Spaceships and mathematics
Spaceships, also known as spacecraft, are vehicles designed for travel beyond Earth's atmosphere, serving various purposes such as exploration, data collection, and communication. They can be classified based on their flight paths—suborbital, orbital, interplanetary, and interstellar—and propulsion methods, including rockets, ion thrusters, and solar sails. Mathematics plays a crucial role in the design and operation of these vehicles, aiding in trajectory planning, collision avoidance, and problem-solving, as demonstrated during historical missions like Apollo 13.
The complexities of propulsion require calculations of forces, fuel efficiency, and thermal dynamics, while guidance systems utilize mathematical models to determine a spacecraft's position relative to celestial bodies. For crewed missions, life support systems must be designed to provide essential resources and protect against environmental hazards, integrating disciplines such as biology and bioengineering.
Furthermore, the challenges of atmospheric flight differ significantly from those of space travel, as they necessitate unique aerodynamic considerations. The intersection of mathematics, science fiction, and computer gaming has also evolved, prompting exploration of fantastical concepts like faster-than-light travel and bioengineered spacecraft. This rich blend of disciplines highlights the integral relationship between mathematics and the future of space exploration.
Spaceships and mathematics
Summary: Every task involving spaceships, from their design to their launch to effective collision avoidance and communication, is mathematically intensive.
Spaceships, also called “spacecraft,” are manned or automatic vehicles for flying beyond planet atmospheres. Different types of spaceships serve different purposes, including scientific or applied observations and data collection, exploration of celestial bodies, communication, and recreation.
According to the routes they take, spaceships can be classified as suborbital, orbital, interplanetary, and interstellar. According to the type of propulsion used, spacecraft engines can be designated as reaction engines, including rockets; electromagnetic, such as ion thrusters; and engines using fields, such as solar sails or gravitational slingshots. Mathematics is fundamental for spaceship design, operation, and evaluation. For example, mathematics is used to plan efficient trajectories, avoid collisions, communicate with satellites, transmit data over vast interplanetary distances, and solve complex problems like those that occurred in the famous Apollo 13 mission.
Mathematics in Spaceship Systems
Propulsion of a spaceship poses scientific and engineering problems that involve balancing forces and computing sufficient fuel, energy, work, and fluid mechanics. For any type of engine, the impulse it gives to the craft has to be calculated and compared to the craft’s tasks such as leaving the gravity well of a planet or maintaining an orbit. For example, calculations for rocket engines involve variables including the changing mass of the craft as its fuel is spent, the efficiency of the engine, and the velocity of the rocket’s exhaust. Solar sail theories involve such variables as radiation pressure of the light, the area of the sail, and the weight of the craft.
Mechanics and material sciences problems involved in the structure of spacecraft include withstanding the forces, temperatures, and electromagnetic fields involved in moving through space. For example, moving through a planetary atmosphere at speeds necessary to leave the planet’s gravity well involves high temperatures from friction.
The guidance and navigation systems of a spaceship collect data and then compute position, speed, and the necessary velocity and acceleration to reach the destination. These systems also determine the relative position of the spaceship to nearby celestial bodies, which influence the craft’s motion by their gravitational and electromagnetic fields. For example, mathematical description of a craft orbiting a planet includes the six Keplerian elements (for example, inclination and eccentricity) defining the shape, the size, and the orientation of the orbit, named for Johannes Kepler.
Most twenty-first-century spacecraft do not carry living organisms, but when they do, life support systems are necessary. Life support systems protect people, animals, or plants in the spaceship from harmful environments and provide air, water, and food. The design of life support systems involves biology, physiology, medical sciences, plant sciences, ecology, and bioengineering. Mathematical models for life support typically include calculations of safety margins, such as maximum allowable radiation doses. All organisms need some inputs (such as food, water, or oxygen) and produce some outputs depending on a variety of variables, such as activity levels. Spaceship ecosystem designers strive to produce waste-free, closed systems where water is reclaimed and plants are used to purify the air. Because of the complexity of the closed ecosystem problem, most current flights employ simpler, machine-driven life support systems.
Atmospheric Flight
Flight within an atmosphere presents very different problems compared to flight in a vacuum. The problems solved by applied mathematicians who study atmospheric flight include friction, turbulence, wing lift, aerodynamic shapes, and control of temperature. Spaceships launching or landing on planets have to be equipped for atmospheric flight. Because of differences in the vacuum and atmosphere flight requirements, many spaceships are designed to change their configuration when they cross atmospheric boundaries. For example, mathematical theories originally developed for origami are used to fold and unfold solar batteries, which can be used only in a vacuum because of their large area.
Science Fiction and Computer Game Mathematics
Space travel frequently appears in science fiction, where plots deal with various existing engineering or physics limitations. Hard science fiction is the more scientifically oriented subgenre, and it frequently includes extensions, discussions, and speculations dealing with the current scientific research. This tradition of blending science and literature started in the late nineteenth century with the works of Jules Verne; many of his then-fantastic devices and ideas (for example, televisions and submarines) were implemented relatively soon after.
As an example of experiments with scientific limits in literature, science-fiction spaceships may travel at superluminal (faster than light) speeds, often through non-physical spaces such as “hyperspace,” “subspace,” or “another dimension.” These are terms from existing mathematical theories, which hard science fiction sometimes discusses.
Sci-fi spaceships may also be living organisms, completely or partially. This idea is a reflection of the current interest in bioengineering and has connections with exciting research in ecology, genetics, cybernetics, and artificial intelligence, as well as social sciences such as philosophy and bioethics.
Computer games and movies about space flight created a demand for applied mathematicians who can model fantastic situations with passable realism. The physics and mathematics of three-dimensional modeling is a fast-growing area, with new courses and programs opening in universities and an expanding job market. What started in the nineteenth century as an exotic occupation for very few writers has become a profession for many programmers and applied mathematicians.
Bibliography
Battin, Richard. An Introduction to the Mathematics and Methods of Astrodynamics. New York: American Institute of Aeronautics and Astronautics, 1987.
National Aeronautics and Space Administration. “Design a Spaceship.” http://www.nasa.gov/centers/langley/news/factsheets/Design-Spaceship.html.
Osserman, Robert. “Mathematics Awareness Month: Space Exploration.” http://www.mathaware.org/mam/05/space.exploration.html.