Keldysh Institute of Applied Mathematics Russian Academy of

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

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

Essential objectives ü Orbit measurements: interpretation, information processing, ballistic and navigational mission support, etc. o ground supported trajectory measurements (GSTM): • range rate o measurements by the strup down ü Orbit determination: determination of all orbital parameters taken into account essential orbit perturbations ü Maneuver optimization: planning the scheme of maneuvers, error estimation of maneuver realization ü 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

Ganymede Lander: Mission Stages q Launching of the spacecraft (SC) Interplanetary flight Earth → Jupiter - gravitational maneuvers about Earth & Venus q Artificial satellite of Jupiter - gravitational maneuvers around Ganymede & Callisto q q 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

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

Scheme of the stage ASG 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 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Perturbing forces Height (km) 1 10 10000 1 E+01 1 E+00 1 E-01 Acceleration (m/s) 1 E-02 Central 1 E-03 c 20 1 E-04 c 22 1 E-05 Jupiter 1 E-06 Io Europa 1 E-07 Callisto 1 E-08 Sun 1 E-09 Saturn 1 E-10 1 E-11 1 E-12 1 E-13 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

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

Preliminary orbit 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 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Evaluation of preliminary orbit (e=0. 5) Altitude of orbit pericenter 100 90 80 Altitude (km) 70 60 50 40 30 20 10 0 0 1 2 Hours 3 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 4

Evaluation of preliminary orbit (e=0. 5) Altitude of ASG orbit 6000 5000 Altitude (km) 4000 3000 2000 1000 0 0 1 2 Hours 3 4 5 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 6

Evaluation of preliminary orbit (e=0. 5) Eccentricity of ASG orbit 0. 52 Eccentricity 0. 515 0. 51 0. 505 0. 5 0 1 2 Hours 3 4 5 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Evaluation of preliminary orbit (e=0. 3) Altitude of orbit pericenter 160 140 Altitude (km) 120 100 80 60 40 20 0 0 10 20 30 Days 40 50 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 60

Evaluation of the polar orbit (e=0. 3) Altitude of orbit pericenter 100 Altitude (km) 80 60 40 20 0 0 10 20 Hours 30 40 50 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 60

Evaluation of preliminary orbit (e=0. 1) Altitude of orbit pericenter 100 Altitude (km) 80 60 40 20 0 0 50 100 Hours 150 200 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 250

Transfer to circular polar orbit Ø 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” “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Circular polar orbit 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 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

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

Evaluation of polar circular orbit Min & Max Altitudes of ASG orbit 200 180 160 Altitude (km) 140 120 Hmax 100 Hmin 80 60 40 20 0 0 20 40 60 80 Days 100 120 140 160 180 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Evaluation of polar circular orbit Eccentricity of ASG orbit 0. 035 0. 03 Eccentricity 0. 025 0. 02 0. 015 0. 01 0. 005 0 0 20 40 60 80 Days 100 120 140 160 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013 180

Prelanding orbit 1) 2) 3) 4) 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 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 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Evaluation of prelanding orbit 100 90 80 Altitude (km) 70 60 50 40 30 20 10 0 0 5 10 Hours 15 20 “Ganymede Lander: scientific goal and experiments”, 5 -7 March 2013

Descent Session • 1) 2) • 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

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

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

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

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

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

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

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

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