Институт прикладной математики им. М.В.Келдыша РАН

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

2. Mathematical Model of the Spacecraft Landing on Ganymede’s Surface Alexey Golikov, Andrey Tuchin Keldysh Institute of Applied Mathematics, Russian Academy of Sciences “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

3. Essential objectives • Orbit measurements: interpretation, information processing, ballistic and navigational mission support, etc. • ground supported trajectory measurements (GSTM): • range • range rate • 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 “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

23. Descent Session • 2 variants depending on the start time of descent: • 24 hours => 16 hours of measurements GSTM • 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 “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

24. Ganymede Lander module • Mass before descent maneuver 900 kg • Mass of propulsion system 215 kg • Total burn 4200 N • Specific thrust 319 s • Dry mass 385 kg “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

25. Example of solution • 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 “Ganymede Lander: scientific goal and experiments”, 5-7 March 2013

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