Download The Laws of Planetary Motion

The Development of
Modern Astronomy
It begins with the introduction of the Suncentered Solar System by Copernicus, and
concludes with Newton's synthesis of the
laws of motion in the heavens and the
Earth, and Einstein's revision of Newton's
ideas in the Relativity Theory.
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The Earth-centered Universe of Aristotle and Ptolemy held sway
on Western thinking for almost 2000 years. Then, in the 16th
century a new idea was proposed by the Polish astronomer Nicolai
Copernicus (1473-1543).
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The Heliocentric System
Copernicus proposed that the Sun, not the
Earth, was the center of the Solar System.
Such a model is called a heliocentric system.
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In this new ordering the Earth is
just another planet (the third
outward from the Sun), and the
Moon is in orbit around the Earth,
not the Sun. The stars are distant
objects that do not revolve around
the Sun. Instead, the Earth is
assumed to rotate once in 24 hours,
causing the stars to appear to
revolve around the Earth in the
opposite direction
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Retrograde Motion and Varying Brightness
of the Planets Explained
1. The planets in such a system naturally vary in brightness
because they are not always the same distance from the Earth.
2. The retrograde motion could be explained in terms
of geometry and a faster motion for planets with
smaller orbits, as illustrated in the following animation.
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The Copernican model, with it assumption of uniform
circular motion, still could not explain all the details of
planetary motion on the celestial sphere without epicycles.
The difference was that the Copernican system required
many fewer epicycles than the Ptolemaic system because it
moved the Sun to the center.
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The Copernican Revolution
3 incorrect ideas held back the development of modern
astronomy from the time of Aristotle until the 16th and 17th
centuries
1) the assumption that the Earth was the center of the Universe
2) the assumption of uniform circular motion in the heavens
3) the assumption that objects in the heavens were
made from a perfect, unchanging substance not found
on the Earth.
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Copernicus challenged assumption 1, but not assumption 2.
The Copernican model implicitly questions the third
tenet that the objects in the sky were made of special
unchanging stuff. Since the Earth is just another planet,
there will eventually be a natural progression to the idea
that the planets are made from the same stuff that we
find on the Earth.
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His ideas remained rather obscure for about 100
years after his death. But, in the 17th century
the work of Kepler, Galileo, and Newton would
build on the heliocentric Universe of Copernicus
and produce the revolution that would sweep
away completely the ideas of Aristotle and
replace them with the modern view of astronomy
and natural science.
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Been There, Done That: Aristarchus of Samos
The idea of Copernicus was not really new! A sun-centered Solar
System had been proposed as early as about 200 B.C. by
Aristarchus of Samos (Samos is an island off the coast of what
is now Turkey). However, it did not survive long under the weight
of Aristotle's influence and "common sense":
history tends to forget that he came
to this conclusion about 1,750 years
before Copernicus did
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The Life and Times of Tycho Brahe
He allegedly challenged a fellow
student to a duel with swords in a
dispute over who was the better
mathematician. Brahe's nose was
partially cut off, and he was said to
wear a gold and silver replacement
He fell out of favor when a new King came to
power in 1588, and moved to Prague shortly
thereafter
Tycho (1546 - 1601)
This is of great historical significance because this move would
eventually make Brahe's data available to Kepler
Brahe is thought to have died when he contracted a urinary
infection while attending a banquet hosted by a baron in Prague in
which he drank extensively but felt that etiquette prevented him
from leaving the table to relieve himself before the host left.
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King Frederick of Denmark gave him the island Hven
where he built (1576) his observatory "Uraniborg"
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For 20 years he made the most accurate
observations possible with the naked eye
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Summary of Brahe's Contributions
1. He made the most precise observations that had yet been made by
devising the best instruments available before the invention of the
telescope.
2. His observations of planetary motion, particularly that of
Mars, provided the crucial data for later astronomers like
Kepler to construct our present model of the solar system.
3. He made observations of a supernova and declared it a stellar
activity and not an atmospheric activity.
4. Brahe made careful observations of a comet in 1577 and
proved that it was moving in space and not the atmosphere
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5. He made the best measurements that had yet
been made in the search for stellar parallax
Brahe did not believe that the stars could possibly
be so far away and so concluded that the Earth
was the center of the Universe and that
Copernicus was wrong.
6. Brahe proposed a model of the Solar System that was
intermediate between the Ptolemaic and Copernican models
(it had the Earth at the center).
It proved to be incorrect !
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Brahe's Data and Kepler
The next great development in the history of astronomy was
the theoretical intuition of Johannes Kepler (1571-1630), a
German who went to Prague to become Brahe's assistant.
Kepler and Brahe did not get along well.
Brahe apparently mistrusted Kepler, fearing
that his bright young assistant might eclipse
him as the premiere astonomer of his day.
He therefore let Kepler see only part of his
voluminous data.
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He set Kepler the task of understanding the orbit of the planet
Mars, which was particularly troublesome. It is believed that
part of the motivation for giving the Mars problem to Kepler
was that it was difficult, and Brahe hoped it would occupy Kepler
while Brahe worked on his theory of the Solar System
In a supreme irony, it was precisely the Martian data that allowed
Kepler to formulate the correct laws of planetary motion, thus
eventually achieving a place in the development of astronomy far
surpassing that of Brahe.
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Kepler and the Elliptical Orbits
Unlike Brahe, Kepler believed firmly in the Copernican system.
Kepler’s work with Mars led to the discovery that planets move
in elliptical not circular orbits.
Some Properties of Ellipses
1. For an ellipse there are two
points called foci (singular: focus)
such that the sum of the
distances to the foci from any
point on the ellipse is a constant.
In terms of the diagram shown to
the left, with "x" marking the
location of the foci, we have the
equation
a + b = constant
that defines the ellipse in terms
of the distances a and b.
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2. The amount of "flattening" of the ellipse is termed the eccentricity
a. The orbits of the planets are ellipses but the
eccentricities are so small for most of the planets
that they look circular at first glance.
b. Pluto and Mercury are
exceptions: their orbits are
sufficiently eccentric that they
can be seen by inspection to not
be circles.
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3. The long axis of the ellipse is called the major
axis, while the short axis is called the minor axis
(adjacent figure). Half of the major axis is termed
a semimajor axis. The length of a semimajor axis is
often termed the size of the ellipse.
It can be shown that the average separation of a planet from
the Sun as it goes around its elliptical orbit is equal to the
length of the semimajor axis. Thus, by the "radius" of a
planet's orbit one usually means the length of the semimajor
axis.
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The Laws of Planetary Motion
Kepler obtained Brahe's data after his death despite the
attempts by Brahe's family to keep the data from him in the
hope of monetary gain. There is some evidence that Kepler
obtained the data by less than legal means; it is fortunate for
the development of modern astronomy that he was successful.
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Kepler's First Law
The orbits of the planets are ellipses,
with the Sun at one focus of the
ellipse.
The Sun is not at the center of the ellipse, but is
instead at one focus (generally there is nothing at
the other focus of the ellipse).
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Kepler's Second Law:
II. The line joining the planet to the Sun
sweeps out equal areas in equal times as the
planet travels around the ellipse
The line joining the Sun and planet sweeps out equal areas
in equal times, so the planet moves faster when it is nearer
the Sun.
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Kepler's Third Law:
III. The ratio of the squares of the revolutionary
periods for two planets is equal to the ratio of the
cubes of their semimajor axes:
In this equation P represents the period of
revolution for a planet and R represents the
length of its semimajor axis. The subscripts
"1" and "2" distinguish quantities for planet 1
and 2 respectively. The periods for the two
planets are assumed to be in the same time
units and the lengths of the semimajor axes
for the two planets are assumed to be in the
same distance units.
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Kepler's Third Law implies that the period for a planet to orbit the Sun
increases rapidly with the radius of its orbit. Thus, we find that Mercury,
the innermost planet, takes only 88 days to orbit the Sun but the
outermost planet (Pluto) requires 248 years to do the same.
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Galileo: the Telescope &
the Laws of Dynamics
Galileo Galilei (1564-1642)
was a pivotal figure in the
development of modern
astronomy, both because of
his contributions directly to
astronomy, and because of
his work in physics and its
relation to astronomy. He
provided the crucial
observations that proved the
Copernican hypothesis, and
also laid the foundations for
a correct understanding of
how objects moved on the
surface of the earth
(dynamics) and of gravity.
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The Telescope
Galileo did not invent the telescope
. His little telescope was poorer than even a cheap modern
amateur telescope, but what he observed in the heavens
rocked the very foundations of Aristotle's universe and the
theological-philosophical worldview that it supported. It is
said that what Galileo saw was so disturbing for some officials
of the Church that they refused to even look through his
telescope; they reasoned that the Devil was capable of making
anything appear in the telescope, so it was best not to look
through it.
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Three Observations That “Rocked” Astronomy:
1. Sunspots
2. The Moons of Jupiter
3. The Phases of Venus
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Sunspots
Galileo observed the Sun through his telescope and saw that the
Sun had dark patches on it that we now call sunspots (he
eventually went blind, perhaps from damage suffered by looking
at the Sun with his telescope). Furthermore, he observed motion
of the sunspots indicating that the Sun was rotating on an axis.
"blemishes"
rotate
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The Moons of Jupiter
Galileo observed 4 points of light that changed their positions with
time around the planet Jupiter. He concluded that these were
objects in orbit around Jupiter.
One of the arguments against the Copernican system (and the
original heliocentric idea of Aristarchus) had been that if the
moon were in orbit around the Earth and the Earth in orbit
around the Sun, the Earth would leave the Moon behind as it
moved around its orbit.
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The Phases of Venus
Galileo used his
telescope to show that
Venus went through a
complete set of
phases, just like the
Moon. This
observation was among
the most important in
human history, for it
provided the first
conclusive
observational proof
that was consistent
with the Copernican
system but not the
Ptolemaic system.
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The Accomplishments of Newton
(1642-1727)
We shall concentrate on three developments
1) Newton's Three Laws of Motion
2) The Theory of Universal Gravitation
3) The demonstration that Kepler's
Laws follow from the Law of
Gravitation.
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The Great Synthesis of Newton
Kepler had proposed three Laws of Planetary motion based
on the systematics that he found in Brahe's data.
The Problem!
1. The laws were only applied to planets.
2. The Laws worked, but no one knew why.
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1. Newton demonstrated that the motion of objects on the
Earth could be described by three new Laws of motion
2. Kepler’s Laws of planetary motion were special cases of
His Three Laws.
3.Neton demonstrated that Kepler was only partly correct
And corrected his mistakes.
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Newton's First Law of Motion:
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.
This we recognize as essentially Galileo's concept of
inertia, and this is often termed simply the "Law of
Inertia".
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Newton's Second Law of Motion:
II. The relationship between an object's mass m, its
acceleration a, and the applied force F is F = ma.
Acceleration and force are vectors (as indicated by their
symbols being displayed in slant bold font); in this law the
direction of the force vector is the same as the direction
of the acceleration vector.
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Newton's Third Law of Motion:
III. For every action there is an equal and opposite reaction.
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What Really Happened with the Apple?
The apple is
accelerated, since
its velocity
changes from zero
as it is hanging on
the tree and moves
toward the ground.
Thus, by Newton's
2nd Law there
must be a force
that acts on the
apple to cause this
acceleration. Let's
call this force
"gravity",
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Sir Isaac's Most Excellent Idea
Now came Newton's
truly brilliant insight: if
the force of gravity
reaches to the top of
the highest tree, might
it not reach even
further; in particular,
might it not reach all
the way to the orbit of
the Moon!
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If we increase the muzzle velocity of
an imaginary cannon, the projectile
will travel further and further
before returning to earth. Newton
reasoned that if the cannon
projected the cannon ball with
exactly the right velocity, the
projectile would travel completely
around the Earth, always falling in
the gravitational field but never
reaching the Earth, which is curving
away at the same rate that the
projectile falls. That is, the cannon
ball would have been put into orbit
around the Earth. Newton concluded
that the orbit of the Moon was of
exactly the same nature
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the Moon continuously "fell" in its path around the Earth because of
the acceleration due to gravity, thus producing its orbit.
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By such reasoning, Newton came to the conclusion that any two objects in
the Universe exert gravitational attraction on each other, with the force
having a universal form:
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The Center of Mass for a Binary System
Newton, largely as a corollary of his 3rd Law, demonstrated that
the sun’s position actually was more symmetrical than Kepler
imagined and that the Sun does not occupy a privileged postion;
in the process he modified Kepler's 3rd Law.
The center of mass is familiar to anyone who has
ever played on a see-saw
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Newton's Modification of Kepler's Third Law
Because for every action
there is an equal and
opposite reaction,
Newton realized that in
the planet-Sun system
the planet does not orbit
around a stationary Sun.
Instead, Newton
proposed that both the
planet and the Sun
orbited around the
common center of mass
for the planet-Sun
system. He then modified
Kepler's 3rd Law to read,
Which turns out to be
With a small correction
Where P = T = period or orbit
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Two Limiting Cases
1. If one object is so massive that the center of mass between
two object is in the larger mass. This would be like earth
and the sun.
2. If two objects have about the same mass they both revolve
around their center of mass. This would be like a binary
star system.
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Weight and the Gravitational Force
We have seen that in the Universal Law of Gravitation
the crucial quantity is mass. In popular language mass
and weight are often used to mean the same thing; in
reality they are related but quite different things.
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Mass and Weight
What we commonly call weight is really just the
gravitational force exerted on an object of a
certain mass.
Mass is a measure of how much material is
in an object, but weight is a measure of
the gravitational force exerted on that
material in a gravitational field; thus,
mass and weight are proportional to each
other, with the acceleration due to
gravity as the proportionality constant.
Therefore, your mass would be the same on earth as it is
on the moon but your weight would decrease by 1/6
because the gravity of earth is six times greater than
the moons gravity.
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