Mathematics of interplanetary travel

Summary: Space exploration requires mathematics to plan trajectories and to navigate in space, as well as to measure and to analyze massive amounts of data.

Interplanetary travel can be defined as any spaceflight—manned or remotely guided—to the various bodies of the solar system, including planets, their satellites, and asteroids. Such space exploration required new mathematics to plan trajectories and navigate in space, as well as to measure and to analyze massive amounts of data. These flights have had a great societal impact and have radically changed human attitudes toward the outer space surrounding the Earth.

History

A scientific possibility of interplanetary travel was discussed for centuries after Isaac Newton wrote Principia in 1687, in which he unified terrestrial and celestial dynamics by discovering the force of gravity as an important source of motion, including the movement of celestial bodies. Step by step, an important new mathematical branch of astronomy emerged and received the title “celestial mechanics.” In its formative days, celestial mechanics played an outstanding role in the progress of mathematics, demanding and inspiring novel and efficient mathematical tools. Among the pioneers of celestial mechanics were prominent mathematicians such as Leonhard Euler (1707–1783), Alexis-Claude Clairaut (1717–1765), and Joseph-Louis Lagrange (1736–1813). Today, the branch of celestial mechanics dedicated to spaceflight is usually termed astrodynamics.

For many years following Newton’s discovery, the topic of interplanetary travels mainly remained the subject of science fiction writers. In the nineteenth century, among the most influential science fiction writers were Jules Verne (1828–1905) with his books From the Earth to the Moon and All Around the Moon and H. G. Wells (1866–1946) with his book War of the Worlds. Verne’s work contained a great deal of mathematics discussion, much of which was reasonably accurate based on the knowledge of the time.

To put interplanetary travel into practice, it was necessary to realize some significant preconditions, including designing spacecraft with the capacity for maneuvering, designing technologies for boosters to reach escape velocity, developing a theoretical base for space navigation, and creating systems for long-distance radio communications. These technological developments were not made until the beginning of the space era in 1957.

Mathematical Development

From a mathematical viewpoint, the most interesting part of interplanetary travel is space navigation. An appropriate example of a solution with respect to navigational problems is the Hohmann transfer orbit. In 1925, Walter Hohmann calculated that the lowest-energy route between any two celestial bodies is an ellipse that forms a tangent to the starting and destination orbits of these bodies. Such a transfer orbit between the Earth and Mars is graphed in the following illustration. A spacecraft traveling from Earth to Mars along the Hohmann trajectory will arrive near Mars’s orbit in approximately 18 months. Just a small application of thrust is all that is needed to put a space probe into a circular orbit around Mars. The Hohmann transfer applies to any two orbits, not just those with planets involved (see Figure 1). In the figure, Hohmann Transfer Orbit (light gray oblong ring), Earth’s orbit is represented by the white circle, and Mars’ orbit is represented by the darker gray circle. A spaceship leaves from point 2 in Earth’s orbit and arrives at point 3 in Mars’s.

Another example of navigational technique is routinely called the “gravitational slingshot.” It utilizes the gravitational influence of planets and their moons to change the speed and direction of a space probe without the application of an engine. In this case, a spacecraft is sent to a distant planet on a path that is much faster than the Hohmann transfer. This would typically mean that it would arrive at the planet’s orbit and continue past it. However, if there is a planetary mass between the departure point and the target, it can be used to bend the path toward the target, and in many cases the overall travel time is greatly shortened. Prime examples of the gravitational slingshot are the flights of the two spacecraft of the American Voyager program, which used slingshot effects to redirect trajectories several times in the outer solar system. Astrodynamics considers many other interesting approaches. Several technologies have been proposed that both save fuel and provide significantly faster travel than Hohmann transfers; most are still theoretical.

Because of astrodynamics limitations, travel to other solar systems bodies is practical only within certain time windows. Outside of such windows, these bodies are essentially inaccessible from Earth using current technology. Mathematicians helped design the Interplanetary Superhighway, a network of low-energy trajectories, in order to find efficient routes through space; these mathematical foundations originated with French mathematician Henri Poincaré.

Achievements and Obstacles

The modern accomplishments in interplanetary travels are extraordinary. Remotely guided space robots have flown past all of the planets of the solar system from Mercury to Neptune, and the National Aeronautics and Space Administration’s (NASA’s) spacecraft New Horizons is scheduled to fly past the dwarf planet Pluto in 2015. The five most distant spacecraft (including the American ships Pioneer-10, Pioneer-11, Voyager-1, and Voyager-2) were scheduled to leave the solar system at the beginning of the twenty-first century. Artificial satellites have orbited Venus, Mars, Jupiter, and Saturn. Spacecraft have landed on the Moon, Venus, Mars, Saturn’s moon Titan, and asteroid 433 Eros. The first probes to comets (European Giotto, Russian Vegas, American Stardust) were fly-by missions. In 2005, the Deep Impact probe hit the comet 9P/Tempel to study the composition of its interior.

Great achievements took place in manned interplanetary travels once mathematicians, scientists, and engineers understood the mathematical principles required to launch spacecraft outside Earth’s atmosphere and to maneuver in the microgravity environment of space. NASA also recruited astronauts with strong academic credentials in science and mathematics. America’s Mercury and Gemini programs put humans into space and Earth orbit and taught them how to change trajectory in space to move to a new orbital altitude or to dock with other spacecraft, while the Apollo program took them to the moon. After missions in which men orbited the moon and returned, Apollo 11 landed astronauts Neil Armstrong and Edwin “Buzz” Aldrin on the moon in 1969. There were six successful manned American expeditions to the moon from 1969 to 1972.

Further development of interplanetary travel has many obstacles that will require a great deal of mathematical analysis to model, simulate, and solve. For example, astronauts must be protected from extreme radiation exposure in the Van Allen belt, a torus-shaped region of space surrounding the Earth and other planets named after geophysicist James Van Allen of Iowa.

The larger outer radiation belt is about four Earth radii (RE) above the surface of the Earth and the inner is about 1.6 RE, with a gap at roughly 2.2 RE. Apollo astronauts were briefly exposed to this radiation on trips to the moon. Conspiracy theorists who disputed the notion that humans landed on the moon cited the Van Allen belt as evidence that the astronauts would have died from radiation, but simple calculations and the data collected by radiation sensors worn by astronauts (similar to those worn by scientists and hospital workers who may be exposed to radiation) demonstrated that the speed and design of the Apollo capsules protected astronauts during these relatively short trips.

If the Earth was the main focus of many sciences (geodesy, geology, geophysics, geochemistry, and oceanography) for millennia, interplanetary travel created a new important branch of research—comparative planetology—which is essential for understanding the history of Earth and its evolution.

Among many other difficult problems of interplanetary travel is developing adequate human life support. A breathable atmosphere must be maintained, with adequate amounts of oxygen, nitrogen, controlled levels of carbon dioxide, trace gases, and water vapor. It is also necessary to solve the problem of food supply.

At some point in time, all of these problems may be overcome. Incentives for future expansion of interplanetary flights include the possibility of colonizing other portions of the solar system and utilizing resources.

Bibliography

Battin, Richard. An Introduction to the Mathematics and Methods of Astrodynamics. New York: American Institute of Aeronautics and Astronautics, 1999.

Benson, Michael. Beyond: Visions of the Interplanetary Probes. New York: Harry N. Abrams, 2003.

Kemble, Stephen. Interplanetary Mission Analysis and Design. Berlin: Springer, 2006.

Launius, Roger D. Frontiers of Space Exploration. Westport, CT: Greenwood Press, 2004.

Launius, Roger D., and Howard E. McCurdy. Robots in Space: Technology, Evolution, and Interplanetary Travel. Baltimore, MD: Johns Hopkins University Press, 2008.

Zimmerman, Robert. Leaving Earth: Space Stations, Rival Superpowers, and the Quest for Interplanetary Travel. Washington, DC: J. Henry Press, 2003.