How is Newton's Law of Gravity related to Kepler's Laws of Planetary Motion?

Question

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Answer 1

Kepler's laws refer to a specific case of an orbiting object, Newtons Law of Gravity covers all objects, those falling within the gravitational field (literally!), those held in orbit and the height at which the field gives out for a particular body. Keplers laws describe the interaction between two massive bodies in space; it is Newtons law of gravity which provides the underpinning theory for the laws of planetary motion. hope this helps.

Answer 2

All of Kepler's Laws (planet orbit is an ellipse, equal area in equal time, etc) can be derived from Newton's Law of Gravititation. Trying to explain Kepler's laws must have helped Newton to arrive at his great (and simplifying) discovery.

Answer 3

Let's see how well I can do by memory here (just a warning for you to double check me!) ...

I'm confident that the "equal areas in equal times" law is a consequence of conservation of angular momentum. That only depends upon the force on the planet being aimed toward the sun. The force wouldn't have to obey an inverse square law for this to still work.

I remember that an orbit would still be an ellipse if the force were like a spring force, getting stronger proportional to distance away, but in that case the sun would be at the center of the ellipse instead of at a focus. I don't know if there's any other force law that would work with an elliptical orbit and the sun at one focus.

I have a feeling that the period^2 and semimajor axis^3 relation is a consequence of the inverse square law for gravitation but here my memory is a little fuzzy.

I hope this is what you are looking for.

Answer 4

We can derived Kepler's from Newton's.

If we make the simplifying assumption that orbits are circular, we can write:

Using Newt's law, we have f = GmM/R^2 = mw^2R and F = GnM/r^2 = nW^2r for gravity forces by M on m and n at distances R and r from M, and for cetrifugal forces on m at w angular velocity and n at W angular velocity.

Then GmM/R^2//GnM/r^2 = mw^2R//nW^2r; so that 1/R^2//1/r^2 = w^2R/W^2r = r^2/R^2 = (R/r)(w^2/W^2). This give us:

r^3/R^3 = w^2/W^2 which, ta da, is a variant Kepler's law. Of course Kepler's law can be more generalized to take elliptical orbits into account. But that's a bit more difficult to do.

Answer 5

Objects subjected to a force that is a inverse-square type travel in ellipses. The fact that angular momentum is conserved (no external torques on the body) will get you the rest of kepler's laws.