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Our planet's radial equation of motion, (228), can be combined with
Equation (247) to give

(250) 
Suppose that . It follows that

(251) 
Likewise,

(252) 
Hence, Equation (250) can be written in the linear form

(253) 
The general solution to the above equation is

(254) 
where and are arbitrary constants. Without loss of generality, we can
set by rotating our coordinate system about the axis. Thus,
we obtain

(255) 
where

(256) 
We immediately recognize Equation (255) as the equation of a conic
section which is confocal with the origin (i.e., with the Sun).
Specifically, for , Equation (255) is the equation of an ellipse
which is confocal with the Sun. Thus, the orbit of our planet
around the Sun in a confocal ellipsethis is Kepler's first law
of planetary motion. Of course, a planet cannot have a parabolic
or a hyperbolic orbit, since such orbits are only appropriate to objects which are ultimately able to escape from the Sun's gravitational field.
Next: Kepler's Third Law
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Previous: Kepler's Second Law
Richard Fitzpatrick
20110331