# The equation of time

The measurement of time no longer uses sundials but relies on devices, such as clocks, to determine a uniform rate. This rate is calibrated using astronomical observations so that clock time is equivalent to time determined by the mean motion of the Earth.

We know, from modern astronomical observations and from observations of artificial satellites, that the Earth's rotation rate is not constant but varies both over the short term and over centuries. These small variations are due to real variations in the rotation of the Earth and are compensated for by inserting leap-seconds as appropriate.

If a sundial is used to determine the time it rapidly becomes apparent that it does not indicate the same time as clock time. The difference amounting to some 16 minutes at certain times of year. This difference is also seen as an asymmetry in the times of sunrise and sunset. It is called the Equation of Time.

The equation of time has two causes:

• The plane of the Earth's equator is inclined to the plane of the Earth's orbit around the Sun
• The orbit of the Earth around the Sun is an ellipse and not a circle

### The equation of time due to obliquity (the Earth's tilt)

The angle between the planes of the equator and the Earth's orbit around the Sun is called the angle of obliquity.

If the Earth's rotation axis was not tilted with respect to its orbit, the apparent motion of the Sun along the the Ecliptic would fall exactly on the equator, covering equal angles along the equator in equal time. We measure apparent solar time, however, as a projection of the Sun's motion onto the equator, and this changes through the year as the Sun moves above and below the equator.

The projection of the Sun's motion onto the equator will be a maximum when its motion along the Ecliptic is parallel to the equator (at the summer and winter solstices) and will be a minimum at the equinoxes.

The Sun will be on the meridian at noon at both solstices and equinoxes and so the equation of time due to obliquity will be zero at these times. Between the solstices and the equinoxes the Sun will be slow relative to clock time with minima near 5 February and 5 August. Between equinoxes and solstices the Sun will be fast relative to clocks with maxima near 5 May and 5 November.

### The equation of time due to unequal motion (the Earth's elliptical orbit)

The equation of time. The variation of the equation of time has two components: first from the Earth's axial tilt (purple), and second from the elliptical shape of the Earth's orbit around the Sun (dark blue). Courtesy of Keith C. Heidorn. Adapted with kind permission.The orbit of the Earth around the Sun is an ellipse. The distance between the Earth and the Sun is a minimum (perihelion) near 31 December and is greatest (aphelion) near 1 July. The Sun's apparent longitude changes fastest when the Earth is closest to the Sun. The Sun will appear on the meridian at noon on these two dates and so the equation of time due to unequal motion will then be zero. Between perihelion and aphelion the Sun will be slow relative to clock time with a minimum around 31 March. Between aphelion and perihelion the Sun will be fast relative to clock time with a maximum around 30 September.

The equation of time. Diagram courtesy of Keith C. Heidorn. Adapted with kind permission.Animations illustrating this effect are available to view at the Analemma website.

### The equation of time

The total of these two effects gives the equation of time, which is formally defined as the difference between clock time and apparent solar time. The equation of time takes the form of the curve sketched below. It is zero on 16 April, 15 June, 1 September and 25 December and has maxima and minima near 12 February, 15 May, 27 July and 4 November.

The top diagram illustrates the variation of the equation of time due to obliquity (purple) and the variation due to unequal motion (dark blue). The lower diagram shows the final equation of time, a combination of the two effects.