Solving Newton’s second law to find the trajectory of a mass moving under the influence of gravity is mathematically beyond this textbook. It turns out that the solution is a set of elliptical orbits, which is Kepler’s first law. Kepler had no reason why orbits should be ellipses rather than some other shape. Newton was able to show that ellipses are a consequence of his theory of gravity.

The mathematics of ellipses is rather difficult, so we will restrict most of our analysis to the limiting case in which an ellipse becomes a circle. Most planetary orbits differ only very slightly from being circular. The earth’s orbit, for example has a (semiminor axis/semimajor axis) ratio of 0.99986—very close to a true circle!

FIGURE 13.16 shows a massive body M, such as the earth or the sun, with a lighter body m orbiting it. The lighter body is called a satellite, even though it may be a planet orbiting the sun. For circular motion, where the gravitational force provides the centripetal acceleration $v^2\text{/}r$, Newton’s second law for the satellite is

Thus the speed of a satellite in a circular orbit is

FIGURE 13.16 The orbital motion of a satellite due to the force of gravity.

A satellite must have this specific speed in order to have a circular orbit of radius r about the larger mass M. If the velocity differs from this value, the orbit will become elliptical rather than circular. Notice that the orbital speed does not depend on the satellite’s mass m. This is consistent with our previous discovery, for motion on a flat earth, that motion due to gravity is independent of the mass.