Starting in the 20th century, the mathematical exploration of the cosmos became inextricably entwined with the physical exploration of space. On the one hand, virtually all the methods of celestial mechanics that had been developed over the centuries were transformed into tools for the navigation of rockets, artificial satellites and space probes. On the other hand, almost all of those space vehicles were equipped with scientific instruments for gathering data about the earth and other objects in our solar system, as well as distant stars and galaxies going back to the cosmic microwave background radiation. Furthermore, the deviations in the paths of satellites and probes provide direct feedback on the gravitational field around the earth and throughout the solar system.
Beyond these direct effects, there are many other areas of interaction between the space program and mathematics. To name just a few:
- GPS: The global positioning system
- Data compression techniques for transmitting messages
- Digitizing and coding of images
- Error-correcting codes for accurate transmissions
- "Slingshot" or "gravitational boosting" for optimal trajectories
- Exploitation of Lagrange points for strategic placement of satellites
- Dynamical systems methods for energy-efficient orbit placing
- Finite element modeling for structures such as spacecraft and antennas
Some of the satellites and space probes that have contributed to cosmology and astrophysics:
- The Hubble space telescope
- The Hipparcos mission to catalog the positions of a million stars to new levels of accuracy
- The COBE and MAP satellites for studying the cosmic microwave background radiation
- The Genesis mission and SOHO satellite for studying the sun and solar radiation
- The ISEE3/ICE space probe to study solar flares and cosmic gamma rays before going on to visit the Giacobini-Zimmer and Halley's comets
- The LAGEOS satellites to test Einstein's prediction of "frame dragging" around a rotating body.
Rather than trying to cover all or even most of the mathematical links, we focus on two that are absolutely essential and central to the whole endeavor: first, navigation and the planning of trajectories; and second, communication and the transmission of images.
Navigation, trajectories, and orbits
When the U.S. space program was set up in earnest — a process described in detail in the recent History Channel documentary, "Race to the Moon" — a notable feature was the introduction of the Mission Control Center. The first row of seats in mission control were known as "the trench" and it is from there that the mathematicians whose specialty is orbital mechanics kept track of trajectories and fed in the information needed for navigation. Their role is particularly important for operations involving rendezvous between two vehicles, in delicate operations such as landings on the moon, and in emergencies that call on all their skills, the most notable of which was bringing back alive the crew of Apollo 13 after they had to abandon the command module and were forced to use the lunar landing module — never designed for that purpose — to navigate back to earth.
The first thing that an astronaut or former astronaut will tell you about navigating in a spaceship is that no amount of experience piloting a plane will be of any help. On the contrary, previous experience may be a hindrance, since it reinforces one's natural intuition that if you want to catch up with an object ahead you go faster, and conversely. But if you are orbiting at a certain speed, and have to rendezvous with something ahead, then "stepping on the accelerator" (translate as "applying a forward thrust") will lift you into a higher orbit where first of all, the vertical distance between you and the object orbiting ahead will increase, and second, your average angular velocity will decrease, by Kepler's third law, and you will find yourself getting further and further behind.
The ability to navigate started with Isaac Newton. Not only did he formulate his laws of motion and of gravity, but he also developed the calculus which allowed him to put those laws into the language of mathematical equations. Today, our knowledge of the physics involved has been improved with the addition of relativity and other factors that may play a role, and calculus has been further developed into many different branches of mathematics.
What kind of mathematics? Newton's equations involve the gravitational forces acting upon one of the participating bodies, arising from all of the other bodies. Since force is mass times acceleration, and since acceleration is simply the second derivative of position with respect to time, it is the differential calculus which describesthe accelerations. Then, once the accelerations are given, it is necessary to use integral calculus in order to get from the second derivatives to the positions.
In a more general context, where the mass may be changing with time, such as happens with an extended application of thrust to a vehicle, with the gradual reduction of weight as fuel is used up, or in cases of relativistic speeds, the force is given by the first derivative of momentum, but the principle is the same.
In the case of the 2-body problem, where the only force involved is the gravitational attraction between the two bodies, it is frequently said that Newton was able to give a complete solution. That is not, strictly speaking, the case, if one means by "a solution" of a differential equation, an expression for the unknown function whose derivatives appear in the equation. In this case, it would mean finding an expression for the position as a function of time. However, what Newton showed was that the orbit of each of the bodies lies on a conic section (in a fixed inertial frame of reference), and in the case considered by Kepler, where the orbit as an ellipse, there is an explicit expression for the timeas a function of the position. Commonly known as "Kepler's Equation," it is of the form: t = x - e sin x, in suitable units of time t, where x is the polar angle from the center of the ellipse and e is the eccentricity. What one wants, of course, is x as a function of t, and much effort and ingenuity has gone into finding effective means of solving Kepler's equation for x in terms of t. Lagrange did extensive work on the problem, in the course of which he developed both Fourier series and Bessel functions, named after later mathematicians who investigated these concepts in greater detail. Both Laplace and Gauss made major contributions, and succeeding generations continued to work on the subject.
When there are more than two bodies involved, the problem cannot be solved analytically; instead, the integration (positions from accelerations) must be done numerically: now, with high-speed computers. So, numerical integral calculus is a major factor of spacecraft navigation.
One may picture navigation as being the modeling of mother nature on acomputer. At some time, with the planets in their orbits, a spacecraft is given a push outward into the solar system. Its subsequent orbit is then determined by the gravitational forces upon it due to the sun and planets. We compute these, step-by-step in time, seeing how the (changing) forces determine the motion of the spacecraft. This is very similar to what one may picture being done in nature.
How does one get an accurate orbit in the computer? The spacecraft's orbit is measured as it progresses on its journey, and the computer model is adjusted in order to best fit the actual measurements. Here one uses another type of calculus: estimation theory. It involves changing the initial "input parameters" (starting positions and velocities) into the computer in order to make the "output parameters" (positions and velocities at subsequent times) match what is being measured: adjusting the computer model to better fit reality.
Also in navigation, one must "reduce" the measurements. Usually, the measurements don't correspond exactly with the positions in the computer; one must apply a few formulae before a comparison can be made. For instance, the positions in the computer represent the centers of mass of the different planets; a radar echo, however, measures the path from the radio antenna to the spot on a planet's surfaces from which the signal bounces back to earth. This processing involves the use of trigonometry, geometry, and physics.
Finally, there is error analysis, or "covariance" calculus. In the initial planning stages of a mission, one is more interested in how accurately we will know the positions of the spacecraft and its target, not in the exact positions themselves. With low accuracy, greater amounts of fuel are required, and it could be that some precise navigation would not even be possible. Covariance analysis takes into account 1) what measurements we will have of the spacecraft: how many and how good, 2) how accurately we will be able to compute the forces, and 3) how accurately we will know the position of the target. These criteria are then used in order to determine how closely we can deliver the spacecraft to the target. Again, poor accuracy will require more fuel to correct the trajectory once the spacecraft starts approaching its final target.
One of the mathematical tools used to optimize some feature of a flight trajectory, such as fuel consumption or flight time, is a maximum principle introduced by Pontryagin in 1962. Pontryagin's theorem characterizes the optimum values of certain parameters, called the controllers, that determine a trajectory.
In recent decades, ingenious new methods have been developed to milk the maximum effect from the least amount of fuel. One such method is known as the "slingshot" or "gravity-assisted trajectory." By aiming a space vehicle in a way that crosses the orbit of another planet or moon just behind that body, the path of the vehicle will be deflected, sending it on its way to the next target with minimum expenditure of fuel. A number of space probes, such as Cassini-Huygens, have benefited from carefully calculated trajectories that make multiple use of the slingshot effect. Gravity-assist methods are equally important for sending a probe toward the inner planets: Venus and Mercury. In that case, one sends the probe to a point on the orbit just in advance of the body it is passing. In both cases, the effect of the fly-by is to alter the velocity, changing the direction of flight, and leaving the end-speed relative to the body it is passing unchanged. However, that body will be moving with considerable momentum relative to the sun, and there will be an exchange of momentum in which the body will be slowed down or speeded up by an infinitesimal amount, while the probe will be speeded up or slowed down by a considerable amount relative to the sun.
More modern, 20th century mathematical methods of dynamical systems have proved invaluable in designing complicated fuel-efficient orbits. These methods include the theory of stable and unstable manifolds, pioneered by Poincaré, leading to the subject now known as chaotic dynamics, and the KAM theory, due to Kolmogorov, Arnold and Moser, of invariant tori and stability. One of the first achievements of the new methods was the 1991 rescue of a Japanese spacecraft Hiten that was stranded without enough fuel to complete a planned mission when a second satellite, intended to work in tandem with the first, failed to operate. The mathematician Edward Belbruno had designed highly fuel-efficient orbits using methods derived from chaos theory, and that turned out to be just what was needed for this rescue operation. Belbruno's methods were incorporated into the design used by Giuseppe Racca and the European Space Agency in sending "SMART-1" - their first satellite to the moon, in 2004. Newspaper headlines trumpeted "spacecraft reaches moon on 5 million miles a gallon" as a dramatic way of underlining the astonishing fuel efficiency of the method.
In the past few years, other applications of the theory of stable and unstable manifolds have been invoked in trajectory planning. Martin Lo and his colleagues at the Jet Propulsion Laboratory developed ways to apply the theory in order to place satellites such as Genesis in an orbit around the Lagrange point between earth and the sun, and then return it to earth. More recently, the same group, together with Jerry Marsden at CalTech, have expanded the method for use in interplanetary travel, along what they call the "interplanetary superhighway," a route derived from the ever-changing configurations of stable and unstable manifolds in the phase space of our solar system or selected parts of it. A beautifully illustrated article by Douglas Smith describing this work can be found in the journal Engineering and Science.
Communication and Image Transmission
For space exploration and interplanetary probes, navigational techniques and orbital mechanics may get you where you want to go, but it is not worth much if the data collected cannot be successfully transmitted back to earth. In the case of the Cassini spacecraft at Saturn, signals have to travel distances on the order of a billion miles or more. Data transmitted from across the planetary system with very limited power are received on Earth as a very faint signal (as low as a billionth of a billionth of a watt) embedded in noise. Only through miracles of modern technology operating in tandem with ever-improving mathematical methods, is one able to receive the striking and detailed images that are now on display.
Two critical processes come into play for transmitting messages of all sorts. The first is compression, to be able to transmit the maximum amount of information with the least number of bits, and the second is the use of error-correcting codes, to overcome problems of noise and distortion. Basically, one wants to eliminate redundancy to obtain compression of the data, and then one has to introduce redundancy in order to catch and correct errors of transmission. The two operations may at first seem to cancel each other out, but in fact the types of redundancies involved in the two cases are quite different.
A variety of mathematical techniques are used to compress the spacecraft data into fewer bits prior to transmission to the ground. A simple one avoids transmitting all 16 bits of every data element of a data stream. The value of the first data element is sent, but for the rest of the elements, only the difference from the first is sent. The value of the first element might require 16 bits, but the differences are so small they might only need 2 or 3 bits. Once on the ground the first value can be added to all the others to restore the original content. For a large data stream, techniques such as this can save hours of transmission time and much storage capacity.
"Entropy coding" is a technique that takes into account the probability distribution of different sets of data in order to encode more probable data with shorter sequences, just as in Morse code, the letter "E" is represented by a single dot. Image data compression techniques rely on mathematical image probability models that exploit the similarities between neighboring small picture elements to minimize the number of bytes needed to describe the image.
The possibility of detecting and even correcting errors in transmission was first pointed out in a ground-breaking paper of Richard Hamming in 1950. Since then, an entire field has grown up in which, on the one hand, ever-more refined methods have been devised for practical applications, and on the other hand, the theory of error-correcting codes has turned out to have fascinating links with sphere packing and simple groups, beautifully described in the book of Thomas Thompson.
More recent mathematical innovations that have proved of both theoretical interest and of great practical use in this connection are the subjects of fractals and wavelets. An excellent survey of all aspects of image analysis, transmission, and reconstruction is the theme essay on Mathematics and Imaging on the 1998 Mathematics Awareness Week website.
The author gratefully acknowledges the input of Myles Standish, Martin Lo, Fabrizio Polara, Susan Lavoie, and Charles Avis of JPL, and Jerry Marsden of CalTech.
Trajectories, Orbits, and Space Navigation:
- Battin, Richard A., An Introduction to the Mathematics and Methods of Astrodynamics, New York, American Institute of Aeronautics and Astronautics 1987.
- Brumberg, V.A., Essential Relativistic Celestial Mechanics, Bristol, Adam Hilger 1991, section 5.1: "Equations of motion of Earth's artificial satellites."
- Colwell, Peter, Solving Kepler's Equation over Three Centuries, Richmond, VA, Whtiman-Bell 1993.
- Case, James, "Celestial Mechanics Theory Meets the Nitty-Gritty of Trajectory Design," SIAM News 37 (2004), 1-3 (Book Review)
- Belbruno, Edward, Capture Dynamics and Chaotic Motion in Celestial Mechanics: With Applications to the Construction of Low Energy Transfers, Princeton Univ. Press 2004
- Racca, Giuseppe D., "New challenges to trajectory design by the use of electric propulsion and other new means of wandering in the solar system," Celestial Mechanics and Dynamical Astronomy 85 (2003), 1-24.
- Szebehely, V, Theory of Orbits, New York, Academic Press 1967.
- Smith, Douglas L., "Next
Exit 0.5 Million Kilometers," Engineering
and Science No. 4 (2002), 6-14.
- Jet Propulsion Laboratory: http://www.nasa.gov/centers/jpl/home/index.html
- Martin Lo main page: http://www.gg.caltech.edu/~mwl/
- Gravity-assisted trajectories: http://saturn.jpl.nasa.gov/mission/gravity-assist-primer2.cfm
DVD's and Videos:
- The Race to the Moon, The History Channel, 2004
- Ring world: Cassini-Huygens mission to Saturn and its moons, NASA JPL 400-1114, 11/03
- Mathematics of Space-Rendezvous, NASA JSC 1801
- Space Flight: Application of Orbital Mechanics, NASA CMP 277
Communication and image transmission:
- Thompson, Thomas M., From Error-correcting Codes through Sphere Packings to Simple Groups, MAA 1983
- Mathematics Awareness 1998
theme essay: Mathematics and Imaging: