# The Equation of Time

By Murray Bourne, 26 Aug 2010

First, let's investigate the properties of an orbiting body.

In the following graph, the green ball represents the Earth rotating around the (red) Sun, which is at a **focus** of the ellipse.

This is greatly exaggerated so you can see what is going on.

Consider how fast the sun would appear to be moving when we are close to it, compared to when we are way out near the second focus of the ellipse (labeled F_{2}.)

[In fact, the Earth's orbit around the sun is very close to a circle, but it is still elliptical. See more on the Ellipse.]

## The Equation of Time

Many of us don't take much notice of the motion of the sun, moon or planets. (Let's face it - most of us live in places that have bad air and light pollution so we can hardly see the sky).

Sundial [Image source]

Consequently, we assume the Earth's motion around the sun is uniform and that it takes exactly 24 hours for the sun to come back to the same spot above us each day.

But this is not so. There is as much as 30 minutes variation between the position of the sun relative to the stars at the same time of day, throughout the year. That is, the sun can be "behind" by as much as 14 min 6 sec (around 12 February each year) and up to 16 min 33 sec "ahead" (around 3rd Nov each year).

There are 2 reasons why this is so:

- The Earth revolves around the Sun along an ellipse, not a circle. As we saw above, this means the Earth does not travel around the sun at a constant speed.
- The Earth is tilted at 23.44 degrees

If the Earth rotated around the sun in a perfect circle and there was no tilt on the Earth's axis, we would see the sun overhead in exactly the same position every day.

The ancient Greeks knew about this issue but because they didn't have accurate clocks, they were not too concerned. By the 17th century, with the invention of pendulum clocks, knowing where the sun should be at any time of the year became critical for accuracy in navigation at sea.

## The Equation of Time

The Equation of Time takes the above 2 factors into consideration and can tell us how far ahead or behind the sun will be relative to the stars.

An explicit function that is a good approximation to the Equation of Time is as follows, where *d* is the day of the year and the resulting value is in terms of minutes of variation):

Time variation = -7.655 sin *d* + 9.873 sin(2*d* + 3.588)

This is made up of 2 components:

Variation due to Earth's tilt = -7.655 sin *d*

This contributes 7.655 minutes to the Equation of Time, and the period is 1 year (365.25 days).

This is also known as the Effect of Orbit Eccentricity. More strictly, adjusting for the correct period, its equation is:

The second component:

Variation due to elliptical orbit = 9.873 sin(2*d* + 3.588)

This contributes 9.873 minutes to the variation in the sun's position, and has period half of one year:

This is also called the Effect of Obliquity. Once again, adjusting for the correct period, its equation more correctly is:

[See more on Period of Trigonometric Graphs.]

When the 2 components are added together, we obtain:

Time variation = -7.655 sin *d* + 9.873 sin(2*d* + 3.588)

When we add ordinates of trigonometric functions like this, we obtain what is called a composite trigonometric curve. (See more on Composite Trigonometric Curves.)

(You can see how the Equation of Time is derived at Equation of Time.)

## Conclusion

The Equation of Time is a neat real-life application of conic sections (the Ellipse) and Composite Trigonometric Curves.

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