We have seen a little of how ancient astronomers tracked and recorded the positions of the stars on the sky and how modern astronomers make similar observations. But knowing the directions in which objects lie is only part of the information needed to locate them in space. Before we can make a systematic study of the heavens, we must find a way of measuring distances, too. One distance-measurement method, called triangulation, is based on the principles of Euclidean geometry and finds widespread application today in both terrestrial and astronomical settings. Today's engineers, especially surveyors, use these age-old geometric ideas to measure indirectly the distance to faraway objects. Triangulation forms the foundation of the family of distance-measurement techniques that together make up the cosmic distance scale.

Imagine trying to measure the distance to a tree on the other side of a river. The most direct method is to lay a tape across the river, but that's not the simplest way. A smart surveyor would make the measurement by visualizing an imaginary triangle (hence triangulation), sighting the tree on the far side of the river from two positions on the near side, as illustrated in Figure 1.22. The simplest possible triangle is a right triangle, in which one of the angles is exactly 90°, so it is usually convenient to set up one observation position directly opposite the object, as at point A. The surveyor then moves to another observation position at point B, noting the distance covered between points A and B. This distance is called the baseline of the imaginary triangle. Finally, the surveyor, standing at point B, sights toward the tree and notes the angle at point B between this sightline and the baseline. No further observations are required. Knowing the value of one side (AB) and two angles (the right angle itself, at point A, and the angle at point B) of the right triangle, the surveyor can geometrically construct the remaining sides and angles and so establish the distance from A to the tree.

Figure 1.22 Surveyors often use simple geometry and trigonometry to estimate the distance to a faraway object.

To use triangulation to measure distances, a surveyor must be familiar with trigonometry, the mathematics of geometrical angles. However, even if we know no trigonometry at all, we can still solve the problem by graphical means, as shown in Figure 1.23. Suppose that we pace off the baseline AB, measuring it to be 450 meters, and measure the angle between the baseline and the line from B to the tree to be 52°, as illustrated in the figure. We can transfer the problem to paper by letting one box on our graph represent 25 meters on the ground. Drawing the line AB on paper, completing the other two sides of the triangle, at angles of 90° (at A) and 52° (at B), we measure the distance on paper from A to the tree to be 23 boxes—that is, 575 meters. We have solved the real problem by modeling it on paper. The point to remember here is this: nothing more complex than basic geometry is needed to infer the distance, the size, and even the shape of an object too far away or too inaccessible for direct measurement.

Figure 1.23 We don't even need trigonometry to estimate distances indirectly. Scaled estimates, like this one on a piece of paper, often suffice.

Triangles with larger baselines are needed if we are to measure greater distances. Figure 1.24 shows a triangle having a fixed baseline between two observation positions at points A and B. Note how the triangle becomes narrower as an object's distance becomes progressively greater. Narrow triangles cause problems because the angles at points A and B are hard to measure accurately. The measurements can be made easier by "fattening" the triangle—in other words, by lengthening the baseline.

Figure 1.24 A triangle of fixed baseline (distance between points A and B) is narrower the farther away the object. As shown here, the imaginary triangle is much thinner when estimating the distance to a remote hill than it is when estimating the distance to a nearby flower.

Now consider an imaginary triangle extending from Earth to a nearby object in space, perhaps a neighboring planet. The imaginary triangle is extremely long and narrow, even for the nearest cosmic objects. Figure 1.25(a) illustrates the case in which the longest baseline on Earth—Earth's diameter, measured from point A to point B—is used. In principle, two observers could sight the planet from opposite sides of Earth, measuring the triangle's angles at points A and B. In practice, though, these angles cannot be measured with sufficient precision. It is actually easier to measure the third angle of the imaginary triangle, namely, the very small one near the planet. Here's how.

Figure 1.25 (a) This imaginary triangle extends from Earth to a nearby object in space (such as a planet). The group of stars at the top represents a background field of very distant stars. (b) Hypothetical photographs of the same star field showing the nearby object's apparent displacement, or shift, relative to the distant, undisplaced stars.

The observers on either side of Earth sight toward the planet, taking note of its position relative to some distant stars seen on the plane of the sky. The observer at point A sees the planet at apparent location A' relative to those stars, as indicated in Figure 1.25(a). The observer at B sees the planet at point B'. If each observer takes a photograph of the appropriate region of the sky, the planet will appear at slightly different places in the two images. In other words, the planet's photographic image is slightly displaced, or shifted, relative to the field of distant background stars, as shown in Figure 1.25(b). The background stars themselves appear undisplaced because of their much greater distance from the observer. This apparent displacement of a foreground object relative to the background as the observer's location changes is known as parallax. The size of the shift in Figure 1.25(b), measured as an angle on the celestial sphere, is equal to the third, small angle shown in Figure 1.25(a).

The closer an object is to the observer, the larger the parallax. To see this for yourself, hold a pencil vertically in front of your nose, as sketched in Figure 1.26. Concentrate on some far-off object—a distant wall, say. Close one eye, then open it while closing the other. By blinking in this way, you should be able to see a large shift of the apparent position of the pencil projected onto the distant wall—a large parallax. In this example, one eye corresponds to point A, the other eye to point B, the distance between your eyeballs to the baseline, the pencil to the planet, and the distant wall to a remote field of stars. Now hold the pencil at arm's length, corresponding to a more distant object (but still not as far away as the distant stars). The apparent shift of the pencil will be less. By moving the pencil farther away, we are narrowing the triangle and decreasing the parallax (and, in the process, making its accurate measurement more difficult). If you were to paste the pencil to the wall, corresponding to the case where the object of interest is as far away as the background star field, blinking would produce no apparent shift of the pencil at all.

Figure 1.26 Parallax is inversely proportional to an object's distance. An object near your nose has a much larger parallax than an object held at arm's length.

The amount of parallax is thus inversely proportional to an object's distance. Small parallax implies large distance, and large parallax implies small distance. Knowing the amount of parallax (as an angle) and the length of the baseline, we can easily derive the distance through triangulation.

Once we know the distance to an object, we can determine many other properties. In particular, by measuring the object's angular diameter we can calculate its size. Figure 1.27 illustrates the geometry involved. Notice that this is basically the same picture as Figure 1.25(a), except that the angle (the angular diameter) and the distance are known, instead of the angle (the parallax) and the baseline. We compute the object's diameter by noting that the ratio of the diameter to the circumference of the circle centered on the observer and passing through the object (2times the distance) must be equal to the ratio of its angular diameter to one full revolution, 360° :

Surveyors of the land routinely use such simple geometric techniques to map out planet Earth (see More Precisely 1-3 for an early example). As surveyors of the sky, astronomers use the same basic principles to chart the universe.

Figure 1.27 If the angular diameter of a distant object can be measured and its distance is known, its true diameter may be calculated by simple geometry.