1. What is daytime? True solar day vs. mean day
According to the simplest definition, the day is the period that separates two successive passages of the Sun at the meridian. This is the definition of the “true” day. True solar noon is the moment when the Sun passes the meridian of the place of observation. Roughly speaking, we then divide this period into 24 hours, or 1,440 minutes, or 86,400 seconds.
But 24 hours is an average (solar mean time). When we measure the day with a precise chronometer (24 hours), we realize that the Sun does not pass the meridian exactly 24 hours after its previous passage. In fact, the true day lasts between 23:59:30 and 24:00:30. The Sun is therefore sometimes “early” and sometimes “late” compared to the average length of 24 hours. Be careful: this delay (or advance) accumulates day by day. The difference between true noon and mean noon can be up to a maximum delay of 14 minutes towards February 10 and a lead of 16 minutes at the beginning of November. The difference between true time and mean time is called the equation of time.
The variation in the length of the true day is due to two combined reasons: the obliquity of the Earth’s orbit and its ellipticity.
NB: Day length vs. daylight length
If we call the length of the day the time interval between two passages of the sun at the meridian of a place, how do we call the time that elapses between sunrise and sunset? The length of the solar day should not be confused with the length of the day. We have seen that the solar day lasts between 23:59:30 and 24:00:30, but the “day” in the sense of the day, has a much more variable duration depending on the season. In Paris, the day lasts just over 8 hours on December 21 and more than 16 hours on June 21.
What is (relatively) constant is the length of the day plus the length of the night. This is another way of defining the solar day.
2. The apparent movement of the Sun
When we carefully observe the position of the Sun in relation to the stars (the “fixed sphere”), we see that if the Sun appears to us in the direction of a star on a given day, it appears slightly shifted the next day. This difference represents almost one degree per day. The Sun therefore seems to “circle” the celestial vault in a year. This is of course the origin of the 360-degree graduation of the circle. 360 is very close to reality, which is 365.24. And it is also a “magic” number, divisible by 2, 3, 4, 5, etc.
This shift of the Sun against the background of the sky can be explained both in a geocentric vision of the world and in a heliocentric vision:
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Geocentric explanation |
Heliocentric explanation |
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When the fixed sphere has completed a revolution on itself (one day), the star (blue) returns in the same direction as the day before for an observer on Earth, but the Sun has moved, it is now in the direction of another star (red). |
When the Earth turns on its axis (one day), it moves back in the direction of the star (blue) in the drawing. But for the Sun to return to the meridian of a given place on Earth, the Earth must rotate a little more. The Sun is now appearing in the direction of the red star. |
Vocabulary: a “real” orbit of the Earth on its axis is called a sidereal day: a star returns in the same direction after 23 hours and 56 minutes. A revolution of the Earth with respect to the Sun is called a synodic day: the sun returns in the same direction with respect to an observer on Earth after 24 hours.
It is this “catch-up” that is at the origin of the equation of time, for two reasons: obliquity and ellipticity of the Earth’s orbit.
2.1 Obliquity
Let’s say that the Sun revolves around the Earth: we have seen that geocentric vision works very well and it makes it easier to understand this point. We will then come full circle with the modern heliocentric vision.
If the Sun rotated in the plane of the Earth’s equator, its shift on the fixed sphere would be constant throughout the year, approximately one degree. But the Sun seems to revolve around the Earth in a plane inclined by 23° to the Earth’s equator: the ecliptic plane.
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If the Sun rotated in the plane of the Earth’s equator, every day it would move at an angle of one degree. |
But the Sun rotates in the ecliptic plane, inclined by 23° to the equator. A degree on the ecliptic does not correspond to a degree on the equator every day. |
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The Earth is at the center. The Sun rotates around it in an inclined plane with respect to the equator.
Thus, it is 23° above the equator in summer and 23° below in winter. |
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For the ancients, the rotation of the Sun on the ecliptic takes place at a constant speed, but the projection of this movement on the plane of the equator does not have a constant speed. The obliquity and unequal length of the day were known to the ancients. According to their vision, the celestial vault (“fixed sphere”) rotates around the axis of the Earth’s poles in a uniform circular motion in 23 hours and 56 minutes.
In the heliocentric vision, the Earth’s axis of rotation is tilted 23° on the plane of its orbit. The additional daily rotation to “catch up” with the Sun varies throughout the year. It is called “reduction at the equator” because it refers to the projection of ecliptic motion onto the equator.
2.2 Ellipticity of the Earth’s orbit
This reason was not known to the ancients. But it is easier to understand. As Kepler discovered, the Earth does not describe a circle around the Sun, but an ellipse. This is Kepler’s first law. And Kepler’s second law says that the speed of the Earth’s movement in its orbit is not constant. What’s constant is the surface swept by the Earth over a given time interval.
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According to Kepler’s second law, the surface swept by the Earth in a time interval is constant. In other words, depending on where it is in its orbit, the Earth’s speed is not constant. It rotates faster when it is closest to the Sun (perihelion), and slower when it is farther from the Sun (aphelion). Attention: on the diagram, the ellipticity of the orbit is strongly accentuated. |
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Since the Earth does not travel its orbit at a constant speed, whereas the sidereal day is constant (the Earth rotates on its axis at a constant speed), the distance traveled during a revolution varies and the additional rotation necessary to see the Sun again in the same direction is not constant. |
In practice, the ellipse formed by the Earth’s orbit is very little different from a circle, which means that the irregularity is not very noticeable. But just like the irregularity due to the obliquity of the Earth’s orbit, this small daily difference tends to accumulate from one day to the next, until it reaches a lead (or a delay) of 8 minutes. And the two reasons can be combined.
3. Mathematization
To find out where the Sun is on the ecliptic, we give ourselves an origin. By convention, the direction in which it is on the day of the spring equinox, March 21, is chosen. This point on the ecliptic circle is called the vernal point, denoted ^. Thus, each day, we can know the angle that the Sun makes in relation to this origin: it is 360/365 * (d-81), where j is the number of the day counted from the first of January (angle measured in degrees).
Equation of time and watchmaking
The difference between the day measured in minutes and seconds (mean day) and the day measured by reading the direction of the Sun (true day) is between plus or minus 30 seconds per day at most. Be careful, however: this difference is cumulative, and the difference can reach about a quarter of an hour. Such a discrepancy was not a problem at the time when clocks and watches were not very accurate: in any case, you had to set your watch every day, and you could do it very easily at noon accurately. The imprecision of the length of the day was confused with the imprecision of the clock. But from the eighteenth century onwards, when watchmaking began to become really precise, while the official time was still that of the sun, this discrepancy ended up becoming a problem. Customers who had purchased a precision watch or clock still had to set it regularly, as it shifted from the official time, struck at the bell tower or checked at the local sundial. Watchmakers then invented equation of time mechanisms, allowing the mechanical clock to adjust and give a calculated noon corresponding to the true local noon.
Even today, mechanical watches with equations of time are part of what is known as fine watchmaking.