The Fall of the Geocentric Theory, and the Rise of Heliocentrism

The geocentric theory reached its pinnacle with the system devised by Ptolemy. Ptolemy's geocentric theory continued to be the system of choice as the Roman empire succeeded that of the Greeks. With the fall of the Roman empire (circa 450 AD), it was left to the Arabs to keep the Ptolemaic view of the universe alive. In 622 AD, the Prophet Mohammed launched his holy war against the infidels. Within a century, the Islamic Empire extended eastwards across northern India to the borders of China, and westwards across Asia Minor, to north Africa. With the Arabic conquest of Spain and Sicily, the empire made it into western Europe. Alexandria (Egypt), the center of classical learning, fell to the Arabs in 642. When the period of military expansion ended, Islamic scholars became enthusiastic students of classical philosophy. Many important manuscripts were translated from Greek into Arabic. In the world of medieval Islam, Aristotle and Ptolemy were the supreme authorities in matters of natural science and astronomy. They continued to make observations, and to refine the data that went into the model while streamlining/advancing some of the calculation techniques. But they did not make any modifications to the underlying philosophy.1

"In 999 AD, Gerbert of Aurillac, the most accomplished mathematician, musician, astronomer and classical scholar in Europe, ascended to the papal throne as Sylvester II, known as 'the magician Pope' to his contemporaries. His papacy, at the symbolic date 1000 AD marks the turning point of the European 'dark ages'. Contact was established with Arabic centers of learning in Spain, where Muslim, Christian and Jewish scholars congregated. During the following centuries, Hebrew and Arabic versions of ancient Greek texts were translated into Latin and began to circulate in Europe. The works of Aristotle were translated from around 1200. The translation of Ptolemy's Almagest into Latin in 1175 re-vitalised European astronomy. King Alfonso X of Castile (1122-1184) commissioned new astronomical tables calculated according to Ptolemy's theory with Arabic mathematical refinements. Completed in 1252, the Alphonsine Tables remained the best astronomical tables available in Europe for the next three centuries. The complexities of the Ptolemaic system exasperated King Alfonso however. When the intricacies of epicycles, deferents and equants were explained to him Alfonso 'the Wise' is said to have remarked that if the Almighty had consulted him on the matter, he would have recommended something a little simpler... "1

Eventually, with the re-emergence of European learning in the 13th century,  the geocentric view of the universe would be adopted by the Catholic church. It is interesting that in the early European universities, Ptolemy's theory was relegated to the world of mathematics, calendars, and astrology. It was purely a mechanical model of the celestial motion, and not the "true" nature of the universe as described by Aristotle. The latter would be given a more revered place in higher learning, and especially in questions of philosophy and theology.

The Copernican Revolution

In the fifteenth century, a reform of European astronomy began. This was due to the fact that predictions made using the Ptolemaic system were less and less accurate. With the advent of ocean-going vessels, and the expansion of global trade and exploration, navigation using astronomical knowledge had become extremely important. Also, the calendar implemented in 44 BC by Julius Caesar was 10 days off, with the spring equinox now occurring on the 11th of March instead of the 21st (used to fix the date of Easter). Astronomical help was needed! Two astronomers, Peurbach and Regiomontanus (who set up the first European observatory at Nuremburg in 1474) took on the task of locating errors in the works of Ptolemy, as well as errors in published tables and in observations. They had refined some of astronomy, but they had not gotten rid of all of the mounting problems.

Copernicus (1473 to 1543) learned of the works of Peurbach and Regiomontanus while an undergraduate at the University of Cracow (Poland), then spent eight years studying in Italy (five years in Bologna studying liberal arts, and three years at the University of Ferrara where he got a degree in canon law) before returning to a position in the cathedral at Frombork, Poland. Here he spent the majority of his time as a physician, lawyer, and church administrator. In his spare time he dabbled in astronomy. Copernicus was very interested in the Pythagorean mathematics, for he also believed in a harmony of the cosmos. He was, however, not apparently very interested in making new astronomical observations.

By 1514 Copernicus had laid out his heliocentric view of the universe in a manuscript entitled Commentariolus. This short work would put forward most of the elements of the heliocentric system some 39 years before his major manuscript on the theory, De revolutionibus orbium coelestium ("On the Revolutions of the Heavenly Spheres"), was published.  While many of the ideas Copernicus was proposing were radical, he still adhered to the Aristotelian view that everything moved in perfect circles, on solid celestial spheres, and moved via the physics espoused by Aristotle. Indeed, he even kept Ptolemy's epicycles in modified form in his new model. But Copernicus knew that his new theory would upset the Church, and he kept it relatively quiet. By moving the Sun to the center of the universe, man was no longer so important, and this could not stand in the view of the Church.

In the intervening years, Copernicus worked very hard at fully developing his view, and this included a number of mathematical devices that greatly advanced the art of calculation. In fact, De revolutionibus closely resembled Almagest in its structure and content. Copernicus feared the repercussions of his work, and was extremely reluctant to publish it. Only after the publication of a summary of Copernicus' system by an enthusiastic supporter named Rheticus in 1540, called Narratio prima, did the aging Copernicus agree to publish his theory. The story goes that Copernicus received a printed copy of his De revolutionibus on his deathbed in 1543.

A diagram from De revolutionibus orbium lays out his system:

Note that this system proposed a radical new idea: the Earth spun on its axis once per day! His theory also gave a simple explanation for the retrograde motions of the planets: the Earth catches up and passes them once each year. He also argued that his system was more elegant than the traditional geocentric system. However, the predictive power of his model was not substantially better than its competition. Besides removing humanity from the central place in God's creation, there were some obvious "problems" with this theory:

1) If the Earth is spinning so fast, how do we remain attached to it--how do birds find their way home again with all of this spinning? [Of course, the counterpoint is what keeps us attached in the Geocentric Universe, where some people must be on "top" or "bottom" of the spherical Earth?]

2) Why do the stars not show a parallax? They have to be at immense distances to show no parallax, and thus there is this great empty space in between the last sphere (Saturn) and that containing the stars. [The counter-argument to this is that if all of the stars are at the same distance fixed to a crystalline sphere, no parallax can be measured since all stars would shift the same amount!]

All in all, the Copernican heliocentric model of the universe was very slow to gain acceptance.  In fact, during his lifetime and after his death, Copernicus was often the subject of ridicule. But he was never formally charged with any heresies, as the heliocentric model would not really make it on the "radar screen" of the church for  another 50 or 60 years (note also that De revolutionibus orbium was dedicated to Pope Paul III!). While the model may not have been accepted, much of the new tabular material and calculation methods were adopted by geocentrically-oriented astronomers for their own calculations on planetary motion.

Tycho, Galileo, and Kepler

The next player in the Copernican revolution was Tycho Brahe (1546 to 1601). Tycho was from a privileged background of Danish aristocracy, and thus had a good education and the resources necessary to engage in a variety of pursuits:

"He was brought up by his paternal uncle Jörgen Brahe and became his heir. He attended the universities of Copenhagen and Leipzig, and then traveled through the German region, studying further at the universities of Wittenberg, Rostock, and Basel. During this period his interest in alchemy and astronomy was aroused, and he bought several astronomical instruments. In a duel with another student, in Wittenberg in 1566, Tycho lost part of his nose. For the rest of his life he wore a metal insert over the missing part.

He returned to Denmark in 1570. In 1572 Tycho observed the new star in Cassiopeia and published a brief tract about it the following year. In 1574 he gave a course of lectures on astronomy at the University of Copenhagen. He was now  convinced that the improvement of astronomy hinged on accurate observations. After another tour of Germany, where he visited astronomers, Tycho accepted an offer from the King Frederick II to fund an observatory. He was given the little island of Hven in the Sont near Copenhagen, and there he built his observatory, Uraniburg, which became the finest observatory in Europe."2

Tycho went about building the finest astronomical instruments that he could (read about them here), he ran his own printing press, and he began to train astronomers from all over Europe. He was an excellent observer, and the quality of his observations were higher than any that came before him, and were much better than his contemporaries. Two of his most important observations were of the new star" of 1572 (now known to be a supernova), and of the comet of 1577. In Aristotle's view, no new stars could occur, the heavens were immutable! He showed that the comet of 1577 had to be further away than the Moon, and that comets were not an atmospheric phenomena as previously believed. These two observations meant that Aristotle's view of the universe was incorrect, the heavenly realm could change. But Tycho could not accept the heliocentric view either, preferring the Aristotelian physics, and the view that man was important and at the center of the universe.

What Tycho did to resolve this was a peculiar halfway step that was a hybrid of the old geocentric view, and Copernicus' new heliocentric view: the "Tychonic" model. In this model, he placed the Earth at the center, and the Moon and the Sun orbited about it. But the other planets all orbited around the Sun!

Johannes Kepler

After a falling-out with the King, Tycho left Denmark and moved to Prague in 1597. There he hired Johannes Kepler (1571-1630) to help him in reducing the data he had collected at Uraniburg. This extensive data base would not get published until many years after Tycho's death (in 1601). Kepler was born in Weil der Stadt in Swabia, in southwest Germany.

"Kepler's teacher in the mathematical subjects was Michael Maestlin (1580-1635). Maestlin was one of the earliest astronomers to subscribe to Copernicus's heliocentric theory, although in his university lectures he taught only the Ptolemaic system. Only in what we might call graduate seminars did he acquaint his students, among whom was Kepler, with the technical details of the Copernican system. Kepler stated later that at this time he became a Copernican for "physical or, if you prefer, metaphysical reasons".

In 1594 Kepler accepted an appointment as professor of mathematics at the Protestant seminary in Graz (in the Austrian province of Styria). He was also appointed district mathematician and calendar maker. Kepler remained in Graz until 1600, when all Protestants were forced to convert to Catholicism or leave the province, as part of Counter Reformation measures. For six years, Kepler taught arithmetic, geometry (when there were interested students), Virgil, and rhetoric. In his spare time he pursued his private studies in astronomy and astrology. In 1597 Kepler married Barbara Müller. In that same year he published his first important work, The  Cosmographic Mystery, in which he argued that the distances of the planets from the Sun in the Copernican system were determined by the five regular solids, if one supposed that a planet's orbit was circumscribed about one solid and inscribed in another."3



Kepler's peculiar construction actually produced very good results when predicting the positions of the planets (except for Mercury). Tycho was impressed with Kepler's mathematical skills, and invited him to Prague to be his assistant. When Tycho died, Kepler assumed the position of Imperial Mathematician. In 1604, just like Tycho had, Kepler discovered his own "New Star" (the last known supernova to occur in our Milky Way galaxy). With Tycho's excellent data base, Kepler had all the data he needed to launch a new heliocentric view of the universe. In 1609 he published Astronomia Nova ("New Astronomy") which contained his first two laws of planetary motion:

1. Planets move in elliptical orbits with the Sun at one of the foci:





2) A planet sweeps out equal areas in equal times:





It was not until 1618 that Kepler came up with his final law (published in 1619 in Third Law in Harmonices Mundi):

3. The square of the planet's orbital period is proportional to the cube of the orbital size (semi-major axis).  In equation form, P2 = K a3, where P is the period, a is the semi-major axis, and K is a constant. In a log-log plot, the relationship looks like this:

The time taken to complete one orbit grows more than in direct proportion to orbital size. The orbital period is the circumference of the orbit divided by the planet's mean velocity. Since the circumference of an orbit increases in direct proportion to its semi-major axis, the Third Law implies that the velocities of planets in larger orbits are slower than for planets nearer the Sun. A planet with an orbital diameter 5 times the Earth's will require 11 Earth years to complete an orbit.

The timing of the publication (in 1609) of Kepler's original two laws could not have been better, for in 1610, Galileo, published the "Sidereal Messenger", describing his discovery of the four main moons of Jupiter (which we call the Galilean satellites after their discoverer). This discovery showed for the first time that not all celestial objects orbit around the Earth (more on this momentarily!). Kepler quickly published a letter supporting the author and the discovery of these moons. He obtained a telescope that year, and begun his own observations of these moons leading to a write-up describing his observations entitled Narratio de Observatis Quatuor Jovis Satellitibus ("Narration about Four Satellites of Jupiter observed"). These two manuscripts were published in Galileo's home city of Florence, and were a great relief to that beleaguered scientist. By 1611 Kepler had published a preliminary description of how a telescope works in Dioptrice.

Kepler's crowning achievements were his publications entitled Epitome Astronomiae Copernicanae ("Epitome of Copernican Astronomy") in 1621, and his Tabulae Rudolphinae ("Rudolphine Tables" ) in 1627. In Epitome Astronomiae Copernicanae, Kepler fully laid out a new heliocentric model for the universe with the planets on elliptical orbits. This was the first accurate description of how the planets moved. Unfortunately, during the period of 1615-16 there was a witch hunt in Kepler's home country, and his mother was accused of being a witch. Kepler spent several years defending her in court, finally winning her freedom in 1620. On top of this, in 1618 the "Thirty Years War"  broke out. Most of Germany and Austria were affected. Kepler suffered some persecution during these years due to the fact that he was a protestant, and during this time the Catholic counter-reformation movement, that was attempting to keep people from leaving the church (as well as attempting to quash the Protestant movement), was raging through Europe. During these years he was attempting to have his new tables on the motions of the planets, the Tabulae Rudolphinae, published in Linz, but a peasant revolt broke out, and the printer's shop where the manuscript was being printed was burned. He left Linz, and finally had the Tabulae Rudolphinae published in Ulm in 1627. The Tabulae Rudolphinae were the most accurate predictions for the motions of the planets that had ever been published, and Kepler had laid the theoretical foundation for the acceptance of the heliocentric theory. This does not mean, however, that Kepler's new theory was quickly adopted as the correct view of the universe--it was not!

Galileo

While Kepler had escaped much of the wrath of the Church, his contemporary, and probably the staunchest supporter of his work, Galileo, would not. Galileo was born in Pisa in 1564, twenty one years after the publication of Copernicus' De revolutionibus orbium, and seven years before Kepler's birth. Galileo's early years would show the personality that would come back to haunt him later in life:

"Galileo received an excellent education at a monastery near Florence, and in 1581, aged 17, showed enough talent for his father to send him to the University of Pisa to study medicine. Though born into the ranks of lower nobility, the Galilei family struggled to make ends meet and were unable to afford the university fees. It was hoped that Galileo would secure one of 40 scholarships available. [he did]

 In the second year he showed enough promise to discover that a pendulum of any given length swings at a constant frequency, which led to the invention of the 'pulsilogium', a medical device used for timing the pulse of patients. However, Galileo was already attracting animosity by his reluctance to accept the common philosophy and the scholarship was refused, forcing him to leave the university without attaining his degree. Naturally, biographers have looked upon this as an early resistance to his liberal ideas, though Arthur Koestler in his book, The Sleepwalkers, writes: "It is more likely that the refusal of the scholarship was not due to the unpopularity of Galileo's views, but of his person - that cold, sarcastic presumption, by which he managed to spoil his case throughout his life".

Excluded from university life, Galileo tutored privately and managed to maintain his studies. He developed his interest in mechanics, writing treatises on his inventions which he circulated in manuscript. Another early invention was a hydrostatic balance which attracted the attention of a number of scholars, one of whom, Marchese Guidobaldo del Monte, befriended Galileo and began a chain of recommendations that brought him to the attention of Ferdinand de Medici, the Duke of Tuscany. In a rather ironic twist of fate the Duke was so impressed with Galileo that he appointed him as a lecturer at the university which four years earlier had refused his scholarship. Just three years on, aged 28, he rose to the position of Chair of Mathematics at the esteemed University of Padua, where he remained until 1610.

In 1597, Galileo was to receive his first contact from Johannes Kepler, then aged 26 and employed as Professor of Mathematics at Gratz in Austria. Kepler had completed The Cosmic Mystery, a treatise which expressed arguments in favor of Copernicus' Sun-centered universe. He sent a copy as a gift to the Chair of Mathematics at Padua, anxious for academic feedback. Galileo replied immediately, and wrote in cordial terms:

                 "I indeed congratulate myself on having an associate in the study of Truth who
                 is a friend of Truth. For it is a misery that so few exist who pursue the Truth
                 and do not pervert philosophical reason.... I adopted the teachings of
                 Copernicus many years ago, and his point of view enables me to explain many
                 phenomena of nature which certainly remain inexplicable according to the
                 more current hypotheses. I have written many arguments in support of him and
                 in refutation of the opposite view - which, however, so far I have not dared to
                 bring into the public light, frightened by the fate of Copernicus himself, our
                 teacher who, though he acquired immortal fame with some, is yet to remain to
                 an infinite number of others (for such is the number of fools) an object of
                 ridicule and derision. I would certainly dare to publish my reflections at once if
                 more people like you existed; as they don't. I shall refrain from doing so."

Although Kepler replied, imploring him to take a different stance and air his views, Galileo ignored his letter and ended the correspondence. It was 16 years before he would produce the first public indication of his beliefs; throughout the interim he continued to teach, and appeared to endorse the traditional Aristotelian arguments that the Earth did not move."6

The letter to Kepler provides us with excellent insight into the political realities of life in the late 16th century--you had to be careful what you said in a public forum, for you could face serious penalties for going against Church doctrine. But Galileo would not remain quiet for too much longer, for he was a brilliant experimentalist, and as his fame and reputation grew, so did his confidence that he could challenge the status quo. The problem was that while the Copernicun system might be more elegant in explaining the motions of the planets than the Ptolemaic system, these were mere mechanical details, and were not sufficient reason for the Church and others to overthrow the established system--what was needed were observations that could not be explained by a geocentric view.

We have already noted the two discoveries by Tycho (the supernova and comet) that showed the "superlunary" world was not immutable, contrary to the view of Aristotle. But again, this did not violate the geocentric model for the motions of the planets. Other evidence was needed. This evidence required a new instrument, the telescope.

The invention of the telescope has been credited to Hans Lippershey, a Dutch spectacle maker. But this may or may not be correct  (go for  here  for a bit more). The story goes that in July of 1609 Galileo had heard that a Dutchman had devised an instrument composed of two lenses that made objects appear closer. Not wanting to be scooped, Galileo quickly came up with his own version and:

"In August 1609 he invited the Venetian senate to inspect his own 'spy-glass' which, through a combination of a convex and concave lens, was able to magnify objects nine times greater than normal vision. The senate was greatly impressed, particularly by his suggestion that in matters of defense it would enable them to see the sails of ships two hours before they could be seen by the naked eye. When he presented his spy glass to them as a present they expressed their appreciation by doubling his salary to a thousand scudi a year and guaranteeing his position at Padua for the rest of his life. No doubt he felt some embarrassment when local spectacle makers were soon able to replicate his instruments for just a few scudi, but he committed himself to improving the power of his instruments and began to turn his attention to the heavens.

In March of the following year, 1610, he published Sidereus Nuncius, the 'Messenger of the Stars', which revealed the fruits of his observations. The tract was kept deliberately short in order to make it widely accessible and the style of language was uniquely devoid of philosophy. Combined with explosive contents, it made a remarkable impact worldwide. He wrote about the surface of the Moon, dismissing the common view that it was perfectly smooth, and  describing it as full of lofty mountains and deep hollows. He wrote about the fixed stars, and told how he had witnessed "other stars, in myriads, which have never been seen before, and which surpass the old, previously known stars in number more than ten times". Most importantly, he wrote of his discovery of  four new planets, the Moons of Jupiter which had never been imagined before.
 

With this, he justified his first ever 'outing' of his heliocentric beliefs, writing:

                 "Moreover, we have an excellent and exceedingly clear argument to put at rest
                 the scruples of those who can tolerate the revolution of the planets about the
                 Sun in the Copernican system, but are so disturbed by the revolution of the
                 single moon around the earth while both of them describe an annual orbit
                 around the Sun, that they consider this theory to be impossible."6

Galileo was assembling the ammunition to mount the final assault on the geocentric model. That objects could orbit Jupiter instead of the Earth, showed that the Earth could in no way be considered the sole center of the universe. A page from Galileo's journal showing some observations of the movements of the moons of Jupiter:

But this, by itself, was not quite sufficient to de-throne the geocentric model, since Tycho had already proposed a hybrid model where Mercury, Venus, Mars, Jupiter and Saturn orbited the Sun, while the Moon and Sun orbited the Earth.

Galileo's observations continued to mount, and one of the most important came in late 1610 when he found that the planet Venus showed phases just like those of the Moon:

[The top part of this figure also shows Galileo's impression of (from left to right) Saturn, Jupiter and Mars.]

This observation finally ruled out the Ptolemaic system (but, unfortunately, not the Tychonic system!).

He also showed that neither the Sun nor the Moon were perfect, as envisioned in the Aristotelian view of the universe:

Galileo published these findings, and the Church found them offensive--as they violated the Aristotelian view. Galileo eventually was punished, and forced to recant his beliefs, and spent the rest of his life under house arrest. You can read more about the trials and tribulations of Galileo at my website for the Astronomy 301 course here.

Newton and His Laws

Kepler had taken the Copernican proposal and provided a mathematical basis for understanding the motions of the planets, and all observations confirmed this model. This was the confirmation and proof that the heliocentric model was correct. But he did not have an understanding of why the planets moved, and what kept them in their orbits. It was Isaac Newton that put forward the first explanation of why the planets moved by invoking a "force" called gravity. Like Kepler, Newton also had three laws:

  • I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
  • II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma; in this law the direction of the force is the same as the direction of the acceleration.
  • III. For every action there is an equal and opposite reaction.

    The origin of this mysterious force called gravity was the subject of intense debate. The true understanding (at least to the point of what is going on, but not why mass actually exerts a force called gravity---a very deep, and difficult subject) had to wait until Einstein's theory of General Relativity in 1916. But Newton's approach, that mathematics can be used to understand the Universe, was the true beginning of modern physics.

    To understand a little about gravity, we need to discuss some new terms. First let's tackle "mass" and "weight". You are probably familiar with these two terms, but they have completely different meanings. In the English system, weight is measured in "pounds". In the metric system "mass" is measured in kilograms. You may have even seen the conversion 1 kg = 2.2 lbs. But this conversion only works at the Earth's surface. Why? Because mass is an intrinsic property of the object--whether a grain of sand, a rock, a human being, a car, a planet, etc. They all have a "mass" that is always the same number of kilograms, no matter where they are located. But an object's weight depends on where it is located. On the Earth's surface, an object with a mass of 100 kg has a weight of 220 lbs. But on the Moon, the weight of that object is only 36.5 lbs. On Jupiter (which has 318 times the Earth's mass) that weight would be 520 lbs! (To see what you would weigh on other worlds go here.)

    So for an astronaut orbiting the Earth in the Space Shuttle, we talk about them being "weightless". But obviously, they have not lost any mass! Weight is what a bathroom scale reads, and it comes about through the gravity from the Earth's mass pulling on objects. We talk about the "acceleration" of gravity. You are familiar with the term acceleration---you hit the gas pedal ("accelerator") in your car and you go faster, and faster. An acceleration is the changing velocity of an object (also read section 4.1 to see the real definitions of "speed" and "velocity"--we will just consider them to be the same thing for our discussion here, but technically they are not quite the same thing). As shown in Figure 4.1, an object that is speeding up is accelerating--but so is a car going around a corner, or one that is slowing down, "decelerating". A car that is speeding up has a positive acceleration, one that is slowing has a negative acceleration.

    So, how do we accelerate a car? We push on the gas, using more and more of our gasoline (energy!) to speed up. To decelerate we push on the brake---we are again using energy, but this time the brake heats up from the friction to dissipate the speed of the car. If you have ever pulled a heavy load, or descended down a very steep hill, you can smell your brakes when you try to slow down---the energy contained in your motion, the "momentum", has to be dissipated, and in slowing it is transfered to heat, and this creates a smell as the brake pads/shoes heat up.

    Ok, now we know from experience that "what goes up, must come down". So, let's climb to the top of a tall building and drop a rock. What happens? Initially, for just an instant, the rock is not moving, but then it falls, and as it falls it speeds up---it is accelerating. Not only that, it is accelerating at a specific rate. This is shown in Fig 4.2:

    After one second (t = 1s), the rock is moving at a velocity of 10 m/s, after two seconds (t = 2 s) the rock is moving with a velocity of 20 m/s. We can write this as an equation: v = 10 x t. Using this equation, we know that if t = 4 s, then v = 40 m/s. Ok, so velocity is measured in meters per second (or kilometers per second, or kilometers per hour, or miles per hour, etc.). The "10" in our equation (which is actually 9.8) has to have the weird units of m/s2 ("meters per second squared") to get the units on the equation to work out. This "10" is called the gravitational acceleration constant, abbreviated "g". So, in actuality, the equation is v = gt. And "g" depends on the local gravity. On Jupiter "g" is bigger, on the moon "g" is smaller. On the Earth g = 9.8 m/s2.

    This gravitational constant also is what determines your weight! In fact, weight = mass x g. Let's be more general now, and revert to calling "g" the acceleration, abbreviated by "a". Now our equation becomes w = ma. And, calling weight the "force of gravity", abbreviated by "F", we get the equation F = ma. This is the second of Newton's laws that we listed above: force = mass x acceleration. A "Force" is something that tries to change the acceleration of a mass. When you jump up into the air (to dunk a basketball), the force of gravity changes your acceleration: you go up for a while all the time slowing, stop, then go back down, accelerating as you fall. This force of gravity is what keeps the planets orbiting around the Sun on elliptical orbits. But this is just part of the explanation (and it was Newton that developed and used the theory of gravity to explain Kepler's laws).

    To understand why planets orbit the Sun (or why the moon orbits the Earth) we have to talk a little more about motion. Newton's first law states that "in the absence of a net force acting upon it, an object in motion moves with constant velocity". This is kind of hard to envision on the surface of the Earth. If we were to throw a baseball, it would go a little ways, but eventually crash into the surface of the Earth. Why? Because the Earth's gravity is a force that changes the motion of the baseball. But, if we could move into deep space and then throw our baseball, then without any nearby masses, it would move at a uniform velocity through space. But near a large mass, the path of the baseball is altered--it is curved.

    We have had to play a thought game to understand this motion---let's make some more mental leaps and imagine shooting a cannonball off a very tall mountain. This is shown in the figure, below. If we use a little bit of powder, and the ball comes out slowly, it will crash close to the mountain. Let's add more powder and blast it out at a higher velocity---the ball travels further. But Earth's gravity wins and the ball eventually crashes back to Earth. Now, what happens if we shoot the ball to a very high speed so much so that it doesn't crash back into the Earth. Is this possible? Yes, it is called orbital velocity! This is what rockets like the Space Shuttle do, they go fast enough to reach orbit. What is an orbit? Well it is the speed at which the cannonball (or spacecraft) is going where it "continuously falls around the Earth" without hitting it. The gravity of the Earth continually curves the motion of the ball, but at just the right rate so that the cannon ball never hits the Earth. Of course we could use so much explosive that the cannon ball is not captured by the Earth but blasts itself into deep space---just like we must do to launch a space probe to another planet. We call the velocity where the object permanently leaves the Earth, the "escape velocity" (this is a very high speed: 40,000 km/hr!):

    So, how can we understand this curvature thing? Let's look at a simple example, a ball attached to a string that is spun around our head, as in Figure 4.6:

    As long as the string holds, the ball will circle our head---but if it breaks, the ball flies off! What is the force acting on the ball? Well it is the string! The string is the force acting to keep the ball in motion around our head (though we supply the energy). If we could keep spinning the ball faster and faster it would eventually break the string--it would "escape". This is just like gravity---gravity is the invisible string that keeps the ball (or satellite) orbiting the Earth.

    We can now expand upon this demonstration to understand Kepler's third law. If we suddenly shorten the string, what happens? The ball goes around faster! If we lengthen the string, the ball goes around more slowly--just like planets. Mercury is close to the Sun, it's "string" is short (in reality the force of gravity of the Sun is larger), so it has to go around fast (to balance the larger gravitational force) or else it will crash into the Sun (Mercury's orbital period is only 88 days). Jupiter is five times further from the Sun then the Earth, its string is long, so it moves slowly--it takes Jupiter more than 11 years to go around the Sun. Kepler's third law, P2 = a3, says the longer the string ("a") the slower the planet moves and the longer the orbital period ("P").

    It is the force of gravity that keeps planets and comets in orbit around the Sun, and the various moons in orbit around their planets. Newton realized that to explain Kepler's third law, the "force of gravity", that is the force that attracts two bodies with "mass", must be directly proportional to their masses, but inversely proportional to the the square of the separation between them:

    Fgrav = G(m1x m2)/r2

    The "G" in this equation is the "gravitational constant" that makes the equation work. It is something that must be experimentally measured. Here is a figure showing the geometry of gravitational attraction:

    The force of gravity is also the cause of the tides. As discussed in section 4.5, the gravities of the Moon and Sun distort the shape of the Earth, and create the rising and falling tides that coastal dwellers are familiar with. There are other forces in nature, and we will encounter them shortly. Gravity is the weakest of all of nature's forces, but because the others are strong, and have both positive and negative "charges", they quickly cancel themselves out. Gravity is an odd force in that it is only attractive--thus even though it is weak, it is the force the controls how matter behaves in our Universe.

    There is a lot more to gravity that we do not have time to discuss, we are glossing-over most of the subject here. To get a full flavor for this subject, read Chapter 4 in its entirety. To get more requires you to take a physics class (for a simple tutorial, go here). Note that Newton's laws work pretty well for navigating through the solar system. But when you get very close to the Sun, or to other large bodies (such as black holes), all sorts of weird affects reveal themselves. Einstein showed that Newton's laws were not the correct description of gravity, and they only worked fairly well for (and near) "low mass" objects like planets. Objects of higher masses and densities (stars and remnants of stars) require Einsten's theory of general relativity where the gravitational force is actually due to the curvature of empty space! This is a subject beyond our abilities to discuss in this class, but a site like Wikipedia can supply additional details. It is interesting to note that without taking the effects of general relativity into effect, the GPS system that we use for navigation would not work. To read more about this, go here.


    1 From http://www.skyscript.co.uk/copernicus.html
    From http://es.rice.edu/ES/humsoc/Galileo//People/tycho_brahe.html
    3  From http://es.rice.edu/ES/humsoc/Galileo//People/kepler.html
    5 From http://www.skyscript.co.uk/copernicus.html
    6 From http://www.skyscript.co.uk/galileo.html