13.17 13.6: Planetary Motion According to Kepler and Newton

In this section, we derive Kepler’s laws of planetary motion, a feat first accomplished by Isaac Newton and published by him in 1687. No event was more emblematic of the scientific revolution. It demonstrated the power of mathematics to make the natural world comprehensible and it led succeeding generations of scientists to seek and discover mathematical laws governing other phenomena, such as electricity and magnetism, thermodynamics, and atomic processes.

According to Kepler, the planetary orbits are ellipses with the sun at one focus. Furthermore, if we imagine a radial vector r(t) pointing from the sun to the planet, as in Figure 1, then this radial vector sweeps out area at a constant rate or, as Kepler stated in his Second Law, the radial vector sweeps out equal areas in equal times (Figure 2). Kepler’s Third Law determines the period T of the orbit, defined as the time required to complete one full revolution. These laws are valid not just for planets orbiting the sun, but for any body orbiting about another body according to the inverse-square law of gravitation.

The planet travels along an ellipse with the sun at one focus.
The two shaded regions have equal areas, and by Kepler’s Second Law, the planet sweeps them out in equal times. To do so, the planet must travel faster going from A to B than from C to D.

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Kepler’s Three Laws

  • Law of Ellipses: The orbit of a planet is an ellipse with the sun at one focus.

  • Law of Equal Area in Equal Time: The position vector pointing from the sun to the planet sweeps out equal areas in equal times.

  • Law of the Period of Motion: , where

    • a is the semimajor axis of the ellipse (Figure 1).

    • G is the universal gravitational constant: 6.673 × 10−11 m3 kg−1 s−2.

    • M is the mass of the sun, approximately 1.989 × 1030 kg.

Kepler’s version of the Third Law stated only that T2 is proportional to a3. Newton discovered that the constant of proportionality is equal to 4π2/(GM), and he observed that if you can measure T and a through observation, then you can use the Third Law to solve for the mass M. This method is used by astronomers to find the masses of the planets (by measuring T and a for moons revolving around the planet) as well as the masses of binary stars and galaxies. See Exercises 2–5.

Our derivation makes a few simplifying assumptions. We treat the sun and planet as point masses and ignore the gravitational attraction of the planets on each other. And although both the sun and the planet revolve around their mutual center of mass, we ignore the sun’s motion and assume that the planet revolves around the center of the sun. This is justified because the sun is much more massive than the planet.

We place the sun at the origin of the coordinate system. Let r = r(t) be the position vector of a planet of mass m, as in Figure 1, and let (Figure 3)

The gravitational force F, directed from the planet to the sun, is a negative multiple of er.

be the unit radial vector at time t (er is the unit vector that points to the planet as it moves around the sun). By Newton’s Universal Law of Gravitation (the inverse-square law), the sun attracts the planet with a gravitational force

where k = GM (Figure 3). Combining the Law of Gravitation with Newton’s Second Law of Motion F(r(t)) = mr″(t), we obtain

Kepler’s Laws are a consequence of this differential equation.