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MATHEMATICS

OF THE CASSINI’S

JOURNEY TO SATURN

Michael P. Wnuk

NASA Visiting Scientist at Jet Propulsion Laboratory/ California Institute of Technology

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

Mathematical representation of rigid body motion with 6 degrees of freedom

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

Orbital and escape velocities

- First two cosmic constants

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

Other constants pertinent to Orbital Mechanics

- Escape from the solar system/ escape from our galaxy

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).

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).

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.

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

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

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

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.

MGB[$] = English pounds MPL[$] = Polish zloties

MDE[$] = German marks MSL[$] = Slovenian tolars

MI[$$] = Italian liras MF[$$] = French francs

MRU[$] = Russian rubles MMX[$] = Mexican dollars

BBlack box representation of the action

oof a function (a), and an operator (b).

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.

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.

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

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

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

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).

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.

Newtonian Orbital Mechanics (1)

- Acceleration vector along trajectory

Newtonian Orbital Mechanics (2)

- Governing differential equations

Newtonian Orbital Mechanics (3)

- Two DE equations reduce to just one equation

Newtonian Orbital Mechanics (4)

- Solutions depend on the eccentricity and they turn out to be conical sections

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.

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.

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).

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.

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.

MATHEMATICS

OF THE CASSINI’S

JOURNEY TO SATURN

(1997 – 2004…2010)

Michael P. Wnuk

NASA Visiting Scientist at JPL/Caltech

July 2000

SSuggested Reading

John A. Wood, 1979, “The Solar System”, publ. by Prentice-Hall, New Jersey.

2 Anthony Bedford and Wallace Fowler, 1995, “Engineering Mechanics – Dynamics”, publ. by Addison-Wesley, USA.

3 David A. Vallado and Wayne D. McClain, 1997, “Fundamentals of Astrodynamics and Applications”, in Space Technology Series, publ. by McGraw-Hill, USA.

4 Michael P. Wnuk and Carl Swopes, 1999, “From Pyramids and Fibonacci Sequence to the Laws of Chaos”, publ. by Akapit Publishers, Krakow, Poland.

5 Levin Santos, 2000, “Weighing the Earth. Physicists Close in on Newton’s Big G” in the “Sciences”, July/August 2000, publ. by New York Academy of Sciences, p.11.

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