Since one satellite can see half of the earth, why do we need more than two satellites in a given network?
Published: May 10, 2013
A satellite in orbit around the earth cannot see half of the earth because of a simple geometric effect known as perspective. As shown in the diagram, when an observer is looking at a sphere, he can only see the portion of the sphere that lies in front of the points where his line of sight is tangential to the sphere. The closer the observer is to a sphere, the less he can see. The line marking the point where the earth begins curving out of view is called the horizon. It is the same horizon that you see when standing on the earth and observing the sky meet the ground in the distance. While hills and other local ground height variations affect the distance to the horizon, these variations are so small compared to the height of satellites that they can be mostly ignored.
If the radius of the sphere is R and the distance of the observer from the sphere's surface is d, then simple geometry reveals that the percent of the sphere's surface A that is visible to the observer is:
As d becomes much larger than R, this equation shows that the visible area approaches fifty percent of the sphere's surface. But to see exactly half of the sphere, the observer would have to be infinitely far away. To a good approximation, the earth is a sphere and the satellites are distant observers. The point where the earth curves away out of view is known as the horizon. The earth's radius is about 6370 kilometers (3960 miles). A person walking around on the surface of the earth has his eyes elevated about 0.002 km (6 feet) away from the surface. Using this equation, if the earth were perfectly spherical, then a person standing on earth's surface can see about 0.000016% of the earth's surface, or about 82 square kilometers (32 square miles). This corresponds to a circle of visibility with a radius of about 5 km (3 miles). In other words, if you stand on the beach at the water's edge and look out over the calm ocean, the farthest you can see boats before they begin to disappear below the horizon is about 5 km. Naturally, getting up away from the earth's surface will increase your view. For instance, standing on top of a typical mountain with an elevation of 1 km will improve your visibility to 0.0078% of the earth' surface, which is about 40,000 square kilometers (15,000 square miles), such that the horizon is about 110 km (70 miles) away. Often, other mountains, trees, and even the atmosphere itself reduces the visibility.
Earth's satellites orbit at a wide range of altitudes, so let us take the network of GPS satellites as an example. The GPS satellites orbit at an altitude of about d = 20,000 km. Using the equation above, each GPS satellite can only "see" about 38% of earth's surface in a given instant. Therefore, you would need a bare minimum of three GPS satellites in order to "see" the entire globe at once. In reality, the earth is not a perfect sphere, mountains can get in the way, and the atmosphere itself bends light, so that four GPS satellites would be needed to "see" the whole earth at once. Additionally, the job of the GPS satellites is not just to see the earth, but to help a ground GPS receiver determine its location through trilateration. This location-finding process in its most accurate form requires eight satellites to be able to see each spot. Because of the limited field of view of each satellite, and the need to have so many satellites overlapping the same view, the GPS network currently contains 32 satellites.