(K1) - First Law The orbit of each planet is an ellipse, with the sun at one focus.
(K2) - Second Law The line joining the planet to the sun sweeps out equal areas in equal times.
(K3) - Third Law The square of the period of a planet is proportional to the cube of its mean distance form the sun.
These three laws governing planetary motion were discovered by Kepler, but we will see how even more general forms of these laws governing the motion of all celestial bodies may be derived from Newton's Laws.
(K1) TRAJECTORY OF AN ORBIT
Start with the two body equation of motion
cross both sides of the equation into
since
and
Integrating ( 1 )
dot both sides with 
Now because
then
which is the polar form of a conic section with the origin at one focus. Note that r is a function of the polar angle 
and that the semi latus rectum
and the eccentricity vector
are defined in the equation. This establishes more than Kepler's first law. We have shown that any orbit determined by the two body equation of motion must be a conic section with the focus located at the central body.
(K2) CENTRAL FORCE
Angular momentum can be written
and
Since is a constant in (2) the the equation states that equal areas are swept out in equal times. is always a constant for a central force. see Angular Momentum
(K3) PERIOD OF AN ELLIPTICAL ORBIT
Integrating (2) over one period yields
but
which proves (K3) .
The period of an elliptical orbit depends only on the size of the semi major axis and the central mass.
VIS VIVA LAW
Take the expression derived for the specific mechanical energy
We can evaluate the expression at any point in the orbit because energy is a constant of the motion. At perifocus r = q and and 
are perpendicular. We find that
