Mercury and Venus orbit inside ours, so neither can ever appear more than a limited angle from the Sun. Drag through an apparition and watch the planet swing out, turn round, and come back.
Greatest elongation is where the line of sight grazes the planet's orbit.
Both orbit inside Earth's orbit, so the line from us to the planet can never swing more than a certain angle away from the line to the Sun. The limit is reached where our line of sight is tangent to the planet's orbit, and for a circular orbit that angle is the arcsine of the planet's distance from the Sun in AU.
Venus's orbit is almost a circle, so the circular formula predicts its greatest elongations to within a degree. Mercury's is not. At an eccentricity of 0.206 the Sun sits well off the centre of its orbit, and the greatest elongation depends on whether the apparition happens near Mercury's perihelion or its aphelion. Quoting one figure for Mercury would be wrong by up to five degrees.
The turning points are the greatest elongations. Zero is conjunction, and the sign says which side of the Sun the planet is on.
Before looking for Mercury or Venus, check which side of the Sun it is on. Greatest eastern elongation favours the western sky after sunset; greatest western elongation favours the eastern sky before sunrise. Greatest elongation marks a local maximum in angular separation, not necessarily the highest altitude or darkest twilight. Latitude and the ecliptic's angle to the horizon still control the useful viewing window.
The circular limit in the table uses only the ratio of semi-major axes; the actual range retains the effect of orbital eccentricity. Mercury differs enough that 22.8 degrees should not be treated as a fixed value when checking a particular elongation.
That Venus shows phases at all is the observation that broke the geocentric model. In Ptolemy's arrangement Venus rides an epicycle that keeps it between us and the Sun, so it could only ever be a crescent. Galileo saw it gibbous.
Mercury does not obey the rule, and a textbook that states it without qualification is quoting the circular case. Mercury's phase angle at greatest elongation runs from 74 to 106 degrees, so its disc there is anything from 37 to 64 per cent lit.
The peak lands near 39 degrees of elongation with about a quarter of the disc lit, roughly five weeks either side of inferior conjunction. It happens twice in each apparition, once in the evening sky and once in the morning, and it is the brightest any planet ever gets: bright enough to cast a shadow from a dark site, and to be found in daylight if you know where to look.
Brightness here is pure geometry: sunlight received, lit fraction, and distance. Real photometry adds a phase function for the atmosphere and the scattering angle, which moves the peak by a few days. Nothing on this page is a magnitude prediction.
| Planet | Eccentricity | Circular-orbit limit | Actual range, 16 years |
|---|---|---|---|
| Mercury | 0.206 | 22.8° | 18°–28° |
| Venus | 0.0068 | 46.3° | 45°–47° |
That is the whole reason neither is ever visible at midnight, and why both are always twilight objects: low in the west after sunset, or low in the east before sunrise, and nowhere else.
If the line of sight is tangent to a circular orbit then the Sun-planet-Earth angle at greatest elongation is exactly 90 degrees, and a planet lit from 90 degrees away shows exactly half a disc. Venus obliges: at every greatest elongation over the last sixteen years its disc is between 49 and 51 per cent lit.
Between greatest elongation and inferior conjunction Venus keeps coming closer, so its disc grows; at the same time the lit crescent keeps getting thinner. What reaches us is the lit area, which is the product of the two, and the product peaks somewhere in between.
Related tools: Opposition and conjunction · Ptolemaic and heliocentric · Live Sky Map
Orbital elements from NASA/JPL, "Keplerian Elements for Approximate Positions of the Major Planets" (Standish & Williams, 1992), valid 1800–2050. Positions are geometric — no light-time or aberration — because this is a diagram of the ecliptic plane, not a simulation of a telescope view.