Editing Ellipse (section)

==== Planetary orbits ====
{{Main|Elliptic orbit}}

In the 17th century, [[Johannes Kepler]] discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his [[Kepler's laws of planetary motion|first law of planetary motion]]. Later, [[Isaac Newton]] explained this as a corollary of his [[Newton's law of universal gravitation|law of universal gravitation]].

More generally, in the gravitational [[two-body problem]], if the two bodies are bound to each other (that is, the total energy is negative), their orbits are [[Similarity (geometry)|similar]] ellipses with the common [[barycenter]] being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.

Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to [[electromagnetic radiation]] and [[quantum mechanics|quantum effects]], which become significant when the particles are moving at high speed.)

For [[elliptical orbit]]s, useful relations involving the eccentricity <math>e</math> are:
<math display="block">\begin{align}
    e &= \frac{r_a - r_p}{r_a + r_p} = \frac{r_a - r_p}{2a} \\
  r_a &= (1 + e)a \\
  r_p &= (1 - e)a
\end{align}</math>

where
* <math>r_a</math> is the radius at [[apoapsis]], i.e., the farthest distance of the orbit to the [[barycenter]] of the system, which is a [[Focus (geometry)|focus]] of the ellipse
* <math>r_p</math> is the radius at [[periapsis]], the closest distance
* <math>a</math> is the length of the [[semi-major axis]]

Also, in terms of <math>r_a</math> and <math>r_p</math>, the semi-major axis <math>a</math> is their [[arithmetic mean]], the semi-minor axis <math>b</math> is their [[geometric mean]], and the [[conic section#Features|semi-latus rectum]] <math>\ell</math> is their [[harmonic mean]]. In other words,
<math display="block">\begin{align}
     a &= \frac{r_a + r_p}{2} \\[2pt]
     b &= \sqrt{r_a r_p} \\[2pt]
  \ell &= \frac{2}{\frac{1}{r_a} + \frac{1}{r_p}} = \frac{2r_ar_p}{r_a + r_p}.
\end{align}</math>