1. Current Trends in Spaceflight Research: From Galileo to Cassini and Beyond Mrinal Kumar Assistant Professor, MAE 306 MAE-A mrinalkumar@ufl.edu http://www.mae.ufl.edu/~mrinalkumar

2. Presentation Outline o PAST o PRESENT (including my research interests) o FUTURE

3. Presentation Outline o PAST Centuries ago Decades ago o PRESENT (including my research interests) o FUTURE

4. Many names, but not for the same thing… Celestial Mechanics: motion of celestial bodies Astrodynamics: astron + dynamics Orbital Mechanics: mechanics in orbit Space Dynamics: motion in space Astronautics: astron + nauticus: navigation through the stars Astronomy: astron + nomos : related field, observation of stars etc. Old school; only natural bodies like stars, planets, asteroids More contemporary; study motion under gravity in general, including natural objects and especially, spacecraft www.fascinatingly.com www.gpsmagazine.com www.nasa.gov IAU

5. The Cradle of Mathematics Geometrical Analysis: Tycho Brahe, Johannes Kepler Calculus: Isaac Newton, Gottfried Leibniz Calculus of Variations: Leonhard Euler, Joseph Lagrange, Pierre- Simon Laplace, William Rowan Hamilton Vector analysis: Josiah Gibbs Linear Algebra: Arthur Cayley Numerical methods: Carl Friedrich Gauss, numerous others

6. Long Long Ago: A History of Celestial Mechanics Galileo Copernicus Brahe Kepler Pre- 1700’s 1700’s Newton calculus (simultaneously /w Leibniz) law of universal gravitation laws of motion Principia Mathematica Euler calculus of variations, PDE theory rigid body dynamics fluid mechanics author of numerous papers Gauss probability theory system of equations boundary value problems Disquisitiones Aritmeticae Theoria Motus “pauca sed matura” Galileo Copernicus Brahe Kepler

7. A While Ago: A History of Celestial Mechanics 1800’s Jacobi calculus of variations rigid body dynamics special functions and PDEs Lagrange calculus of variations analytical mechanics Mecanique Analytique Hamilton calculus of variations canonical equations of mechanics quaternions, rotational dynamics Laplace special functions and PDEs linear equations potential theory

8. Problems of Interest: Two examples The Lambert’s Problem Kepler’s Time Equation

9. Not So Long Ago: Celestial Mechanics Late 1800’s – Early 1900’s Gibbs vector analysis matrix theory thermodynamics Einstein quantum mechanics general and special relativity modern physics Cayley matrix analysis differential equations linear algebra Since 1900’s No giants in particular, but numerous smaller contributions leading to development of the field Numerical methods Estimation theory: Kalman Filter Optimization theory and control Trajectory design and Navigation Sensor technology

10. Contemporary Celestial Mechanics (USA) UCLA: Sam Herrick (1911 – 1974) Purdue: James Longuski Kathleen Howell MIT: Richard Battin; Jonathan How Boulder: George Born, Robert Culp; Dan Scheeres, Hanspeter Schaub Texas A&M: John Junkins (Sam Herrick’s student at UCLA) Daniele Mortari, Kyle Alfriend Malcolm Shuster: Last academic appointment at UF

11. Some Current Problems of Interest o Deep Space Exploration/Advanced Mission Design o Aerocapture and Aerobraking Technologies o Formation Flying: Spacecraft Constellations o Novel Methods of Control o Space Debris Management o Uncertainty Handling in Space

12. Some Current Problems of Interest Explore: Gravity assists, Patched conics Deep Space Exploration/ Advanced Mission Design: www.nasa.gov

13. Some Current Problems of Interest Deep Space Exploration/ Advanced Mission Design: An Interplanetary Superhighway Objectives: Minimize fuel weight Maximize Solar system exploration Explore: Lagrange points, Halo orbits, Lissajous Orbits People: Dan Scheeres, Boulder Martin Lo, JPL Shane Ross, VTech www.nasa.gov

14. Some Current Problems of Interest Aerocapture/Aerobraking Technologies:

15. Some Current Problems of Interest Aerocapture/Aerobraking Technologies: Explore: Mars Global Surveyor, Mars Odyssey Entry corridor Atmospheric modeling Ablatives People: James Longuski (Purdue)

16. Some Current Problems of Interest Formation Flying: Spacecraft Constellations Simple formation: Docking Complex formation: Maintaining cluster shape for max coverage

17. Some Current Problems of Interest Flower constellations 4-1-4 3-1-50 Explore: Clohessy-Wiltshire Eqns. Discrete number theory GPS constellations Flower constellations People: Daniele Mortari (TAMU) John Junkins (TAMU) Hanspeter Schaub (Boulder) Jonathan How (MIT) Formation Flying: Spacecraft Constellations

18. Some Current Problems of Interest Novel Control Methods: High Earth Orbits Control Neighboring S/C within 10-1000 m Nearly “propellant-less control” Coulomb Spacecraft Formation Control People: Hanspeter Schaub (Boulder)

19. Some Current Problems of Interest Novel Control Methods: Navigation with Solar Sails Fragile spacecraft Operates on radiation pressure People: Dan Scheeres (Boulder) NASA/MFSC (Alabama)

20. Some Current Problems of Interest Space Debris Management: View of debris in LEO Expanded view of debris to include HEO Defunct spacecraft Broken up spacecraft New collisions Threat to active spacecraft Threat to astronauts www.nasa.gov

21. Some Current Problems of Interest Space Debris Management: Space shuttle window damage Collision between Iridium 33 and Cosmos 2251 on Feb 10, 2009, 490 miles over Siberia Explore: Cloud propagation Tethered Spacecraft People: Mrinal Kumar(UF) David Spencer (Penn State) Dan Scheeres (Boulder) NASA Orbital Debris Program www.nasa.gov

22. Some Current Problems of Interest Uncertainty in Space: Space collisions: Asteroid + Planet Debris + Spacecraft Space object tracking: Nonlinear Filtering Theory Asteroid 4581-Asclepius (1989-FC) Near Earth Asteroid (NEA) Program Potentially Hazardous Asteroids (PHA’s) Project LINEAR Project Spacewatch Project Neat www.nature-talk.com

23. Some Current Problems of Interest Uncertainty in Space: Space collisions: Asteroid + Planet Case in point: 99942 Apophis Case in point: 99942 Apophis

24. Some Current Problems of Interest Uncertainty in Space: Explore: Stochastic Systems Probability Flow Fokker-Planck equation Nonlinear Bayes’ Filtering People: Mrinal Kumar (UF) Don Yeomans, JPL Suman Chakravorty (TAMU) Dan Scheeres (Boulder)

25. Summary Spaceflight research has a LONG history It has continuously spurred development in mathematics Spaceflight research is extremely rich in mathematics Current space research is inherently multi-disciplinary Please see me if you want to work on one/more of the described problems!!