Saturday, March 12, 2005


Dear Reader,

I have been learning to tell time.

For our D&D games, I have been researching how the heathen Anglo-Saxons told time. They used two calendars in parallel: the Julian calendar, which was an early draft of ours, and their own lunar calendar. Learning about them in enough detail to use them in our games has taught me how much I took for granted about how we tell time today.

Designing a calendar system is not easy. The different units by which we measure time do not divide into one another evenly, and we have no fixed frame of reference for measurement.

For example, what is a day?

One naive answer is twenty-four hours, but that turns out to be circular reasoning because an hour is defined as a fraction of a day, and has no other natural meaning. The day is the natural unit, but what is it?

A second naive answer is that it is the span of time from midnight one day to midnight the next, but this turns out to be the first answer in disguise. After all, when is midnight? It is a repeating point in time spaced out every twenty-four hours, which takes us back to an hour being an arbitrary fraction of a day, and back to the original question, what is a day?

A third naive answer is that it is the time from sunset one day to sunset the next, or from sunrise to sunrise. This captures an important element of the definition of a day, but not all of it. We do know the sun rising and setting is involved, but the time of sunrise and sunset ebbs and flows through the seasons, with the daylight longer in the summer and shorter in the winter. Yet we know the answer we are looking for describes something of approximately constant length throughout the year. This answer does not satisfy that criterion, but it does correctly identify that our definition is related to the relationship between the sun and the earth, particularly to the cycles of day and night. Any definition of a day that runs out of sync with the cycles of day and night cannot be right. So again I ask what is a day?

A fourth answer, a more astronomical answer, is one complete rotation of the planet Earth, but this turns out to be subtly inaccurate. What counts as a full rotation? If we watch the second hand move from 12 around back to 12 on a clock, we can see that we are thinking about a complete rotation as returning to the starting point on a fixed reference. We have no fixed reference for the Earth's rotation, but if we approximated one, say by using the stars, or by projecting an imaginary frame of reference on space around us, the answer is still wrong. The result of doing this would be that over the course of a year, midnight would rotate around through the day as the earth revolves around the sun. Midnight would sometimes be at night, sometimes at sunrise or sunset, sometimes during the day, gradually shifting with the seasons. This seemingly reasonable answer, which is the one given by all the people I know who think they are smart people, is mathematical and astronomical but false.

A more correct answer (but no promises) is that a day is slightly more than one complete rotation of the planet Earth. The line of the Earth's orbit around the sun is our guide to the length of a day. In the almost-day it takes for the Earth to spin completely around, the Earth has also moved just under one degree around its orbit line, changing the sun's position with respect to the Earth by one degree. For the times of day to stay stable, for example for midnight to remain in the middle of the night and noon to remain in the middle of the day, the start and end of each day must remain oriented toward the sun. Since during the course of one day the Earth's revolution around the sun changes its orientation by almost one degree, the Earth must rotate just a little bit more (almost one degree) than one complete rotation to compensate, to line back up with the orbit line and with the sun. That way, noon stays mid-day and midnight stays...well, you get the picture.

If this explanation does not make sense to you, ask me in the comments and I will explain it differently.

It is easy to explain what a day is with a picture but even with words it is pretty easy to explain. So why do we not explain it this way in first grade? Everyone should know what a day is. Why do even most intelligent people define it incorrectly?

Sincerely yours,

Extra credit: Explain why a day is a about a degree more than one complete rotation instead of about a degree less. What would have to be different for a day to be about a degree less than a complete rotation?


Linda M.R. Yaw said...

How about the definition of a day being the midpoint between a sunset and a sunrise to the next midpoint between a sunset and sunrise?

An interesting discussion.

Thank you!

Beverly Marshall Saling said...

Ack--a science test! And I didn't study!

I *think* Linda's definition would have a similar problem as going sunset to sunset or sunrise to sunrise; the times would still shift a bit seasonally, wouldn't they?

On the extra credit question, I have a guess: a day is about a degree more than a complete rotation instead of less because the earth has to rotate and then it takes a degree more to get itself back into the same orientation toward the sun. So for a day to be a degree less than a complete rotation, the earth would have to be closer to the sun, so its orbital path would be shorter and it would get back to the same orientation before it could finish rotating. Is that right?

If not, can you please not tell Mr. Tomlin? He would be so disappointed :)

Rick Marshall said...

Beverly is partly correct about the problem with Linda's answer. Sunset and sunrise are not only not constant, because they shift with the seasons, they also accelerate. That is, whichever one happened most recently is more affected by the shift than the one before.

For example, during the spring the days are getting longer and the nights shorter. Each day, sunset is a little later, and sunrise a little earlier, than the day before, because of the Earth's revolution around the sun overnight, sunrise is affected a little more than sunset was. If we simply took the midpoint as midnight, the earlier sunrise would make us guess wrong and put it a minute or two early. During the fall we would make the opposite mistake and put midnight a little too late.

At the solstices your strategy would work pretty well. At the equinoxes they work the least. This is not what a day is, but it is more or less how we figured out the length of a day.

As for Beverly's answer to the extra credit question, it sounds reasonable but is not right. The reason a day is about a degree more than a complete rotation is not related to how far from the sun the Earth is.

Rick Saling said...

Good one Rick! I hadn't ever really thought about that extra degree before!

Here's another problem for you: when you see a crescent moon, it is either waxing or waning. Which side is which? I.e. when the lit up part is on the right, is it waxing or waning? You can figure this out with simple geometry, but it's remarkably hard to do in your head. And I don't remember the answer, but we have all the information needed to figure it out.

PS: I'm reading all your blogs now, which is why the comments are so late...

Rick Marshall said...

The key to your question about the waxing crescent moon is indeed geometry.

First, the motion. Looking down on the Earth-Moon system from celestial North, the Earth rotates on its axis counter-clockwise (west to east, making the sun appear to travel east to west), and the Moon revolves around the Earth counter-clockwise (yet so slowly compared to the Earth's rotation that the Moon still rises in the east and sets in the west). As the Moon revolves around the Earth, it also rotates at more or less the same speed, so that we always see the same half (although slight changes in speed cause a slight rocking motion from our perspective, called libration, so that over time we see 59% of the Moon's surface, not 50%).

Second, the light. Half the Moon is always lit up by the Sun, and half dark (except during lunar eclipses). As the Moon revolves around the Earth, slowly rotating to keep the same face toward the Earth, the Moon's rotation gradually shifts which part of the Moon is illuminated. From the Moon's perspective, the Sun rises and sets as it does on the Earth but much more slowly.

Third, the quarter phases. The waxing and waning phases of the Moon are an optical illusion generated by the motion within the Earth-Moon system. When the Moon is between the Earth and the Sun, the half of the Moon we can see is dark, since the Sun is illuminating the far side of the Moon. When the Earth is between the Sun and Moon, the half of the Moon we can see is lit. When the Moon is beside the Earth in its orbit, we see the Moon half full.

Finally, the waxing and waning. The lunar cycle begins with the new Moon, with the Moon between the Earth and Sun. As the Moon orbits, moving counter-clockwise, the lit side of the Moon slowly begins to appear on the right side of the Moon face, then appears to spread from right to left as our perspective on the Moon shifts until the Moon is full. Likewise, from a full Moon, the shadow also appears to spread from the right side until the Moon is new again. The alternating waves of light and shadow always appear to pass across the face of the Moon from right to left.

You can find a good explanation, including pictures and movies that show the phases (and the Moon's libration, the rocking motion, is visible in the movies), here.