Lander: Mission Stages  gravitational maneuvers about Earth & Venus “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013  gravitational maneuvers around Ganymede & Callisto  preliminary elliptical orbit  circular polar orbit at the altitude of 100 km  prelanding orbit with low pericenter  session on Ganymede’s surface  Artificial satellite of Jupiter  Artificial satellite of Ganymede (ASG)  Launching of the spacecraft (SC)  Interplanetary flight Earth → Jupiter

5. Scheme of the stage ASG “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Preliminary orbit Orbital corrections GSTM Orbit period Inclination Eccen tricity Descent Prelanding orbit

6. Scheme of the stage ASG “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 1.Transition to preliminary elliptical orbit after braking at approach to Ganymede 2.Series of GSTM for orbit determination 3.Orbital corrections of orbit period & inclination to form circular polar orbit at the altitude of 100 km 4.Series of GSTM within 2 days for orbit determination 5.Bound orbital corrections (consisting of 2 corrections of the orbit period) to precise circular polar orbit 6.Circular polar orbit with science experiments 7.Orbital maneuver to form prelanding orbit 8.Series of GSTM on 2-3 adjacent circuits of prelanding orbit 9.Descent maneuver into given point on the surface of Ganymede

7. Perturbing forces “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

8. Essential perturbating factors “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Gravitational field of Ganymede (2×2): 2 nd zonal harmonics 2 nd sectorial harmonics Jupiter’s gravity attraction: circular equatorial orbit Rotation of Ganymede is synchronized with its orbit around Jupiter, there are resonance effects

9. Preliminary orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 1)Near equatorial and high eccentric orbit 2)Take into account the orbit evolution (perturbations) 3)Preliminary orbit with high eccentricity is very unstable: for e=0.5 it will destroy in 2 hours 4)For eccentricity e<0.3 equatorial elliptical orbits are stable 5)Polar elliptical orbits are unstable for e>0.01

10. Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

11. Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

12. Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

13. Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.3) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

14. Evaluation of the polar orbit Evaluation of the polar orbit (e=0.3) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

15. Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.1) “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

16. Transfer to circular polar orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013  Series of maneuvers to change the orbit period & inclination 1)Maneuver optimization by using Lambert’s problem with unfixed finite constraints 2)Solution of this problem is achieved by iterative procedure 3)Take into consideration an essential condition: polar orbit at high altitudes is unstable! 4)Supplementary constraint: to form polar orbit only on low heights & using “quasiequilibrium points”

17. Circular polar orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 1)Altitude 100 km 2)Series of GSTM within 2 days for orbit determination 3)Bound orbital corrections (consisting 2 corrections of the orbit period) to precise circular polar orbit 4)Science experiments (with orbit keeping corrections) 5)It needs to take into account the orbit evolution (perturbations) 6)Orbital maneuvers to form prelanding orbit with low pericenter

18. Circular polar orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Long-periodic perturbations of the orbit: where

19. Evaluation of polar circular orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

20. Evaluation of polar circular orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

21. Prelanding orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 1)Altitude of the pericenter 15 km 2)Altitude of the apocenter 100 km 3)Eccentricity 0.0158 4)Series of GSTM on 2-3 adjacent circuits of prelanding orbit to precise orbital parameters 5)Limit errors of GSTM are non greater than 0.2 mm/s and 20 m 6)Preliminary estimated errors of orbit prediction at the start of descent are non greater 2.5 m/s and 5 km

22. Evaluation of prelanding orbit “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

23. Descent Session “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 2 variants depending on the start time of descent: 1)24 hours => 16 hours of measurements GSTM 2)12 hours => 6 hours of measurements GSTM Nominal program of the thrust direction corresponds to the solution of the problem optimization Using Pontryagin’s principle of maximum Constraints depend on the problem definition Navigation is provided by the strup down

24. Mass before descent maneuver900 kg Mass of propulsion system215 kg Total burn4200 N Specific thrust319 s Dry mass385 kg Ganymede Lander module “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

25. Example of solution “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013 Solution by Pontryagin’s principle of maximum First stage of the descent session: from 15 km to 2 km Results of solution:  vertical velocity: 10 m/s forward to center of Ganymede  descent duration: 320 sec  fuel expenses: 422 kg  angle distance of descent: 7.4 deg

26. Direction of the Thrust “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

27. Altitude vs. Distance “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

28. Velocity vs. Time “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

29. Radial velocity “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

30. Transversal velocity “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

31. Contacts Alexei R. Golikov golikov@keldysh.ru Andrey G. Tuchin tag@keldysh.ru Keldysh Institute of Applied Mathematics, Russian Academy of Sciences “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013