Orbiting and Flight

Definition: Sustained repetitive motion about a gravitating body, generally consisting of closed circles or ellipses.

Significance: Orbiting is the type of motion exhibited by both natural and artificial satellites.

The Laws of Motion

Isaac Newton, in 1684, published three laws of motion that put its study on a firm scientific basis for the first time.

The first law states that a body at rest will remain at rest, and a body in motion will remain in motion in a straight line at constant speed unless acted on by an outside force. Sometimes referred to as the law of inertia, this law first articulated the principle that motion, not rest, is the natural state of objects. Contrary to commonsense observation and the beliefs of many philosophers before 1684, a force is not necessary to keep objects moving; rather, a force is necessary to bring them to a halt once they are moving.

The second law states that applying a force to an object will cause it to accelerate in the direction of the force, with the magnitude of the acceleration equal to the strength of the force divided by the mass of the object. Acceleration is defined as a change in velocity; velocity covers both speed and direction of motion. Acceleration can be a change in speed in a constant direction, or it can be a change in the direction of motion at a constant speed, or it can be a simultaneous change in both speed and direction. An airplane increasing its speed from 100 to 200 miles per hour is accelerating. So is an airplane banking in a tight circle to the left at a constant 100 miles per hour.

The third law states that for every action, that is, a force exerted by one body on a second, there will be an equal and opposite reaction, that is, a diametrically opposite force exerted on the first body by the second.

In addition to the three laws of motion, Newton also discovered the law of gravitation, which states that any two objects will attract each other with a force proportional to the product of their masses divided by the square of the distance between their centers. The gravitational attraction of Earth and an object is the force of weight.

Free Fall

Objects solely under the influence of gravity are said to be in free fall. In such a situation, weight is the only force acting on the object, and the law of gravitation states that the direction of this force is toward the center of Earth. From the second law, it follows that the object will accelerate toward the center of Earth. If the object is initially at rest or in pure vertical motion, then the resultant acceleration will be a change in speed only: an object traveling vertically upward will slow to a halt and then begin to travel downward at an ever-increasing rate, or if initially traveling vertically downward, will simply increase its downward speed.

An object traveling horizontally will also accelerate downward, but in this case, the acceleration will include a change in direction. The initial horizontal velocity will accumulate a downward component in addition to the initial horizontal component, and combining the two will result in a curved path. The object will travel on a parabola. In both cases, the force is the same—the object’s weight does not change—and the acceleration is the same. The effect of the acceleration is different because of the different initial velocities of the two situations.

Orbiting

The force of gravity extends to infinity and cannot be canceled or screened. At altitudes where Earth’s atmosphere is too thin to exert the aerodynamic forces of lift and drag, the motion of an object is governed solely by the force of gravity: it is in free fall. Such is the case for the Moon. It accelerates toward the center of Earth, but because its initial velocity is horizontal, the acceleration results in a curved trajectory. Because the gravitational force decreases with distance, the curvature of the trajectory is shallow, and the path of the Moon does not bend enough to intersect the surface of Earth. Instead of falling toward Earth, the Moon falls around Earth and circles it repeatedly. The horizontal velocity and downward acceleration are delicately matched to give the Moon a trajectory, which is almost a perfect circle.

Johannes Kepler discovered three laws of planetary motion between 1601 and 1618. The first law expresses the discovery that contrary to all previous expectations, the orbits of the planets are ellipses instead of circles. The Sun occupies a special position at one focus of the ellipse, placing it offset from the geometric center. As a result, the distance from the planet to the Sun changes from a minimum (perihelion) to a maximum (aphelion) and back to a minimum as the planet completes an orbit. The orbit lies entirely in one plane containing the Sun's center.

The second law expresses Kepler’s discovery that a planet's speed varies along its orbit, being greatest at perihelion and smallest at aphelion. The speed variation is such that a line drawn from the Sun to the planet will sweep out equal areas at equal times.

The third law expresses Kepler’s discovery that the size of an orbit is related to the time a planet takes to complete one orbit, called the period. The size of an orbit is indicated by the planet's average distance (mean radius) from the Sun. The cube of the mean radius divided by the square of the period is the same for all planets.

Newton’s demonstration that all three laws follow mathematically from the three laws of motion and the law of gravitation was a magnificent scientific triumph that marked the beginning of the modern scientific age.

Earth Orbit

Kepler’s three laws apply to satellites in Earth orbit with minor changes. The closest approach of a satellite to Earth is called perigee. The most distant point is called apogee. The orbits are still ellipses with Earth at one focus, and they lie in a plane that contains the center of Earth. The cube of the orbit's mean radius divided by the period's square is a constant for all satellites but is not the same constant associated with orbit around the Sun.

Ellipses vary from near-circular to very long and narrow. The degree of narrowing is referred to as the eccentricity. A circle is an ellipse of zero eccentricity. As ellipses get longer and narrower, the eccentricity approaches one.

The plane of the ellipse may be tilted with respect to the equator. The angle between the plane of the orbit and the plane of the equator is the inclination zero. Inclinations from 0 to 90 degrees are associated with satellites orbiting Earth counterclockwise, as seen from a vantage point over the North Pole. Inclinations from 90 to 180 degrees are associated with satellites orbiting clockwise, as seen from above the North Pole.

Positive-inclination orbits are called prograde, and negative-inclination orbits are called retrograde. Prograde orbits are easier to attain because the counterclockwise rotation of Earth adds a free contribution to the velocity of the satellite. Satellites destined for retrograde orbit must launch to the west against the rotation of Earth, making orbit harder to achieve.

Inclinations near 90 degrees are referred to as polar orbits. Satellites in polar orbit will eventually pass over every spot on Earth, making them extremely useful for scientific, remote sensing, and photographic missions.

A satellite in low-Earth orbit has an orbital period of just over ninety minutes. As altitude decreases, the orbital period increases. At an altitude of 35,780 kilometers (22,360 miles), the orbital period is exactly twenty-four hours. At this altitude, a satellite in circular equatorial orbit (zero eccentricity, zero inclination) will travel along its orbit at the same rate Earth turns beneath it. The satellite appears to have a fixed position in the sky, as seen from Earth. These geostationary orbits are particularly advantageous for communications satellites. Since satellites in these orbits never change their apparent position, no antenna tracking is necessary, and the satellites are always available since they never go below the horizon.

Earth Orbit Decay

Ideally, orbits are perfect ellipses that never change. In reality, complications due to the irregular shape of Earth, aerodynamic drag from the thin residual air at orbital altitudes, and the extra gravitational tug of the Sun and Moon continually change the shape and size of satellite orbits.

For low-Earth orbits, the predominant effect is aerodynamic drag. Drag is a dissipative force that converts an object’s energy of motion into heat. Ordinarily, it slows things down, but as a satellite loses kinetic energy, it drops closer to Earth. When this happens, gravitational potential energy is converted into kinetic energy, and a satellite gains more kinetic energy this way than it loses due to drag. The paradoxical result is that a satellite actually ends up going faster (albeit at a lower altitude) due to the drag. Since drag increases with speed, so does the loss of altitude, and eventually, the satellite reenters the atmosphere. The resulting high speeds through dense air create a powerful shock wave in front of the satellite, which compresses and heats the air to the point of incandescence. The satellite burns up like a meteor.

At higher altitudes, aerodynamic drag is negligible, and the change in size, shape, and orientation of the orbit due to the irregular shape of Earth and the extra gravitational tug of the Sun and Moon predominate. These orbital changes can be measured with such accuracy that they can be used to refine knowledge of the shape of Earth and the distribution of mass within its interior. Geology now looks at the motion of objects in the sky to find out what is buried in the ground beneath.

Escape Orbits

The apogee height of a satellite increases as the total energy of the satellite increases. If the total energy is great enough, the apogee height becomes infinite, and the satellite is in an escape orbit. The eccentricity of an escape orbit is greater than one, and the orbit is an open curve called a hyperbola rather than a closed ellipse. The minimum velocity required to put a satellite on an escape trajectory is called the local escape velocity. For low-Earth orbit, the local escape velocity is about eleven kilometers per second (seven miles per second). Satellites with this velocity or greater will leave Earth forever and become artificial planets, satellites of the Sun.

The Solar System and Beyond

Kepler’s third law may be rephrased as the principle that the cube of the mean orbital radius divided by the square of the orbital period equals a constant value multiplied by the gravitating object's mass at the orbit's focus. For Earth orbits, this formula gives the mass of Earth. For the solar system, it provides the mass of the Sun. The principle can be used to determine the mass of any object in the universe with detectable satellites whose orbital radius and period can be measured. Thus, astronomers know the mass of distant objects ranging from tiny asteroids to immense galaxies.

Bibliography

"Chapter 5: Planetary Orbits." NASA, 2024, science.nasa.gov/learn/basics-of-space-flight/chapter5-1. Accessed 20 July 2024.

Hall, Nancy. "Flight To Orbit." NASA, 20 Nov. 2023, www1.grc.nasa.gov/beginners-guide-to-aeronautics/flight-to-orbit. Accessed 20 July 2024.

Layzer, D. Constructing the Universe. New York: Scientific American Library, 1984.

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

Sellers, J. Understanding Space: An Introduction to Astronautics. New York: McGraw-Hill, 1994.