Newton’s Law is not just a good idea

In Principia, Newton proposed that the force of gravity exerted by objects upon one another is proportional to the mass of the two objects, and weakens as the square of the distance between those objects.

Specifically, he postulated that the gravitational force between two objects is directly proportional to the product of their masses (mass of object A times mass of object B). So two objects, one very massive and the other with very little mass, will “feel” the same mutual attraction. In addition, he claimed that the force between two objects will decrease in proportion to the square of the distance. This “inverse-square law” means that the force of gravity mutually exerted by two objects, say 10 units of distance apart, is 100 times (102) weaker than that exerted by objects only 1 unit apart—yet this force never reaches zero.

The most distant galaxies in the universe exert a gravitational pull on one another. These relations between mass, distance, and force comprise what we call Newton’s Law of Universal Gravitation. Consider the solar system, with the planets moving in elliptical orbits around the sun. Newton’s Principia explained not only what holds the planets in their elliptical orbits (an “inverse-square” force called gravity), but also predicted that the planets themselves (massive Jupiter in particular) would have a small but measurable effect on each other’s orbits.

Like any good scientific theory, Newton’s laws not only explained what was already observed (the motion of the planets), but was able to make testable predictions. The orbit of Saturn, for example, was known to deviate slightly from what one would expect if it were simply in orbit around the sun (with no other planets present). The mass of Jupiter has a small, but measurable, effect on its orbital path. Newton noted with a sense of humor that the effect of Jupiter on Saturn’s orbit made so much sense (according to his theory) that “astronomers are puzzled with it.” For the first time, a scientist had claimed that the rules of motion on the earth were no different from the rules of motion in the heavens.

The moon was just a big apple, much farther away, falling to the earth in its own way. The planets orbit the sun following the same rules as a baseball thrown up into the air, and the pocket watch of the earth is held in its orbit by a chain called gravity. Did Newton bring the celestial sphere down to Earth, or elevate us all to the status of planets? Whatever you think, we have never looked at the solar system or the universe in the same way since.

The Weighty Matters

Throw a ball up into the air, and you will observe that it travels in a familiar curved (parabolic) trajectory: first up, up, up, leveling off, then down, down, down. Common sense tells us that the force of gravity pulls the ball back to Earth.

Newton’s brilliance was in postulating not only that there is a force, gravity, that pulls the ball (or apple) back to the earth, but that such a force applies to everything in the universe that has mass. The gravitational force due to the mass of the earth also pulls on the moon, holding in its orbit, and pulls on each of us, keeping us in contact with the ground. Finally there was an answer to those who thought the earth could not be spinning and orbiting the sun. What was there to keep us firmly footed on the earth? Newton had the answer: the force of gravity.

Newton’s Three Laws of Motion

Newton’s first law of motion states that, unless acted upon by some external force, a body at rest remains at rest and a moving object continues to move forever in a straight line and at a constant speed. This property is known as inertia. The measure of an object’s inertia is its mass (in effect, the amount of matter the object contains). The more massive an object, the greater its inertia.
Newton’s first law explains why the planets move in nearly circular orbits—essentially because an external force (gravity) acts on each planet. Without gravity, the planets would all fly off in straight lines, like so many pocket watches.

Newton’s second law states that the acceleration of an object is directly proportional to the force applied to the object and inversely proportional to the mass of the object. Pull two objects with the same force, and the more massive object will accelerate more slowly than the less massive one. We all know this intuitively. Your subcompact car’s engine would have a much harder time accelerating than an 18-wheel truck!
Newton’s third law of motion states that forces do not act in isolation. If object A exerts a force on object B, object B exerts an equal but opposite force on object A. A hammer, for example, exerts a force on the nail, driving it into the wall. The nail exerts an equal and opposite force on the hammer, stopping its motion.

What Hold The Universe Together?

One of us knew a young man who owned a pocket watch, which he would habitually twirl, holding the end of the chain and allowing it to orbit around the focus of his finger. One day, the chain broke, sending the costly timepiece flying off into space—and against a wall, with predictably catastrophic results.

Why don’t the planets suffer the same fate?
Despite his brilliant explanation of planetary motion, Kepler had not explained how the planets orbited the sun without flying off into space, and why they traveled in ellipses.
The answer came in the late seventeenth century when an Englishman, Isaac Newton (1642–1727), one of the most brilliant mathematicians who ever lived, formulated three laws of motion and the law of universal gravitation in a great work, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), better known simply as Principia.

Understanding Galileo's Work

While Kepler was theorizing from Copernicus’s data, an Italian astronomer, Galileo Galilei (1565–1642), directed his gaze skyward, amplifying his eyesight with the aid of a new Dutch invention (which we’ll meet in the next chapter), the telescope. Through this instrument, Galileo explored the imperfect surface of the moon, covered as it was with craters, “seas,” and features that looked very much like the surface of the earth. Galileo was even more surprised to find that the surface of the sun was blemished. These “sunspots”, he noted, changed position from day to day. From this fact, Galileo did not conclude that the spots changed, but that the sun was rotating, making a complete revolution about once each month.

His telescope also revealed for the first time that moons orbited Jupiter—another observation that strongly supported the notion that the earth was not the center of all things. He observed that the planet Venus cycled through phases, much like the moon, and that the size of the planet varied with its phase. From this, he concluded that Venus must orbit not the earth, but the sun.

Galileo published these many independent experimental proofs of a heliocentric solar system in 1610.Six years later, the Catholic Church judged the work heretical and banned them, as well as the work of Copernicus. Galileo defied the ban and, in 1632, published a comparison of the Ptolemaic and Copernican models written as a kind of three-way discussion. He was so bold as to write in Italian, instead of learned Latin, which meant that common folk (at least those who were literate) were being invited to read a theory that challenged the teaching of the Church.

Galileo was silenced by the Holy Inquisition in 1633, forced to recant his heresy under threat of death, and placed under house arrest for the rest of his life.

Three Kepler Laws

Kepler had predicted that with Tycho’s data, he would solve the problem of planetary motion in a matter of days. After almost eight years of study, trial, and error, Kepler had a stroke of genius. He concluded that the planets must orbit the sun not in perfect circles, but in elliptical orbits (an ellipse is a flattened circle). He wrote to a friend: “I have the answer … The orbit of the planet is a perfect ellipse. Kepler was able to state the fundamentals of planetary motion in three basic laws.

That planets move in elliptical orbits, with the sun at one focus of the ellipse, is known as Kepler’s First Law. It resolved the discrepancies in observed planetary motion that both Ptolemy and Copernicus had failed to explain adequately. Both of those great minds had been convinced that in a perfect universe, the orbits of planets had to be circular.

Another apparent attribute of elliptical orbits determined Kepler’s Second Law. It states that an imaginary line connecting the sun to any planet would sweep out equal areas of the ellipse in equal intervals of time. The Second Law explained the variation in speed with which planets travel. They will move faster when they are closer to the sun. Kepler did not say why this was so, just that it was apparently so. Later minds would confront that why. The first two of Kepler’s laws were published in 1609.

The third did not appear until ten years later and is slightly more complex. It states that the square of a planet’s orbital period (the time needed to complete one orbit around the sun) is proportional to the cube of its semi-major axis. Since the planets’ orbits, while elliptical, are very nearly circular, the semi-major axis can be considered to be a planet’s average distance from the sun.

Understanding Kepler's work

When King Frederick died in 1588, Tycho lost his most understanding and indulgent patron. Frederick’s son Christian IV was less interested in astronomy than his father had been, and, to Christian, Tycho was an unreasonably demanding protégé, who repeatedly sought more money. At last, the astronomer left Denmark and ultimately settled in Prague in 1599 as Imperial Mathematician of the Holy Roman Empire. In the Czech city, he was joined by a persistent younger German astronomer, Johannes Kepler (1571–1630), who, after writing several flattering letters to Tycho, became his student and disciple.

Tycho Brahe, a colorful character, who lost part of his nose in a duel (he replaced it with a golden prosthesis), died ingloriously in 1601, apparently from a burst bladder after drinking too much at a dinner party. After a bit of a struggle, Kepler got a hold of the mass of complex observational data Tycho had accumulated. While Tycho had been a brilliant observer, he was not a particularly good theoretician. Kepler, a sickly child, had grown into a frail adult with the mind of a brilliant theorist—though with very little aptitude for close observation, since he also suffered from poor eyesight. Indeed, Tycho and Kepler were the original odd couple, who argued incessantly; yet their skills were perfectly complementary.

And when Tycho died, instead of using his instruments to make new observations, Kepler dived into Tycho’s data, seeking in its precise observations of planetary positions a unifying principle that would explain the motions of the planets without resorting to epicycles. It was clear, especially with Tycho’s data, that the Copernican system of planets moving around the sun in perfectly circular orbits was not going to be sufficient. Kepler sought the missing piece of information that would harmonize the heliostatic solar system of Copernicus with the mountain of data that was Tycho’s planetary observations.

The Man with the Golden Nose

The astronomer Tycho Brahe (1546–1601) was born in Denmark just three years after the publication of De Revolutionibus. As a youth, he studied law, but was so impressed by astronomers’ ability to predict a total solar eclipse on August 21, 1560, that he began to study astronomy on his own. In August 1563, he made his first recorded astronomical observation, a conjunction (a coming together in the sky) of Jupiter and Saturn, and discovered that the existing ephemerides were highly inaccurate. From this point on, Tycho decided to devote his life to careful astronomical observation.

On November 11, 1572, he recorded the appearance of a new star , brighter than Venus, in the constellation Cassiopeia. The publication of his observations (De Nova Stella, 1573) made him famous.

King Frederick II of Denmark gave him land and financed the construction of an observatory Tycho called Uraniborg (after Urania, the muse of astronomy). Here Tycho not only attracted scholars from all over the world, but designed innovative astronomical instruments and made meticulous astronomical observations—the most accurate possible before the invention of the telescope.

A Revolution of Revolutions

For centuries, astronomy had been a science in which errors of a few degrees in planetary position on the sky were acceptable. But to Copernicus, Kepler, and others who would follow, errors of that magnitude indicated that something was seriously wrong in our understanding of planetary motion. They were driven to discover their origin.

Now we turn to the details of Copernicus’s model. The first of the six sections of De Revolutionibus sets out some mathematical principles and rearranges the planets in order from the sun: Mercury is closest to the sun, followed by Venus, Earth (with the Moon orbiting it), Mars, Jupiter, and Saturn.

The second part applies the mathematical rules set out in the first to explain the apparent motions of the stars, planets, and sun. The third section describes Earth’s motions mathematically and includes a discussion of precession of the equinoxes, attributing it correctly to the slow gyration of the earth’s rotational axis. The last three parts of the book are devoted to the motions of the moon and the planets other than Earth.

While Copernicus’s purpose in reordering the planetary system may have been relatively modest, it soon became apparent that the new theory required the most profound revision of thought.

First: The universe had to be a much bigger place than previously imagined. The stars always appeared in the same positions with the same apparent brightness. But if the earth really were in orbit around the sun, the stars should display a small but noticeable periodic change in position and brightness. Why didn’t they? Copernicus said that the starry celestial sphere had to be so distant from Earth that changes simply could not be detected.

Building on this explanation, others theorized an infinite universe, in which the stars were not arranged on a celestial sphere, but were scattered throughout space.
The second required revision, while not as obvious, was even more basic.
Why do things fall? Aristotle explained that bodies fell toward their “natural place,” which, he said, was the center of the universe.

That explanation worked as long as the earth was considered to be the center of the universe. But now that it wasn’t the center, how could the behavior of falling bodies be explained? The answers would have to wait until the late seventeenth century, when Isaac Newton published his Principia, including a theory of universal gravitation.

Profound as were the astronomical and other scientific implications, the emotional shock of the Copernican universe was even greater. Suddenly, the earth, with humankind upon it, was no longer at the center of all creation, but was instead hurtling through space like a ball on a string. Why did we stay on the ball? What was the string? These were all unanswered questions that must have been very unsettling for those who thought about them.

“More Pleasing to the Mind”

This uncertainty in the calendar bothered Copernicus deeply. Many astronomers thought that any errors in the Ptolemaic system might be due to the many small “typos” that had crept into the manuscript with centuries of copying by scribes. However, connections between east and west at this time meant that Copernicus was able to have a nearly pristine copy of Ptolemy’s work Almagest. Any errors had to be errors in the model itself.

Ptolemy’s complex geocentric system of epicycles Copernicus questioned the Ptolemaic system on the very basis that a modern scientist might. The model had become too complicated, and scientists tend to seek simplicity (where possible) in their models of the universe. The printing presses that were firing up across Europe at the time made it possible for many more scholars to read good copies of ancient works.

As Copernicus started reading Greek manuscripts that had been long neglected, he rediscovered Aristarchus’s old idea of a heliocentric (sun-centered) universe. He concluded that putting the sun at (or near) the center of a solar system with planets in orbit around it created a model that was “more pleasing to the mind” than what Ptolemy had proposed and medieval Europe accepted for so many centuries.

But he did not rush to publish, only after much hesitation privately circulating a
brief manuscript, De Hypothesibus Motuum Coelestium a Se Constitutis Commentariolus
(A Commentary on the Theories of the Motions of Heavenly Objects from Their Arrangements)

in 1514. He argued that all of the motion we see in the heavens is the result of the earth’s daily rotation on its axis and yearly revolution around the sun, which is motionless at the center of the planetary system.
Sound familiar?

Aristarchus had suggested it almost 2,000 years earlier, but no one had listened. The earth, Copernicus explained, was central only to the orbit of the moon. For almost two more decades he refined his thought before consenting, in 1536, to publish the full theory. But largely because of opposition from Martin Luther and other German religious reformers, De Revolutionibus Orbium Caelestium (On the Revolutions of the Celestial Spheres) wasn’t actually printed until 1543.

As close as Copernicus’s model came to representing the motion of planets in the solar system, it insisted on the perfection of circular orbits, so that it actually had no better predictive ability for planetary motions than the Ptolemaic model it replaced. For all its creakiness, the Ptolemaic model still predicted, for example, where Mercury would be on a particular night about as well as Copernicus’s model did.

The ‘Heretical’ Polish Priest

In Europe, astronomy—as a truly observational science—did not revive until the Middle Ages had given way to the Renaissance. The German mathematician and astronomer Johann Müller (1436–1476) called himself Regiomontanus, after the Latinized form of Königsberg (King’s Mountain), his birthplace. Enrolled at the University of Leipzig by the time he was 11, Regiomontanus assisted the Austrian mathematician Georg von Peuerbach in composing a work on Ptolemaic astronomy.

Regiomontanus took his job seriously and, in 1461, journeyed to Rome to learn Greek and collect Greek manuscripts from refugee astronomers fleeing the Turks, so that he was able to read the most important texts, including the Greek translation of Almagest. In the meantime, his mentor Peuerbach had died and Regiomontanus completed the master’s work in 1463. Three years later, he moved to Nürnberg, where a wealthy patron built him an observatory and gave him a printing press. Beginning in 1474, he used the press to publish ephemerides, celestial almanacs giving the daily positions of the heavenly bodies for periods of several years.

The publications of Regiomontanus, which were issued until his death in 1476, did much to reintroduce to European astronomy the practice of scientific observation.
He was so highly respected that Pope Sixtus IV summoned him to Rome to oversee revision of the notably inaccurate Julian calendar then in use.

Regiomontanus began this work on the calendar, but then died mysteriously—possibly from plague or from poison, perhaps administered by enemies resentful of his probing the cosmos too insistently. At that time, it could be dangerous to question accepted ideas, especially where the heavens were concerned. Nikolaus Krebs (1401–1464), known as Nicholas of Cusa, wrote a book called De Docta Ignorantia suggesting that the earth might not be the center of the universe.

Fortunately for Nicholas (who was a cardinal of the Catholic church), few paid attention to the idea. Another Nicholas (actually spelled Nicolaus)—Copernicus—was born in eastern Poland in 1473, almost a decade after Krebs’s death. A brilliant youth, he studied at the universities of Kraków, Bologna, Padua, and Ferrara, learning just about everything that was then known in the fields of mathematics, astronomy, medicine, and theology.

Copernicus earned great renown as an astronomer and in 1514 was asked by the church for his opinion on the vexing question of calendar reform. The great Copernicus declared that he could not give an opinion, because the positions of the sun and moon were not understood with sufficient accuracy.

Arabian Astronomers

From our perspective just beyond the cusp of the millennium, it is easy to disparage Ptolemy for insisting that the earth stood at the center of the solar system. But we often forget that we live in a unique age, when images of the earth and other planets are routinely beamed from space. These stunning pictures of our cosmic neighborhood have become so familiar to us that commercial TV networks wouldn’t think of elbowing aside this or that sitcom to show the images to the viewing public.

Informed as we are with “the truth” about how the solar system works, we wonder how Ptolemy’s complicated explanation could have been accepted for so long. There is no doubt that his model of the solar system was wrong, but, wrong as he was, his book contained the heart and soul of classical astronomy and survived into an age that had turned its back on classical learning. During the early Middle Ages, Ptolemy’s work remained unread in Europe, but his principal book found its way into the Arab world, and in 820 it was translated into Arabic as Almagest (roughly translatable as The Greatest Book). The circulation of Ptolemy’s work renewed interest in astronomy throughout Arabia, with centers of learning being established in both Damascus and Baghdad.

Abu ‘Abd Allah Muhammad Ibn Jabir Ibn Sinan Al-battani Al-harrani As-sabi’, more conveniently known as al-Battani (ca. 858–929), became the most celebrated of the Arab astronomers, although it took many years before his major work, On Stellar Motion, was brought to Europe in Latin (about 1116) and in Spanish translation (in the thirteenth century). Al-Battani made important refinements to calculations of the length of the year and the seasons, as well as the annual precession of the equinoxes and the angle of the ecliptic. Moreover, he demonstrated that the Sun’s apogee (its farthest point from Earth) is variable, and he refined Ptolemy’s astronomical calculations by replacing geometry with sleek trigonometry.

Another Arab, Al-Sûfi (903–986), wrote a book translated as Uranographia (in essence, Writings of the Celestial Muse), in which he discussed the comparative brilliance of the stars. Like the scale of the Greek astronomer Hipparchos, the system of Al-Sûfi rated star brightness in orders of magnitude. Relative star brightness is still rated in terms of magnitude.

Arab astronomers like Al-Sûfi also contributed star maps and catalogues, which were so influential that many of the star names in use today are of Arab origin (such as Betelgeuse, Aldebaran, and Algol), as are such basic astronomical terms as azimuth and zenith.

The prelude for Modern Astronomy

The “Dark Ages” weren’t dark everywhere, and astronomy didn’t exactly wither and die after Ptolemy. Outside of Europe, there were some exciting discoveries being made. The Indian astronomer Aryabh¯atta (born ca. 476) held that the earth was a rotating globe, and he correctly explained the causes of eclipses.

The Maya in particular created a complex calendar, prepared planetary tables, and closely studied Venus, basing much of their system of timekeeping on its movements. In Europe, however, astronomy was no longer so much studied as it was taught. Much Medieval learning discouraged direct observation in favor of poring over texts of recorded and accepted wisdom. Despite the work of Bishop Isidorus of Seville (570), who drew a sharp distinction between astronomy and astrology, medieval astronomy was mired in the superstition, which has its apologists and practitioners to this very day. But this chapter says little about astronomy’s “Dark Ages” and turns, instead, to its rebirth in the Renaissance.