Mechanical Principles › Complications › Astronomical Indications

THE EQUATION OF TIME

1. Astronomy

The Equation of time is an astronomical concept that quantifies the difference between true solar noon (the sun’s passage at zenith) and mean solar noon over the course of the year. This difference results from the irregularity of the Earth’s trajectory around the Sun, combined with the tilt of its axis. This concept is essential to understanding why noon (when the sun reaches its highest point in the sky, the zenith) does not exactly correspond to the time shown by our watches and clocks.

Influenced by these two factors, the length of an astronomical day (true solar day) is not constant and fluctuates throughout the year depending on the Earth’s position around the Sun. Our calendars and our time measurement scale, on the other hand, consider days of a constant average length (24 h) throughout the year (mean solar day). The Equation of time expresses the difference between these two values.

2. Factors

1. The Earth’s elliptical orbit: it follows an elliptical path around the Sun. It is known that the Earth moves faster when it is closer to the Sun (perihelion, around January 3) and slower when it is farther from the Sun (aphelion, around July 4) (Kepler’s Law). As a result, the interval between two occurrences of the Sun at the zenith of a point lengthens when the Earth moves away from the Sun and shortens when the Earth approaches the Sun.
2. The tilt of the Earth’s Axis : The Earth’s Axis is tilted relative to its orbital plane by about 23.5°, which causes seasonal variations in the angle of illumination from the Sun, thus affecting its apparent position.

The Equation of time is the result of combining these two effects, and it manifests as a function that varies throughout the year. It can be expressed by a mathematical relationship giving the difference (in minutes) between mean solar time and true solar time.

3. Value of the Equation of time

Figure 1

Value of the equation of time

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The amplitude of the equation of time reaches about 16 minutes of difference. This difference varies throughout the year, reaching maximum values in spring and autumn, and a minimum around the winter (December 21) and summer (June 21) solstices. For example, in February, the equation of time can be +14 minutes (the sun appears “ahead” of mean solar time), while in November, it can reach about -16 minutes (the sun appears “behind”).

4. Practical applications

The equation of time is important for several applications, including:

• Astronomical navigation: when their GPS fails, sailors use true solar time to determine their geographic position, and the equation of time makes it possible to correct the time shown by a watch.
• Sundials must take into account the equation of time in order to provide an accurate transcription of true solar time.
• Astronomy and solar observation: the equation of time allows astronomers to correct the time shown by a standard watch or chronometer in order to obtain an accurate solar time during observations.

In summary, the equation of time is an astronomical phenomenon resulting from the combination of the Earth’s elliptical orbit and the tilt of its axis, causing variations in true solar time compared to mean solar time. Although the term “equation of time” may suggest a complex mathematical problem, it is actually a periodic function that finds its use in fields such as navigation, astronomy, and timekeeping in general.

5. Construction

In practice, an equation of time mechanism is relatively simple and does not require many components. However, its driving mode deserves attention and its components must be designed and manufactured with great precision to guarantee the accuracy of the information.

The astronomical tab of this chapter showed us that the annual curve of the equation of time value is this one:

Figure 2

Equation of time value

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5.1 Design of the equation of time cam

To mechanically translate this information, it is necessary to transcribe this curve in the form of a circular expression in order to draw a cam which will drive the equation of time indicator. The circular projection of this curve corresponds to the following pattern:

Figure 3

Circular projection of the equation of time curve

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This curve can thus be used directly to trace the exact shape and outline of the equation of time cam. The cam drawn in this way will serve as mechanical programming for displaying the function.

6. Operating principle

The equation of time cam is fixed to the equation of time wheel, generally driven by an intermediate wheel.

The equation of time wheel must make one revolution in one astronomical year, i.e. 365.25 days. It is therefore incorrect to drive an equation of time mechanism with a perpetual calendar based on the Gregorian calendar, which breaks down the leap year cycle into three years of 365 days and one year of 366 days. Driving an equation of time mechanism with a perpetual calendar generates an error of 0.25 day per year, i.e. 0.75 day of error after three years. This difference can lead to a relatively significant reading error. It is therefore preferable to use an annual calendar or, better still, to specifically calculate a gear to guarantee the correct rotation speed of the equation of time wheel (1 rev/365.25 days).

A feeler is held in contact with the periphery of the cam by a spring. The feeler is located at the end of a large pivoting lever.

As the cam turns, it moves the feeler, which pivots in one direction or the other as it follows the contour of the cam.

The other end of the feeler, located opposite its pivot point, ends in a toothed sector (rack). This meshes with the equation of time pinion which carries the equation of time hand. The angular value of the display is determined from the gear ratio between the feeler’s sector and the equation of time pinion.

Figure 4

Equation of time mechanism

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