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Orbital parameters

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

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

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

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

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

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

The parameter

$\displaystyle e = \frac{r_a-r_p}{r_a+r_p}$ (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: 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.
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As is easily demonstrated from the preceding analysis, Kepler laws of planetary motion can be written in the convenient form

    $\displaystyle r$ $\displaystyle = \frac{a\,(1-e^{\,2})}{1+e\,\cos\theta},$ (4.38)
    $\displaystyle h\equiv r^{\,2}\,\skew{5}\dot{\theta}$ $\displaystyle = (1-e^{\,2})^{1/2}\,n\,a^{\,2},$ (4.39)
and   $\displaystyle G\,M$ $\displaystyle = n^2\,a^{\,3},$ (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.


next up previous
Next: Orbital energies Up: Keplerian orbits Previous: Kepler's third law
Richard Fitzpatrick 2016-03-31