Copyright © Karl Dahlke, 2022
When there were no lights to distract us, people would gaze at the night sky in wonder. A blanket of stars passed overhead, unchanging and eternal. It looked like a sphere with points of light, slowly spinning around the earth. The sun also goes around the earth, but just a hair slower than the stars. Each night you see a slightly different portion of the celestial sphere. In exactly one year, the same stars are back in the night sky, and the cycle repeats. As the years rolled by, people were familiar with all the stars in the sky, the ones they could see at night, and the ones that were hidden by the glare of the noonday sun, that would become visible in 6 months. Groups of stars looked like people, or animals, and were called constellations. There are many, but 12 were chosen to represent the 12 months of the year, as they are evenly spaced around the sky. These are the signs of the zodiac: Aries the ram, Taurus the bull, gemini the twins, cancer the crab, Leo the lion, Virgo the virgin, Libra the scales, Scorpio the scorpion, Sagittarius the archer, Capricorn the goat, Aquarius the water bearer, and Pisces the fish. The sun would pass through each of these 12 constellations in turn, for each of the 12 months.
Against the unchanging backdrop of the stars, 7 objects moved in the sky. These objects were called planets, an Arabic word for wanderer. Planet has a different meaning today, which is still being refined, thus Pluto's demotion to dwarf planet, but 2,000 years ago, the sun and the moon were called planets as well. Thus we have the sun, the moon, Mars, Venus, Mercury, Jupiter, and Saturn. Everything else was fixed in space.
Why did these objects move in the sky? Every culture assigned supernatural forces, i.e. gods, that moved the planets. In Greek mythology, Helios pulled the sun across the sky in a chariot. This gives us words like heliocentric (going around the sun), perihelian (closest to the sun), and even helium, which we'll get to later. The Romans took most of the Greek gods and gave them new, Latin names, thus Helios became Apollo, god of the sun. In the same way, the Greek god Zeus, the god of Jupiter, became, well, Jupiter, for the Romans, and Thor in Norse mythology, etc.
There were dozens of gods, but the planetary gods had a special significance, moving the planets stately across the sky. For this reason, many cultures established the 7 day week, a day to honor each god, a day to honor each planet. Three days are obvious to an English speaker: Sunday, day of the sun, Monday, day of the moon, and Saturday, day of Saturn. If you speak French, a couple more days might jump out at you: Mardi, day of Mars, Mercredi, day of Mercury, and Jeudi, day of Jupiter. French is a romance language, derived from Latin, the language of the Roman empire, thus the planetary correspondence is clear. English is a Germanic language, and our middle days come from Norse gods, rather than Roman gods, but they are gods of the planets nonetheless. Tuesday, Tue, god of Mars, Wednesday, Woden, god of Mercury, and Thursday, Thor, god of Jupiter.
Along with the predictable planets, certain transients appeared in the sky, and often struck terror in the hearts of men. When a comet whizzed past the earth, was it an omen? We call the fiery trail a tail, but the Greeks thought it looked like hair streaking behind a heavenly creature as she raced by. The Greek word kometes means long-haired.
The stars are the only thing you can truly count on, and yet, sometimes a new star would appear in the sky. This was called a nova, the Latin word for new. Was it an omen? Nobody knew. It didn't move, like the planets, or even a comet. It held a fixed position relative to the other stars, but it wasn't there yesterday. In a few weeks it would go away, but people were always on the lookout for another one, because everyone knew the stars, and everyone watched the stars.
On the fourth of July, in the year 1054, a nova appeared in the night sky, but not like any that had been seen before. It was 4 times brighter than Venus, and could even be seen during the day. This was truly a supernova. It gradually faded over time, but could be seen in the night sky for almost 2 years. A thousand years later, our telescopes can see the Crab Nebula, the ashes of this particular supernova.
The Greeks believed in their gods, but they also believe the universe was mechanical in nature, and could be explained by material processes. This was an odd mix of science and religion that persisted for 2000 years, all the way to Isaac Newton, and even Charles Darwin, the founder of evolution. The gods put planets in the sky, but the motion of the planets could be explained by wheels within wheels within wheels. In other words, the gods set things up, but they didn't have to push the planets along, inch by inch, every day. Each planet was on its own track within the celestial sphere. Understanding those tracks was the purview of science, and it represented the birth of astronomy.
The sun was easy. It trekked along at a constant speed, just a hair slower than the stars. It was on a circular track just inside the sphere of stars, and it moved a little slower, and everything is explained. The moon was also pretty steady on its track, going around the earth once a month instead of once a day. The planets were quite another matter however, especially Mercury and Venus, which never strayed far from the sun. Sometimes they would pull ahead, sometimes they would lag behind. Venus is never more than 33 degrees from the sun, and Mercury is never more than 18 degrees from the sun. With this back and forth motion, they could not sit on a circular track running at a fixed speed.
A good scientist begins by making careful observations, and the Greeks did just that, recording the locations of the planets relative to the stars, day by day, month by month, year by year. With the planetary paths established, the next task is to search for an explanation. Unfortunately the Greeks made two crucial mistakes. They assumed everything went around the earth, and, they were certain the answer was based on circles, since the circle was the divine shape of the gods. Since Venus always stays close to the sun, put Venus on a circular track that travels at exactly the same speed as the sun, then put a wheel on this track, so that Venus spins around in its wheel as the wheel moves across the sky on its track. This would produce the back and forth motion of Venus as seen from Earth. It pulls ahead of the sun, then lags behind the sun, then pulls ahead, then lags behind, in a repeating pattern. It made sense, but since the wheels were perfect circles moving at constant speeds, the predicted path didn't match the observed path at all times. Sometimes Venus moved too quickly toward or away from the sun; sometimes too slowly. So they put another wheel inside the first wheel, which traveled along the track. Give the wheels just the right diameter, and just the right speed, and the path is closer, but still not right. It would never be right, no matter how many wheels they used, because the premise was wrong.
Ptolemy, a second century Greco Roman astronomer and mathematician, spent years working on this problem, perhaps decades, using careful mathematics and trigonometry. Even though he couldn't explain the paths mathematically, he documented them with great precision, and since they repeat like clockwork, his charts and tables were used for centuries to come, and became the gold standard of astronomy. We couldn't take the next step without Ptolemy.
In 1532, Nicolaus Copernicus, a Polish astronomer, demonstrated, rather conclusively, that the earth goes around the sun, rather than the sun revolving around the earth. The apparent motion of the sun in the sky is caused by earth's rotation, i.e. the earth spins as it travels around the sun. He was reluctant to publish his work however, cognizant of the inevitable backlash from religious organizations and even the general public. This was a much bigger paradigm shift than the round earth. The shape of the earth is not critical, as long as we are still the center of the universe. God made the heavens and the earth, and just 5 days later, he made man in his own image. Copernicus wanted to move the center of the universe away from the earth and over to the sun, and the hubris of mankind would not swallow that bitter pill easily. In 100 years this would come to a head, as Galileo battled the Catholic church.
Heliocentricism was not an entirely new idea. Some of the early Greeks, including Pythagoras, proposed a sun-centered system, but the idea was considered ludicrous, even by other Greek philosophers. Copernicus thought it made sense, and his reasons were compelling. His planetary paths were more accurate, as he showed using methodical calculations - and the model made intuitive sense. For example, you don't need wheels within wheels to explain the motion of Venus; it goes around the sun, just like the earth, but it is closer to the sun, thus it always appears near the sun in the sky. It doesn't just happen to travel at the same speed as the sun while wabbling back and forth, it circles the sun in a tight orbit. That is a simpler explanation, and the simpler explanation is usually right. Other planets are farther away from the sun, outside the orbit of earth, and can be seen anywhere in the sky - close to the sun (on the left), or high overhead in the night sky (on the right). Planets are not drawn to scale.
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Copernicus knew which planets were close to the sun and which planets were far away. Earth was number 3 on the list: Mercury Venus Earth Mars Jupiter Saturn.
In 1605, Johannes Kepler took the next step. Orbits were not defined by circles, or even circles within circles, but by ellipses. The sun is not at the center of the ellipse, but rather one focus. An ellipse can be very close to a circle, or it can be long and narrow. Planetary orbits are nearly circular, while cometary orbits are long and narrow. Still, each planet travels along its own ellipse, and the math would never be right until this fact was recognized. Here is an elliptical orbit around the sun, more eccentric than the planets for illustration. Note that the sun is at one focus, not at the center.
The green planet, or whatever it is, is at the top of its orbit, farthest from the sun. This is called aphelion. When it swings around to the other end of the ellipse, it will be closest to the sun, called perihelion. The planet moves slowly at aphelion, and quickly at perihelion. In fact, Kepler derived a formula that describes the speed of a planet at each point in its orbit. this was a game changer - Kepler's ellipse described the path perfectly, including the speed of the planet at each point in its orbit, down to the arc-minute. It could not be denied!
It took the genius of Isaac Newton, 70 years later, to explain the ellipse in terms of universal gravity. He had to invent differential and vector calculus in rectangular and polar coordinates to make these calculations, but that hardly slowed him down. Entire books have been written about Isaac Newton, perhaps the smartest man who ever lived. I'm going to skate past all that, and apologize to Newton along the way, and make his theory sound simpler than it really is. Before Newton, everyone thought the earth was gravity and everything was pulled towards the earth. Newton said no - everything has gravity in proportion to its mass. You and I, standing next to each other, experience a slight gravitational attraction, but it is too small to measure, because we don't weigh enough. The sun is millions of times heavier than the earth, so it has much more gravity; in fact it dominates everything else in our solar system. It pulls the planets, including the earth, around in their orbits.
Why do planets move faster as they get closer to the sun? This is also intuitive. Remember the girl on the swing in the park? She traded kinetic energy for potential energy in a repeating cycle, so that the total amount of energy was conserved. Each planet is doing the same thing. When it is high in its orbit, it isn't moving very fast, because its upward motion has been converted into potential energy. After aphelion, the planet falls toward the sun, so to speak, just as the girl falls toward earth in her swing. The planet picks up speed, until it is traveling quickly as it zips around perihelion, like some kind of crack-the-whip ride. Then the cycle repeats, but unlike the girl on the swing, there is no air to slow things down. An orbit can continue for a billion years, for 10 billion years, for a trillion years. Jupiter will remain in its orbit around the sun, long after the sun has burned out and become a dark cinder, a compact remnant of its former self.
In high school, a friend told me that if gravity were an inverse seventh force, rather than an inverse square force, i.e. a stronger force, planets would orbit in a figure 8. It's such a cool thought, it rattled around in my head for 50 years. But think about it for five minutes and it's wrong. 😞
Draw a big figure 8 on a piece of paper, this is the orbit of your planet. Put the sun in the center of the bottom circle. Look at the very bottom of the orbit, the bottom of the figure 8. The planet is moving right to left. It has some lateral velocity, from somewhere, from the day it was formed, or the day it passed by the sun and was captured. So it is moving clockwise around the sun. The sun pulls on the planet, but never imparts a lateral force. It can only pull, like gravity. The planet swings around the left half of the bottom circle, crosses the center of the figure 8, and starts to swing up the right half of the top circle. At the right edge, something magical happens. Draw a line from the sun tangent to the right edge of the top circle. When the planet reaches this point it stops moving clockwise and starts moving counterclockwise. What imparts that lateral force? It can't be the sun. Therefore, such an orbit is impossible. Any radial force, similar to gravity, produces an orbit that looks somewhat like an ellipse.
We all know the sight of a full moon - it's bright, big, and beautiful! The entire disk is visible, thus it is a "full" moon. A half moon has half its disk illuminated, and is only half as bright. A crescent moon presents a thin crescent of its disk, and isn't even bright enough to cast a shadow. Every 29.5 days, the moon passes through all its phases and starts over again. In other words, there are 29.5 days from full moon to full moon. You've seen all these phases, but have you noticed the geometry? When the moon is full, it is opposite the sun. It rises in the east when the sun sets in the west, and it sets in the west when the sun rises in the east the next morning. A half moon hangs directly overhead when the sun rises or sets. A crescent moon is always close to the sun, and can be seen just before sunrise or just after sunset. Let's see why this is so.
Go into a dark room with a flashlight and a basketball, or some other round object. Stand a good distance from the light, and make sure that is the only source of light in the room. Hold the ball between you and the light and you are looking at the dark side of the ball; there is nothing to see. this is a new moon. Slowly turn, holding the ball at arm's length in front of you. One edge of the ball is lit, a crescent if you will. Gradually turn further until the light is at your side. You are the corner of a right triangle, with the sun and the moon forming the legs. From your point of view, half the ball is lit and half is in darkness. This is a half moon. Continue turning, until you are between the light and the ball. At this point your shadow is in the way, and that's not very realistic. Sometimes the earth's shadow falls across the face of the moon, but not very often, so duck down - duck down so the light shines directly on the ball. The entire surface is lit, corresponding to a full moon.
Notice that the geometry is correct. The moon is full when it is opposite the sun, and is half when it is 90 degrees from the sun, and so on.
If you wish, take a break and watch the moon each night for a month. Document its phase and its geometry relative to the sun. Verify that the basketball experiment is accurate; it confirms what you see in the sky. The moon goes around the earth, and the sun shines on the earth moon system from far away.
The outer planets do not have phases, because they can never get between us and the sun. Whenever you look at Jupiter, the entire disk is visible, or nearly so. (Well, it would be if you look through a telescope; Jupiter is just a dot to the unaided eye.) Venus is an inner planet however, and it has phases.
Go back into your dark room, with an omnidirectional light source that represents the sun. Hold the basketball between you and the sun and move it back and forth. Other than an occasional crescent on the left or the right, the ball is dark.
Now try something else. Move the ball slowly around the light source while you stand in place. When the ball is to the left of the light, half the surface is lit. This is a half Venus, if you will. Move the ball further around the light, so that it is almost behind the light source. Almost all of the disk is lit. This is nearly a full Venus. Of course, when Venus is actually full, it is hidden behind the glare of the sun and the bright blue sky, so you will never see its full disk. However, just before opposition, or just after opposition, most of the disk is visible, or would be through a telescope.
Three objects are bright enough to cast shadows: the sun, the moon, and Venus. You'll probably never see a Venusian shadow, a street light 3 blocks away will overwhelm it. Even if you go out to the country, conditions must be just right. Venus is at ¼, and it's just before sunrise or after sunset, and the moon is nowhere to be seen, and there's not a cloud in the sky. Venus shines like a beacon on the horizon, bright enough to cast a faint shadow.
As the phase of Venus moves from ¼ to ¾, so that more of its disk is lit, it also moves farther away from us, circling around to the far side of the sun. When is it brightest? This is an interesting calculus problem, maximizing an expression in θ, But this isn't a math book, so I'll jump ahead to the answer. Proximity to us is more important than the phase, so Venus is brightest when it is about ¼, or 27% illuminated. It is only 15 degrees away from the sun in the sky, so you want to catch it just before the break of dawn, or just after the red and orange sunset fades.
In 1609, Galileo invented the first practical telescope. This led to 3 important discoveries in rapid succession.
Every town had a craftsman who could cast glass into lenses, and within a couple decades, ordinary people could see what Galileo saw. This was threatening to the Catholic church, and they forced Galileo to recant under pain of death. Galileo made the following statement in 1633, then settled for house arrest for the next 8 years, until his death in 1642. That's better than being burned alive.
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I, Galileo, son of the late Vincenzo Galilei, Florentine, aged seventy years, arraigned personally before this tribunal, and kneeling before you, Most Eminent and Reverend Lord Cardinals, Inquisitors-General against heretical depravity throughout the entire Christian commonwealth, having before my eyes and touching with my hands, the Holy Gospels, swear that I have always believed, do believe, and by God's help will in the future believe, all that is held, preached, and taught by the Holy Catholic and Apostolic Church. But whereas -- after an injunction had been judicially intimated to me by this Holy Office, to the effect that I must altogether abandon the false opinion that the sun is the center of the world and immovable, and that the earth is not the center of the world, and moves, and that I must not hold, defend, or teach in any way whatsoever, verbally or in writing, the said false doctrine, and after it had been notified to me that the said doctrine was contrary to Holy Scripture -- I wrote and printed a book in which I discuss this new doctrine already condemned, and adduce arguments of great cogency in its favor, without presenting any solution of these, and for this reason I have been pronounced by the Holy Office to be vehemently suspected of heresy, that is to say, of having held and believed that the Sun is the center of the world and immovable, and that the earth is not the center and moves: Therefore, desiring to remove from the minds of your Eminences, and of all faithful Christians, this vehement suspicion, justly conceived against me, with sincere heart and unfeigned faith I abjure, curse, and detest the aforesaid errors and heresies, and generally every other error, heresy, and sect whatsoever contrary to the said Holy Church, and I swear that in the future I will never again say or assert, verbally or in writing, anything that might furnish occasion for a similar suspicion regarding me; but that should I know any heretic, or person suspected of heresy, I will denounce him to this Holy Office, or to the Inquisitor or Ordinary of the place where I may be. Further, I swear and promise to fulfill and observe in their integrity all penances that have been, or that shall be, imposed upon me by this Holy Office. And, in the event of my contravening, (which God forbid) any of these my promises and oaths, I submit myself to all the pains and penalties imposed and promulgated in the sacred canons and other constitutions, general and particular, against such delinquents. So help me God, and these His Holy Gospels, which I touch with my hands. I, the said Galileo Galilei, have abjured, sworn, promised, and bound myself as above; and in witness of the truth thereof I have with my own hand subscribed the present document of my abjuration, and recited it word for word at Rome, in the Convent of Minerva, this twenty-second day of June, 1633. |
In 1983, 350 years later, the Catholic church issued a formal apology to Galileo, for the earth does indeed move. Better late than never.
A computer controlled clockdrive telescope can cost $5,000, but you don't need one of those. You can purchase a hand-held telescope for $50, and that's all you need. Every home should have one. You can see everything Galileo saw, including the phases of Venus. Be patient however; the moon passes through its phases in 29 days, Venus takes 225 days. If you want to watch Venus change from ¼ to ¾, a convincing demonstration of its phases, you need to view the planet, perhaps at an inconvenient hour of the early morning, for 55 consecutive days. Well scientists are patient - some of them search for decades for that rare fossil on the African plain, or a gram of radium hidden in a ton of pitchblend. They're patient - as long as the funding holds out. So after 2 months of observations you have confirmed, with your small telescope, that Venus goes around the sun. You didn't have to take my word for it.
Now pass the telescope over to your children and let them explore the heavens; it's never too early for science. Be sure to warn them, and remind yourself, never to point the telescope toward the sun. There's probably no reason to use it during the day anyways, there's nothing but blue sky, but if your kids insist on looking at clouds, be sure to supervise their activities. A short glance at the magnified sun can lead to permanent blindness.
Finally we come back to the speed of light. The moons around Jupiter implement a clock of sorts, more reliable than any on earth. Io, the innermost moon, orbits Jupiter once every 42.45930686 hours, without fail. It is precise and predictable, like the rest of astronomy. And yet, as the earth pulled away from jupiter, the moons would lose ground, as though the clock was running slow. Clearly the moons were marching along as they should, so something else was causing the delay. The light from Jupiter's moons traveled at a finite speed, and took longer to reach earth when earth and Jupiter were far apart. Indeed, at farthest separation, the moons were 16 minutes behind schedule. This is a significant delay in the context of 42 hours. When earth was once again close to Jupiter, the moons were back where they belong, each one circling Jupiter on its timetable. Here are two pictures, with earth close to Jupiter and with earth far away. The difference is the diameter of earth's orbit around the sun, or twice the distance from the earth to the sun.
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We have our hands on an important relationship, the earth is 8 light minutes from the sun. If we know the size of earth's orbit, we can compute the speed of light. Conversely, if we know the speed of light, we can compute the size of earth's orbit. There was no practical way to measure the speed of light in the 1600's, so astronomers tried to measure the solar system instead. Cassini made good progress in 1672, and other astronomers refined his distances a century later. You can review this process here. With the earth-sun distance in hand, the speed of light is 186,000 miles per second, or 300,000 kilometers per second. This in turn can be used to refine the paths of the other planets; after all, we are seeing them from afar. It also establishes the exchange rate between mass and energy, as described in an earlier chapter.