Kepler's First Law

Our planet's radial equation of motion, (1.293), can be combined with Equation (1.312) to give

$$\skew{3}\ddot{r} -\frac{h^{\,2}}{r^{3}}= - \frac{G\,M}{r^{2}}.$$ (1.315)

Suppose that $r = u^{-1}$, where $u=u(\theta)$. It follows that

$$\skew{3}\dot{r} = - \frac{\dot{u}}{u^{2}} = - r^{\,2}\,\frac{du}{d\theta}\,\frac{d\theta}{dt} = - h\,\frac{du}{d\theta}.$$ (1.316)

Likewise,

$$\skew{3}\ddot{r} = - h \,\frac{d^{2} u}{d\theta^{2}}\,\skew{5}\dot{\theta} = - u^{2}\,h^{2}\,\frac{d^{2} u}{d\theta^{2}}.$$ (1.317)

Hence, Equation (1.315) can be written in the linear form

$$\frac{d^{\,2} u}{d\theta^{\,2}} + u = \frac{G\,M}{h^{\,2}}.$$ (1.318)

The general solution to the previous equation is

$$u(\theta) = \frac{G\,M}{h^{2}}\left[1 - e\,\cos(\theta-\theta_0)\right],$$ (1.319)

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

$$r(\theta) = \frac{r_c}{1 - e\,\cos\theta},$$ (1.320)

where

$$r_c = \frac{h^{\,2}}{G\,M}.$$ (1.321)

We immediately recognize Equation (1.320) as the equation of a conic section that is confocal with the origin (i.e., with the Sun). Specifically, for $e<1$, Equation (1.320) is the equation of an ellipse that is confocal with the Sun. Thus, the orbit of our planet around the Sun is a confocal ellipse. This is Kepler's first law of planetary motion. Of course, 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.

For the case of an elliptic orbit, the eccentricity, $e$, measures the displacement of the Sun from the geometric center of the orbit; in fact, according to Equation (1.303), this displacement is $e\,a$, where $a=r_c/(e^2-1)$ is the major radius. The eccentricity also measures the elongation of the orbit; in fact, according to Equations (1.301) and (1.302), $b/a=\!\sqrt{1-e^2}$, where $b$ is the minor radius. Note that the displacement is first order in $e$, whereas the elongation is second order. As is clear from Table 1.4, the planets in the solar system all have orbits characterized by small eccentricities, which implies that these orbits are actually all quite close to being circular.