# How does Kepler's laws and Newtons universal gravitation for two bodies come together?

Does Newton's laws of gravitation and Kepler's laws give exactly the same orbit for two bodies?

Could someone please explain the derivation if so?

## 2 Answers

Kepler's 3 laws are descriptions of how 2 bodies move if the force between them is an inverse square attraction.

He derived them from experimental observations of planets and so they are 'correct'. Newton later calculated that an inverse square law would explain these properties of orbits and so decided that gravity followed this equation.

So except that the only apply to a 2 body system (they don't take into account the effect of other planets) they agree with Newton's law of gravitation.

• I think that the OP is asking if the two sets of laws are equivalent in the sense that Kepler's laws imply $F=ma$ for that particular problem. Dec 26 '13 at 21:07
• I do not think that Kepler knew that his laws implied aforce/acceleration. He made a model which fitted his observations, so only came up with a trajectory but. I believe Newton made the connection between the resulting acceleration and therefore the force. But by reading the Wikipedia page it seems that Newton did derive his law of gravitation from Keplers laws. Dec 27 '13 at 16:02

For two bodies, Newton's laws and Kepler's laws agree if and only if the mass of one body (the "planet") is negligible compared with the mass of the other (the "sun"), so that the latter can be considered stationary. I've written an elementary derivation of Kepler's laws from Newton's laws for that special case. On the relationship between that special case and the general two-body problem, see e.g. Peraire & Widnall.