Mathematics of satellites

Summary: Mathematics is fundamental to the design, function, and launch of satellites.

Astronomy and mathematics have long developed together. Many early mathematicians studied the motion of celestial objects. The term “satellite” comes from the Latin satelles (meaning “companion”), which was used by mathematician and astronomer Johannes Kepler to describe the moons of Jupiter in the seventeenth century. Mathematician Giovanni Cassini correctly inferred that Saturn’s rings were composed of many small satellites in the seventeenth century. Mathematicians Jean Delambre and Cassini Jacques both published books of astronomical tables, including planetary satellites, in the eighteenth century. When artificial satellites were developed, the term “satellite” largely came to refer to those in common speech, while “moon” was applied to natural bodies orbiting planets. Mathematicians like Michael Lighthill and engineers like John Pierce helped develop satellites in the 1960s.

By the first decade of the twenty-first century, there were several hundred operational satellites orbiting the Earth to facilitate communication, weather observation, research, and observation. The advantage of satellites for communication are that signals are not blocked by land features in the same manner as a lower-altitude signal would be, making long-distance communication possible without multiple ground-based relays. Early communication satellites simply reflected signals back to Earth to broaden reception. Modern satellites use many different kinds of orbits to facilitate complex functioning, including low Earth orbit; medium Earth orbit; geosynchronous orbit; highly elliptical orbit; and Lagrangian point orbit, named for mathematician Joseph Lagrange. Mathematics is involved in the creation and function of such satellites, as well as for solving problems related to launching satellites, guiding movable satellites, powering satellite systems, and protecting satellites from radiation in the Van Allen belt, named for physicist James Van Allen. For example, graph theory is useful in comparing satellite communication networks. Techniques of origami map folding, researched by mathematicians like Koryo Miura, have been used in satellite design. Chaos theory has been used to design highly fuel-efficient orbits, derived in part from mathematician Henri Poincaré’s work in stable and unstable manifolds. Government agencies like the U.S. National Aeronautics and Space Administration (NASA) and private companies like GeoEye employ mathematicians for research and applications. The Union of Concerned Scientists (UCS) maintains a database of operational satellites.

Orbits

The orbit of a satellite about the Earth determines when it will pass over various points on the Earth’s surface and how high it is above the Earth. In general, orbits are characterized by altitude, inclination, eccentricity, and synchronicity. As defined by NASA, low Earth orbits have altitudes of 80–2000 kilometers. This orbit includes the majority of satellites, the International Space Station, and the Hubble Space Telescope. Statistical estimates at the start of the twenty-first century suggest that the number of functional satellites and nonfunctional debris in low orbit ranges from a few thousand (tracked by the U.S. Joint Space Operations Center) to millions (including very small objects). Objects in low orbit must travel at speeds of several thousand kilometers per hour, so even a small object can cause damage in a collision. Medium Earth orbit extends to about 35,000 kilometers (21,000 miles), the altitude determined by Kepler’s laws of planetary motion for geosynchronous orbits. Inclination is an angular measure with respect to the equator, while eccentricity refers to how elliptical an orbit is. Geosynchronous satellites rotate at the same rate as the Earth spins, so they appear stationary relative to Earth. They usually have inclination and eccentricity of zero; they circle the equator to balance gravitational forces. The Global Positioning System (GPS) is one example of satellites at this orbital level. Sun synchronous orbits are retrograde patterns that allow a satellite to pass over a section of the Earth at the same time every day. They have an inclination of 20–90 degrees and must shift by approximately one degree per day. These orbits are often used for satellites that require constant sunlight or darkness. The maximal inclination of 90 degrees denotes a polar orbit. A halo or Lagrangian orbit is a periodic, three-dimensional orbit near one of the Lagrange points in the three-body problem of orbital mechanics, which was used for the International Sun/Earth Explorer 3 (ISEE-3) satellite.

Signals

Antennas and satellite dishes are used to receive satellite signals on Earth. Most satellite dishes have a parabolic shape. A signal striking a planar surface reflects directly back to the source. If the surface is curved, the reflection is in the plane tangent to the surface. A parabola is the locus of points equidistant from a fixed point and a plane, so a parabolic dish focuses all incoming signals to the same point at the same time, increasing the quality of the signal. Mathematics is used to compress, filter, interpret, and model vast amounts of data produced by satellites. Reed–Solomon codes, derived by mathematicians Irving Reed and Gustave Solomon, are widely used in digital storage and communication for satellites. Much of the data from satellites is images, which utilize mathematical algorithms for rendering and restoration. One notable case that necessitated mathematical correction is the Hubble Space Telescope. An incorrectly ground mirror was found to have a spherical aberration, which resulted in improperly focused images. Mathematical image analysis allowed scientists to deduce the degree of correction needed. Some of the mathematical concepts involved in these corrections include the Nyquist frequency, which is a function of the sampling frequency of a discrete signal system named for physicist Harry Nyquist, and the Strehl ratio, named for mathematician Karl Strehl, which quantifies optical quality as a fraction of a system’s theoretical peak intensity.

Bibliography

Montenbruck, Oliver, and Gill Eberhard. Satellite Orbits: Models, Methods and Applications. Berlin: Springer, 2000.

Whiting, Jim. John R. Pierce: Pioneer in Satellite Communication. Hockessin, DE: Mitchell Lane Publishers, 2003.