The Motion of satellites

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Artificial satellites orbiting the earth are a familiar part of technology (Fig. 13.13). But how do they stay in orbit, and what determines the properties of their orbits? We can use Newton’s laws and the law of gravitation to provide the answers. In the next section we’ll analyze the motion of planets in the same way.

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In Example 3.6 a motorcycle rider rides horizontally off the edge of a cliff, launching himself into a parabolic path that ends on the flat ground at the base of the cliff. If he survives and repeats the experiment with increased launch speed, he will land farther from the starting point. We can imagine him launching himself with great enough speed that the earth’s curvature becomes significant. As he falls, the earth curves away beneath him. If he is going fast enough, and if his launch point is high enough that he clears the mountaintops, he may be able to go right on around the earth without ever landing.

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Figure 13.14 shows a variation on this theme. We launch a projectile from point A in the direction AB, tangent to the earth’s surface. Trajectories 1 through 7 show the effect of increasing the initial speed. In trajectories 3 through 5 the projectile misses the earth and becomes a satellite. If there is no retarding force such as air resistance, the projectile’s speed when it returns to point A is the same as its initial speed and it repeats its motion indefinitely.

Trajectories 1 through 5 close on themselves and are called closed orbits. All closed orbits are ellipses or segments of ellipses; trajectory 4 is a circle, a special case of an ellipse.  Trajectories 6 and 7 are open orbits. For these paths the projectile never returns to its starting point but travels ever farther away from the earth.

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