1. University of Colorado Boulder ASEN 6008 Interplanetary Mission Design Spring 2015 Kate Davis Tisserand Plots 1

2. University of Colorado Boulder  Gravity assists are obviously powerful.  They do have their limitations.  This lecture develops an understanding of the powers and limits of gravity assists ◦ Interplanetary transfers ◦ Moon tours ◦ Tisserand Plots 2

3. University of Colorado Boulder  Flyby Vector Diagram 3 Sun Planet

4. University of Colorado Boulder  What value of  provides the highest energy heliocentric orbit? 4  = 0 o Where on the heliocentric orbit does the flyby occur? Periapsis

5. University of Colorado Boulder  What value of  provides the lowest energy heliocentric orbit? 5  = 180 o Where on the heliocentric orbit does the flyby occur? Apopasis

6. University of Colorado Boulder  Locus of orbits with the same energy relative to the planet 6 Image credit: Strange, N., and Longuski, J., “Graphical Method for Gravity-Assist Trajectory Design,” Journal of Spacecraft and Rockets, Vol. 39, No. 1, Jan-Feb 2002.

7. University of Colorado Boulder  Assumes circular orbit, coplanar with spacecraft.  What if orbits are not in the ecliptic? 7 V ∞ contours become surfaces

8. University of Colorado Boulder 8 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune What type of transfer is represented by the red dots?

9. University of Colorado Boulder 9 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Hohmann Transfer from Jupiter to Uranus Hohmann Transfer from Earth to Jupiter

10. University of Colorado Boulder 10 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Hohmann Transfer from Jupiter to Uranus Hohmann Transfer from Earth to Jupiter V ∞ into Jupiter: 5.643 km/s V ∞ out of Jupiter: 3.324 km/s

11. University of Colorado Boulder 11 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Hohmann Transfer from Jupiter to Uranus V ∞ out of Jupiter: 3.324 km/s Where can a spacecraft COME from and hit this energy at Jupiter?

12. University of Colorado Boulder 12 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Hohmann Transfer from Jupiter to Uranus V ∞ out of Jupiter: 3.324 km/s Where can a spacecraft COME from and hit this energy at Jupiter? r p ~ 2.002 AU r a ~ 5.204 AU r p ~ 2.002 AU r a ~ 5.204 AU

13. University of Colorado Boulder 13 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Where can a spacecraft go if it takes a Hohmann to Jupiter? Hohmann Transfer from Earth to Jupiter V ∞ into Jupiter: 5.643 km/s

14. University of Colorado Boulder 14 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Where can a spacecraft go if it takes a Hohmann to Jupiter? Hohmann Transfer from Earth to Jupiter V ∞ into Jupiter: 5.643 km/s Well beyond Neptune!

15. University of Colorado Boulder 15 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune V ∞ Curves for JGA 1 3 km/s 5 km/s 7 km/s 9 km/s

16. University of Colorado Boulder 16 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune V ∞ Curves for JGA 1 3 km/s 5 km/s 7 km/s 9 km/s JGAs that will reach Uranus JGAs that can originate from Earth

17. University of Colorado Boulder  There are many ways you can plot a Tisserand Plot: ◦ Rp vs. Ra ◦ Rp vs. Period ◦ Rp vs. Specific Energy ◦ Some are more useful than others for different reasons.  The following Tisserand Plots may be found in: Strange, N., and Longuski, J., “Graphical Method for Gravity-Assist Trajectory Design,” Journal of Spacecraft and Rockets, Vol. 39, No. 1, Jan-Feb 2002. 17

18. University of Colorado Boulder  The following Tisserand Plots may be found in: Strange, N., and Longuski, J., “Graphical Method for Gravity-Assist Trajectory Design,” Journal of Spacecraft and Rockets, Vol. 39, No. 1, Jan-Feb 2002. 18 GOAL: Find missions that use VGAs, EGAs, and MGAs to reach Jupiter

19. University of Colorado Boulder  Illustrating a VEEGA to Jupiter 19 Launch places you on this orbit VGA EGA1 EGA2 JOI Resonant Orbit (separated by 200 km flybys)

20. University of Colorado Boulder 20

21. University of Colorado Boulder 21 The Launch Places you on this orbit. JGA Transfer to Saturn SGA Transfer to Uranus

22. University of Colorado Boulder 22

23. University of Colorado Boulder 23 Io Europa Ganymede Callisto Notice the amount you can move per flyby. Notice the resonant orbits. Notice the amount you can move per flyby. Notice the resonant orbits.

24. University of Colorado Boulder  Practical Benefits ◦ Illustrates how to move around the solar system with different V ∞ values. ◦ Illustrates the amount of change permitted with a minimum altitude flyby 24

25. University of Colorado Boulder  Drawbacks ◦ Does not illustrate phasing. ◦ Assumes coplanar motion. 25

26. University of Colorado Boulder  Félix Tisserand’s work.  The idea is that two observations of an asteroid or a comet may indeed be observations of the SAME object if they satisfy Tisserand’s criteria.  If the comet/asteroid likely encountered Jupiter between the observations then they may be the same object if their Sun-Jupiter Jacobi constant is the same.  Tisserand’s criteria: 26