How Big is Big and How Small is Small

by Smith, Timothy Paul

  Figure 12.2 The transit of Venus as seen by two separate observers. Two observers, 10,000 km apart on the Earth, will see Venus crossing a different part of the Sun. The black spots show Venus every hour during the transit. One observer sees the transit in just over 5 h, the other in just under 6 h. In this case the difference was a half hour, and with an uncertainty of 1 s that is one part in 1800.

  Halley did not live to see either of his predictions confirmed, but his comet returned in 1758–9 and in 1761 and 1769 the scientific community of Europe mounted a worldwide effort to observe the Transit of Venus. The story of that measurement is a heroic adventure, because you need people to trek across the Earth to widely spaced positions to obtain the greatest possible baseline. Expeditions were mounted by the French Academy of Science, the British Royal Society, as well as numerous individuals across the globe. Captain James Cook with the astronomer Charles Green went to Tahiti, where the sailors nearly traded away the nails that held together their ship, the HM Bark Endeavour. Father Maximilian Hell travelled above the Arctic circle to Vardø in Norway. William Wales and Joseph Dymond spent the preceding winter at Churchill on the Hudson Bay in Canada, waiting for the great event, watching the thermometer drop to −43°F and seeing brandy freeze.

  One of the most disappointing expeditions was that of Guillaume Le Gentil, who traveled to Pondicherry in India. His travels were interrupted by war between England and France and so he was afloat for the 1761 transit. He decided to stay there until 1769 and was ashore and prepared for the second encounter, but that morning the Sun was obscured by clouds. When he returned to France he had been gone so long that he had been declared legally dead, his wife had remarried, he had lost his seat at the French Academy, and his estate had been plundered by relatives.

  Abbé Chappe d’Auteroche travelled across Russia on the “highway” of the frozen Volga River to Tobolsk in Siberia for the 1761 transit. Then, eight years later he traveled to San José del Cabo, on the Baja California peninsula of Mexico. Here he made a detailed measurement, but unfortunately everyone except one person in his expedition died of “epidemical distemper.” However, his notes and measurements still made it back to Paris.

  These were the most extraordinary efforts and took place in the midst of England and France struggling for world domination. The Seven Year War (1756–1763), or the French and Indian War, as it is known in North America, was in full fury. Yet there was a unique camaraderie among scientists internationally. The Royal Society secured letters of passage for French astronomers in the war zones and the French Academy of Science reciprocated. Data from both empires and dozens of other places was collected and shared. In the end the final and best analysis was done in Germany. Johann Encke of the Seeberg Observatory in Gotha published his analysis in 1824. Among his results were a number of new longitudes, because you need to know where you are when making these measurements. He reported a solar parallax of 8.58 arc-seconds and a distance of 153, 340,000 ± 660, 000 km. That reported uncertainty was just under half a percent. In reality his estimates were a bit more than 2% off.

  The solar parallax is the angle that the Sun would appear to shift in the sky if you instantly moved from the Earth’s equator to a pole. An arc second is 1/60th of an arc-minute, and an arc-minute is 1/60th of a degree. So 8.58 arc-seconds is exceedingly small. It is the angle between two pins in a cork that are 4 mm apart and viewed from 100 m away.

  Scientists did not achieve the 0.2% that Halley had hoped for primarily due to a phenomena called the black drop effect. As Venus approached the edge of the Sun, the edge seemed to bleed into the Sun, making the moment of contact and the start of the transit hard to identify. This meant that the exact second of contact could generally not be determined.

  Of course, the expeditions of 1761 and 1769 were not the last word on the astronomical unit. In 1874 and 1882 another worldwide effort was mounted. This time the United States was a major participant. There was also a widespread use of photography. This meant that photographs could be taken as Venus approached the Sun’s edge, and even if the moment of contact was still blurred by the block drop effect, astronomers could deduce that moment by extrapolating between photographic images. With the aid of trains and steam ships the expeditions took less time and the data returned faster than in the previous century. Their final result was a figure of 149,840,000 km. This is about a 0.2% difference from modern measurements, a result that I think would have satisfied Halley.

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  There was a recent pair of transits in 2004 and 2012, but they were more curiosities than important measurements. Our best measurements of the AU now come from using radar. We bounce radio waves off of nearby planets, Mars, Venus and Mercury, and time how long it takes until the echo is received. Ironically, instead of measuring the distance to the Sun and using Kepler’s laws to get the distance to the planets, we do it the other way around. Radar off of a rocky planet is a cleaner signal than off of a gaseous planet, so we use Mercury, Venus and Mars. Finally, our best number for the AU is 149, 597, 870.691 ± 0.030 km. That means that we actually know the AU to ten digits of accuracy, and to within 30 m.

  In some sense the way we measure the distance to the planets and the Sun now seems too easy. We do not mount expeditions to far-off continents, risking life and limb and taking years. All we do now is bounce radio waves off of Venus or Mercury and time how long it takes for the signals to return. But we stand on the shoulders of giants. The technique only works because we know the speed of light so very well, and that number has also taken centuries to establish.

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  Once we have measured the AU and the size of the Sun, the dimensions of planets and various satellites are much easier to determine. To measure the size of the orbit of Mars or Uranus, we need only measure its orbit time and apply Kepler’s third law. If we know how far away Neptune is, and measure the angle it subtends in our telescope, we know its size. So now we can march out to the edge of the solar system, naming the sizes of things as we go. But if we did that we would miss a lot of interesting stuff. Instead I want to view the sizes of the planet’s orbits as a frame, a canvas upon which we can now go back in sketch in details. I want to look at the sizes and shapes of planets, moons and asteroids.

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  One of the most striking things about the solar system is that the Sun, the planets and many of the moons are spherical. In space it is so common that it is hard to think that it could be otherwise. All the large objects, the things that catch your eyes, are spherical. But here on the surface of Earth it is a rare natural object that is spherical. Sometime fluids, like water droplets, may be spheres but not rigid bodies.

  Figure 12.3 Some of the moons in our solar system. The images are arranged by size, from the spherical Enceladus (radius 252 km) to the irregular Hyperion (radius 133 km). Nereid orbits Neptune and is difficult to photograph. Courtesy of NASA.

  Some of the natural satellites, or moons, of the planets are spherical and some are not (see Figure 12.3). At present there are over 180 natural satellites catalogued in our solar system, and that number keeps growing. Of all the moons, 19 are spherical. For example, the two moons of Mars, Phobos and Demos (named after the sons of Mars from mythology), are irregular lumps of rock, reminding one of potatoes with dimples and pock marks. Moons such as Prometheus and Calypso of Saturn are even elongated. And then there are the asteroids, most of which are very far from spherical. So why is it that some are globes and others are potatoes? Why spherical or irregular?

  The trend here is that the larger the moon, the more likely it is to be spherical. All the moons larger than, and including, Miranda are spherical. Miranda is a moon of Uranus and has a radius of 236 km. All the moons smaller than Nereid are irregular. Nereid orbits Neptune and has a radius of 170 km. Between these two are Proteus, which is irregular and has a radius of 210 km, and Mimas, which is spherical with a radius of 200 km. There really is a threshold somewhere about 200 km, above which moons form into spheres. To understand th
is, we will first look at the two extremes.

  A star, which is clearly spherical, is held together by gravity. A boulder or cobble need not be spherical, and is held together by chemical bonds. We have now seen that chemical bonds are a result of the electromagnetic force. So at first we might reason that an object shaped by gravity or the electromagnetic force would have the same shape since both forces have the same form. They extend over all space and get weaker as 1/r2. Even though the electromagnetic force is so much stronger, the fact that it comes with both a positive and negative charge means that Calypso can have an odd shape.

  We have already discussed chemical bonds in Chapter 10 and also the fact that electrically neutral atoms can still feel electromagnetic forces when they are close to each other because one part of the atom may be more positive and one part more negative. That is the source of chemical bonds, and that means that by their very nature chemical bonds are short ranged and essentially work only between neighbors.

  A rough rock that I might pry out of a cliff face is jagged and stays jagged. This is because of the powerful chemical bonds between the atoms. Each atom is tightly bound to its neighbor. But all the atoms are ignorant of any distant atom, where “distant” can mean a few angstroms away.

  Gravity, by contrast, is the great shaper of the cosmos and it is a lot different than the electromagnetic force in two major ways. Firstly, it is a lot weaker. Secondly, the charge is mass and only comes in one type. Mass attracts mass and there is no “anti-mass” that can repel or neutralize that attraction. Therefore an atom on one side of the moon gravitationally interacts with—and attracts—atoms on the other side of the moon. The bigger the object, the more mass, the greater the forces that pull that body into a sphere. At some size one crosses a threshold. The global gravitational forces overwhelm the local chemical forces and reshape that body into a sphere.

  There is in fact, no simple threshold. We cannot derive a line between Miranda and Proteus, the spherical and the irregular, by simply balancing chemical and gravitational forces. If our Moon was split in half, each piece would be big enough to eventually form new spherical moons. But it would not reform into spheres in an instant, and it would never be a perfect sphere. Even on our home planet, geology is raising up mountains, whereas erosion and landslides (gravity), are pulling them down. Still, given enough time and enough mass, a large irregular moon or asteroid will become more spherical.

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  One other place where these two forces compete is something called the Roche limit or Roche radius. Named after the French astronomer Édouard Roche (1820–1883), it was inspired by the observation that the rings of Saturn are inside the orbits of the spherical moons (see Figure 12.4) Since Roche’s days we have observed that his limit also applies to all plants with rings: Jupiter, Saturn, Uranus and Neptune. After his observation, Roche calculated the forces within a moon. He balanced the tidal forces, which distort, against the gravitational forces, which shape a moon into a sphere. He found that if a moon was to venture too close to its primary (the planet it is orbiting), the tidal forces from the primary will exceed the moon’s self-cohesion and the moon would break-up. Roche proposed that this is how the rings of Saturn were formed.

  Within the Roche limit you can have satellites, but they are rigid; that is, they are held together by their chemical bonds and not gravity. For example, the Roche limit for the Earth is between 9,000 and 18,000 km, depending upon the rigidity of the satellite. Our moon is well beyond that limit, orbiting at about 400,000 km; it is safe from tidal break-up. On the other hand, man-made satellites, such as the International Space Station and the Hubble telescope, which are between 400 and 600 km up or a bit less than 7000 km from the center of the Earth, are well within the Roche limit. But they are held together primarily by chemical bonds and not gravity.

  Figure 12.4 The rings of Saturn. A moon within the Roche limit would break up due to tidal forces. Courtesy of NASA.

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  Whereas much of the structure of the solar system is determined by the balance between chemical bonds and gravity, other aspects are governed by history and the initial conditions of the region of space where our solar system evolved. Thus there are parts of our description that are unique to the solar system and parts that we can expect to find in any planetary system, even around stars in distant galaxies.

  The size of the bodies in our system are a result of how much gas and dust was in the region when the Sun and planets were forming, about four and a half billion years ago. This region of space was then filled with material that must have included the ashes of a supernova, since it contained elements heavier than iron. Gold in a wedding band is proof that the Earth contains remnants of older stars. The gasses and dust in the embryonic nebula attracted each other due to gravity, and they started to coalesce. Once the core of this cloud was dense enough the Sun formed and started to burn. This young star radiated a powerful solar wind that pushed a lot of the light elements out and left the heavier elements near to the Sun, in the inner solar system. So the planets that formed close to the Sun contain rocks and metals. These are the terrestrial planets: Mercury, Venus, Earth and Mars. Farther out, the gasses or Jovian planets formed: Jupiter, Saturn, Uranus and Neptune.

  Planets range in size from Mercury, with a radius of 2440 km, to Jupiter, with a radius of 69,920 km (see Table 12.2). This observation leads us to the question of whether planets could be of a different size? At the lower end, a planet cannot be much smaller than Miranda if it is going to be spherical. At the other end we think a planet cannot be more than 40% bigger than Jupiter. We will see why in the next chapter. In the last few years we have started to observe exoplanets, planets that orbit other stars. We wait to see if our theories, based upon what we see with our eight local planets, holds true when we measure hundreds of new planets.

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  So now we can start our tour of the solar system, starting with the inner planets and working our way out to the cold, dark regions of space.

  Table 12.2 Some properties of the planets.

  One curious trend that was noted over 200 years ago is that not only does the spacing between planets increase as you move out from the Sun, but it seems to do so in a regular pattern. The trend was quantified as the Titus-Bode law:

  where n doubles beyond 1. The law is also illustrated in Figure 12.5. It is just an observational trend, which fits the data pretty well, except for the exceptions. Where is the fifth Titus-Bode planet? When Uranus was discovered it fell where number eight had been forecast. However, Neptune is not near any predictions (although Pluto was also close). The equation was formulated to fit the data and has no deep theoretical basis. However, it may be more than just complete coincidence.

  Figure 12.5 The Titus–Bode law. Top: Inner planets. Bottom: Outer planets. The circles are the law’s prediction. The labeled triangles are the position of the actual planets.

  There is something called orbital resonance. Two planets that are orbiting the Sun will push and pull on each other every time they pass each other. If two orbits are in resonance, the smaller planet can be pushed out. A way to see this is to imagine pushing a child on a swing. If the period of the push and the period of the swing are in sync, in resonance the swing will go higher and higher. The gaps in the rings of Saturn are a good example; the Cassini division or gap is caused by a resonance with the moon Mimas. A rock in orbit in one of these gaps gets pushed in the same direction every time Mimas goes by, and soon the gap is cleared. Rocks not in that gap get pushed randomly and so are mixed but not cleared out.

  But what of the gap between Mars and Jupiter? This is the region where the asteroid belt is found. We now think there was enough material in this area to form a planet, but because of the gravitational disturbance caused by Jupiter that planet never completely formed. Jupiter would swing by and stir things up before that planet could really congeal. The asteroid belt also gives us another place to look for orbital resonances and it is full of them. The Kirkwood
gaps are orbits that would be in resonance with Jupiter. There are no asteroids that have an orbital period of that of Jupiter’s. There are also none at Jupiter’s period either. So the shape of the asteroid belt is determined by Jupiter.

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  Beyond the asteroid belt are the great Jovian planets, Jupiter, Saturn, Uranus and Neptune. Neptune is the one that really broke the Titus-Bode law. At 30 AU, it was too close and just did not fit. These planets have swept up most of the stray hydrogen and helium in the solar system. In fact, there is so much mass in Jupiter that it is close to becoming a star.

  But the Jovian planets are not the end of the solar system. We all know that out there in the cold dark hinterlands of the system lies Pluto, formally the ninth planet.

  I find it curious how many people were disturbed when Pluto was “demoted” from a planet to a dwarf planet. It seemed as if this mysterious, distant, ice ball was being “relegated,” especially when it had not done anything wrong. It just continued to be what it always was. True, it had a tilted orbit that was also so eccentric that it was sometimes closer to us than Neptune. But that was not really the problem. The problem was that Pluto was not really unique. In 2005 Eris was discovered. It is bigger than Pluto but we just had not seen it before because it is much farther away. It became apparent at that time that we were soon going to have a long list of new planets if we accepted everything. In fact Eris was named after the goddess of discord, because it was recognized that its discovery would cause a disruption in the astronomical community.