Mathematics and missile accuracy

Summary: Mathematicians have long worked on improving missile accuracy and performance.

Stone or arrow missiles have been used for thousands of years. Missiles with explosives can be traced back to China following the Song dynasty. Mathematical and technological advances have led to countless improvements in missile design, trajectory, range, and accuracy and have continually revolutionized warfare. Aristotle theorized on laws governing projectile motion, as did mathematicians like Leonhard Euler and Daniel Bernoulli, who derived or refined mathematical principles of projectile motion using geometry, calculus, and differential equations. In the nineteenth century, mathematicians Alfred Freenhill and Percy MacMahon worked on a missile trajectory model that related resistance to the cube of the velocity, suggested from experimental data. During World War I, mathematics took on an increasingly important role. John Littlewood created techniques to reduce the work required for accurate missile trajectory calculations, and Gilbert Bliss used the calculus of variations to account for variables like wind and the rotation of the Earth. During the 1950s, mathematician John von Neumann headed the committee that led to the development of U.S. intercontinental ballistic missiles. During the space age, mathematicians made a breadth of contributions, like Evelyn Boyd Granville, who worked on the development of missile fuses at the National Bureau of Standards.

Mathematician Peter Swerling, known for his theory of radar, also researched optimal estimation of satellite and missile orbits and trajectories. Missiles of the twenty-first century can be defined as weapons that follow a trajectory for the purpose of delivering explosive warheads to targets by means of lift and rocket propulsion. They may be launched from ground, submarines, and airplanes to nearly any target on the face of the Earth. Mathematicians working in government, industry, and academia continue to contribute to the development of all types of missiles and missile defense systems.

Trajectory and Guidance

The basic flight path of a missile is a parabolic arc. Sixteenth-century mathematician Niccolo Tartaglia described cannonball flight paths. Seventeenth-century mathematician Evangelista Torricelli published a geometric method for computing projectile range. Benjamin Robins, an eighteenth-century mathematician, invented the ballistic pendulum. His experiments, later expanded by Euler, demonstrated that air resistance could not be ignored in calculating trajectories. Scientist Heinrich Magnus showed that other forces could affect spinning spheres and cylinders; this effect is now known as the Magnus Effect. The importance of higher mathematics, like calculus, in computing trajectories contributed to the inclusion of these topics in many military school curricula in the nineteenth century.

In the early twenty-first century, mathematics continues to play a key role in missile accuracy. Most modern guidance systems use mathematical methods to determine the trajectory needed, such as angular coordinates between the missile and the target or the distance between the target and the missile. Sometimes computations are done ahead of time and the missile follows a predetermined path. Other times, the missile can make adjustments to the flight path in order to correct the trajectory as needed and may follow a path that is very different from the basic parabola. Some systems utilize astronomy—the accuracy of a missile is determined by examining the relationship of the missile to a fixed start position. Others employ altitude maps and compute the missile’s distance from the ground to determine the path of the missile. These systems, however, are subject to error. Navigation systems that utilize a path calculated prior to launch may be influenced by instrument errors, while systems that utilize flight path data are more accurate but are subject to the effects of countermeasures such as radar decoys or infrared flares.

Advanced missiles are propelled by an internal combustion mechanism and guided by radiation, lasers, radio waves, or computers. Guidance often involves the use of mathematical techniques, like Kalman filtering, named after Rudolf Kalman, which allows a missile’s course to be manipulated. Many of these latest-generation weapons come complete with cameras that record visual and spatial location information to aid human operators in their direction. Other missiles are guided by locations systems, such as INS, TERCOM, or GPS, which are programmed to recognize the weapon’s global positioning at its origin and use it to calculate the distance, trajectory, and course to the target. These modern flight systems use positioning, targeting, and guidance data, along with thrust and aerodynamics, to maneuver missiles while they are in flight, even allowing them to seek and destroy moving targets.

Defensive Systems

With the development of more advanced missiles has come the need for more advanced defense systems. For example, satellites could measure the missile’s trajectory and speed to determine a probable impact point and relay this information to an interceptor vehicle. The interceptor might initially utilize celestial guidance to track the incoming missile, and then use preset guidance to collide with the incoming missile. The U.S. Missile Defense Agency employs many engineers, scientists, and mathematicians to work collaboratively on defense solutions.

Bibliography

National Aeronautics and Space Administration. “Beginner’s Guide to Rockets.” http://exploration.grc.nasa.gov/education/rocket/bgmr.html.

Shneydor, N. A. Missile Guidance and Pursuit: Kinematics, Dynamics and Control. Cambridge, England: Woodhead Publishing, 1998.

Van Riper, Bowdoin. Rockets and Missiles: The Life Story of a Technology. Westport, CT: Greenwood Press, 2004.