B00B7H7M2E EBOK (18 page)

Standing at Y, the next step is to build the second imaginary triangle. One corner of this triangle is the point where you now stand (Y). One side of it is an imaginary line drawn from you at Y
through
the cactus off to the horizon. There is a convenient landmark there as well, a particularly precipitous crag. You don’t have to know the length of the line to the crag. A second side of the triangle is an imaginary line drawn from you at Y through the peak. Again, you don’t have to know the length of the line. See
Figure 4.1b.
What you do need to know is the ‘angular distance’ between the peak and the crag.

Angular distance is the number of ‘degrees’ between two lines pointing from an observer (you at Y) to two distant objects whose separation the observer wishes to know (the peak and the crag). It works like this: if you are at the centre of a circle, and the crag and the peak are on the rim of the circle, how many degrees apart on that rim are they? A circle has 360°. A quarter way around a circle (from 12 o’clock to 3 o’clock) is
90°.
(Lines drawn from 12 and 3 o’clock to the centre of the circle – where the observer is supposed to be – meet there at a 90° angle.)The angular distance between the peak and the crag is obviously not that great. It’s more like 12 o’clock to 1 o’clock. That’s 30°.

Figure 4.1

a. The cactus is near the road and there is a ring of mountains on the horizon. From X, on the road, the cactus is directly lined up with a snowcapped peak. Y is a little to the left of X. There is an imaginary triangle whose sides are a line from X to Y, a line from X to the cactus, and a line from Y to the cactus.

b. Draw lines from Y to the peak and from Y through the cactus to the horizon. The second line is a continuation of the line previously drawn from Y to the cactus, and it leads to a crag on the horizon.

Now pretend you are hovering in a helicopter over this scene, for it’s necessary to draw an idealized picture of it from that vantage point. On
Figure 4.2,
the large circle is the horizon, with the mountains. The dot in the centre of the circle is the cactus. The segment of the road between X and Y is part of a smaller circle, centred on the cactus and near to it. We don’t yet know the size of that circle, but that doesn’t matter at the moment. We’ve drawn it in as a broken line. In
Figure 4.2,
the huge circle of the horizon becomes a giant clock face, aligned so that the snowcapped peak is at twelve o’clock. The two lines that meet at Y, on the road, end up widely separated at the horizon. One line, of course, still leads to the peak, the other leads to ‘one o’clock’, where the precipitous crag is. The clock face gives us a way to measure the separation between the two landmarks, and, hence, the angle of separation between the two lines. Twelve o’clock to one o’clock on a clock face (viewed from the centre, where the hands are attached) is a shift of 30°, meaning that the angle at Y (close to the centre), where those two lines meet, is approximately a 30° angle.

So far you may not seem to have found out very much. Actually you have all the information necessary to calculate the distance to the cactus from the road.

Taking stock: you can pace off the distance between X and Y on the road – an easily measurable distance. You know the angular separation between the two lines leading from Y to the peak and to the crag on the horizon (30°). The rules of geometry say that you, standing at Y, will measure almost exactly the same angular separation between the peak and the crag as the cactus, looking back at the road, would measure between X and Y. On the clock-face drawing, the lines drawn
from
the cactus through X and Y pass approximately through six o’clock and seven o’clock, and that means that the angular separation between those two lines, also, is approximately 30°. If the horizon were as far away from the cactus, X, and Y as the stars are from Mars and the Earth, the angles would be even closer to identical.

Figure 4.2

An idealized drawing, from the air, shows the cactus at the centre of a huge circle (the horizon), visualized as a clock. The part of the road between X and Y is a segment of a much smaller imaginary circle centred on the cactus. Lines drawn from Y to the peak and through the cactus to the horizon form approximately a 30° angle at Y.

Continuing, then, with the measurement: there is only one distance the cactus
could
be from the road at which the cactus would measure a 30° angular separation between X and Y, and at which the
actual
distance between X and Y along the road would be the distance you paced off.

To apply the same method to the distance to Mars: the cactus is Mars, and the horizon with the ring of mountains is the distant stars. It’s necessary to find two viewing positions analogous to X and Y. Astronomers must be able to get to both positions to take measurements and they must know the distance between the viewing positions. The two positions must be far enough apart so that a parallax shift can be detected. The distance between them – called the ‘base line’ for the measurement – is a tiny part of an imaginary circle analogous to the little one in
Figure 4.2.
In the Mars measurement, this imaginary circle is so minuscule compared to the ring of distant stars that Mars, X and Y can be thought of as all being, essentially, at the centre of the star ring. The parallax shift of Mars as viewed from two ends of any base line possible on Earth is not anywhere near as large as 30°.

It should be clear now why Cassini’s project required having observers in two widely separated locations on the face of the Earth. There had been vague plans for the Observatory to send an expedition to the tropics for other astronomical research, and this rare opportunity to make the first-ever measurement to another planet was an incentive to proceed posthaste with those plans. Fortuitously, Colbert was hoping to establish a colony at the mouth of the Cayenne River in South America in what is now French Guiana, where there had been French settlers earlier in the century. Ships were sailing there regularly. Cassini sent a young colleague, Jean Richer, and an assistant, a M Meurisse, off to Cayenne equipped with several measuring devices and telescopic sights. They were instructed by the Academy to make observations leading to the measurement of the parallaxes of the Moon, the Sun, Venus, and especially of Mars.

In Cassini’s triangle (analogous to the ‘first imaginary triangle’ in
Figures 4.1a
and
4.1b
) the three corners were Paris, Cayenne and Mars. He knew approximately the distance from Paris to Cayenne (the longitude of Cayenne was not certain, but that was something the expedition hoped to remedy), taking into account the curvature of the Earth. That distance was the base line, Cassini’s ‘distance from X to Y’. It was possible, though problematical with the instruments available to Cassini, to detect the parallax of Mars from that base line. Parallax measurement can, in principle, be made to work for anything that can be seen to have a parallax shift.
Figure 4.3
demonstrates (though not to scale or with the angles that really exist) parallax measurements of the Moon and Mars.

Figure 4.3

Note: This figure is not drawn to scale.

Cassini’s measurement to Mars was not as simple as finding the distance to a cactus, for several reasons. A cactus isn’t going anywhere. A planet is. If Cassini and Richer couldn’t time their measurements precisely and know how the time of a measurement in Cayenne compared with the time of a measurement in Paris, the resulting data would be worthless. Cassini couldn’t merely give the order ‘synchronize watches’. The best clocks available were pendulums. They could be synchronized but they wouldn’t stay synchronized while one of them was taking a sea voyage to the other side of the world.

During his years at the University of Bologna, Cassini had studied the eclipses of Jupiter’s moons and their shadows as they crossed the body of the planet, and he had drawn up tables of their motions. In 1666 he’d noticed that the moons were close enough to Jupiter so that a moon’s appearance from behind Jupiter would be seen simultaneously from any point on Earth, and he realized that Jupiter and its moons could provide a way to determine the time difference between widely separated points on the Earth.

Other problems couldn’t be so handily solved and caused some later experts to be critical of Cassini’s blunt announcement of a definite parallax for Mars, for Cassini was well aware of the
inevitability
of a large margin of error in his measurements. In fact, probably no one was better able to appreciate that margin of error than he, because of his earlier study of the refraction of light
by
the Earth’s atmosphere and his previous efforts to measure the Sun’s parallax.

Nevertheless, in the summer of 1673, Cassini and the Academy awaited Richer’s return from Cayenne with intense excitement. When he arrived, in August, Cassini set immediately to work analysing the data from Cayenne and Paris and additional locations in France where there had been observations. He found most of it not too revealing, but when he had completed his analysis, he reported that from the parallax of Mars he had derived a distance from Earth to the Sun of 87 million miles (140 million kilometres).

There were others besides Cassini who took advantage of Mars’s unusual proximity. In England, a young, largely self-educated astronomer named John Flamsteed had been studying the Sun, Moon and planets. He decided that an old method Tycho Brahe had used to measure parallax would work much more successfully now that there were telescopes. Tycho’s method didn’t require observers at widely separated locations. One observer would suffice to measure a ‘diurnal parallax’, which means a change in position as observed from one single location on the Earth’s surface at two different times of day. The rotation of the Earth carries the observer from one end of the base line to the other. Though Flamsteed’s father sent him on a business trip at just the wrong time, Flamsteed did manage to observe on one clear night and he also came up with parallax calculations for Mars. His results were largely in agreement with Cassini’s.

Cassini’s and Flamsteed’s findings did not, as is often supposed, bring immediate consensus among astronomers about the distances to the Sun and the planets. Others besides Cassini were aware of the large margin of error. However, though these measurements were still imprecise and somewhat in dispute, they became the key to the solar system. Now it was possible to use Kepler’s laws to calculate distances from the Sun to all the known planets. It was a source of amazement how far away the Sun was from Earth. Copernicus had calculated the distance from Earth to
the
Sun as two million miles; Tycho Brahe, five million; Kepler, no more than 14 million. The new measurement was 87 million miles or 140 million kilometres. (The modern measurement is 93 million miles or 149.5 million kilometres.) In the late 1670s, men and women thus for the first time became aware of the size of the solar system and found it enormous beyond anyone’s previous imagining. The universe beyond must be inconceivably vast. In the Middle Ages the distance to the ‘fixed stars’ had been illustrated by how many years Adam, starting his journey on the day of Creation, would still have to walk at a rate of 25 miles per day to reach them. We hope he brought provisions for a long hike. The new illustration had a cannon ball travelling at 600 feet a second (considerably faster than Adam, and presumably passing him as it went) taking 692,000 years.