Sunday, June 7, 2015

The problem of planets: from Aristotle to Newton

The ancient Greeks, along with a number of other civilizations, noticed five “wandering stars” that over many nights appeared to travel against the background of fixed stars in the sky. These “stars” went along the same path as the Sun and the moon, but in the opposite direction. They named the wandering stars Hermes, Aphrodite, Ares, Zeus, and Cronos.  The Romans translated these names into Mercury, Venus, Mars, Jupiter, and Saturn.  The names of the wandering stars, along with the Sun and the moon, became the names of the 7 days of the week. Saturday, Sunday, and Monday are associated with Saturn, the Sun, and the moon. Tuesday is thought to be associated with the Germanic god Tyr (Mars), Wednesday with Wotan (Mercury), Thursday with Thor (Jupiter), and Friday with Frigga (Venus). 

The wandering stars, of course, were no stars at all, but planets.  It took about two thousand years to understand why the planets appeared to wander. The story begins around the time of Aristotle, and ends with Newton. Along the way, humans learned how to use mathematics to represent observations in nature, and this led to the birth of science. In a recent book titled “To Explain the World”, Stephen Weinberg, a physicist and Noble Laureate, tells this story. Here, I simplify his eloquent and thorough text, and highlight the key ideas.

Aristotle and Ptolemy

Anaxagoras, an Ionian Greek born around 500 BC, reasoned that the earth is spherical because when the Sun placed the earth’s shadow on the moon, one could see the round outline of the earth. Aristotle repeated this idea in his book “On the Heavens”, writing: “In eclipses the outline is always curved, and, since it is the interposition of the Earth that makes the eclipse, the form of the line will be caused by the form of the Earth’s surface, which is therefore spherical.” But he also argued that the earth must be stationary and not moving, because if it were moving a rock thrown upward would not fall straight down, but to one side. He wrote: “heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an unlimited distance.” 

Given that the earth is not moving, how does one explain the fixed and the wandering stars (the planets)? Aristotle, citing an earlier work by Eudoxus of Cnidus, suggested that the fixed stars are carried around the earth on a sphere that revolves once a day from east to west, while the sun and moon and planets are carried around the earth on separate (and transparent) spheres. Now there were lots of problems with this scheme. For example, because the planets were thought to shine with their own light, and the spheres were always the same distance from the earth, the brightness of the planets should not change, which disagreed with observations.

This issue remained unresolved until 650 years later, with Claudius Ptolemy, who in AD 150, working in Alexandria, Egypt, wrote Almagest. Ptolemy gave up on the notion that earth was the center of rotation for the planets, and instead suggested that each planet had a center of rotation that itself went around the earth. For the nearby planets of Venus and Mercury, he proposed that the centers of rotation were always along a line between the earth and the sun, and went around the earth in exactly one year. For Mars, Jupiter, and Saturn, the centers of rotation were beyond the sun.  

Ptolemy's planetary model
Ptolemy wrote: “I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods.”

Copernicus and Tycho Brahe

For centuries the idea that the earth was stationary remained, so that even in the middle ages, scholars like Jean Buridan would reject the idea that the earth could be rotating, not realizing that if earth rotated, then its rotation would give everything, including an arrow that was shot straight up, an impetus. Like all good mentors, Buridan had a student who thought independently. His name was Nicole Oresme. Oresme studied with his mentor Buridan in Paris in 1340s.  In his book “On the Heavens and the Earth”, Oresme rejected Aristotle’s arguments for a stationary earth, stating that when an archer shoots an arrow vertically, the earth’s rotation carries the arrow with it (along with the archer). Therefore this observation is not a demonstration of an immovable earth, but also consistent with a rotating earth.  Aristotle’s argument on a stationary earth took its first major blow.

The idea that the earth might be rotating took center stage with Nicolaus Copernicus, who in 1510 wrote a short, anonymous book titled “Little Commentary”. The book was not published until after the author’s death, but in it he put forth a new theory. He began by asserting that there is no center for the orbits of the celestial bodies: the moon goes around the earth, but all other heavenly bodies go around a point near the sun. He further asserted that the night sky has fixed stars that are much farther away than the sun, and appear to move around the earth only because the earth is rotating on its axis and about the sun.

Tycho Brahe was impressed with the simplicity of Copernicus’ theory, but pointed out a huge problem:  if the earth is moving, what is moving it? After all, earth was made of rocks and dirt, materials that would make something the size of earth weigh an enormous amount. In contrast, ever since Aristotle it was thought that the heavenly bodies were nothing like earth, made of some kind of substance that gave them a natural tendency to undergo rapid circular motion. The problem was, if earth was moving around the sun, what was pushing it, and what was keeping it there in its orbit?

In an ironic twist, to explain motion of the earth it was the Copernican astronomers who called on divine intervention. In a letter to Brahe, Copernican Christoph Rothmann wrote: “These things that vulgar sorts see as absurd at first glance are not easily charged with absurdity, for in fact divine Sapience and Majesty are far greater than they understand.”

Being unimpressed with divine intervention, in 1588 Tycho Brahe pointed out that if one took Ptolemy’s theory and put the moving center of all the planets (except earth) on the sun, and have the sun go around the stationary earth, then much of the observed data would fit just as well as Copernicus’ theory. This “Tychonic” system kept the advantage of a stationary earth, and was mathematically identical to the model of Copernicus.

Tycho Brahe's planetary model
In January of 1610, Galileo used his newly built telescope to look at Jupiter, and saw that “three little stars were positioned near him, small but very bright.” The next night he noticed that the little stars seemed to have moved, and eventually he concluded that the little stars were actually satellites of Jupiter, its moons. This observation was critical, as it was the first discovery of heavenly objects that circled something other than earth. They were a miniature example of what Copernicus had proposed. But Tycho Brahe’s model remained a viable alternative, because the fundamental question for a sun-centric theory remained that if the earth is moving, what could be so powerful as to move it?  

Newton and calculus

In 1665, Issac Newton asked a simple question: how does one compute speed of some object if the distance traveled as a function of time is not constant (or uniform). Suppose x(t) represents position as a function time t. Newton argued that in order to calculate speed, we need to think of an infinitesimally small period of time, which he called o.  Speed becomes:
For example, suppose that
For o an infinitesimal period of time, we can ignore terms that include squared and cubic powers of o. This means that:

Newton called this the "fluxion" of x(t). We now call it the derivative of x(t).  

Newton was considering this question because he wanted to ask about the acceleration that a body would experience as it travels in constant speed about a circle. At any time t, the velocity of this body is a vector tangent to the circle, with amplitude v.  

Suppose that the circle is radius r. After an infinitesimal time o, the body will have traveled by a distance vo, and angle q about the circle. At this new location the speed would still be v, but the velocity vector will have rotated by an angle q. We now have two isosceles triangles that are scaled versions of each other. Therefore, the ratio of the short side to the long side of the two triangles is equal: 
We can re-write the above equation as follows:
Eq. (1)
The term on the left of the above equation is a derivative. It represents the length of the acceleration vector that the body experiences (pointing to the center of the circle) as it rotates with constant speed around the circle.  

Newton realized that this acceleration toward the center is due to a force that is pulling the body toward the center of the circle (otherwise, it would fly off in a straight line, tangent to the circle). That force, he assumed, is proportional to square of the velocity v, divided by radius r.  

Next, Newton considered Kepler’s observation (his third law) that the square of the period of a planet in its orbits is proportional to the cube of the radius of its orbit. The period of a body moving with speed v around a circle of radius r is the circumference  2pr divided by speed v.  And so Kepler’s third law says that 
We can re-write the above equation as follows: 
Eq. (2)
If we now compare Eq. (1) with Eq. (2), we see that the acceleration that was keeping the body moving in circular motion, is also proportional to the reciprocal of squared r. This means that the force that is pulling the body toward the center is proportional to the inverse of the squared distance of the body from the center. This is the inverse square law of gravity.

But the incredible discovery was still one step away. Newton now asked whether the acceleration of the moon in its orbit around the earth is the same acceleration that a body undergoes when it is falling here on earth. To calculate moon’s acceleration, he estimated the distance of the moon to the center of the earth to be around 60 times the radius of earth, or around 314 million meters. Next, he computed the speed of the moon by dividing the circumference of one orbit around the earth by its period of travel (27.3 days, or 2.36 million seconds):
He then used Eq. (1) to compute the acceleration of the moon toward the earth:
This is the moon’s acceleration toward earth. It is quite small, but Newton understood that the acceleration is small because the moon is very far away from earth. An object on the surface of earth accelerates faster because it is at a distance of one radius of earth away from the earth’s center. The moon is 60 times farther. Therefore, using Eq. (2), he argued that the moon’s acceleration should be 1/60^2 that of an object on the surface of the earth.  

Multiplying moon’s acceleration by 60^2 we find the result that a body on the surface of earth should accelerate at around 8 m/s^2.  (The actual value is 9.8 m/s^2. The greatest source of error in Newton’s calculations was that the distance of moon from earth, which he underestimated by around 15%). He then writes:  

“I began to think of gravity extending to the orb of the moon and (having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule of the periodical times of the plants being in sesquilterate proportion of their distances from the center of the orbs, I deduced that the forces which keep the planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolved and thereby compared the moon in her orb with the force of gravity at the surface of the earth and found them answer pretty nearly.”

So what Newton had done was to show that the motion of the moon around the earth described an acceleration toward earth that was due to a force quite identical to the force that acts on an apple on the surface of the earth. The only reason that the moon accelerates much slower toward earth is because the moon is much farther, and therefore the force that it feels from earth is much weaker.  

The acceleration of the apple, the moon, and the planets around the sun, are all governed by the same rules: force grows weaker as the squared distance of one body from another.

Dennis Danielson and Christopher M. Graney (2014) The case against Copernicus.  Scientific American, January 2014, pp. 74-77.

Stephen Weinberg (2015) To Explain the World: The discovery of modern science. HarperCollins.

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