Orbital parameters

For an elliptical orbit, the closest distance to the Sun—the so-called perihelion distance—is [see Equation (4.30)]

$$r_p = \frac{r_c}{1+e} = a\,(1-e).$$ (4.34)

This equation also holds for parabolic and hyperbolic orbits. Likewise, the furthest distance from the Sun—the so-called aphelion distance—is

$$\index{orbital parameter!aphelion distance}\index{aphelion} r_a = \frac{r_c}{1-e} = a\,(1+e).$$ (4.35)

It follows that, for an elliptical orbit, the major radius, $a$, is simply the mean of the perihelion and aphelion distances,

$$a = \frac{r_p+r_a}{2}.$$ (4.36)

The parameter

$$e = \frac{r_a-r_p}{r_a+r_p}$$ (4.37)

is called the eccentricity and measures the deviation of the orbit from circularity. Thus, $e=0$ corresponds to a circular orbit, whereas $e\rightarrow 1$ corresponds to an infinitely elongated elliptical orbit. Note that the Sun is displaced a distance $e\,a$ along the major axis from the geometric center of the orbit. (See Section A.9 and Figure 4.3.)

Figure: 4.3 A Keplerian elliptical orbit. $S$ is the Sun, $P$ the planet, $F$ the empty focus, ${\mit \Pi }$ the perihelion point, $A$ the aphelion point, $a$ the major radius, $b$ the minor radius, $e$ the eccentricity, $r$ the radial distance, and $\theta$ the true anomaly.

As is easily demonstrated from the preceding analysis, Kepler laws of planetary motion can be written in the convenient form

$$r = \frac{a\,(1-e^{\,2})}{1+e\,\cos\theta},$$ (4.38)

$$h\equiv r^{\,2}\,\skew{5}\dot{\theta} = (1-e^{\,2})^{1/2}\,n\,a^{\,2},$$ (4.39)

$$G\,M = n^2\,a^{\,3},$$ (4.40)

where $a$ is the mean orbital radius (i.e., the major radius), $e$ the orbital eccentricity, and $n=2\pi/T$ the mean orbital angular velocity.