## The Moon

The moon is our nearest neighbour in space. It does not emit light; we see it by reflected sunlight, but we can only see the part which happens to be illuminated.

### 4.2.1 The phases of the moon

Ancient astronomers knew that the moon shines by reflected sunlight, and that it orbits the earth approximately once every four weeks. We can observe the changing phases, which depend precisely on how much of the sunny side of the moon can be seen from the earth at a given time, as illustrated in Figure 4.2.

Figure 4.2 The four phases of the moon. Phase 1 (new moon): The sunny side of the moon is facing away from the earth, and the moon is not visible from the earth. Phases 2 and 4 (first and last quarters): We see half the illuminated side. Phase 3 (full moon) : The sunny side of the moon is facing the earth. We see the complete illuminated hemisphere.

Figure 4.2 The four phases of the moon. Phase 1 (new moon): The sunny side of the moon is facing away from the earth, and the moon is not visible from the earth. Phases 2 and 4 (first and last quarters): We see half the illuminated side. Phase 3 (full moon) : The sunny side of the moon is facing the earth. We see the complete illuminated hemisphere.

### 4.2.2 A lunar eclipse

Occasionally the moon, or part of it, is in shadow for a completely different reason. The sun's rays may be obscured because the earth gets 'in the way'. This is called a lunar eclipse. During a total eclipse there is not quite total darkness; we see a slight glow due to sunlight scattered by the earth's dusty atmosphere.

It might appear that a lunar eclipse always occurs when there is a 'full moon' (phase 3). In fact, this is usually not the case because the earth, moon and sun are rarely directly in line. Figure 4.2 is just a representation in two dimensions, whereas the real situation is one of three dimensions, as illustrated in Figure 4.3.

Here, however, it can be seen that the plane in which the moon orbits the earth is not the same plane as that of the

Figure 4.3 Planes of orbit of the earth and the moon.

Figure 4.3 Planes of orbit of the earth and the moon.

orbit of the earth around the sun (the plane of the ecliptic). The angle between the two planes is about 5°. A lunar eclipse occurs only when the moon is full and happens to be crossing the plane of the ecliptic when the sun, earth and moon are in a straight line.

During a lunar eclipse the moon passes across the earth. From the shape and size of the shadow we can deduce the shape and size of the earth. In Figure 4.4,

we are looking into space in the direction directly opposite to the sun. The shadow of the earth stretches out from us directly in front. The moon comes into view. We see its complete sunny side — a full moon. As it passes through our shadow it is plunged into almost complete darkness!

It is remarkable how well the Greeks were able to observe eclipses of the moon and to make measurements. They also realised that the size of the shadow was not the same as the size of the earth, and made the necessary corrections, which we will describe in the next section.

### 4.2.3 A solar eclipse

A solar eclipse occurs when the moon is directly in line between the earth and the sun, and casts its shadow on the earth. The situation is now reversed in the sense that the moon obscures the sun from us rather than the earth obscuring the sun from the moon. When we are at the centre of the shadow we experience almost complete darkness; all light from the sun is obscured.

During a total eclipse, the sun is almost completely hidden by the moon, showing that the angular size (~ 0.5°) of the sun and of the moon are approximately equal when viewed from the earth, as illustrated in Figure 4.5.

From similar triangles it is clear that the ratio of the diameter of the sun to the diameter of the moon is almost exactly

Figure 4.5a Ray diagram of a solar eclipse.

equal to the ratio of our distance from the sun to our distance to the moon. The moon's shadow on the earth is just a point, or at least very small compared to the size of the moon or the earth.

The result of a solar eclipse is clearly seen in the photograph (Figure 4.5b), taken from the Mir space station. It shows the shadow of the moon, with a diameter of about 100 km, as it moves across the earth. As the earth rotates, the shadow travels at about 2000 km/h. Had satellites been available to the ancient astronomers, they would, one would imagine, have been impressed!

The moon's orbit around the earth is not exactly circular, and the size of the shadow depends on the exact distance of the moon at the moment of the eclipse. Sometimes the moon may be just too far, and the rays drawn in Figure 4.7 may meet at a point above the earth. There is then no region of complete totality on the earth's surface, and a thin ring of sunlight can be seen surrounding the moon. Such an occurrence is called an annular eclipse.

It is interesting to point out a major difference between lunar and solar eclipses. A lunar eclipse is seen at the same time from everywhere on the earth. A solar eclipse occurs at different times along the band covered by the shadow, as the earth rotates, and the shadow travels across the earth's surface.

4.3 Sizes and distances

4.3.1 Relative sizes of the sun and the moon

We already have one piece of information from Figure 4.5a in the previous section, namely:

diameter of the sun _ distance from the earth to the sun diameter of the earth distance from the earth to the moon

This ratio was derived from the fact that the moon and the sun happen to look about the same size in the sky when viewed from the earth. The equation itself does not provide any information about their actual size, which one is smaller and nearer and which one bigger and further away. Of course it is obvious that, since it is the sun which is eclipsed, the sun must be the more distant object.

Aristarchus (~ 310-230 BC) devised very clever ways of determining astronomical sizes and distances. The size of the shadow of the earth on the moon was his next clue in a remarkable chain of logic, and led to a comparison of the actual size of the earth and of the moon. As we shall see below, this second step is not quite as straightforward as the first.

### 4.3.2 The shadow of the earth on the moon

By measuring the time for the moon to go through the earth's shadow in a lunar eclipse, Aristarchus estimated that the diameter of the shadow cast by the earth on the moon is 2Y

times the diameter of the moon.

First piece of information: Size of the shadow of the earth at the moon = 2.5 x size of the moon. How is this related to the size of the earth itself?

4.3.3 Shrinking shadows

The size of the shadow cast by an object depends on the direction of the light rays which illuminate it.

When light comes from an extended source like the sun, the region of the shadow can be divided into two areas: the umbra, which is an area not illuminated by light from any point on the sun, and the penumbra, which is illuminated by light from some but not all points on the sun (Figure 4.6). Usually, the umbra is quite well defined and it is that area of complete shade to which we refer in our discussion of both lunar and solar eclipses.

The same definitions apply in more common situations, such as the shade under a parasol on a beach. In most cases, owing to the brightness of the sun, the penumbra is quite bright and barely distinguishable from the sunlit area.

earth

Figure 4.6 Umbra and penumbra.

If the sun were a point source we would expect the shadow cast by the earth on an imaginary screen situated behind the earth to spread out as the screen is moved further away from the earth. However, the sun is an extended source and the size of the umbra decreases as the screen is moved further away, as we may see from Figure 4.6. When the angular size of the screen becomes less than the angular size of the sun, the umbra disappears.

Common experience tells us that a parasol placed too far above the ground will not cover the whole sun and will offer little protection. In practice, scattered light will make it even less useful!

### Reverting to astronomy

When the moon is obstructing the sun, the umbra of the moon's shadow has shrunk to a point by the time it reaches the earth. Similarly, when the earth is in front of the moon, the shadow of the earth gets smaller by the time it reaches the moon, but because the earth is bigger, the shadow will not be reduced to a point.

### Comparing shadows

Assuming the sun is so far away that its angular size does not change over the earth-moon region, the angles of the light rays from the edge of the sun will be the same in Figure 4.7. Since the shadow of the moon tapers down to a point over the earth-moon distance, the diameter of the earth's shadow will decrease by the same amount over the same distance. Therefore

the shadow of the earth on the moon is narrower than the actual size of the earth by about one moon diameter. Aristarchus was able to conclude that:

Diameter of the earth = size of the shadow of the earth on the moon + 1 moon diameter = 2.5 x diameter of the moon + 1 moon diameter = 3.5 x diameter of the moon

The circumference of the earth ~ 40,000 km (Eratosthenes)

The circumference of the moon « 40,000 „ 11,500 km

and the diameter of the moon « 3,650 km.

### 4.3.4 The distance to the moon

Aristarchus knew that the sun is not a point source. Using the information that the angular sizes of the moon and the sun as viewed from the earth are about the same, he made a correction to take into account the taper of the shadow:

Once we know the actual size of the moon, Figure 4.8 illustrates how it is easy to calculate the distance from the earth to tan(0.25o) = 1825 x = 1825 400,000 km

0.00436

Figure 4.8 Distance from the earth to the moon.

0.00436

Figure 4.8 Distance from the earth to the moon.

the moon from the angle subtended by the moon at some point on the earth.

4.3.5 The distance to the sun

Just one more piece of information remains to complete the jigsaw.

We have the absolute size of the moon and its distance from the earth. If we also knew the ratio of the distance from the earth to the moon and the distance from the earth to the sun, everything would fall nicely into place. Aristarchus had a very clever idea about how to obtain this information:

If we go back to Figure 4.2, which illustrates the phases of the moon, and look at it more closely, we will see that it is not completely accurate. When the moon is exactly in its first and last quarters, the sunny side in each case is tilted inwards slightly as it faces the sun. The situation is made clearer in Figure 4.9.

The line from the moon to the sun is tangential to the moon's orbital circle, and the line from the moon to the earth is a radial line. They are mutually perpendicular and the marked angles are each 90°. It is easily seen that the orbital arc from the first to the

last quarter is longer than that from the last quarter back to the first. Assuming that the moon moves at a uniform speed, there will be a difference in time between the two journeys. From this time difference all the angles in the diagram can be determined, and hence the ratio of the distance between the earth and the sun and the distance between the earth and the moon. Once we know that ratio we have all the information we require.

### 4.3.6 A practical problem

The argument of Aristarchus is correct, but there was an experimental problem which he was not quite in a position to solve. As one would imagine, it is very difficult to determine the exact moment at which the moon is at a given phase. The total time taken for the moon to go through all four phases is about 30 days. Aristarchus' visual estimate was that it takes one day longer to go from the first quarter to the last quarter than from the last back to the first.

Using the data of Aristarchus, let us follow his reasoning to calculate the distance of the earth from the sun in terms of the earth-moon distance:

The total travel time = 30 days; the long segment = 15.5

days; the short segment = 14.5 days.

ES = distance from the earth to the sun EM = distance from the earth to the moon

Figure 4.10 The geometry of Aristarchus' calculations.

ES = distance from the earth to the sun EM = distance from the earth to the moon

Figure 4.10 The geometry of Aristarchus' calculations.

The first quarter is 1/4 day early and the last quarter is 1/4 day late.

30 days corresponds to 360° (1 complete revolution), so 1/4 day corresponds to 3°.

This means that size of the angle d in Figure 4.10 is 3°.

According to Aristarchus, the sun is about 20 times more distant than the moon.

While the method was correct, the result (which is extremely sensitive to the measurement of the time of the phases) was much too small. We now know that the distance from the earth to the sun is about 390 times larger than the distance from the earth to the moon. (The angle d is not 3°, but 0.15°.)

Despite the fact that the final experimental answer was out by a factor of 20, one cannot but be very impressed by the reasoning of the ancient astronomers. They started with the one absolute measurement, made by Eratosthenes — that of the distance from Alexandria to Syene. There followed the determination of a series of ratios, from which they determined the sizes and distances from the earth, of the moon and the sun. All were nearly correct, except for the distance to the sun. The estimated size of the sun, based on this distance, showed the same degree of inaccuracy.

Aristarchus' value of the distance to the sun was accepted, until the late 17th century. He was far ahead of his time; he even suspected that the sun did not rotate about the earth. He reasoned that since the earth was much smaller than the sun, surely it was reasonable for it to orbit the sun rather than the other way around. This view, as we know, was not accepted for 2000 years.

4.3.7 A summary concerning the earth, moon and sun

The diameter of the earth is 12,750 km (a modern value, little different from the value obtained by Eratosthenes). Relative diameters of the moon, earth and sun.

Object |
Moon |
Earth |
Sun |

What the Greeks thought |
G.3 |
1 |
6 |

Modern values |
G.27 |
1 |
1G9 |

4.3.8 Astronomical distances

To help obtain an intuitive feeling for these distances, let us construct a timetable for non-stop travel by a Concorde supersonic aircraft at a maximum cruising speed that is twice the speed of sound (~ 2150 km per hour or Mach 2, as in Table 4.1).

Journey |
Distance in km |
At speed of light |
Concorde (one way) |

Around the earth |
4G,GGG |
0.13 seconds |
18.5 hours |

Earth-moon |
3.84 x 1G5 |
1.28 seconds |
7.4 days |

Earth-sun |
1.5G x 1G8 |
8.33 minutes |
7.9 years |

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