How Big is Big and How Small is Small

by Smith, Timothy Paul

  Consider a Gedanken: a thought experiment. If I am trying to interact with a particle that is a Planck length across I will need a photon with a wavelength of about a Planck length. A photon with that wavelength will have an energy of about 1028 eV or 0.1 J. That is not a lot of energy. But when it is crammed into such a tiny particle it would make it into a black hole, which will swallow up any photon that is trying to escape. In other words, if there is a particle as small as the Planck length it will be invisible. We can never, even in principle, interact with it.

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  One other important feature of the Planck length is that this is the scale of the world where string theory operates. String theory was created to deal with a number of problems in particle physics theories. One of the first problems was the fact that in most theories forces become infinite as distances go to zero. We have said that the forces of gravity and electromagnetism decrease as 1/r2 as the distances become large. But that also means they increase as the separation between charges approach zero. In fact, at zero the force would be infinite Mathematically this is called a singularity. Physically it is unpalatable. String theory says that at a very small scale particles are not truly point-like, rather they are loops, described as strings. It also suggests that different types of particles may just be the same loop of string oscillating in different modes. That would help the standard model by explaining all of its parts as just simple modes of oscillation. But string theory has yet to get to that stage.

  It has also been proposed that not only particles, but the very nature of space and time are different on this scale. Instead of space and time as a continuous fabric, it is sometimes described a space-time foam.

  I like the motivation behind string theory, but to me it has one major difficulty. After several decades of development it has yet to make any unique and testable predictions. Actually, that is not really surprising when you remember that the size of these strings are ten quadrillion (1016) times smaller than the scales probed by our biggest experiments. This is like starting with quarks and QCD and trying to figure out which way a butterfly will flutter. It may work in principle, but it may be very difficult in practice.

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  When Johann Loschmidt first measured the size of an atom in 1865 he wrote:

  An imposing string of numbers such as our calculation yields, especially when taken into three dimensions, means that it is not too much to say that they are the true residue of the expectations created when microscopists have stood at the edge of the bottomless precipice and described them as “infinitesimally small.” It even raises the concern that the whole theory, at least in its present form, might wreck on the reef that the infinitesimally small cannot be confined by experiment.

  When Loschmidt looked into the bottomless precipice he could see ten orders of magnitude. We can now see nearly twice that. Yet if the true bottom is down there at 35 orders of magnitude we might never clearly see it. However, maybe there is something between 10−18 and 10−35 m. That is a large region and nature may yet hold more twists and surprises. Or maybe space and time at the Planck scale are not quite what we think they are.

  We stand at the edge of the precipice with Loschmidt and wonder.

  12

  Stepping Into Space: the Scales of the Solar System

  The night of 7 January 1610, was clear above the city of Padua, in northern Italy. Galileo Galilei was out that night looking at the stars. Later he wrote about that evening, “And since I had prepared for myself a superlative instrument, I saw (which earlier had not happened because of the weakness of the other instruments) that three little stars were positioned near [Jupiter].” With his simple hand-ground lens and homemade telescope, the bright disk of Jupiter would have been distorted and the colors would bleed, with white separated into blues and reds. But what caught his attention was that these three new stars formed a straight line, two on the east side of Jupiter and one on the west. The orientation of his diagram, with east (Ori.) to the left and west (Occ.) to the right seems backwards at first. But he was drawing a map of the sky overhead and not a map of the land underfoot.

  Galileo also noted that this line of stars were parallel to the plane of the eclipse, which is the path of the planets as they move across the sky. But to him, on that night, this was just a coincidence.

  “… guided by I know not what fate …”Galileo pointed his spyglass again at Jupiter the next night and saw all three stars to the right.

  At this time Galileo assumed that the newly seen, tiny stars were fixed in the sky like all other stars. So he interpreted the rearrangement of these stars relative to Jupiter to mean that Jupiter was moving to the left, or east. But that was contrary to the calculated motion of the planet for that day.

  On 9 January Galileo was prepared to look at Jupiter again, for what he had observed did not fit into his understanding of planetary motion. But the sky was cloudy and did not cooperate. I have always liked that night’s report. For weather is very real, and real stories, real discoveries can be interrupted by very real clouds.

  The next night, 10 January, was again clear over Padua and again Galileo trained his telescope upon the great planet. This time there were two of his new “stars” to the left of Jupiter. Galileo thought that the third may have been hidden behind the planet. The mystery was deepening, but Galileo was also starting to see a possible interpretation: “… now, moving from doubt to astonishment, I found that the observed change was not in Jupiter but in the said stars.”

  Galileo knew that he was seeing something important and so he started making a series of meticulous observations. A few months later, in March 1610, he published his results in a book, Sidereus Nuncius or Starry Messenger. In this book he includes 65 sketches of Jupiter and its satellites, whereas in the rest of the book he has only five diagrams of our moon, two of star clusters, and two of nebulae. That is because there was something unusual, even extraordinary, going on around Jupiter. His drawings became more and more detailed, with one “star” soon marked as larger. On 12 January, while he was watching at 3:00 in the morning, one of the stars came out from behind Jupiter. On the next night he spotted a fourth “star.”

  Galileo soon realized that these were not stars. As Jupiter traveled across the winter sky the four companions followed. His later sketches even marked “fixed” stars in the background, as Jupiter and its entourage drifted across the sky. Within a year, Johannes Kepler had read Galileo and coined a new word for these companions; his word satellite is based on the Latin word for “attendant.”

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  There are a lot of good stories about the “Galilean moons.” The reason I wanted to introduce them, and Galileo, is that so many of the puzzles and solutions encountered in trying to figure out the size of the solar system are illustrated by these four satellites. At about the same time as Galileo was playing with his telescope, Johannes Kepler was deriving his laws of planetary motion. These laws apply to satellites like Jupiter’s as well as planets. They gave a relationship between orbit periods and size. Orbit periods are easy to measure, but you also need one distance in the solar system to set the scale for everything. If we knew the distance to the Sun, we would know the size of the orbits of all the planets and the size of the solar system.

  Our best measurement of the Sun–Earth distance are based upon knowing the speed of light, and the first good measurement of the speed of light was based upon the Sun–Earth distance and the moons of Jupiter. In fact one of the first attempts to measure the speed of light was also made by Galileo.

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  At one time René Descartes, a contemporary of Galileo’s, said, “I confess that I know nothing of Philosophy, if the light of the Sun is not transmitted to our Eyes in an instant.” Does light have a finite speed, or does it travel instantly? This had been an open question for centuries. But it was Galileo, more the experimentalist than philosopher, who first tried to bring the question into the laboratory, or at least out of the philosopher’s salon.


  In his book Dialogues Concerning Two New Sciences (1638) he proposed a method. To start with, two people, each with a lantern with a shutter face each other. Initially the shutters are closed, to cut off the lanterns’ light. Then the first person opens their shutter. When the second person sees the light, they unshutter the second lantern. The first person will note the delay time between when they opened their shutter and when they saw the other lantern’s light. Galileo reasoned that at short distances most of the delay was due to the reaction time of the two people. The measurement was then repeated over a much longer distance with any additional time being due to the time it takes light to travel between them. Galileo reported, in the voice of Salviati:

  In fact I have tried the experiment only at a short distance, less than a mile, from which I have not been able to ascertain with certainty whether the appearance of the opposite light was instantaneous or not; but if not instantaneous it is extraordinarily rapid – I should call it momentary; and for the present I should compare it to motion which we see in the lightning flash between clouds eight or ten miles distant from us.

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  The next attempt to measure the speed of light takes us back to watching the moon of Jupiter. Within a year of serving those moons, Galileo realized that there was a potential to use them as a great universal clock hung in the sky for everyone to see. Or at least it offered a way of synchronizing clocks. For instance, if the observatory at Paris or Greenwich announced that Io, one of those moons, would slip behind Jupiter at exactly ten minutes after midnight, the whole world could watch the event, the occultation, and synchronize their clocks. This would have beneficial consequence for navigation and commerce.

  One of the great technical challenges of the seventeenth century was how to measure longitude, especially at sea. A mariner could measure the elevation of the Sun above the horizon and then by knowing the tilt of the Earth on that day, they could calculate their latitude. But longitude was much more difficult. A solution that people understood was based on knowing local and standard time. If you know the time at a standard location, such as the observatory at Greenwich, and you can determine your local time, you can then calculate your longitude. For example if one day you know it is local noon, since the Sun is due south (with corrections based on the equation of time or the analemma) and at the same moment it is 9 AM in Greenwich, you know that you are 3 h before Greenwich. You are an eighth of the way around the world to the east. You are at 45° east, which is the longitudinal line that runs near Volgograd in Russia (48°42′ N, 44°31′ E) or Baghdad (33°20′ N, 44°25′ E). The problem with this technique was that no seventeenth century clock was reliable enough to tell you the time in Greenwich, especially as ships would have to carry these clocks and would be bobbing on the ocean for months or even years. If you are at the equator, a clock that is off by a single minute means your longitude calculation is off by 27 km (17 miles). If you are off by an hour, you are in the wrong time zone.

  Galileo, and many other astronomers, realized that if someone could hand you a chart or table that listed the moment that Io was eclipsed by Jupiter as seen in a standard location, you could synchronize your clock. You could be sailing in the Indian Ocean, or the South Pacific, and you would see that event exactly at the same time as it was seen in Padua or Greenwich or Paris. And then you could determine your longitude.

  Galileo worked for years trying to devise such a table, but never succeeded; the orbits did not seem to be regular enough. He also was not the only one to recognize the utility of such a table. In fact the Paris Observatory and the observatory at Greenwich were started in 1667 and 1674 in part to create such a table. As these countries sought to build far-flung empires and rule the waves, it was a concern of national importance and pride and so deserved public financial support.

  The first director of the Paris Observatory, which is still located at the southern tip of the Jardin du Luxembourg (Luxembourg Gardens), was Giovanni Domenico Cassini (1625–1712). Cassini was a great Italian astronomer and it must have been quite the coup for the organizers of the observatory to get him to come north. In fact he made so many discoveries, especially concerning Saturn, that a recent space probe sent to that planet was named after him.

  Cassini had a young Danish assistant, Olaf Rømer (1644–1710) who was set to work on the moons of Jupiter problem. Most of the work concentrated on Io, whose period would appear to get shorter for half the year, and then longer. Rømer realized that it got shorter as the Earth approached Jupiter and longer as we backed away, and so in 1675 he suggested an audacious solution. He proposed that light from Io and Jupiter took longer to reach the Earth the farther away they were. We do not see the eclipse at the moment Io passes behind Jupiter. Instead we see it a few minutes later when light from the event has crossed the millions of black leagues between us. This meant that light took time to travel.

  Rømer, upon examining all the data that he and Cassini had collected, calculated that it took about 22 min for light to travel across the Earth’s orbit. Actually this time was later corrected to a value closer to the 16½ min we measure today. This was combined with a measurement of the Earth–Sun distance and a speed of 2.75 × 108 m/s was determined, only about 8% slower than modern measurements. The problem was not with the reasoning or the time, but with the fact that the diameter of the Earth’s orbit was not known very well.

  The distance from the Sun to the Earth is called an astronomical unit (AU), and it becomes the basis upon which all astronomical measurements are made, because it is our baseline for surveying the stars and planets. But it is not an easy measurement and things had not changed much since the time of the Greeks until the middle of the seventeenth century.

  In 1672 Cassini sent the Jesuit astronomer Jean Richer to Cayenne in South America. Cassini and Richer both measured the position of Mars at the same time. They had to do it at the same moment because the Earth is in motion, both spinning and orbiting. After a long voyage and a delicate measurement, they derived a value of 138 million km (compared to a modern measurement of 150 million km), which is what Rømer used.

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  Table 12.1 Radius of the orbits of the planets: Copernicus vs modern measurements.

  The importance of the astronomical unit really came to light a hundred years before Galileo. When Nicolaus Copernicus (1473–1543) was putting together his model of the solar system he could not see details of the planets: he preceded the telescope by a century. But he could measure angles well. He could plot the paths of the planets in the sky and fit them to his ideas about circular orbits. He even published estimates of the radius of the planets’ orbits in terms of the AU, which agreed remarkably well with modern numbers (see Table 12.1). The fact that Copernicus got the Earth’s orbit precisely right is not surprising. An AU was and is defined as the distance between the Sun and the Earth. But Copernicus could go no farther. He knew the size of the other planet’s orbits in terms of the AU, but he did not know how far an AU was in terms of distances used on Earth, like the mile or the meter.

  The AU shows up in Kepler’s third law of planetary motion. In that law he states that the orbital time of a planet (T = period) squared, is equal to the radius (more precisely, the semi-major axis) of the orbit cubed

  T2 =r 3.

  That means that if you could measure a planets orbit time in years, you could calculate its orbital radius in AU. Kepler not only could check this against the planets but since this law emerged after 1610, he could also check it against the satellites of Jupiter (see Figure 12.1). If only he could stretch a steel tape measure from here to the Sun, he could map the whole solar system.

  So Copernicus and Kepler reported the size of the orbits of the planets in terms of the AU, but the AU—the baseline of astronomical measurement—was elusive. Cassini and Richer measured by observing Mars from Paris and Cayenne, using the width of the Atlantic as a baseline, but their estimate had a lot of uncertainty. Cassini and Rømer made another measurement. This time they both s
tayed in Paris, but they let the Earth’s rotation move them to a second location. It was a simple measurement to perform, but a difficult one to analyze. But the most important measurement of the AU was the transit of Venus, made a century later in 1769. It was an event with which the names Captain Cook and Point Venus in Tahiti are forever linked.

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  Figure 12.1 Kepler’s third law demonstrated by Jupiter’s moons and the planets. The top row shows orbit radius vs period for Jupiter’s moons, the inner planets and the outer planets. The second row shows the square of the period versus the cube of the radius for these three cases. Since the moons and planets lay on a straight line this means T2 = r3.

  The transit of Venus measurement was conceived of by Edmond Halley. Like Halley’s comet, the transit of Venus is a very rarer event. A transit is when Venus (or Mars) passes in front of the Sun, appearing from Earth as a black spot in front of the glowing orb. It is rare because the orbits of the planets are not in the same plane. In fact it only happens if the Earth and Venus both cross the line where their planes intersect, an alignment that happens only about once every 130 years, and then again eight years later. After Halley conceived of the measurement the next transits would be in 1761 and 1769, then again in 1874 and 1882, and recently in 2004 and 2012.

  The reason there was such optimism for this technique is that it depended upon the measurement of time instead of angles. The amount of time it takes for Venus to transit the Sun depends upon the distance to the Sun and your position on Earth. For instance, two observers 10,000 km apart on Earth will see Venus crossing slightly different parts of the Sun. The transit time they measure will typically differ by a half hour or so (see Figure 12.2). A remote observer with a moderately accurate watch can measure the transit to within a second, which should lead to a distance measurement error of less than 1%. Halley had measured the transit of Mars in 1677 and felt that with a coordinated effort the AU could be determined to 0.2%.