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Our planet's radial equation of motion, (228), can be combined with
Equation (247) to give
![\begin{displaymath}
\ddot{r} -\frac{h^2}{r^3}= - \frac{G\,M}{r^2}.
\end{displaymath}](img743.png) |
(250) |
Suppose that
. It follows that
![\begin{displaymath}
\dot{r} = - \frac{\dot{u}}{u^2} = - r^2\,\frac{du}{d\theta}\,\frac{d\theta}{dt} = - h\,\frac{du}{d\theta}.
\end{displaymath}](img745.png) |
(251) |
Likewise,
![\begin{displaymath}
\ddot{r} = - h \,\frac{d^2 u}{d\theta^2}\,\dot{\theta} = - u^2\,h^2\,\frac{d^2 u}{d\theta^2}.
\end{displaymath}](img746.png) |
(252) |
Hence, Equation (250) can be written in the linear form
![\begin{displaymath}
\frac{d^2 u}{d\theta^2} + u = \frac{G\,M}{h^2}.
\end{displaymath}](img747.png) |
(253) |
The general solution to the above equation is
![\begin{displaymath}
u(\theta) = \frac{G\,M}{h^2}\left[1 - e\,\cos(\theta-\theta_0)\right],
\end{displaymath}](img748.png) |
(254) |
where
and
are arbitrary constants. Without loss of generality, we can
set
by rotating our coordinate system about the
-axis. Thus,
we obtain
![\begin{displaymath}
r(\theta) = \frac{r_c}{1 - e\,\cos\theta},
\end{displaymath}](img751.png) |
(255) |
where
![\begin{displaymath}
r_c = \frac{h^2}{G\,M}.
\end{displaymath}](img752.png) |
(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 ellipse--this 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
Up: Planetary Motion
Previous: Kepler's Second Law
Richard Fitzpatrick
2011-03-31