ASTR 1210 (O'Connell) Study Guide
7. THE DISCOVERY OF GRAVITY
Thomas Digges' version of the
Following the early work of Copernicus and his contemporaries,
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, understanding, and quantification of gravity.
Here, we describe the work of four astronomers/physicists whose work
was pivotal in understanding gravity and founding modern science. To
keep track of who's who and when, you might want to consult
timeline chart linked here.
Copernican Universe (1576)
A. Tycho (d. 1601)
Observer--greatest before 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 is 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 rotated and all the other planets
orbited the Sun).
- Tycho uses a triangulation, or "parallax", technique to
show that the supernova is more distant than Saturn and therefore must
reside in the cosmic region the Greeks believed to be unchanging.
- 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 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.
B. Galileo (d. 1642)
Galileo (picture here) played a pivotal role in the transition from
medieval to modern science. He gave experiment and observation
explicit precedence over authority:
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
As an astronomer:
- Experimentally demonstrates that objects falling in response to
gravity accelerate---that is, increase their velocity---in a
highly systematic way and that their acceleration is independent
of their mass.
- This, and many other of G's
experimental results, flatly contradict the principles of
Aristotlean physics, which, owing to Greek disdain for experiment,
had never been subjected to empirical tests.
- 1609: first astronomical use of
One of Galileo's telescopes. (Click for info.)
- G's telescopes were small (above), with relatively crude lenses
only a few inches in diameter. But they yielded the first
fundamentally new astronomical data in over 1500 years. They utterly
- Discovers mountains/craters on the Moon; and spots on the
- Discovers the existence of thousands of stars fainter than the eye
- Shows that stars and planets are different kinds of objects:
planets show perceptible disks, but stars are points of light.
- Discovers the 4 brightest
satellites of Jupiter, the first new planetary bodies
discovered in recorded history.
- Discovers the phases of Venus
- These discoveries strongly support the Copernican interpretation,
though don't actually prove it:
- The craters on the Moon and spots on the Sun contradict the
claims of the Greeks (and also the Catholic Church) about the
perfection of the heavenly bodies.
- Venus' phases prove that Venus
orbits the Sun, not the Earth.
- Jupiter's satellites show the existence of a second
"center of motion" in the solar system and that planets can move
without "losing" their satellites. (It had been argued that if the
Earth moved around the Sun, the Moon could not be bound to it.)
- 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.
C. Kepler (d. 1630)
Mathematician (see his picture
Analyzes Tycho's data: all obtained without telescopes but
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".
- 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.
(This is 8 times
the observational precision of Tycho's data.)
- He finally realizes that Mars' orbit is an ellipse, not a
- The methods Kepler used to achieve this breakthrough are
re-created in the ASTR 1210 optional laboratory on the Orbit of Mars.
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 uniform, circular motion.
- 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 at the second focus of the orbital ellipse.
- The Sun is in the same plane as the ellipse for a given
planet, but the orbits of different planets can lie in different
- 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.
Kepler's Second Law -- Click for
This implies a given planet moves faster when it is nearer the Sun,
with a specific relationship between its sideways motion and its
[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.
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 complex epicycles in Ptolemy's model were necessitated by two
factors: the fact that the Earth moves, whereas it was assumed to
be stationary, and the fact that actual orbits are elliptical, whereas
they were assumed to be circular.
Note that Kepler's Laws were derived empirically from
Tycho's data. They are not "theoretical." They simply summarize the
Kepler was a very smart person, but his breakthrough was entirely
dependent on a large body of highly accurate data.
The concept of force
- In the early Greek cosmologies, there was no need for a
special "force" to control the planets in their orbits or for
interactions between the planets or with the Sun, 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, 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. Kepler therefore postulated that
the Sun exerts a force on the planets---e.g. one which either pulls or
pushes planets along their orbits. Although he was
unable to formulate this idea properly, Newton did so with phenomenal
D. Newton (d. 1727)
Mathematician & physicist. His picture on the British Pound note is
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 draws directly from Galileo's experiments.
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 speed.
- 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 can occur 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 would 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. The force declines with distance.
Quantitatively, the force is an "inverse square law":
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 gravitational force diminishes rapidly with distance. 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 25.
Importantly, however, the force only becomes very
small with distance, it never actually goes to zero. So, all the
stars in our Galaxy, for example, despite being on average 1000's
of light years away, combine to affect the motion of the Sun around
- 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.
- 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.
- 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.
- [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 resulting
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 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 were complete, quantitative, and
predictive. They were amazingly 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.
Puzzlah for Wednesday, Feb. 13:
You have two objects, A and B, both of which are the same shape.
B weighs twice as much as A. You drop both simultaneously
from a height of 3 feet. What happens?
You should attempt your own experiments to determine the
answer...don't just take the word of the readings!
Reading for next lecture:
- A (the lighter object) hits the ground first.
- B (the heavier object) hits the ground first.
- They hit at the same time.
Bennett textbook: Ch. 4.1, 4.2, 4.3, 4.4 (Newtonian dynamics & gravitational orbits)
Study Guide 8
February 2013 by rwo
Text copyright © 1998-2013 Robert W. O'Connell. All
rights reserved. 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.