Kepler's First Law

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

$$\ddot{r} -\frac{h^2}{r^3}= - \frac{G\,M}{r^2}.$$ (250)

Suppose that $r = u^{-1}$. It follows that

$$\dot{r} = - \frac{\dot{u}}{u^2} = - r^2\,\frac{du}{d\theta}\,\frac{d\theta}{dt} = - h\,\frac{du}{d\theta}.$$ (251)

Likewise,

$$\ddot{r} = - h \,\frac{d^2 u}{d\theta^2}\,\dot{\theta} = - u^2\,h^2\,\frac{d^2 u}{d\theta^2}.$$ (252)

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

$$\frac{d^2 u}{d\theta^2} + u = \frac{G\,M}{h^2}.$$ (253)

The general solution to the above equation is

$$u(\theta) = \frac{G\,M}{h^2}\left[1 - e\,\cos(\theta-\theta_0)\right],$$ (254)

where $e$ and $\theta_0$ are arbitrary constants. Without loss of generality, we can set $\theta_0=0$ by rotating our coordinate system about the $z$-axis. Thus, we obtain

$$r(\theta) = \frac{r_c}{1 - e\,\cos\theta},$$ (255)

where

$$r_c = \frac{h^2}{G\,M}.$$ (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 $e<1$, 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.