Kepler's second Law

Figure 13.19 shows Kepler’s second law. In a small time interval dt, the line from the sun S to the planet P turns through an angle du. The area swept out is the colored triangle with height r, base length r du, and area dA = 1/ 2 r² du in Fig. 13.19b. The rate at which area is swept out, dA/dt, is called the sector velocity:

The essence of Kepler’s second law is that the sector velocity has the same value at all points in the orbit. When the planet is close to the sun, r is small and du/dt is large; when the planet is far from the sun, r is large and du/dt is small.

To see how Kepler’s second law follows from Newton’s laws, we express dA/dt in terms of the velocity vector v S of the planet P. The component of v S perpendicular to the radial line is v. = vsinØ. From Fig. 13.19b the displacement along the direction of v. during time dt is r du, so we also have v# = r du/dt. Using this relationship in Eq. (13.14), we find

Now rvsinØ is the magnitude of the vector product r X S : v X S , which in turn is 1/m times the angular momentum L X S = r X S : mv S of the planet with respect to the sun. So we have

Thus Kepler’s second law—that sector velocity is constant—means that angular momentum is constant!

It is easy to see why the angular momentum of the planet must be constant. According to Eq. (10.26), the rate of change of L S equals the torque of the gravitational force F S acting on the planet:

In our situation, r S is the vector from the sun to the planet, and the force F → is directed from the planet to the sun (Fig. 13.20). So these vectors always lie along the same line, and their vector product r S : F S is zero. Hence dL → /dt = 0. This conclusion does not depend on the 1/r² behavior of the force; angular momentum is conserved for any force that acts always along the line joining the particle to a fixed point. Such a force is called a central force. (Kepler’s first and third laws are valid for a 1/r² force only.)

Conservation of angular momentum also explains why the orbit lies in a plane. The vector L S = r S : mv S is always perpendicular to the plane of the vectors r S and v S ; since L S is constant in magnitude and direction, r → and v → always lie in the same plane, which is just the plane of the planet’s orbit.

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