Newton, Ellipse and acceleration

Using the Kepler's second law (areas law) , Newton proved that the acceleration of the planet in its orbit was always pointed at the sun. Using Kepler's third law (radius of orbit cubed over period of orbit squared is the same for all planets) and assuming circular orbits, Wren and Hooke and Newton (who did it first is still hotly debated) showed that at least on average, the acceleration of the planets in their orbits was proprotional to the inverse of the distance between the planet and the sun squared.

Newton used Kepler's second law and first law to argue that even in the single orbit, of a planet, the acceleration of the planet around the orbit, which by the second law always pointed at the sun, fell off by the square of the inverse distance. The complete proof that this is true can be done purely by geometry, and was done so by Newton, but the demonstration is exceedingly complicated. It was also done by Feynman in a different way in some rediscovered lectures Again this is very complicated and requires some real fascility with geometry and the properties of ellipses.

However, Newton in the {\it Principia} shows fairly simply that Kepler's first law implies that the accelerations at the two ends of the ellipse must be related as the ratio of the squares of the distance from the focus of the ellipse where the sun is to where the planet is at the two ends. (the acceleration at the nearer end to the sun being larger than the one at the other end).

The key feature is that the ellipse is symmetric. Ie, if you fold over the ellipse across either axis, the folded part will lie on top of the other half. This means that the way in which the line curves at the two ends of the ellipse are identical. Now, use the same argument that Huygens used for the centrifugal acceleration of an object going around the orbit. Just as for the circle, if we look at the object going around the end the of the ellipse, and draw the stright line that the object would go along at the end if there were nothing that made it go in an ellipse, one has what looks like Galileo's free fall trajectory with a sideways combined motion. The Galilean acceleration is, just as in Huygen's argument, proportional to the velocity squared divided by some "radius" of the curve (this "radius" happens to be the distance from the focus of the ellipse (the sun) vertially to the ellipse but its value does not really matter). What is important is that because the ellipse at the ends are identical, that "radius" is also identical at the two ends. Thus the accelertions at the two end are just proportional to the the velocities squareds at the two ends. But by Kepler's second law, the area of a tiny triangle formed over the same small interval of time at the two ends is proportional to the one half of the velocity there times the distance to the sun (the area swept out is the same for the same amount of time). In the diagram, the base of that triangle is curved since I took a large enough value of the time interval so you could see the triangle. If you imagine taking it much much smaller, the curve of the ellipse will become flat and the area will really be that of a triangle. Thus the velocity is proportional to one over the distance to the sun, and the acceleration, which is proportional to the velocity squared, is proportional to the distance squared.

Thus the acceleration at each end times the distance of the end to the sun squared is the same at both ends. This suggests but does not prove that the same would be true anywhere on the orbit-- the acceleration is proportional to the distance squared.

Since, from Newton's second law, the force is proportional to the acceleration, the force required to keep the planet in its orbit must be proportional to one over the distance to the sun squared.

Newton, in the Principia, also proved by similar, but much more complex arguments that the force must be an inverse square force everywhere in the orbit if the orbit is an ellipse and Kepler's area law is true.

None of this was even dreamed of by Kepler himself. It would have astonished him. He was still influenced by Aristotle, and believed that the planets moved because the sun was rotating, and that rotation had flails which reached out to the planet and swept them along, with the strength of the flails weakening the further from the sun they stretched. It was a picture which was almost impossible to mathematize. However his three laws of motion were possible to mathematize.

Copyright W Unruh (2018)