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.

By the mid-2020s, GPS was shown to have limitations as a military application. This was evident following the Russian invasion of Ukraine in 2022. The Russians employed types of GPS jammers that could interfere with the ability of GPS systems to direct precision munitions to a target on a battlefield. The future of both civilian and military navigation began to move beyond dependence on GPS. In 2024, the U.S. Air Force began experimenting with artificial intelligence (AI) as an alternative. Theoretically, an aircraft or missile navigational system empowered with AI could determine its global position without receiving inputs from GPS. It would instead rely on terrestrial features and celestial objects like stars and follow these landmarks from its current position to its target. This is essentially how humans performed navigation before the advent of GPS. 

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.

Defensive systems evolved tremendously in the half-century between the 1970s and the mid-2020s. Defensive systems were originally developed to intercept ballistic missiles and warheads during the latter part of Cold War. The Anti-Ballistic Missile treaty, signed by the Soviet Union and the United States in 1972, allowed for the deployment of two Anti-Ballistic Missile (ABM) systems per country. The United States attempted to develop such a system in the 1960s called the Safeguard but later ceased its efforts after concluding the complexity of the problem set was too great. Such a system was likened to trying to “hit a bullet with a bullet.”

The United States enjoyed greater success with the development and deployment of its Patriot missile system. In the 1991 Persian Gulf War, the Patriot achieved a level of success against Iraqi short-ranged ballistic missile systems nicknamed the Scud. By the 2020s, Israel had developed remarkably effective missile systems. The first was its “Iron Dome” defensive missile system that was employed against short-ranged, unguided rockets. The second was its Arrow and “David’s Sling” systems, designed to counter ballistic missile attacks. Initiated by the October 7, 2023, massacre of 1,200 Israeli citizens by Hamas, Israel later began offensive operations against the Lebanese non-state group Hezbollah. Hezbollah had earlier attacked Israel to demonstrate its support of Hamas. After a series of spectacular Israeli successes against Hamas, including the successful targeting of Hezbollah leader Hassan Nasrallah, Iran intervened against Israel. On October 1, 2024, Iran launched a volley of almost two hundred ballistic missiles against Israel. Israel countered with its defensive missile systems, assisted by United States military forces.

Bibliography

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