ASTR 1210 (O'Connell) Study Guide
7. THE DISCOVERY OF GRAVITY
Thomas Digges' version of the
Copernican Universe (1576)
A. Expanding Horizons
Copernicus' heliocentric universe sparked immediate controversy
because it contradicted both the scientific and religious conventions
of the times. It was hotly debated in the 150 years following
publication of De revolutionibus, the most famous episode
of heliocentrism forced upon
Galileo by the Catholic Church (1633).
But a number of thinkers quickly and enthusiastically embraced the
Digges (d. 1595) in England
Bruno (d. 1600) in Italy. Even though Copernicus himself had pictured
the stars as lying in a shell at a fixed distance from Earth, both
Digges and Bruno realized that the model allowed the stars to be
arbitrarily far away. In the first published description of
heliocentrism in English, Digges drew the stars as stretching away
to infinity (see the picture at the top of this page), and his
inscription ( "This orb of stars fixed infinitely
up extendeth itself in altitude spherically...with perpetual shining
glorious lights innumerable...replenished with perfect endless
joy...") shows that he was thrilled by this prospect. His
concrete depiction of an infinite universe had lasting influence on
later English scientists. Bruno emphasized the possibility that the
stars were other Suns and that an unbounded universe was filled with
inhabited planets orbiting other stars (a "plurality of
The parallax technique introduced
by the heliocentric model provided, for the first time, a practical
method to estimate the distances to stars by triangulation and
hence their intrinsic brightnesses. Well before the first actual
measurement of a stellar parallax, the improving lower limits on
stellar distances possible with telescopes demonstrated that stars
might well be, as Bruno and others believed, as bright intrinsically
as the Sun. The case that the Sun is a star gradually grew
After 1600, scientific discoveries about the natural world progressed
rapidly, at least by earlier standards. Scientific information and
the standards of the "scientific method" were quickly disseminated by
printed books. The next key development for physics & astronomy was
the discovery that what we call "gravity" is really only the local
manifestation of a universal force between all forms of matter.
Here, we describe the work of four astronomers/physicists whose work
was pivotal in understanding and quantifying gravity and founding
modern science. This period includes the watershed first use
of telescopes in astronomy (1609). The timeline chart below
will help you keep track of who's who and when. Click for a larger
B. Tycho (d. 1601)
A Danish astronomer. An observer -- the greatest before the
invention of telescopes. See his picture
His observations of the "supernova"
of 1572 (an exploding star) demolish the Aristotlean doctrine of
heavenly perfection & permanence.
At his magnificent, state-sponsored observatory (see picture at
right), Tycho compiled a massive set of unprecedentedly accurate
(uncertainties less than about 1 arc-minute) data on planetary
motions, later analyzed by his assistant, Kepler. The accuracy of
Tycho's data was the best possible without optical instruments.
Although Tycho died before he was able to analyze his data, he
favored a geocentric universe
(albeit one in which the Earth spun on its axis and all the other planets
orbited the Sun).
- Tycho uses a version of the parallax
technique, based on observations taken 6-8 hours apart, to show
that the supernova is more distant than Saturn and therefore
must reside in the cosmic region the Greeks believed to be
- Similarly, Tycho showed that the bright comet of 1577 lay beyond the Moon. Previously, it
was assumed that comets (obviously changing from day to day) had to be
phenomena inside Earth's atmosphere.
- The idea that there were changes in the distant cosmos
other than the serene circular motions favored by astronomers up to
this time was startling. Not only did this fatally damage the cozy
medieval picture of a tranquil, eternal universe beyond Earth, but it
obviously made the distant universe of much greater intellectual
interest than it may have been before.
- Primary reason? Because he did not believe the stars could be so
distant that he could not measure their parallaxes. This is a
sound scientific argument and a major obstacle to earlier acceptance
of the Copernican model.
- If Tycho could have measured stellar parallaxes, he
would have instantly become a Copernican. But the distances between
stars would have had to be about 100 times smaller than they
are for that to be possible without telescopes. (Interestingly, that
would have been the case had the Earth been situated in the central
core of our Galaxy rather than in its outskirts.)
- Here is an animation of the
stellar parallax effect as it would be observed with a modern telescope.
Galileo's notes on the discovery of the
satellites of Jupiter.
C. Galileo (d. 1642)
The Italian scientist Galileo
(picture here) played a
pivotal role in the transition from medieval to modern science. He
made fundamental contributions in three separate areas: experimental
physics, astronomy, and popularizing science. Ironically, it was his
success as a popularizer, more than as a scientist, that embroiled him
in political difficulties with Church authorities.
Galileo gave experiment and observation explicit precedence over
authority. His many amazing discoveries made with small
telescopes were a resounding demonstration of the superiority
of empiricism in learning about nature. His disdain for a
reliance on authority and his devotion to mathematical empiricism are
often expressed in his writings:
As a physicist:
"You must read the book of Nature... In other words, observe and do
experiments. This is against the medieval idea of scholasticism--that
all wisdom and knowledge are best found in ancient authorities."
"Truth cannot be found in the book of Aristotle but in the book of
Nature; and the book of Nature is written in the language of
"...without [mathematics] we are wandering in vain through a dark
As an astronomer:
- Experimentally, he demonstrates that objects falling in
response to gravity accelerate---that is, increase
their velocity---in a highly systematic way (directly proportional to
time) and that their acceleration is independent of their
Among other things, this means that two objects
if dropped simultaneously from any height will hit the ground at
the same time (ignoring extraneous effects like atmospheric
drag) no matter how much they differ in mass.
- He realizes that these results, and those of many other of his
experiments, flatly contradict the principles of Aristotlean
physics. Aristotle had naively assumed, as do most people, that
heavier objects fall faster than lighter ones and that downward
velocities are constant. Owing to the Greek distrust of experiment,
Aristotle's assumptions had not been subjected to empirical tests, nor
had most of the other details of
Isn't it amazing that you, if you did
the "object drop"
assignment in Guide 6, were able to make a very simple experiment
in your dorm room that invalidates the claims of one of the most
famous philosphers in history and that were assumed for be true for
1900 years? That's the power of empirical exploration of the real
At least one Greek scientist, Strato, was aware of gravitational
acceleration: he had observed the increasing separation between
water droplets as they fall from a spout. But as in the case
of Aristarchus, this important insight was submerged by the
overwhelming influence of Aristotle on later thinking.
- Galileo made the first astronomical use of a
telescope in 1609. He had learned of the invention of telescopes
in Holland and quickly decided to make his own.
One of Galileo's telescopes. (Click for details.)
Galileo demonstrating telescope to his
- Galileo's telescopes were small (above), with relatively crude lenses
only a few inches in diameter. But they yielded the first
fundamentally new astronomical insights in over 1500 years. They utterly
transformed astronomy. They allowed Galileo to discover:
- These discoveries strongly support the Copernican interpretation,
though they don't actually prove it:
- The craters on the Moon and spots on the Sun contradict the
claims of the Greeks and the Catholic Church about the
perfection of the heavenly bodies.
- The fact that Venus displays an almost full range of phases from
"crescent" to nearly "full" proves that Venus orbits the Sun,
not the Earth. In Ptolemy's model, the epicycles of both Mercury and
Venus lie between Earth and the Sun, so Venus can never appear in
other than a crescent phase. See this illustration of the difference between the Copernican and
- Jupiter's satellites show the existence of a second "center
of motion" in the solar system. This ruins the symmetry of the
geocentric model, in which the Earth lies at the only center of motion
in the universe. Galileo's observations also showed that planets can
move without "losing" their satellites. It had been argued that if
the Earth moved around the Sun, the Moon would be left behind. Not
- In another blow to conventional wisdom, the discovery of vast
numbers of previously invisible stars shows that the stars are not
simply some kind of celestial decoration, "put there" for the benefit
or instruction of human beings. This is the first evidence of a huge
physical system of stars surrounding the Sun (confirming the
speculations of Digges and Bruno) and the most important contribution
to that time to the argument that the Sun itself is a star.
- The telescope is sufficiently cheap and easy to build
that anyone can test claims made about the nature of
astronomical objects. Somewhat like the Internet in our time, this
encourages the questioning of traditional authorities.
- The subsequent history of astronomy was revolutionized at
intervals by the advent of new
telescope designs and auxiliary equipment.
D. Kepler (d. 1630)
A German mathematician. His picture is at the right.
He analyzes Tycho's data, all obtained without telescopes
but much more accurate than any previous.
Without any deliberate intent, Kepler introduces
the conceptual foundation of modern empirical science:
Kepler quickly discovers that models based on pure circular
motions could not fit Tycho's data for Mars.
- Because of his great respect for Tycho's precise observational technique,
Kepler insisted that interpretive models agree with the observations
within observational uncertainty or "error." Where there is a disagreement,
it is the model, not the data, that must be revised.
- This is the basis of modern empiricism. Today, this
requirement for acceptable interpretations is taken to extend to
all relevant data. It is a demand for consistency with all
interpreted phenomena and established physical principles. This is the
"cumulative" aspect of science described in Study Guide
- The term "error" here does not imply some kind of
mistake. Instead, it refers to the uncertainties that are inherent
in any measuring process, no matter how careful. An essential part of a
scientist's job is always to make good estimates of the uncertainty in
any published data.
Kepler reinterprets the data for all planets and condenses his conclusions
to three "Laws of Planetary Motion."
- He works eight years to resolve an 8 arc-min discrepancy between
the models and data.
(The discrepancy was 8 times
the observational uncertainty of Tycho's data.)
- He finally realizes that Mars' orbit is an ellipse, not a
Ellipses are perfectly well-defined geometrical figures, studied by
the ancient Greek mathematicians. But because they are less
symmetrical than circles, Ptolemy did not attempt to incorporate them
in his cosmological model.
- The methods Kepler used to achieve this breakthrough are
re-created in the ASTR 1210 optional
laboratory on the Orbit of Mars.
Kepler's Second Law -- Click for
- Planetary orbits are ellipses with the Sun at one focus
- Note that the Sun is not at the center of the ellipse and
that there is nothing there or at the second focus of the
orbital ellipse. The distance between any planet and the Sun will
vary as it moves around its orbit.
- The Sun is in the same plane as the ellipse for a given
planet, but the orbits of different planets lie in different
The fact that the planes of the orbits of the other planets always
include the Sun but do not include the Earth is a simple but important
objection to geocentric interpretations of the Solar System.
- The planetary orbits are not very elliptical, which is why
circles are fair approximations, as in Copernicus' model.
- For a given planet, a line joining the planet as it moves and the
Sun sweeps out equal areas in the orbital plane in equal times
This implies a given planet moves faster when it is nearer the
Sun, with a specific (inverse) relationship between its sideways
motion and its distance.
[This behavior is also the first hint of a universal physical principle
not recognized until after Newton:
the conservation of angular momentum.]
The squares of the orbital periods of different planets are
proportional to the cubes of the orbital sizes (semi-major axes).
In equation form, P2 = K a3, where P is
the period, a is the semi-major axis, and K is a constant.
The time P taken to complete one orbit is therefore proportional to (a x
a1/2) and grows more than in direct
proportion to orbital size.
Java illustrations of Kepler's three laws are available
at this web
Net result: A tremendous simplification. Tens of thousands of
individual observations have been reduced to
set of simple geometric and arithmetic relationships. All the
arbitrary complexity ("wheels within wheels") of Ptolemy has vanished.
So, too, however, has the perfection of uniform, circular motion and
the beautiful symmetry so admired by the Greeks.
A planet with an orbital diameter 5 times the Earth's will
require 11 Earth years to complete an orbit.
The easiest way to think about the Third Law is that it
implies that the velocities of planets in larger orbits are
slower than for planets nearer the Sun:
A planet's mean velocity in its orbit is equal to the circumference
of the orbit divided by its orbital period. Since the circumference
of an orbit increases in direct proportion to its semi-major axis, but
the period increases more than in direct proportion, the mean
velocity of planets in larger orbits is slower. See graph above right.
The complexities such as epicycles in Ptolemy's model were
needed because of three factors: the fact that the Earth moves,
whereas Ptolemy assumed it to be stationary; the fact that actual
orbits are elliptical, whereas he assumed them to be circular;
and the fact that the velocity of a planet in its orbit
varies, whereas he assumed constant circular velocities.
Note that Kepler's Laws were derived empirically from
Tycho's data. They are not "theoretical." They simply summarize the
central observational facts.
Kepler was a very smart person, but his breakthrough was entirely
dependent on the large body of highly accurate data compiled
The concept of force
- In the early Greek cosmologies, there was no need for a
special "force" to control the motions of planets in their orbits,
since all the cosmic bodies were thought to be affixed to crystalline
spheres, and the tendency of things to fall downward was thought to be
the product of Earth's special location at the center of the
universe, not something caused by the Earth itself.
- But in the real world revealed by Kepler's work, there are no
crystalline spheres; rather, the Sun is the key to the
planetary motions. It is not just the center of the solar system but
is also directly linked to the geometry of planetary orbits
and to motions within those orbits by precise but
non-intuitive mathematical relations. Kepler therefore postulated
that the Sun exerts a force on the planets---e.g. one which
either pulls or pushes planets along their orbits. Another kind of
force was required in order to keep the Moon and the satellites of
Jupiter bound to their parent planets as they moved around the Sun.
Although Kepler was unable to formulate this idea properly, Newton
did so with phenomenal success.
E. Newton (d. 1727)
Isaac Newton was an English mathematician and physicist and ranks
as one of the two or three most important people in human history
because of the profound influence of his work on all later science and
technology. His picture on the British Pound note is shown at the
Attempting to understand Kepler's Laws, Newton develops the basic
principles of dynamics---i.e. the methods needed
to predict how objects move in response to forces.
Newton's First Law of Motion
His formulation of dynamics (his "laws of motion") draws directly from
Newton's Second Law of Motion
- In the absence of a (net) force, an object will remain at
rest or in straight-line motion with no change in velocity.
- The presence of a net force produces a
change in velocity (i.e. an acceleration).
- This law directly contradicts Aristotle, who believed that objects
cannot move at all unless subjected to a continuous force. Newton
argues that continuous, uniform motion only occurs in the absence of a
- The quantitative relationship between applied force and resulting
(Force) = (Mass) x (Acceleration)
Or, solving for the acceleration: (Acceleration) = (Force)/(Mass)
There is a directionality inherent in this formulation. Mass has
no "direction," but forces do. The Second Law implies that the acceleration
will occur in the same direction as the applied force.
- The second law is covered in more detail in Guide 8 and in
- Kepler had introduced the idea of a special force, exerted by the
Sun on the planets to keep them moving in their elliptical orbits.
Newton brilliantly adapted this concept, realizing that all bodies
could exert a force on one another. In particular,
the Earth could exert a force on the Moon to keep it in
its orbit around the Earth. But in that case the Earth should also
exert a force on objects near the Earth's surface.
- Newton argued that our everday experience of "gravity"
(our sense of a downward pull or the downward acceleration of falling
objects) was a manifestation of exactly the
same kind of force, exerted by the Earth, that kept the Moon in
its orbit. He sought a formulation that would simultaneously
explain local gravity and planetary orbits.
- So Newton postulated the existence of a universal
gravitational force in the following form:
Every object with mass exerts an attractive force on every
other object with mass, and the force declines with distance.
Fgrav = G M1 M2 / R2
where G is a universal constant, the M's are the masses of the two
objects, and R is the distance between the two masses
The same expression applies to the Earth's influence on a falling
apple, to the Earth's influence on the Moon, to the Sun's influence on
the Earth, and so on.
The gravitational force diminishes rapidly with distance.
Quantitatively, it is an "inverse square law". The Sun's
gravitational force on two identical planets, one of which is 1 AU
from the Sun and the other of which is 5 AU distant would differ by
a factor of 52 or 25.
Importantly, however, the force only becomes very small with
distance, it never actually vanishes (or goes to zero). So,
all the stars in our Galaxy, for example, despite being on average
tens of thousands of light years away, combine gravitationally to
determine the motion of the Sun around the Galaxy.
- Gravity acts along the (radial) line connecting the two bodies.
It therefore does not (as Kepler thought) necessarily act to
"push" or "pull" objects along their current directions of motion. In
fact, for a circular orbit, the gravitational force acts
perpendicular to the direction of motion.
- Newton's formulation of gravity would have been conceptually
impossible without the elimination by Copernicus of the assumption of
the special, central location of the Earth.
And conversely, the geocentric, circular-motion universe becomes
impossible if one accepts the reality of Newtonian gravity.
- [Modern footnote: Gravity was only the first universal
force to be recognized. Physics now recognizes four types
of force, the other three all much stronger than gravity but having
much shorter ranges in practice.
See Supplements 2 & 3.]
- Newton combined his gravitational force law with his Second Law
of motion in order to predict the motion of an object in an external
gravitational field. To solve the equations, Newton had to
- For two gravitating objects (e.g. one planet and the Sun) the predicted
orbit satisfies all three of Kepler's Laws. Details are given
in Study Guide 8.
- Newton's theory explained all the known properties of the
planetary orbits and also the motion of objects (like apples, bullets,
and pendulums) moving in Earth's gravity.
Remarkably, it was quickly extended to determine the orbit
of Halley's Comet (which was correctly predicted
to return in
1759) and to explain the ocean tides, the
Earth's precession, the Earth's
oblate shape, and many other formerly mysterious phenomena.
- In 1846 Newton's theory was used to correctly predict the
location of a new planet, Neptune,
based on unexpected residuals in the orbit of the planet Uranus. This
was regarded as the
"greatest triumph" of Newtonian mechanics.
Newton's Laws and theory of gravity were complete, quantitative,
and predictive. They were profoundly clarifying, and their
importance extended vastly beyond their original frame of reference.
They represent the first generalized physical laws. From them emerged
modern physics, mathematics, and engineering. They became the
cornerstone of the "Scientific Revolution."
Reading for this lecture:
Bennett textbook: Ch. 3.3 (Copernicus, Tycho, Kepler, Galileo)
Study Guide 7
Further optional reading: Arthur Koestler, The Sleepwalkers; Timothy
Ferris, Coming of Age in the Milky Way; J. L. E. Dreyer,
A History of Astronomy from Thales to Kepler; Richard
S. Westfall, Never at Rest: A Biography of Isaac Newton.
Reading for next lecture:
Bennett textbook: Ch. 4.1, 4.2, 4.3, 4.4 (Newtonian dynamics & gravitational orbits)
Study Guide 8
April 2018 by rwo
Text copyright © 1998-2018 Robert W. O'Connell. All rights
reserved. Timeline chart copyright © by Cengage Learning, Inc. .
Illustrations of Kepler's laws
by Nick Strobel. Falling
apple animation from ASTR 161 UTenn at Knoxville. These notes are
intended for the private, noncommercial use of students enrolled in
Astronomy 1210 at the University of Virginia.