13.6.2 Kepler’s Second Law

FIGURE 13.18a shows a planet moving in an elliptical orbit. In Chapter 12 we defined a particle’s angular momentum to be

L = m r v sin β

(13.26)

where $β$ is the angle between $\vec{r}$ and $\vec{v}$ . For a circular orbit, where $β$ is always $90 °$ , this reduces to simply $L = m r v$ .

The figure labeled ("a") illustrates the elliptical orbit of a planet depicted as a small sphere around the Sun.

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The figure labeled (b) illustrates the elliptical orbit of a planet depicted as a small sphere around the Sun.

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The only force on the satellite, the gravitational force, points directly toward the star or planet that the satellite is orbiting and exerts no torque; thus the satellite’s angular momentum is conserved as it orbits.

The satellite moves forward a small distance $Δ s = v Δ t$ during the small interval of time $Δ t$ . This motion defines the triangle of area $Δ A$ shown in FIGURE 13.18b. $Δ A$ is the area “swept out” by the satellite during $Δ t$ . You can see that the height of the triangle is $h = Δ s sin β$ , so the triangle’s area is

Δ A = \frac{1}{2} \times base \times height = \frac{1}{2} \times r \times Δ s sin β = \frac{1}{2} r v sin β Δ t

(13.27)

The rate at which the area is swept out by the satellite as it moves is

\frac{Δ A}{Δ t} = \frac{1}{2} r v sin β = \frac{m r v sin β}{2 m} = \frac{L}{2 m}

(13.28)

The angular momentum L is conserved, so it has the same value at every point in the orbit. Consequently, the rate at which the area is swept out by the satellite is constant. This is Kepler’s second law, which says that a line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. We see that Kepler’s second law is a consequence of the conservation of angular momentum.

Another consequence of angular momentum is that the orbital speed is constant only for a circular orbit. Consider the “ends” of an elliptical orbit, where $r$ is a minimum or maximum. At these points, $β = 90 °$ and thus $L = m r v$ . Because $L$ is constant, the satellite’s speed at the farthest point must be less than its speed at the nearest point. In general, a satellite slows as $r$ increases, then speeds up as $r$ decreases, to keep its angular momentum (and its energy) constant.

Kepler’s laws summarize observational data about the motions of the planets. They were an outstanding achievement, but they did not form a theory. Newton put forward a theory, a specific set of relationships between force and motion that allows any motion to be understood and calculated. Newton’s theory of gravity has allowed us to deduce Kepler’s laws and, thus, to understand them at a more fundamental level.