MATHEMATICS OF THE CASSINI’S JOURNEY TO SATURN

1. MATHEMATICS OF THE CASSINI’S JOURNEY TO SATURN Michael P. Wnuk NASA Visiting Scientist at Jet Propulsion Laboratory/ California Institute of Technology

3. Launch of Cassini on October 15, 1997Two-stage rockets Titan IV-B and Centaur are used

7. Assembling the spacecraft at JPL/Caltech, Pasadena, California

8. Cassini spacecraft

9. CASSINI AT SATURN

11. Mathematical representation of rigid body motion with 6 degrees of freedom

12. CCONTENTS 1.     Introduction. An Overview of Cassini Mission -        Cassini as a Link Between Newton’s Orbital Mechanics and the SpaceExploration Program in 21st Century  2.     Numbers, Functions and Operators -        Numbers, -        Functions,-        Operators,-        Differential Equations 3.     Calculus Underlying Orbital Mechanics -        Motion in the central force field -        Orbits of planets and spaceships -        Navigating the Spaceship  4.     Scalars, Vectors, Quaternions, Matrices and Tensors -        Scalars that describe Cassini mission -        Vectors and quaternions pertinent to the mission -        Matrices and tensors applicable to the mission

13. Orbital and escape velocities • First two cosmic constants

14. Decrease of the orbital and escape velocity with an increasing distance from Earth • Velocity (orbital or escape) at the Earth’s surface is assumed as the normalization constant

15. Other constants pertinent to Orbital Mechanics • Escape from the solar system/ escape from our galaxy

16. The original sketch drawn by Jean-Dominique Cassini in 1676 (top), and the picture of Saturn received via radio signal from Voyager 2 in 1981 (bottom).

17. The original sketch drawn by Jean-Dominique Cassini in 1676 (top), and the picture of Saturn received via radio signal from Voyager 2 in 1981 (bottom).

18. Saturn as seen by Cassini’s camera, December 2007

21. A swing-by event by the Cassini spacecraft required an encounter with the planet Earth. Cassini’s altitude at the point of closest approach was 1176 km above the surface of Earth, less than the altitude of geostationary communication satellites that orbit Earth at 35,786 km above the sea level. The speed of Cassini spacecraft, though, equaled then 19.03 km/s, which exceeded the escape velocity at this particular height (10.29 km/sec) by a factor of 1.85.

22. Our own planet and the Cassini spaceship during the “swingby” maneuver on August 18, 1999.

23. Closest approach to Venus for a gravitational assist maneuver on April 26, 1998.

24. Jupiter Fly-by on December 30, 2000

25. Transfer maneuver using Carl elliptical orbit involving eccentricity that equals the second Carl constant, e. Radius of the outer circular orbit, which is the geostationary orbit, Rg= 42,164.1 km. Radius of the parking orbit, Rp= 14,984.3 km. Eccentricity of the transfer orbit is defined by the second Carl constant. Ratio of radii of geostationary and parking orbits is very close to the Euler number e=2.718281828. Equation of the transfer ellipse: e=second Carl constant

26. Examples of application of first two Carl constants. The shape of the transfer orbits is determined by the Carl eccentricities, S and e. Eccentricities of the transfer orbits are either S or e.

28. MGB[$] = English pounds MPL[$] = Polish zloties MDE[$] = German marks MSL[$] = Slovenian tolars MI[$$] = Italian liras MF[$$] = French francs MRU[$] = Russian rubles MMX[$] = Mexican dollars

31. BBlack box representation of the action oof a function (a), and an operator (b).

32. W INPUT OUTPUT Operator W (wicked witch in bad mood) W–1 Operator W–1 (wicked witch in good mood) Concept of an operator, W, and an inverse operator, W–1.

33. Population increase over three characteristic time intervals according to Eq. (6). Note that the starting number was 10, while the characteristic time T = 9 months.

34. A decaying wave-form is the solution of the differential equation subject to the initial conditions x(0)=0 and v(0)=1. Note that the wave is contained within an exponentially decreasing envelope

35. Velocity of the vibratory system consistent with the solution of the differential equation, shown here as a function of time.

36. Damper c k cdx/dt kx C Mass C m m x Fext x Fext Free body diagram revealing all forces Block of mass m suspended on a spring and a viscous damper and set into a vibratory motion

37. Two functions are shown x1(t) and x2(t). They resulted as the solutions to the initial value problem and the boundary values problem, respectively. Note that they both satisfy the second order differential equation (8).

38. Polar coordinates used to describe motion under central force condition. Quantities shown in (a) are used to define the initial conditions, while (b) shows two unit vectors aligned with the radial and transverse axes, er and e.

39. Newtonian Orbital Mechanics (1) • Acceleration vector along trajectory

40. Newtonian Orbital Mechanics (2) • Governing differential equations

41. Newtonian Orbital Mechanics (3) • Two DE equations reduce to just one equation

42. Newtonian Orbital Mechanics (4) • Solutions depend on the eccentricity and they turn out to be conical sections

43. =0 circle, =1 parabola, 0<<1ellipse, >1 hyperbola

44. Example of perturbation of an elliptical orbit of a planetoid (or a spacecraft) circling the Sun caused by an interaction with Earth’s gravitational field. This so-called “close encounter” event visibly alters the original orbit, as seen by the segment ABCD of the trajectory depicted in the figure. A “three body problem” has to be considered between points A and D, where an exchange of the mutual forces between three objects (Sun, Earth and the planetoid) must be accounted for. The closed form solution to such a problem is not available. In the Cassini mission this situation occurs each time the spaceship enters the “sphere of influence” of another planet on its path, such as Earth, Venus and Jupiter, which are used to accomplish a gravitational assist maneuver.

45. Transition from order to chaosDEMO

46. Phase diagram for a dynamic system consisting of a nonlinear pendulum subjected toviscous damping , and governed by the following nonlinear differential equation of the second order: d2/dt2 = -sin- d/dt. The graph is “well-behaved” and there is no indication of any instabilities or chaotic behavior.

47. This viewgraphs shows the phase diagrams when we deal with chaos, i.e., when the amplitude f is greater than the critical value of 1.87. Yet, in this totally chaotic type of motion, it is possible to find order at a deeper level (as revealed by the existence of an attractor, see the next Viewgraph).

48. Existence of the attractor, though, is indicative of the certain rules that apply to this chaotic motion. Attractor shown here is an example of a Poincare section, which has a fractal dimension of 2.52.

50. YAW h q k P’ O j i PITCH P y x ROLL Cartesian coordinate system (x, y, z) with the corresponding unit vectors (i, j, k). A vector PP’ can be represented by its components [PP’x , PP’y , PP’z], or by this equation: PP’ = (PP’x)i + (PP’y)j + (PP’z)k. If plane (x, y) is chosen as the plane in which Earth circles the Sun, the (x, y, z) coordinates shown here represent J2000 inertial reference frame.Unit vector h and the rotation q are used to define a quaternion.