Elongation (astronomy)

In astronomy, a planet's elongation is the angular separation between the Sun and the planet, with Earth as the reference point. The greatest elongation of a given inferior planet occurs when this planet's position, in its orbital path around the Sun, is at tangent to the observer on Earth. Since an inferior planet is well within the area of Earth's orbit around the Sun, observation of its elongation should not pose that much a challenge (compared to deep-sky objects, for example). When a planet is at its greatest elongation, it appears farthest from the Sun as viewed from Earth, so its apparition is also best at that point.
When an inferior planet is visible after sunset, it is near its greatest eastern elongation. When an inferior planet is visible before sunrise, it is near its greatest western elongation. The angle of the maximum elongation (east or west) for Mercury is between 18° and 28°, while that for Venus is between 45° and 47°. These values vary because the planetary orbits are elliptical rather than perfectly circular. Another factor contributing to this inconsistency is orbital inclination, in which each planet's orbital plane is slightly tilted relative to a reference plane, like the ecliptic and invariable planes.
Astronomical tables and websites, such as Heavens-Above, forecast when and where the planets reach their next maximum elongations.
Elongation period.
Greatest elongations of a planet happen periodically, with a greatest eastern elongation followed by a greatest western elongation, and "vice versa". The period depends on the relative angular velocity of Earth and the planet, as seen from the Sun. The time it takes to complete this period is the synodic period of the planet.
Let "T" be the period (for example the time between two greatest eastern elongations), "ω" be the relative angular velocity, "ω"e Earth's angular velocity and "ω"p the planet's angular velocity. Then
where "T"e and "T"p are Earth's and the planet's years (i.e. periods of revolution around the Sun, called sidereal periods).
For example, Venus's year (sidereal period) is 225 days, and Earth's is 365 days. Thus Venus's synodic period, which gives the time between two subsequent eastern (or western) greatest elongations, is 584 days.
These values are approximate, because (as mentioned above) the planets do not have perfectly circular, coplanar orbits. When a planet is closer to the Sun it moves faster than when it is further away, so exact determination of the date and time of greatest elongation requires a much more complicated analysis of orbital mechanics.
Of superior planets.
Superior planets, dwarf planets and asteroids undergo a different cycle. After conjunction, such an object's elongation continues to increase until it approaches a maximum value larger than 90° (impossible with inferior planets) and typically very near 180°, which is known as "opposition" and corresponds to a heliocentric conjunction with Earth. In other words, as seen by an observer on the superior planet at opposition, the Earth appears at conjunction with the Sun. Technically, the exact moment of opposition is slightly different from the moment of maximum elongation. Opposition is defined as the moment when the apparent ecliptic longitudes of the superior planet and the Sun differ by 180°, which ignores the fact that the planet is outside the plane of the Earth's orbit. For example, Pluto, whose orbit is highly inclined to the Earth's orbital plane, can have a maximum elongation significantly less than 180° at opposition.
All superior planets are most easily visible at their oppositions because they are near their closest approach to Earth and are also above the horizon all night. The variation in magnitude caused by changes in elongation are greater the closer the planet's orbit is to the Earth's. Mars' magnitude in particular changes with elongation: it can be as low as +1.8 when in conjunction near aphelion but at a rare favourable opposition it is as high as −2.9, or seventy-five times brighter than its minimum brightness. As one moves further out, the difference in magnitude caused by the difference in elongation gradually fall. The maximum and minimum brightness of Jupiter differ by only a factor of 3.3 times, whilst those of Uranus – which is the most distant Solar System body visible to the naked eye – differ by a factor of 1.7 times.
Since asteroids travel in an orbit not much larger than the Earth's, their magnitude can vary greatly depending on elongation. Although more than a dozen objects in the asteroid belt can be seen with 10x50 binoculars at an average opposition, only Ceres and Vesta are always above the binocular limit of +9.5 at small elongations.
A quadrature occurs when the position of a body (moon or planet) is such that its elongation is 90° or 270°; i.e. the body-earth-sun angle is 90°
Of moons of other planets.
Sometimes elongation may instead refer to the angular distance of a moon of another planet from its central planet, for instance the angular distance of Io from Jupiter. Here we can also talk about "greatest eastern elongation" and "greatest western elongation". In the case of the moons of Uranus, one often talks about "greatest northern elongation" and "greatest southern elongation" instead, due to the very high inclination of Uranus' axis of rotation..