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The Equation of Time

Today, 13 June, is one of only four days in the year when the time as read on a sundial will be exactly correct.

Sundials usually tell the time using the shadow of the gnomon as cast by the Sun. This is possible as the Sun appears to move across the sky at an approximately constant speed, and so the shadow of the gnomon also moves at an approximately constant speed. The inconstancy of the Sun’s apparent motion in the sky – and therefore of the gnomon’s shadow on a sundial – is the subject of this article, and is calculated using the Equation of Time.

A Sundial

If you look at the shadow of a sundial’s gnomon it will fall onto a curve of numbers, along hour lines indicating local solar time. This is not equal to the official clock time until three important corrections are made:

1. Correct for daylight saving time. “The Sun reaches it highest point due south at noon” is only true in the UK between the last Sunday in October and the last Saturday in March, when the UK operates on Greenwich Mean Time (GMT). For the rest of the year we use British Summer Time, or BST, which is GMT+1. Therefore if you’re using a sundial in the UK during BST you will need to add one hour one to the sundial time to correct for this.

2. Correct for longitude (the G in GMT). In the UK our time is based on the time in Greenwich, London, hence the G in GMT. At noon in Greenwich the Sun will be due south at 12 noon GMT. At locations east of Greenwich (Norwich, Ipswich, Cambridge) the Sun will have reached its highest point due south earlier than 12 noon, and for locations west of Greenwich (which is most of the UK) the Sun will reach its highest point due south later than 12 noon. Given that most sundials are fixed in location this correction can be built into the sundial itself (i.e. the hour lines can be moved through an angle equal to the difference between the sundial’s longitude and the meridian that mean solar time is calculated).

3. Correct for the average (“mean”) position of the Sun (the M in GMT). If the Earth orbited the Sun in a perfectly circular orbit, and if its axis of rotation was perfectly perpendicular to the plane of its orbit, then the Sun would always be in the same point of the sky at noon each day. The fact that the Earth’s orbit is not a perfect circle, and the Earth’s axis is titled, means that the position of the noon Sun varies over the year.

Orbital Eccentricity

The Earth orbits the Sun in an elliptical path, meaning that at some times of the year it is further from the Sun than at other times. Crucially though, this also means that the Earth orbits the Sun at different speeds depending on where it is in its orbit. At its closest approach to the Sun (called perihelion) in early January the Earth is moving fastest in its orbit, and at its furthest from the Sun (called aphelion) in early July it is moving slowest. The Earth’s orbit isn’t drastically elliptical (it is almost circular) and so the differences are not great, but at perihelion we’re moving at 30.3 km/s and at aphelion we’re moving at 29.3 km/s (a 3.4% difference).

This means that if we view the Sun in the sky it will appear to move against the background stars at different speeds depending on where in its orbit the Earth is. At perihelion, in early January, the Sun speeds up by 7.9 seconds per day from the average, or mean; at aphelion is slows down by 7.9 seconds per day. These few seconds per day accumulate over the year, and mean that the amount by which the Sun might deviate from being at due south at noon, due solely to the effects of our elliptical orbit, varies from zero (early Jan and early July) up to 7m 40s (early April, when it is negative, and early October, when it is positive). It can be drawn as a sine wave with an amplitude of  7m40s and a period of 1 year.

The component of the Equation of Time due to Earth's Elliptical Orbit (y-axis in minutes)

Axial Tilt

There is a second factor that affects the speed at which the Sun appears to move in the sky: the tilt of the Earth’ s axis with respect to the plane of the Earth’s orbit. This means that the Sun is sometimes north of the celestial equator (the projection of the Earth’s equator out into space) and sometimes south of it. Clock time is measured along the celestial equator; in effect an imaginary Sun is drawn on this line and time calculated from that. The speed of this imaginary Sun varies over the year, and reaches a maximum at the solstices, when the major component of the real Sun’s motion due to the Earth’s orbit is parallel to the celestial equator, and reaches a minimum at the equinoxes when the major component of the real Sun’s motion due to the Earth’s orbit is mainly perpendicular to the celestial equator. At the solstices (late June and late December) the Sun appears to speed up by 20.3 seconds every day, and at the equinoxes (late March and late September) it appears to slow down by the same amount. These few seconds per day accumulate over the year, and mean that the amount by which the Sun might deviate from being at due south at noon, due solely to the effects of our axial tilt, varies from zero (solstices and equinoxes) up to 9m 52s (early February and early August, when it is negative, and early May and November when it is positive). It can be drawn as a sine wave with an amplitude of 9m52s and a period of 0.5 years.

The component of the Equation of Time due to Earth's Axial Tilt (y-axis in minutes)

 The Equation of Time

Adding these two effects together gives you the Equation of Time, that is the amount by which you need to adjust the time as read on a sundial in order for it to read the correct clock time. Or in other words the amount of time by which the Sun is off being due south at noon.

The Equation of Time (y-axis in minutes)

The maximum amount of time that a sundial is ever ahead of clock time occurs on 03 November, when the sundial will read 16m25s fast. The maximum amount of time that a sundial is ever behind clock time occurs on 11 February, when the sundial is 14m15s slow.

There are four dates each year when the Equation of Time is zero, and the sundial reads correct clock time, and those dates are 15 April, 1 September, 25 December, and today 13 June.

  1. Colin Anderson
    June 17, 2011 at 08:21

    Steve,

    A great explanation, very clearly described on your graphs. These fluctuations are also clearly illustrated in the figure of the analemma, which used to be printed on globes of the world. Plotting the analemma for a given location is a good science project for any classroom, providing there are sufficient clear days to plot the sun’s position at noon over the course of a year!

    Colin

    • June 18, 2011 at 21:46

      That’s right, I plan to write a post about the analemma soon too. That shows very nicely the max and min midday altitude variation and the equation of time variation, with respect to one another. In fact I started writing this post about the analemma, and soon realised that I had enough content using just the EoT itself!

  2. October 27, 2016 at 01:36

    Why does the axial-tilt component of the equation of time, reach its biggest positive and negative values at the cross-quarters? What kind of “turning point” is reached at these times of years?

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