Solar elongation
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 the planets able to elongate are well within the area of Earth's orbit around the Sun, observation of such a phenomenon should not pose that much a challenge (compared to deep-sky objects, for example). When a planet is at its greatest elongation, it is 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°. This value varies because the orbits of the planets are elliptical, rather than perfect circles. Another factor contributing to this inconsistency is orbital inclination, in which each planet's orbital plane is slightly tilted.
Astronomical tables and websites such as Heavens-Above forecast 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
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