satellites: Circular orbits
A circular orbit is the simplest case. It is also an important case, since many artificial satellites have nearly circular orbits and the orbits of the planets around the sun are also fairly circular. The only force acting on a satellite in circular orbit around the earth is the earth’s gravitational attraction, which is directed toward the center of the earth and hence toward the center of the orbit (Fig. 13.15). As we discussed in Section 5.4, this means that the satellite is in uniform circular motion and its speed is constant. The satellite isn’t falling toward the earth; rather, it’s constantly falling around the earth. In a circular orbit the speed is just right to keep the distance from the satellite to the center of the earth constant.
Let’s see how to find the constant speed v of a satellite in a circular orbit. The radius of the orbit is r, measured from the center of the earth; the acceleration of the satellite has magnitude arad = v2/r and is always directed toward the center of the circle. By the law of gravitation, the net force (gravitational force) on the satellite of mass m has magnitude Fg = GmEm/r2 and is in the same direction as the acceleration. Newton’s second law then tells us that
Solving this for v, we find
This relationship shows that we can’t choose the orbit radius r and the speed v independently; for a given radius r, the speed v for a circular orbit is determined
The satellite’s mass m doesn’t appear in Eq. (13.10), which shows that the motion of a satellite does not depend on its mass. An astronaut on board an orbiting space station is herself a satellite of the earth, held by the earth’s gravity in the same orbit as the station. The astronaut has the same velocity and acceleration as the station, so nothing is pushing her against the station’s floor or walls. She is in a state of apparent weightlessness, as in a freely falling elevator; see the discussion following Example 5.9 in Section 5.2. (True weightlessness would occur only if the astronaut were infinitely far from any other masses, so that the gravitational force on her would be zero.) Indeed, every part of her body is apparently weightless; she feels nothing pushing her stomach against her intestines or her head against her shoulders (Fig. 13.16).
Apparent weightlessness is not just a feature of circular orbits; it occurs whenever gravity is the only force acting on a spacecraft. Hence it occurs for orbits of any shape, including open orbits such as trajectories 6 and 7 in Fig. 13.14. We can derive a relationship between the radius r of a circular orbit and the period T, the time for one revolution. The speed v is the distance 2pr traveled in one revolution, divided by the period:
We solve Eq. (13.11) for T and substitute v from Eq. (13.10)
Equations (13.10) and (13.12) show that larger orbits correspond to slower speeds and longer periods. As an example, the International Space Station (Fig. 13.13) orbits 6800 km from the center of the earth (400 km above the earth’s surface) with an orbital speed of 7.7 km/s and an orbital period of 93 min. The moon orbits the earth in a much larger orbit of radius 384,000 km, and so has a much slower orbital speed (1.0 km/s) and a much longer orbital period (27.3 days).
It’s interesting to compare Eq. (13.10) to the calculation of escape speed in Example 13.5. We see that the escape speed from a spherical body with radius R is (2)1/2 times greater than the speed of a satellite in a circular orbit at that radius. If our spacecraft is in circular orbit around any planet, we have to multiply our speed by a factor of 21/2 to escape to infinity, regardless of the planet’s mass
Since the speed v in a circular orbit is determined by Eq. (13.10) for a given orbit radius r, the total mechanical energy E = K + U is determined as well. Using Eqs. (13.9) and (13.10), we have
The total mechanical energy in a circular orbit is negative and equal to one-half the potential energy. Increasing the orbit radius r means increasing the mechanical energy (that is, making E less negative). If the satellite is in a relatively low orbit that encounters the outer fringes of earth’s atmosphere, mechanical energy decreases due to negative work done by the force of air resistance; as a result, the orbit radius decreases until the satellite hits the ground or burns up in the atmosphere.
We have talked mostly about earth satellites, but we can apply the same analysis to the circular motion of any body under its gravitational attraction to a stationary body. Figure 13.17 shows an example.