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One of the most fundamental problems facing early astronomers was the scale of the universe. How big were the Earth, Sun, and Moon, and how far away were they? It seemed evident that Earth was huge compared with every other object, and since it was the home of humanity, it was assumed that Earth was pre‐eminent.

The question of the size of the spherical Earth was solved in the 3^rd century BCE by Eratosthenes, who compared the length of shadows made at different locations at the time of the spring equinox (see Chapter 3). Some facts were also known about the relative sizes and distances of other objects.

Since its shadow easily covered the entire Moon during lunar eclipses, Earth had to be substantially larger than its satellite. During a solar eclipse, the Moon passed in front of the Sun, so the latter had to be further away. However, since their apparent sizes were identical, the Sun must be considerably larger than the Moon. Similarly, the Moon sometimes occulted or passed in front of stars and planets, so these, too, had to be much more remote.

Calculations by the Greek astronomers Aristarchus (c.310–c.230 BCE) and Hipparchus (c.190–120 BCE), based on the size of Earth's shadow, suggested that the Moon's diameter is about one third that of Earth and that its distance is nearly 59 times Earth's radius. This established the scale of the Earth–Moon system with a fair degree of accuracy. However, their simple geometric methods grossly underestimated the Sun's distance.

Figure 1.2 All the major planets follow orbits that lie within 8° of the Sun's path across the sky – the ecliptic. This narrow celestial belt is known as the zodiac. In this image from the SOHO spacecraft, four planets appear close to the Sun (whose light is blocked by an occulting disk). Also in view are some background “fixed” stars, including the Pleiades cluster.

(ESA‐NASA)

Determination of the planetary distances remained problematic for a long time. It soon became clear to observers in the classical world that some planets move more slowly through the constellations of the night sky. Since a slow‐moving planet such as Saturn was also fainter than the faster‐moving objects, Mars and Jupiter, it seemed logical that Saturn was further away from Earth.

It was also clear that the Sun, Moon, and planets did not move at uniform speeds or follow simple curved paths across the sky. One of the most difficult observations to explain was an occasional “loop” in the motions of the more distant planets. This occurred when Mars, Jupiter, and Saturn were shining brightly around midnight (Figure 1.3 and Box 1.1). At such times, the planet's nightly eastward (“prograde”) motion would gradually come to a stop. It would then reverse direction toward the west, becoming “retrograde,” before resuming its general movement toward the east.

The explanation for this motion had to wait until astronomers realized that the Sun was at the center of the planetary system, and that Earth orbited the Sun (see The Central Sun). The loops could then be accounted for by Earth traveling along a smaller orbit so that it would catch up with, then overtake, the outer planets (see Figure 1.3) – like an athlete on an inside track.

Accurate calculations of planetary distances also had to wait until the 17^th century, when observers were able to measure angular distances with reasonable accuracy. The basic geometrical method they used was called parallax (Figure 1.4).

Figure 1.3 The apparent retrograde (“backward” or east–west) motions of Mars, Jupiter, and Saturn are now known to be caused by the relative orbital movement of the planets and Earth. Since Earth moves faster along its orbit than the more distant planets, it overtakes them on the inside track. As Earth approaches and passes Mars, the slower moving outer planet appears to move backward for a few months against the backcloth of “fixed” stars.

(After NASA)

Figure 1.4 The distance of a planet such as Mars can be calculated by measuring its angle of sight – its location against the background of fixed stars – from two or more places on Earth. If the length of the baseline (e.g. the distance between two viewing sites, A‐B) is known, the distance can be found by using simple trigonometry.

(ESO)

This involved measurement of the apparent shift in position of an object when viewed from two different locations. To illustrate this, hold one finger upright in front of your nose and close first one eye and then the other. The finger seems to shift position against the background, although it is, of course, stationary. When the finger is moved closer, the shift appears larger, and vice versa.

Astronomers realized that, if a parallax shift in a planet's position could be measured from two widely separated locations, then its distance could be calculated. This method was first used by a French astronomer, Jean Richer, working in Cayenne (French Guiana), together with Giovanni Domenico Cassini and Jean Picard in Paris. They made simultaneous parallax observations of Mars during its closest approach in 1671, using the recently invented pendulum clocks to ensure that the measurements were made at precisely the same moment.³

Cassini's calculations led to a value of about 140 million km for the astronomical unit (AU) – the mean Sun–Earth distance. Now that this distance was known with reasonable accuracy, Kepler's third law (see Box 1.2) could be used to calculate the distances of the Sun and planets for the first time.

During the 18^th century a great deal of time, money, and effort was spent in attempting to refine these figures. One method was to observe rare transits of Venus across the face of the Sun from many different locations. The most famous transit observations took place in 1761 and 1769 when the British explorer, Captain James Cook, sailed to the Pacific as part of an army of 150 observers scattered across the globe, but these gave very inaccurate results (see Chapter 6).

More successful was the worldwide effort to determine the parallax of the asteroid Eros when it passed close to Earth in 1931. Highly accurate measurements were possible since Eros has no atmosphere and appears as a mere point of light in even the largest telescopes. The value of the astronomical unit turned out to be 149.6 million km.

Since then, more sophisticated techniques have been introduced to refine the scale of the Solar System. One of the most successful is radar, when radio signals are reflected from the surfaces of distant objects (see Chapters 5, 6, and 13). Since the velocity of these microwaves is known and the time taken between emission and reception can be measured to a fraction of a second, the distance can be readily calculated. (Radar has also revealed the sizes and shapes of hundreds of asteroids.) A similar technique used to calculate changes in the Earth–Moon distance involves the use of laser pulses bounced off special reflectors left on the lunar surface.

Once an object's distance is accurately known, the diameter can be determined from its apparent angular size, as seen in a telescope. Unfortunately, this is very difficult for the smaller or more distant members of the Solar System, particularly if their albedo, or surface reflectivity, is uncertain.

In general, the larger an object, the more light its surface reflects. However, some objects are much better mirrors than others. A small, reflective object can have the same apparent brightness as a large, dark object. For example, observations of some Kuiper Belt objects, beyond the orbit of Pluto, indicate that their albedos are greater than previously believed. Since they are more reflective than anticipated, astronomers have revised their diameters downwards.

Another method, involving the occultation of a star by a planet or other object, is especially valuable in relation to objects which are normally difficult to observe. The object's diameter is calculated from the length of time during which it hides the star from view. This technique has been used to discover the rings of Uranus and Neptune, and to study Pluto's largest moon, Charon, for example. It is also invaluable for the detection and observation of exoplanets in orbit around distant stars (see Chapter 14). Unfortunately, if the object possesses a dense, cloudy atmosphere, the occultation only gives the diameter at the cloud tops.