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.
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| 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.
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| 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
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.
Sources:
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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