### K3 implies the Inverse Square Law. Johannes Kepler. Stamp issued by the German Democratic Republic in 1971, the 400th anniversary of Kepler’s birth.

Kepler formulated three remarkable laws of planetary motion. He deduced them directly from observations of the planets, most particularly of the motion of Mars. The first two laws appeared in 1609 in Kepler’s Astronomia Nova. The first law (K1) describes the orbit of a planet as an ellipse with the Sun at one focus. The second law (K2) states that the radial line from Sun to planet sweeps out equal areas in equal times; we now describe this in terms of conservation of angular momentum.

The third law (K3), which appeared in 1619 in Kepler’s Harmonices Mundi, is of a different character. It does not relate to a single planet, but connects the motions of different planets. It states that the squares of the orbital periods vary in proportion to the cubes of the semi-major axes. For circular orbits, the period squared is proportional to the radius cubed.

K3 implies the Inverse Square Law of Gravity

For simplicity, we will consider only circular orbits, although all results hold in the more general case of elliptical orbits. For circular motion, it is a simple kinematical result that the centripetal acceleration varies in proportion to the square of velocity and inversely as the radius. Thus, for a circular orbit, the force towards the centre must be proportional to this: $\displaystyle F \propto \frac{V^2}{R}$

Now comes K3: it states that ${T^2 \propto R^3}$. The distance around the orbit is ${2\pi R}$ and the period is ${T}$, so the speed is ${V = 2\pi R / T}$. Using these results in the expression for the force, we get $\displaystyle F \propto \frac{(2\pi R/T)^2}{R} = 4\pi^2 \frac{R}{T^2} \propto \frac{R}{R^3} = \frac{1}{R^2} \,.$

This is the inverse square law. Thus was Newton able to make use of Kepler’s empirical laws to formulate his Law of Gravity.

Feynman’s Lost Lecture

We have considered circular orbits, but Kepler found after a long struggle that he could not reconcile the observations with such simple orbits. He eventually discovered his law of the ellipse (K1) and his law of equal areas (K2). But how do these `square’ with Newton’s law?

This is the topic of Feynman’s Lost Lecture. On 13 March, 1964, Richard Feynman gave an additional lecture to the students, that was not included in the three-volume collection, The Feynman Lectures on Physics, published in 1963–65. The record of this lecture was misplaced and it was only years later that Feynman’s rough notes surfaced again. The full story is told in the book by David and Judith Goodstein (1996).

Feynman showed how elliptical orbits emerge as a consequence of the inverse square law of gravitation. He tried to follow Newton’s method, as expounded in the Principia Mathematica, but found it so obscure that he “cooked up” a method of his own. His reasoning was elementary, requiring no advanced knowledge or skill; but it was intricate and demanding to follow in detail. For that reason, David Goodstein reconstructed the proof, expanding it to provide clarity.

The entire reconstruction of the Lost Lecture, together with Feynman’s proof, is presented in the book, Feynman’s Lost Lecture (Goodstein and Goodstein, 1996). it is also the subject of an excellent 3Blue1Brown Youtube video by Grant Sanderson.

Carving up the Ellipse

Kepler’s second law states that the radial vector sweeps out equal areas in equal times. This leads us to consider diagrams like this: Diagram for Kepler’s Second Law [Image from 3Blue1Brown]

The two sectors are of equal area. However, Feynman found it advantageous to divide the ellipse into sectors having equal central angles, as shown here: Diagram as used by Feynman for Kepler’s First Law [Image from 3Blue1Brown].

This allowed him to use known geometric properties of the ellipse to arrive at the desired result. For the full story of Feynman’s Lost Lecture, see the references below.

Sources ${\bullet}$ Goodstein, David L and Judith R Goodstein, 1996: Feynman’s Lost Lecture. Jonathan Cape, London, 191pp. ${\bullet}$ Grant Sanderson 2018: {Feynman’s Lost Lecture}. Youtube 3Blue1Brown Video.