How do kepler's laws of planetary motion relate to newton's law of universal gravitation?

were obtained from direct observation of planetary motion, but can be derived from Newton's universal law.

You might want to look at:http://www.physicsclassroom.com/class/circles/Lesson-4/Kepler-s-Three-Laws

A quick example is as follows:the universal law is

##F={GMm}/R^2##,

where ##M## is the mass of the Sun say, ##m## of a planet and ##R## the radius of its orbit.

Now, since ##F=ma## we can rewritte this as

##a={M}/R^2##,

At this point we can apply dimensional analysis, by noting that is a length divided by a time squared and thus must be given by

## a \propto R/T^2##,

where ##T## is the period of the orbit - the only thing we don't know is the constant of proportionality (but that's not part of Kepler's law). So we have

## {GM}/R^2 \propto R/T^2##,

which, dropping the unimportant constant ##GM## and rearranging, gives

## T^2 \propto R^3##.