B00B7H7M2E EBOK, page 12
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.
Cassini went on attempting to measure parallaxes and to tease fresh results out of the data from the Cayenne expedition. His work continued to fascinate the King, the court (especially the faithful Colbert) and the public. He became, for a while, the dominant figure in astronomy in Europe. Like Galileo, Cassini was astute when it came to self-promotion. More than anyone else in his time, he used the publication of papers in scientific periodicals as a way of establishing priority and of letting the educated world hear of his successes.
In 1676, Jupiter’s moons made possible another measurement that would be essential to later astronomy. Ole Roemer, a Danish astronomer also working at the Royal Observatory in Paris, studied the eclipses of Jupiter’s moons and noticed that the time that elapsed between their disappearances behind Jupiter varied with the distance between Jupiter and the Earth, a distance that changes as the two planets move in their orbits. Roemer speculated that the velocity of light was responsible for what looked like delays in the eclipses. When Jupiter was further from the Earth, it took longer for the picture of the eclipse to reach eyes and telescopes on the Earth. Timing the delays, Roemer proceeded to calculate the speed of light at about 140,000 miles per second. That figure fell short of the 186,282 miles per second that we assign to the speed of light today. The discrepancy in the measurement was due in part to Roemer’s less-than-precise knowledge of the distance to Jupiter. The modern calculation is made with an atomic clock and a laser beam. (This book uses rounded-off figures of 186,000 miles or 300,000 kilometres per second for the speed of light.)
The Royal Observatory in England was established somewhat later than the Paris Observatory, and the impetus for its founding came from an unlikely source. In 1674, Louise de Kéroualle – a Bretonne who had recently been made Duchess of Portsmouth and was one of King Charles II’s mistresses – brought a Frenchman named Sieur de St Pierre to the King’s attention. St Pierre had discovered, so he said, a secret method for finding longitudes. Determining longitude requires having some form of universal clock – such as Jupiter’s moons were for Cassini and Richer – that will allow comparison of celestial phenomena as seen from different locations. If the Sun is directly overhead in New York, what is its position in Greenwich, England? Answering such questions is one way of finding longitude.
Flamsteed and others thought that St Pierre’s ‘secret method’ must involve using the Moon as a time-keeper. Who needed St Pierre? King Charles told the Royal Society, then known as the ‘Royal Society of London for Improving Natural Knowledge’, to collect whatever lunar data were necessary to determine whether the Moon could serve as a universal clock. Flamsteed soon reported that lunar and stellar positions were not known sufficiently well to make the method reliable. Nevertheless, the bee was in the King’s bonnet and he founded the Observatory, designating Flamsteed as ‘astronomical observator’, a position that would later become ‘astronomer royal’. A new building, designed by Sir Christopher Wren, was ready for occupancy in July 1676, and Flamsteed and his associates began the task of producing correct star positions and tables for the Sun, Moon and planets – for the express purpose, as those who governed England saw it, of aiding navigation, but also for the furthering of astronomy.
So things stood in the last quarter of the 17th century. Kings were footing the bill for observatories in Paris and at Greenwich. There was also royal patronage for such bodies as the French Academy and the Royal Society, where gentlemen – expert and not – got together and shared knowledge of the wonders of science. They had a fairly good idea of the dimensions of the solar system. Astronomy seemed poised and ready to measure much greater distances. But a base line from Paris to Cayenne isn’t long enough to measure the parallax shift of stars; in fact, no possible base line on the face of the Earth would suffice. Was there any way to make the parallax method work for stars?
The clue was there long before Ptolemy. One objection other Hellenistic scholars raised to Aristarchus’s suggestion that the Earth moves and orbits the Sun was that, if it does, we ought to be able to observe stellar parallax as we travel from one extreme of the orbit to the other. In other words, the stars ought to shift position relative to one another in the sky, just as the cactus and the mountain peak shifted. If we’re viewing the cactus and the mountains from a car window and they don’t seem to be changing positions relative to one another, either our car is standing still or both the cactus and the mountains are very far away. Likewise either the Earth does not move and orbit the Sun, or the stars are extremely far away – further away than most ancient scholars, except for Aristarchus, were willing to conceive.
By 1700, nearly all astronomers agreed that the Earth rotates on its axis and orbits the Sun. They also recognized the potential of this orbit for a base line. Even so, at that time and even much later, no one was able to detect an annual shift in a star’s position. That discovery had to wait for considerable improvement in telescopes and the precision instruments used with them – advances that would eventually come with the Industrial Revolution. But there was also meanwhile more to be learned from a theoretical point of view, in order to be able to recognize stellar parallax and not confuse it with other effects.
Parallax wasn’t the only hope for measuring the distance to the stars. According to the ‘inverse square law’, the brightness of a light falls off with distance in a mathematically dependable way. The measured intensity of light diminishes by the square of the distance to its source. Imagine that you have two 100-watt lightbulbs. Place one of them twice as far away as the other. The further bulb will appear to be only a fourth as bright as the nearer. It will seem to you that it must be a 25-watt bulb. Turning that exercise around: suppose you keep one of the 100-watt bulbs nearby and ask an assistant to carry the second to an unknown distance. If the more distant bulb appears to be a 25-watt bulb, then you can be sure it’s twice as far away as the first bulb. Comparing their apparent brightness gives the distance of the second light. How does this apply to stars? If stars all have the same close-up brightness and we know the distance to one star, then in principle we can find the distance to any other star by comparing its apparent brightness (how it looks to us from the Earth) with the brightness of the first star.
The reasons why measuring the distance of stars has turned out to be far more complicated than that become clearer if we come in out of the dark and note that the same inverse square law that works with the brightness of light works with the size of objects. Even without knowing the mathematics, you and I use the method instinctively, if imprecisely. Suppose you are overlooking a vast expanse of land with a great number of elephants grazing on it. Some of the elephants look tiny, but because you have previous experience of elephants and know how large an elephant would be if you were standing near it, you make an educated guess that these aren’t really tiny at all, just far away. From the difference between their apparent sizes and the normal close-up size of an elephant, you judge how far away.
Your success relies upon: (1) having some way of knowing that elephants are all approximately the same size – some reason for assuming there aren’t elephants that are pygmies and others that are giants; and (2) having experience of what that size actually is – that is, how big an elephant looks when it’s standing a known distance away.
With a light seen at night at an unknown distance, on the Earth or in the sky, we lack those essential ingredients. Experience tells us that lights can’t be depended on all to have approximately the same close-up brightness, so knowing the close-up brightness of one light is of no use when studying the brightness of a distant light. A comparison is meaningless. We’re left with such questions as: Is it a dim bicycle headlamp over there, just a few yards up the road, or is it the high-beam headlight of a car a mile away? Is it a meteor suffering a fiery death in the upper atmosphere, or is it a firefly no further away than the treetops?
The situation is puzzling but not completely hopeless, nor did it seem so at the end of the 17th century. Astronomers did know how far it is to one star – the Sun. Stars might all be equally bright, and it might be safe to assume the Sun is a typical star. It would seem highly desirable to know the brightness or the distance of at least one star other than the Sun, but failing that, the Sun would have to serve as a standard. On the assumption that it could, Englishman Isaac Newton set out to measure the distances to the nearest stars.
Newton, born in 1642, the year that Galileo died, was 30 years old when Cassini and Flamsteed measured the distance to Mars. Except for his Principia Mathematica, which appeared in 1687, unarguably one of the most important achievements in the history of human knowledge, Newton published almost nothing. However, he obsessively researched an astounding variety of subjects including optics, theology, alchemy and calculus, and was responsible for significant advances in many of the fields he investigated. He contributed to the development of a new type of telescope that used a mirror rather than a lens to focus incoming light. (See Figure 4.5 on here.)
Newton probably wouldn’t even have published the Principia had his friend Edmund Halley not goaded him sufficiently. Publication meant public notice, and contact and correspondence with other scholars. It meant he would be expected to take part in the discussions at the Royal Society. There would be invitations to perform experiments for that body and to watch others do the same. Though honours were tempting and sometimes Newton succumbed, those distractions more often seemed anathema to him. They took precious time away from his research. It is surprising therefore that he agreed to become head of the Royal Society. Unfortunately, he used the position of power in an extremely unpleasant, autocratic manner, bringing misery to other fine scientists (including the elderly Flamsteed). Newton also eventually accepted the job as Master of the Mint and thoroughly enjoyed his rather pedestrian duties there.
Whatever his personal failings, it was Newton who capped the Copernican revolution with his discovery of laws of gravity. In Newton’s description, each body in the universe is attracted towards every other body by the force called gravity. How much bodies are influenced by one another’s gravitational attraction depends on how massive the bodies are and how near they are to one another. For instance, any change in the mass of the Earth or the Moon, or in their distance from one another, would change the strength of the gravitational attraction between them. If the mass of the Earth were doubled, the attraction between the Earth and the Moon would double. If the Moon were twice as far from the Earth as it is, the attraction of gravity between the Earth and the Moon would be only a quarter as strong.
