Works and Days

I have taken the title of this post from Hesiod's book. Although I won’t talk about works here, I will talk about days. The word day has two different meanings: the whole day (24 hours) and the part of the day when there is sunlight. Thus, it could be said that

day = day + night

which seems absurd, for a mathematician could deduce that the night does not exist. Instead of that equation, and to make it clear that there are two kinds of days, we should use this expression:

day₁ = day₂ + night

Here, day₁ is a natural cycle, the period of the Earth's rotational motion around its axis. But then a problem arises: when can we say that the Earth has made one complete rotation around its axis? The problem is, to define the period of a moving object, it is necessary to have a reference point. The results will be different depending on which point is chosen.

Suppose we take the sun as a reference point. A complete rotation would be the time elapsed between two successive passes of the sun through the same meridian. But we can also take as reference any of the stars, for example, Sirius. In this case, a period of rotation would be the time elapsed between two successive passes of that star through the same meridian. The two results are not the same, because in addition to rotating on its axis, the Earth revolves around the sun.

If the day is measured as the time elapsed between two consecutive passages of the sun through the same meridian, a longer duration is obtained than if the star Sirius or any other star is taken as a reference. The stars are so far away that they would all give results almost identical. There will therefore be two different days: solar day and sidereal day.

Things get even more complicated, because the solar day is not constant. Earth's orbit is not exactly circular, but a little elliptical. The revolution of the Earth around the sun is sometimes faster, sometimes slower, and the difference between the sidereal day and the solar day changes around the year. The longest solar day lasts about half an hour longer than the shortest. Since we do not want the length of the day to change around the year, what we do is use the average, ignoring daily differences. Thus, we get a mean solar day equal to 24 hours. In contrast, the sidereal day is shorter: almost four minutes less.

We could also take the moon as a reference and consider two successive passes of our satellite through the same meridian. Each day, the moon is 48 minutes and 45.78 seconds ahead of the sun (that's the time difference between two high tides on consecutive days). Therefore, the lunar day lasts 24 hours less that period, a little more than 23 hours (see the attached table).

Type of day	Duration
Lunar	23 hours, 11 minutes, 14,22 seconds
Sidereal	23 hours, 56 minutes, 4,06 seconds
Mean Solar	24 hours

 

From our point of view, the natural day is the solar day, since the sun provides us with the alternation of light (day₂) and darkness (night). The mean solar day is the basis for the measurement of time. Thus, day₁ (the mean solar day) is divided into twenty-four hours, each hour into sixty minutes, and each minute into sixty seconds, a way of measuring time that we have inherited from the Babylonian civilization. That is why the X General Conference on Weights and Measures (G.C.W.M.) of 1954 defined the second as the mean solar day/86,400 (24×60×60), although a later conference changed the definition.

But the Earth's rotation is slowing down slowly. The length of both the solar day and the sidereal day increases at the rate of one second every 62,500 years: 44 billionths of a second per day. It doesn't sound like much, but as each successive day introduces a small error, the cumulative effect is quickly noticeable. I am going to explain it with a much greater difference, so as to be clearer.

Assume that a given day lasts exactly 24 hours and that each successive day is lengthened by exactly one second. 30 days later, the day will no longer be 24 hours, but 24 hours plus 30 seconds. But now let's see when each day begins. Day 2 will start a second later than it should; day 3, three seconds late (1 + 2); day 4, six seconds late (1 + 2 + 3); day 31 will start (1 + 2 + 3 +… + 30) = 435 seconds late, i.e. 7 and a quarter minutes late, since the delays from all the previous days have accumulated. This sum grows very fast.

Using the real values, it is easy to see that the cumulative delay adds up to one second after 6,743 days (about eighteen and a half years), although the length of that day will exceed 24 hours by just about three ten thousandths of a second. This, then, would be the time elapsed between two leap second adjustments in atomic clocks, if other factors were not involved.

The same post in Spanish

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Manuel Alfonseca