Elongation

Definition

The elongation of a planet or of the Moon is its angular distance from the Sun, as seen from the Earth. It goes from 0° at conjunction to 180° at opposition. It is called eastern when the body is east of the Sun, visible in the evening after sunset, and western when it is west of the Sun, visible in the morning before sunrise.

With the geocentric ecliptic longitude λ and latitude β of the body and the longitude λ0 of the Sun, the elongation ψ follows from:

cos ψ = cos β · cos(λ − λ0)

Since planetary latitudes are small, the elongation is almost equal to the difference in longitude. The sign of λ − λ0 tells whether the elongation is eastern (positive) or western (negative).

Greatest elongations of Mercury and Venus

The inner planets can never move far from the Sun. Their elongation reaches a maximum, the greatest elongation, when the line of sight from the Earth is tangent to their orbit. For a circular orbit of radius a (in AU), sin ψmax = a, which gives 22.8° for Mercury and 46.3° for Venus. Because the orbits are eccentric, the actual values vary:

Greatest elongation is the usual best time to see Mercury, but not all elongations are equal: for observers at mid-northern latitudes, evening elongations in spring and morning elongations in autumn are the best, because the ecliptic then stands steeply to the horizon and the planet is higher in the twilight.

Upcoming greatest elongations

Venus: greatest elongation is not greatest brilliancy

Venus is brightest not at greatest elongation but about five weeks before and after inferior conjunction, at an elongation of about 39°. The planet then shows a thick crescent: less of its disc is lit than at greatest elongation, but it is much closer and its disc much larger. At greatest brilliancy it reaches magnitude −4.6 to −4.9, bright enough to cast shadows on a dark site.

In a telescope, Venus at greatest elongation should show a half-lit disc, but the observed half phase (dichotomy) often comes a few days earlier at evening elongations and later at morning ones. This Schröter effect is attributed to the planet's dense atmosphere.

Elongation of the Moon and of the outer planets

For the Moon, elongation is the phase: 0° at new Moon, 90° at first and last quarter, 180° at full Moon. The illuminated fraction of the disc follows from the phase angle i, which is close to 180° − ψ: k = (1 + cos i) / 2. Meeus uses exactly this route in his chapter on the illuminated fraction of the Moon's disc.

Outer planets can take any elongation. At 90° they are at quadrature, east in the evening sky or west in the morning sky. At quadrature Mars shows its most noticeable phase, about 85 to 90 % lit, the only outer planet where the phase is obvious in a small telescope.

Explore it on AstroAlgos

• All-sky disk — Mercury and Venus near the Sun
• Night sky — the evening and morning planets
• Conjunction — elongation zero
• Magnitude — how bright Venus gets