Why is Sundial Time and Clock Time Different?

The annual, cyclic variation between mean solar time shown on uniformly running clocks and apparent solar time displayed on sundials.analemma_dates

In the course of the Sun’s daily east-to-west transit of the sky, the Sun crosses the meridian, an imaginary line running from north to south that passes overhead and divides the sky into equal halves. An observer in the middle of a time zone generally thinks of noon as being the moment that the Sun reaches the meridian (directly overhead). This event, however, corresponds to noon recorded by mechanical or electronic clocks on only four dates each year (approximately April 16, June 14, September 1, and December 25). On all other dates the Sun reaches the meridian either early or late, with the extremes being 16.3 min early around November 3 and 14.3 min late around February 12. This difference is the equation of time, and results from the combined effects of Earth’s axis of rotation being tipped 23° relative to Earth’s orbital plane and the elliptical rather than circular shape of the orbit. See also Meridian.

The elliptical orbit brings the Earth closest to the Sun in January. This proximity, as J. Kepler discovered in the sixteenth century, causes the Earth to move more rapidly in its orbit than in July, when the Earth is farthest from the Sun. The changing orbital speed varies the Sun’s apparent rate of motion along the ecliptic and is in part responsible for the equation of time.

Slightly more influential is Earth’s tipped axis, which varies the Sun’s position north and south of the celestial equator according to the season. Around the time of the spring and autumn equinoxes the Sun moves at a steep angle relative to the celestial equator. Its daily motion projected onto the equator is less than at the solstices, when the Sun travels parallel to the equator. This situation also creates a departure between the Sun’s actual position and that of a mean Sun moving uniformly along the celestial equator. The effects of the inclination of the Earth’s axis and the eccentricity of its orbit are combined.