Spaceflight, Mathematics of

The mathematics of spaceflight involves the combination of knowledge from two different areas of physics and mathematics: rocket science and celestial mechanics. Rocket science is the study of how to design and build rockets and how to efficiently launch a vehicle into space. Celestial mechanics is the study of the mathematics of orbits and trajectories in space. Celestial mechanics uses the laws of motion and the law of gravity as first stated by Isaac Newton.

A Short History of Rockets

People have wondered about the possibility of space flight since ancient times. Centuries ago the Chinese used rockets for ceremonial and military purposes. In the latter half of the twentieth century, we have been able to develop rockets powerful enough to launch vehicles into space with sufficient energy to achieve or exceed orbital velocities .

The rockets mentioned in the American national anthem, the "Star Spangled Banner," were British rockets fired at the American fortress in Baltimore. However, these so-called "Congreves," named for their inventor, Sir William Congreve, were not the first war rockets. Some six centuries before, the Chinese connected a tube of primitive gunpowder to an arrow and created an effective rocket weapon to defend against the Mongols.

The Physics Behind Rockets and Their Flight

The early Chinese "fire arrows," the British Congreves, and even our familiar Fourth of July skyrockets, are propelled by exactly the same principles of physics that propelled the Delta, Atlas, Titan, and Saturn rockets, as well as the modern space shuttle. In each case, some kind of fuel is burned with an oxidant in the combustion chamber, creating hot gases accompanied by high pressure. In the combustion chamber, these gases try to expand, pushing equally in all directions. At the bottom of the combustion chamber, there is an exhaust nozzle that flares out like a bell. The expanding gases push on all sides of the combustion chamber and the nozzle, except at the bottom. This results in a net force on the rocket, accelerating it in a direction opposite to the nozzle.

This idea is summarized in Newton's Third Law of Motion: "To every action, there is an equal and opposite reaction." The gases "thrown" out of the bottom of the nozzle constitute the action. The reaction is the rocket's acceleration in the opposite direction. This system of propulsion does not depend on the presence of an atmosphere—it works just as well in empty space.*

*One of the early astronauts answered a question from mission control asking who was flying with the words, "Isaac Newton is flying this thing."

According to Newton's First Law of Motion, "A body at rest will remain at rest unless acted on by an unbalanced force." The unbalanced force is provided by the rocket motor, so the rocket motor and attached spacecraft are accelerated. Because Newton's laws were postulated in the seventeenth century, some people have argued, "All the basic scientific problems of spaceflight were solved 300 years ago. Everything since has been engineering." However, many difficult problems of mathematics, chemistry, and engineering had to be solved before humans could apply Newton's basic principles to achieve controlled spaceflight.

Using Rockets in Spaceflight

Robert Goddard was an early American rocket pioneer, who initially encountered resistance to his ideas. Some people, not understanding Newton's laws very well, insisted rockets would not operate in the vacuum of space because there would be no air for them to push against. They were thinking in terms of a propeller-driven airplane. But a rocket needs no air to push against; action-reaction forces propel it. Nor does a rocket need air to burn the fuel. An oxidizer is mixed with the fuel in the engine.

In 1926 in Massachusetts, Goddard successfully launched the world's first liquid-fuel rocket. By modern standards it was very rudimentary: It flew 184 feet in two and a half seconds. But the mighty Atlas, Titan, and Saturn liquid-fueled rockets are all direct descendents of Goddard's first rocket— even the Space Shuttle uses liquid fuel rockets as its main engine.

At the same time that Goddard was doing his pioneering work on rockets in the United States, two other pioneers were beginning their work in other countries. In the early Soviet Union, Konstantin Tsiolkovski was conducting experiments very similar to Goddard's. In Germany, Hermann Oberth was studying the same ideas. Military leaders in Nazi Germany recognized the possibilities of using long-range rockets as weapons. This led to the German V-2 missiles that could be launched from Germany to a height of 100 kilometers across the English Channel. These missiles could reach speeds greater than 5,600 kilometers per hour.

After World War II, the United States and the Soviet Union used captured German V-2 rockets, as well as the knowledge of German rocket scientists, to start their own missile programs. In October 1957, the Soviets propelled a satellite named Sputnik into space. Four years later, the first human orbited the Earth.

The first U.S. satellite, Explorer I, was launched into orbit in January of 1958. On May 5, 1961, Alan Shepard became the first American to fly into space with a fifteen-minute flight of the Freedom 7 capsule. Four years later, on February 20, 1962, John Glenn became the first American to orbit Earth in the historic flight of the Friendship 7 capsule.

In 1961, President John F. Kennedy had set a national goal of sending and landing a man on the Moon and then returning him safely to Earth within the decade. In July 1969, Astronaut Neil Armstrong took "a giant step for mankind" as he stepped onto the surface of the Moon. It was an event few Americans living at the time will ever forget.*

*In total, six Apollo expeditions landed on the Moon and returned to Earth between 1969 and 1972.

Navigating in Space

Getting to the Moon, Mars, or Venus is not simply a matter of aiming a rocket in the desired direction and hitting a launch button. The process of launching a spacecraft on a trajectory that will take it to another planet has been compared to trying to shoot at a flying duck from a moving speedboat. Launching a spacecraft into orbit is relatively simpler. The spacecraft is launched toward the east so that it can take advantage of Earth's rotation, and gain some extra speed. For the same reason, satellites are usually launched from sites as close to the equator as possible.

On the Apollo missions to the Moon, the spacecraft was boosted into an Earth orbit where it coasted until it was in the proper position to start the second leg of the trip. At the appointed time, the Apollo spacecraft was boosted out of its parking orbit and into a trajectory or path to the Moon. This meant firing the spacecraft's rocket engines when it was at a point in orbit on the opposite side of Earth from the Moon. Orbital paths are ellipses , parabolas , or some other conic section.

In the case of a spacecraft launched from Earth into orbit, the spacecraft follows an elliptical orbit with the center of Earth at one focus. The point in the orbit closest to Earth is known as the "perigee" and the point on the elliptical orbit farthest from Earth is called the "apogee." The Apollo's engines were fired when it was at perigee.

For Apollo to reach the Moon, the point of maximum altitude above Earth, the apogee, had to intersect the orbit of the Moon and the arrival had to be timed so that the Moon would arrive at the same position when the spacecraft appeared.

When the Moon's gravitational pull on the spacecraft was greater than Earth's pull, it made sense to shift the points of reference. Thus, the position of the Apollo spacecraft began to be considered in terms of its where it was relative to the Moon. At that point, the spacecraft was more than half way to the Moon. Its orbit was now calculated in relation to the Moon's orbit. As the spacecraft approached the Moon, it swung around the Moon in an elliptical orbit, with the Moon at one focus. Apollo then fired its engines again to slow down and enter orbit.

Orbital Velocity

The velocities needed for specific space missions were calculated long before space flight was possible. To put an object into orbit around the Earth, for instance, a velocity of at least 7.9 kilometers/second (km/s) (about 18,000 miles per hour), depending on the precise orbit desired, must be achieved. This is called "orbital velocity."

In effect, an object in Earth orbit is falling around the Earth. A satellite or missile with a speed less than orbital velocity moves in an elliptical orbit with the center of Earth at one focus. However, the orbit intersects Earth's surface. With a slightly increased speed, the satellite still falls back toward Earth. In simply moves sideways fast enough that it does not actually hit the Earth. It keeps missing.

To leave Earth orbit for distant space missions, a velocity greater than 11.2 km/s (25,000 miles per hour) is required. This is called "escape velocity." In Earth orbit, the downward force of Earth's gravity maintains the satellite's acceleration towards Earth, but Earth keeps curving away. At still higher speeds, the orbit becomes more elliptical and the apogee point is at a greater distance. If the launch speed reaches escape velocity, the apogee is infinitely far away and the ellipse has become a parabola in Earth's frame of reference. At even greater speed, the orbit is a hyperbola in Earth's frame of reference.

see also Solar System Geometry, Modern Understandings of; Space, Commercialization of; Space Exploration; Space, Growing Old in; Spaceflight, History of.

Elliot Richmond

Bibliography

Chaisson, Eric, and Steve McMillan. Astronomy Today, 3^rd ed. Upper Saddle River, NJ: Prentice Hall, 1993.

Giancoli, Douglas C. Physics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1991.

WHAT EXACTLY IS "FREE-FALL"?

A satellite in orbit around Earth is falling towards Earth; it is accelerating downward. Since there is nothing to stop its fall, it is said to be in "free-fall." Astronauts on board the satellite are also falling downward, and this produces a sensation of weightlessness.

However, it would be incorrect to say there is no gravity or that the astronauts are weightless, since the satellite is only a few hundred kilometers above Earth's surface. NASA likes to use the term "microgravity," which refers to the fact that the objects in the spacecraft actually attract each other. Astronauts feel weightless not because they have "escaped gravity" but because they are freely falling toward the surface of Earth.