Dynamics of Satellite Orbits
Leon Blitzer
University of Arizona
Equations 6.3 and 6.4 for the velocity and period of a satellite in a circular orbit about an assumed spherical earth have long been known:
(6.3)
(6.4)
However, the advent of the Space Age had to await the development of rockets with sufficient thrust to propel the missile into orbit. With the launch of Sputnik I on October 4, 1957, artificial earth satellites became a reality, and since then numerous satellites and space probes have been sent into orbit. These manned, as well as unmanned, space explorations have captured the interest and imagination of the entire world. The unique character of the satellite is that it provides a sustained observing platform outside the atmosphere for studying the earth and its environs, as well as outer space. Today, hundreds of satellites and space probes are in orbit, and their applications encompass just about the entire range of science and engineering research: astronomy, global communication, navigation, weather reconnaissance, planetary studies, geodesy, studies of biological organisms in a weightless environment, cosmic ray and solar physics, ocean and land surveillance for marine and mineral resources, agricultural surveys, location of sites for archaeological explorations, etc. There is no doubt that the future will see an increasing number of satellites and space probes being used for ever-expanding applications.
You should note that Equations 6.3 and 6.4 have application beyond artificial earth satellites, for they are valid for any satellite (moon) moving in a circular orbit about its parent planet, or any planet moving in a circular orbit about the sun. More generally, satellite and planetary orbits are elliptical, the circle being a special case of the ellipse.*
The Satellite Orbit Paradox
Clearly, forces are required to overcome the earth's gravitational attraction and to propel a satellite into orbit, with higher orbits requiring greater forces. Consider a satellite moving in a given circular orbit. If the satellite is subjected to some force in the same direction as its motion, it will be propelled into a higher orbit and will travel at a slower speed according to Equation 6.3. Conversely, if the satellite is subjected to some force in a direction opposite to its motion, it will be driven into a lower orbit and will move at a greater speed according to Equation 6.3.
It is known that even at altitudes of 500 km and more above the earth there are sufficient atmospheric particles to create a significant frictional (drag) force on the rapidly moving satellite. Since the frictional force is in a direction opposite to its motion, it will cause the satellite to shift into a lower orbit and move with greater velocity. Hence, we have the so-called "drag paradox"; namely, atmospheric friction causes the orbital velocity to increase. Indeed, all satellites moving within the earth's atmosphere slowly spiral inward at ever-increasing speed until they burn up or impact the earth (see Fig. 1). Moreover, this is also the case for satellites moving in elliptical orbits.º
Note that the paradox is not limited to drag, for any force acting in the same direction as the motion of the satellite causes the missile to shift into a higher orbit and move with slower speed, while any retarding force actually results in an increase in speed.º
Geostationary SYNCOM Satellites
Consider a satellite moving in a circular orbit in the plane of the equator at such a distance that its orbital period is synchronous with the rotational period of the earth, namely one sidereal day. Such a satellite will then be at a fixed geographic longitude, and is referred to as geostationary. Figure 2 shows three uniformly spaced satellites in synchronous equatorial orbits. This configuration of three geostationary satellites, when equipped with radio transponders, can provide line-of-sight global communication between any two points on earth. Practically all satellites currently used for communication are in such 24-hour synchronous orbits and hence referred to as SYNCOMS.
Libration of Synchronous Satellites
Because of its nonuniform shape and mass distribution÷÷oceans, mountains, variations in density÷÷the earth is far from spherical. In fact, it is flattened at the poles and bulges at the equator, with the equatorial cross-section being very nearly elliptical. (On the basis of satellite tracking data, the difference between the major and minor axes of the equatorial ellipse is about 130 m.)
Let us examine the motion of a 24-hour satellite in a frame of reference rotating with the earth (see Fig. 3). It is clear from symmetry that when the satellite is on the extension of either principal axis of the equatorial ellipse, at positions S or U, the gravitational force is purely radial. These must then be equilibrium positions, or stationary points, in the rotating frame. On the other hand, at off-axis positions, the greater gravitational attraction will be toward the nearest major axis. Hence, there is a net tangential force FT toward the nearest major axis, as indicated for various positions of the satellite in Figure 3.
At first thought, one might expect the satellite to accelerate in the direction of FT. However, in accordance with the orbit paradox, the satellite will actually drift in the opposite direction toward the nearest equilibrium position S on the minor axis. Since it acquires momentum, the satellite will drift past S, the direction of FT will then be reversed, and the drift will gradually be reversed. Hence the satellite will oscillate, or librate, about the equilibrium position S on the minor axis. Contrary to the drag paradox, there is no friction involved in this process.
The path of one such 24-hour satellite in the rotating earth frame is shown as a dashed curve in Figure 3. The period of libration depends on the amplitude, which, in turn, is determined by the initial position. For small-amplitude librations, the period is approximately 2.1 years. Unless provided with propulsion for repositioning, all SYNCOM satellites experience such librations.
Suggested Readings
Blitzer, L., "Satellite Orbit Paradox: A General View," Amer. J. Phys. 39:882, 1971.
"Basic Facts About The Satellite Orbits," Sky and Telescope 15:408, 1956.
Dubridge, L.A., "Fun In Space," Amer. J. Phys. 28:719, 1960.
Blitzer, L., "Equilibrium and Stability of a Pendulum in an Orbiting Spaceship," Amer. J. Phys. 47:241, 1979.
Footnotes
* This is apart from perturbations due to the attraction of other planets, tidal forces, radiation pressure, particle drag, electromagnetic forces, nonspherical shapes, etc.
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Figure Captions
Figure 1 Shrinking of the orbit under atmospheric drag.
Figure 2 Three satellites uniformly spaced in synchronous equatorial orbits. (Not drawn to scale.)
Figure 3 Equatorial section of Earth (looking south along polar axis). FT is the net tangential force on the satellite at the positions shown. S is the stable equilibrium position; U, unstable. The blue dashed curve shows the librational path of a 24-hour satellite.