In astronomy, a planet's elongation is the angle between the Sun and the planet, as viewed from Earth.
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 value of the greatest elongation (west or east), for Mercury, is between 18° and 28°; and for Venus between 45° and 47°. This value varies because the orbits of the planets are elliptical, rather than perfect circles. Another minor contributor to this inconsistency is orbital inclination: each planet's orbit is in a slightly different plane.
Refer to astronomical tables and websites such as heavens-above to see when the planets reach their next maximum elongations.
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, ωe Earth's angular velocity and ωp the planet's angular velocity. Then
where Te and Tp 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' 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.
Elongation of superior planets
Superior planets, dwarf planets and asteroids undergo a different cycle. After superior 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 from an observer on the superior planet at opposition, the Earth appears at inferior 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.
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