THE LAW OF GRAVITY
Energy Considerations in Planetary and Satellite Motion
Consider a body of mass m moving with a speed v in the vicinity of a massive body of mass M, where M >> m.The system might be a planet moving around the Sun, a satellite in orbit around the Earth, or a comet making a one-time flyby of the Sun. If we assume that the body of mass M is at rest in an inertial reference frame, then the total mechanical energy E of the two-body system when the bodies are separated by a distance r is the sum of the kinetic energy of the body of mass m and the potential energy of the system, given by Equation 14.15
This equation shows that E may be positive, negative, or zero, depending on the value of v. However, for a bound system, such as the Earth–Sun system, E is necessarily less than zero because we have chosen the convention that U → 0 as r → ∞.
We can easily establish that E < 0 for the system consisting of a body of mass m moving in a circular orbit about a body of mass M >> m (Fig. 14.16). Newton’s second law applied to the body of mass m gives
Figure 14.16 A body of mass m moving in a circular orbit about a much larger body of mass M.
Multiplying both sides by r and dividing by 2 gives
Substituting this into Equation 14.17, we obtain
This result clearly shows that the total mechanical energy is negative in the case of circular orbits. Note that the kinetic energy is positive and equal to one-half the absolute value of the potential energy. The absolute value of E is also equal to the binding energy of the system, because this amount of energy must be provided to the system to move the two masses infinitely far apart.
The total mechanical energy is also negative in the case of elliptical orbits. The expression for E for elliptical orbits is the same as Equation 14.19 with r replaced by the semimajor axis length a. Furthermore, the total energy is constant if we assume that the system is isolated. Therefore, as the body of mass m moves from P to Q in Figure 14.13, the total energy remains constant and Equation 14.17 gives
Combining this statement of energy conservation with our earlier discussion of conservation of angular momentum, we see that both the total energy and the total angular momentum of a gravitationally bound, two-body system are constants of the motion.
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