Part 1 - Sun, Moon & Planets

Part 2 - The Stars

Part 3 - Seasons & the Celestial Sphere

Part 4 - Conclusion - Teaching Science Inductively

The Inductive Way to Understand the Sky

"This course is an unusual combination of elementary science and advanced epistemology."


Part 3 - Seasons & the Celestial Sphere

The key idea that is crucial to understanding the seasons is called the “ecliptic.” I want to spend some time discussing this idea, because it provides a good example of the difference between induction and the usual dogmatic approach to science education.

So what is the ecliptic? Well, here’s the definition that is usually given at the outset: The ecliptic is the plane of the Earth’s orbit around the sun. Of course, that’s true, but it’s an advanced definition; at the beginning, it’s a complete inversion of the proper hierarchy. Astronomers knew all about the ecliptic more than 2000 years before they knew that the Earth orbits the sun. They had to know about the ecliptic long before they could ever hope to arrive at the correct theory of the solar system.

So that definition is out; it’s pure dogma at this early stage. Let’s try going back to a definition that doesn’t presuppose the heliocentric theory. Here’s a typical formulation: The ecliptic is the yearly path of the sun against the background of stars. Of course, this is also a valid description; the ecliptic is the annual path of the sun. Don’t confuse it with the daily path of the sun across the sky, which is directly observable over the course of a single day. The ecliptic is a bit more elusive to grasp; it is the path that explains how the sun’s daily path changes over the course of a year.

Now, I guarantee that those of you who were not already clear on the idea of the ecliptic are still not clear on it. The definition I just gave is confusing unless you know all the observations that led to it. In fact, it’s pure rationalism to start with a definition, although that is what textbooks generally do. Instead, we need to start with observations and see how the idea of the ecliptic emerges from them. In essence, the inductive approach treats the material like a detective story; we discover a series of facts, and then the path of the ecliptic is the solution that explains all of them.

So here’s your first fact: the noon sun varies in altitude by a large amount over the course of a year. It is much higher in summer than in winter. The change is more than an eighth of a circle.

If this were the only difference that we noticed in the sun’s path, then we could model it just by changing the tilt of the circular path to get the right altitude at every time of year. But this doesn’t work. If all we do is change the tilt, then the rising and setting positions of the sun don’t change, and the length of day is always the same. But we know that isn’t true.

So, the second fact: summer days are much longer than winter days. And the third fact: rising and setting positions change. During the summer, the sun rises in the northeast and sets in the northwest; during the winter, the sun rises in the southeast and sets in the southwest.

How can we account for all three changes—the changes in altitude, rising and setting positions, and length of day? Well, instead of changing the tilt of the path, what if we slide it? In other words, keeping the tilt constant, we slide the whole path north in the summer and south in the winter. As it turns out, this works perfectly. We can make the rising and setting positions, the length of day, and the altitude all change in just the right way.

Now let’s consider the moon. We see the same type of variations in its altitude, its time above the horizon, and its rising and setting positions. But the moon does not just slavishly follow the sun. It is not simply low in the winter sky and high in the summer sky. Instead, the moon changes according to the beat of its own drummer: that is, the changes occur on its monthly cycle rather than the yearly cycle of the sun. But, in an interesting way, the changes are related to the seasons.

We know that the full moon is opposite in the sky from the sun. It’s opposite in other ways, too. In the winter, when the sun is low in the sky, the full moon is high; at this time, the full moon follows the summer path of the sun. In the summer, when the sun is high in the sky, the full moon is low; at this time, it follows the winter path of the sun. Only around the time of early spring or fall does the full moon take about the same path across the sky as the sun.

Now consider the thin crescent moons. In these cases (waxing or waning), the moon rises in about the same place as the sun, travels across the sky on roughly the same path, and then sets in about the same place. So the variations in the moon’s path are similar to those of the sun, but the pattern is this: Whenever the moon is far from the sun, and therefore nearly full, it takes the high path when the sun takes the low path, and it takes the low path when the sun takes the high path. When the moon is close to the sun in the sky, however, it follows almost the same path.

One more clue will put us on the track to understanding all of these facts. Every year, the sun travels through the same constellations, which are called the zodiac constellations. Now, how do we know this, since we cannot see the stars during the day when we see the sun?

We can know it by observing immediately after sunset. For instance, say we observe a sunset on May 1. As soon as it gets dark, the first star seen in the direction of the sunset is Aldebaran, the brightest star in Bull. But you will see it for only a short time before it disappears below the horizon in about the same place as the sun set.

We know that the stars move a little faster than the sun, so Aldebaran and the rest of Bull are catching up to the sun. A month later, on June 1, they have caught up and the sun is in Bull. So Aldebaran sets with the sun and cannot be seen at all. In early June, the first bright stars to appear in the direction of the sunset are Pollux and Castor, the heads of the Twins. By July, the Twins have caught up with the sun and you cannot see them anymore. And so on for all the other zodiac constellations. By knowing that the stars move a little faster than the sun, and observing which constellations appear directly above the sunset, we can know the path of the sun through the stars.

Now, what about the moon’s path through the stars? This is easy to identify, since we can see the moon and the stars at the same time. When the sun is in Bull, we see the full moon in Scorpion, and when the sun is in Twins, we see the full moon in Archer. The reverse is also true; around December 1, when the sun is low in the sky in Scorpion, the full moon is high in Bull; a month later, when the sun is low in Archer, the full moon will be seen high in Twins. The moon makes a complete circuit through the zodiac constellations during its monthly cycle, just as the sun does during its yearly cycle!

Okay, so we are going to call this path the “ecliptic.” But how exactly do we describe it? How is it oriented?

Recall that the stars all turn together as if attached to a big sphere. The axis of this celestial sphere goes through Polaris, the North Star. So Polaris is the one star that doesn’t move; it is always in the direction of north, and always the same altitude above the horizon (when observed from a particular location). We can define the “celestial equator” as the middle of the sphere—in other words, it is the arc across the sky that is a quarter circle (90 degrees) south of Polaris.

When you are outside at night, you can easily find the celestial equator. Extend one arm and point at Polaris. Extend your other arm and sweep it across the sky while keeping your arms a quarter circle apart. You have just swept out the celestial equator.

If you do this when you can see Orion, you will find that the celestial equator goes right through Orion’s belt—that is, the stars of the belt are south of Polaris by almost exactly a quarter circle. But the zodiac constellations Bull and Twins are seen well to the north of Orion’s belt. The sun is in Bull and Twins in June and July, and this part of the ecliptic passes high over the celestial equator.

If you do the same thing when you can see Altair (the bottom of the summer triangle), you will find that the celestial equator goes through the constellation Eagle. But the zodiac constellations Scorpion and Archer are well to the south of Eagle. The sun is in Scorpion and Archer in December and January, and this part of the ecliptic passes far below the celestial equator.

There are two zodiac constellations that are on the celestial equator: they are called Fishes and the Virgin. If you point at Polaris and then at Fishes or Virgin, you will find that your extended arms are a quarter circle apart. The sun passes through Fishes in early spring and through Virgin in early fall. These are the places where the ecliptic coincides with the celestial equator.

Now, finally, we have all the observations that we need in order to understand the path of the ecliptic. We know that the annual path of the sun reaches its maximum altitude in late June (when it is in the vicinity of Bull and Twins), and it descends to its minimum altitude in late December (when it is in the vicinity of Scorpion and Archer). And we know that the sun is near the celestial equator in late March (when it is in Fishes) and late September (when it is in Virgin).

We also know the amount that the ecliptic is tilted relative to the celestial equator. The difference in the altitude of the noon sun between its high point in late June and its low point in late December is a little more than an eighth of a circle. If you think in terms of the “sun clock,” this is about three hours (except the change is in the north-south direction, rather than the east-west direction). This means that the sun is above the celestial equator by about a sixteenth of a circle (an hour and a half) in late June, and below the celestial equator by the same amount in late December.

It is impossible to understand and remember all these facts unless we have a simple way to visualize the ecliptic and the celestial equator. And now there is such a way. We call it the Horizon Globe, and it’s an amazing teaching device—it knows how to point at Polaris, it shows you the ecliptic and the celestial equator, and it happily accepts any celestial body that you wish to put on it. The flat plate in the globe is the observer’s horizon. You can imagine yourself standing in the center of the plate, and then turn the globe and see how any celestial body moves across the sky at any time. The horizon globe makes astronomy easy for us.

Here in San Diego, Polaris is about 33 degrees above the horizon, so I’ve set the axis of the globe at that angle.

Let’s start by putting the sun on the ecliptic in the late March position when it is in Fishes. If you are looking at the globe from the viewpoint of Polaris, the late March position is a quarter turn clockwise from the highest point of the ecliptic (at one intersection with the celestial equator). Attach the sun, turn the globe clockwise, and see what happens. The sun rises due east, it sets due west, and it is above the horizon for exactly half a turn. When the sun is in this position on the ecliptic, it is the “spring equinox.” The term “equinox” derives from the Latin words meaning “equal night”—that is, the night and the day are both 12 hours. We define this as the beginning of spring.

The sun moves eastward (counterclockwise) along the ecliptic during the course of the year. Three months after the spring equinox, in late June, it reaches its highest position on the ecliptic. This is called the “summer solstice”; it is time when we enjoy the longest days and the shortest nights of the year. We define this point as the beginning of summer.

With the sun at the summer solstice position, turn the horizon globe and observe what happens. Notice that the sun rises in the northeast and sets in the northwest. At noon, it is high in the sky, but not directly overhead; in the continental United States, the noon sun is always in the south, even at summer solstice. Also notice that the sun is above the horizon for much more than half a turn; in other words, the day is much longer than the night.

About three months later, in late September, the sun has reached the “fall equinox” position on the ecliptic. Again, just as in the case of the spring equinox, the sun rises due east and sets due west, and the day and the night are both 12 hours long.

Throughout the fall, the sun is moving lower on the ecliptic. In late December, it reaches its lowest point, which is called the “winter solstice.” With the sun at this position, turn the horizon globe and observe what happens. It rises in the southeast, follows a low path across the southern sky, and sets in the southwest. Notice that it is above the horizon for much less than half a turn; in other words, the day is much shorter than the night.

You now understand the seasons. The differences in the sun’s altitude, in its rising and setting positions, in its time above the horizon—all of this is explained by the sun’s yearly path around the ecliptic. And the seasonal variations in temperature are caused by changes in the sun’s position on the ecliptic, not by variations in its distance from Earth. Since the apparent size of the sun remains nearly constant, we know that the distance to the sun doesn’t change very much.

Now let’s revisit the moon, and see how the ecliptic enables us to understand its path across the sky.

Again, we will start with the sun at the spring equinox. Where on the ecliptic is the first quarter moon? We know that it is a quarter turn counter-clockwise from the sun, and that puts it in the summer solstice position. In late March, a first quarter moon takes the high “road”; it goes across the sky on about the same path that the sun takes in late June. At this time of year, a full moon is in the fall equinox position, so it will follow the same path as the sun and be above the horizon for 12 hours. A third quarter moon is in the winter solstice position, so it will be low in the sky and above the horizon for less time.

The horizon globe can show you the path of any phase of the moon for any time of year. At the four places where the sun is at an equinox or a solstice, use the globe to show the path of the first quarter moon, the full moon, and third quarter moon. After you grasp the pattern, you will never be surprised by where you see the moon in the sky—even at the times when it is much lower or much higher than the sun!

Now let’s review the unique features of our inductive approach to understanding this topic.

First, the ecliptic is a simple model that explains many observed facts. We can see the daily path of the sun across the sky, but we can’t directly see the path of the sun relative to the stars over the course of a year. We have to figure it out from a large range of observations. Once we do figure it out, we grasp that this path enables us to understand all of our observations regarding the seasons and the major variations in the moon’s path. Then the idea of the ecliptic is full of content—it’s the integration of all those observed facts we talked about.

In contrast, if you are merely told a definition at the outset, then in your mind the ecliptic just stands for a floating geometrical abstraction that is not clearly related to the observations. And then, whenever you think about anything that requires the idea of the ecliptic, you get a vaguely uncomfortable feeling, which amounts to: “Well, I sort of know what it is.” And that isn’t good enough. We need to be sensitive to that uncomfortable feeling, and give it the respect that it deserves: It’s the signal that you don’t really understand something from the ground up; in other words, you don’t understand it inductively; or, in other words, you don’t understand it.

The horizon globe serves a crucial function. Once we get the idea of the ecliptic, we need a device that demonstrates how this path explains the observations. It’s too difficult to simply picture it in our minds; we need to see it in concrete form. That’s the purpose of the horizon globe; we can point it at Polaris, turn it, and see exactly how any celestial body is observed to move from any place on Earth and at any time of year.

Devices that show the revolving celestial sphere are not uncommon. Instead of a horizon plate, however, a spherical Earth is put in the center. This ruins the device as a teaching tool for demonstrating what an observer actually sees. What we see around us is the horizon, not the spherical Earth. We don’t look around and say: “Here I am, standing on this little spherical ball and observing the big celestial sphere.” What we observe is celestial bodies rising above the horizon, traveling across the sky on a particular path, and then sinking below the horizon. And that is what you see with the horizon globe.

[Describe the standard way of teaching the seasons, and explain why it fails so badly. Revisit the issue of quantitative precision. At the outset, we do not give the exact angle of the ecliptic, or the exact dates of the equinoxes and solstices, or the exact inclination of the moon’s path.]

Book 3: Seasons & the Celestial Sphere

Part 4 - Teaching Science Inductively