Kepler's first law
Our planet's radial equation of motion, Equation (4.18), can be combined with
Equation (4.22) to give

(4.25) 
Suppose that
, where
and
. It follows that

(4.26) 
Likewise,

(4.27) 
Hence, Equation (4.25) can be written in the linear form

(4.28) 
As can be seen via inspection, the general solution to the preceding equation is

(4.29) 
where and are arbitrary constants. Without loss of generality, we can
set
by rotating our coordinate system about the axis. We can also assume that . Thus,
we obtain

(4.30) 
where

(4.31) 
Equation (4.30) is the equation of a conic
section that is confocal with the origin (i.e., with the Sun). (See Section A.9.)
Specifically, for , Equation (4.30) is the equation of an ellipse; for , Equation (4.30) is the equation of a parabola; finally, for , Equation (4.30) is the equation of a hyperbola. However, a planet cannot have a parabolic
or a hyperbolic orbit, because such orbits are only appropriate to objects that are ultimately able to escape from the Sun's gravitational field.
Thus, the orbit of our planet
is an ellipse that is confocal with the Sun; this is Kepler's first law
of planetary motion