The solution of the two-body problem
 
It is not within the scope of this text to show how equations (1)
(1)
And (2)
(2)
may be solved. The solution, however, may be written as
(3)
where h is a constant which is twice the rate of description of area by the radius vector, e is the eccentricity of the orbit and θ is the true anomaly .
Equation (3) is, in fact, the polar equation of a conic section. In obtaining this solution, Newton generalized Kepler’s first law, for a conic section can not only be an ellipse but also a parabola or a hyperbola.
We will, however, obtain an important relation, the energy integral. This leads to a useful expression giving the velocity of a planet at any point in his orbit.