Commons

stated roughly as a whole number, for convenience in calculation).What Newton discovered was the reasons why Kepler’s laws weretrue: he showed that they followed from his laws of motion. From amodern point of view, conservation laws are more fundamental thanNewton’s laws, so rather than following Newton’s approach, it makesmore sense to look for the reasons why Kepler’s laws follow fromconservation laws. The equal-area law is most easily understood asa consequence of conservation of angular momentum, which is a newconserved quantity to be discussed in chapter 3. The proof of theelliptical orbit law is a little too mathematical to be appropriate forthis book, but the interested reader can find the proof in chapter 5of my online book Conservation Laws.v / Connecting Kepler’s law of periodsto the laws of physics.The law of periods follows directly from the physics we’ve alreadycovered. Consider the example of Jupiter and Uranus. We want toshow that the result of example 13 is the only one that’s consistentwith conservation of energy and momentum, and Newton’s law ofgravity. Since Uranus takes eight times longer to cover four timesthe distance, it’s evidently moving at half Jupiter’s speed. In figurev, the distance Jupiter covers from A to B is therefore twice thedistance Uranus covers, over the same time, from D to E. If therehadn’t been any gravitational force from the sun, Jupiter wouldhave ended up at C, and Uranus at F. The distance from B to C isa measure of how much force acted on Jupiter, and likewise for thevery small distance from E to F. We find that BC is 16 millimeterson this scale drawing, and EF is 1 mm, but this is exactly whatwe expect from Newton’s law of gravity: quadrupling the distanceshould give 1/16 the force.58 Chapter 2 Conservation of Momentum