parabolic case, remains 1). For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury (e = 0.2056), one must simply calculate the inverse sine to find the projection angle of 11.86 degrees. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity. Etymology The word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ekkentros "out of the center", from ek-, "out of" + kentron "center". "Eccentric" first appeared in English in 1551, with the definition "...a circle in which the earth, sun. etc. deviates from its center". In 1556, five years later, an adjectival form of the word had developed. Calculation The eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector: where: is the eccentricity vector ("Hamilton's vector"). For elliptical orbits it can also be calculated from the periapsis and apoapsis since and where is the length of the semi-major axis, where: is the radius at apoapsis (also "apofocus", "aphelion", "apogee"), i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus of the ellipse. is the radius at periapsis (or "perifocus" etc.), the closest distance. The eccentricity of an elliptical orbit can also be used to obtain the ratio of the apoapsis radius to the periapsis radius: For Earth, orbital eccentricity apoapsis is aphelion and periapsis is perihelion, relative to the Sun. For Earth's annual orbit path, the ratio of longest radius () / shortest radius () is Examples The eccentricity of Earth's orbit is currently about ; its orbit is nearly circular. Venus and Neptune have even lower eccentricities. Over hundreds of thousands of years, the eccentricity of the Earth's orbit varies from nearly to almost 0.058 as a result of gravitational attractions among the planets. The table lists the values for all planets and dwarf planets, and selected asteroids, comets, and moons. Mercury has the greatest orbital eccentricity of any planet in the Solar System (e = ). Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion. Before