1. Method of Virtual Trajectories for the Preliminary Design of Multiple Gravity-Assist Interplanetary Trajectories Sergey Trofimov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of Physics and Technology Michael Ovchinnikov Keldysh Institute of Applied Mathematics, RAS Maksim Shirobokov Keldysh Institute of Applied Mathematics, RAS Moscow Institute of Physics and Technology

2. Contents Motivation for inventing a method Method of virtual trajectories (MVT) Benefits and flaws of the MVT Test case: Flight to Jupiter Conclusions 2/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

3. Mission feasibility study When studying the mission feasibility, a designer wants: To quickly estimate the best  V, the transfer time and launch windows for a number of planetary sequences To have an option of varying some mission constraints and imposing new ones (ideally without repeating the whole optimization procedure) To do all of this without involving skilled specialists in astrodynamics These demands are difficult to meet in case of multiple gravity-assist (MGA) trajectory design 3/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

4. Method of virtual trajectories Based on the fact that the orbits of planets are changing very slowly For a given planetary sequence, a database of all “geometrically feasible” trajectories can be constructed once and for all (“for all” means at least for several decades) The second, fast computing step: to screen and refine such a database of virtual trajectories 4/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

5. Classes of trajectories considered Basic class of trajectories: Coast heliocentric conic arcs Powered gravity assists (single impulse at the pericenter) Method of VT was also adapted to the trajectories with non-powered gravity assists deep space maneuvers (DSMs) At most one DSM is allowed on each heliocentric arc 5/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

6. Some basic concepts and assumptions 1)The orbits of planets: assumed to be closed curves fixed in space are discretized (i.e., represented as a 1D mesh) 2)Virtual trajectory (VT): consists of heliocentric conic arcs sequentially connecting the mesh points on the orbits of planets included in the planetary sequence chosen 3)A virtual trajectory is referred to as near-feasible if a spacecraft moving along it would fly by the mesh node on the planet’s orbit approximately (within some time tolerance) at the same time with the planet itself 6/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

7. Discretization of planetary orbits and beams of virtual trajectories 7/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

8. Patching of incoming and outgoing planetocentric hyperbolic arcs 8/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Powered GA maneuversUnpowered GA maneuvers

9. Screening of a VT database and refinement of near-feasible trajectories 9/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China Pruning infeasible trajectoriesRefinement of near-feasible ones

10. Comparison of computational costs Number of gravity assists CPU time for VT database screening and refinement, min* CPU time for standard Lambert-based approach, min* 10.5-22-3 23-610-15 38-1560-80 420-40>200 *All values of computational time are relative to a PC with 2.13 GHz CPU and 2Gb RAM 10/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

11. Benefits and flaws of the VT method +One and the same set of databases can be used many times for the design of various missions +Easy handles with imposing different additional constraints, without extra computational cost −Sensitive to step sizes during the discretization of planets’ orbits −Requires considerable hard disk space for saving all the VT databases (from 10 MB up to 1 GB for a long planetary sequence with 5 GAs) 11/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

12. Sample problem: Transfer to Jupiter Objective function: Constraints: No conjunctions during performing GAs or DSMs To check some standard planetary sequences: EVJ, EVEJ, EEVJ, EVEEJ 12/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

13. EVEEJ with powered GA maneuvers 13/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

14. EVEEJ with DSMs and unpowered GAs 14/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China This trajectory is similar to the baseline trajectory of Jupiter Ganymede Orbiter (JGO) mission

15. Comparison of trajectories obtained using the MVT with DSMs and in the JGO mission JGO trajectoryMVT with DSMs Launch11/03/202013/03/2020 Venus flyby01/07/202030/06/2020 First Earth flyby27/04/2021 Second Earth flyby28/07/2023 Jupiter approach04/02/202625/03/2026 00.01 00.07 3988 00.4 Escape velocity, km/s3.393.41 Approach velocity, km/s5.505.58 Duration, year5.96.0 15/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

16. Conclusions Based on a number of beforehand computed databases of virtual trajectories, a mission designer can: quickly estimate the possible mission timeline options (planetary sequence, launch date, transfer time) pick and choose the planetary sequence which is best suited to various constraints and scientific requirements change his mind and impose new constraints without a serious increase in time of mission feasibility analysis 16/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

17. Acknowledgments Russian Academy of Sciences (RAS), Presidium Program “Fundamental Issues in Investigation and Exploration of Solar System”, Subprogram “Mission Scenarios and Trajectory Design” Russian Foundation for Basic Research (RFBR), Grant No. 13-01-00665 17/17 64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China