Download Outline of Lecture on Copernican Revolution: 1. Source of word

Outline of Lecture on Copernican Revolution:
1. Source of word “revolution” with its present meaning.
2. Why did people care where the planets were?
a. Agriculture.
b. Auguries.
g
c. Cyclic behavior suggests some law, difficult to divine.
3. Why the sun, stars, and planets all should revolve around us.
a. We are the center of everything, of course.
b. Most observed motions immediately explained.
c. Seasons caused byy inclination and speed
p
of sun’s motion.
d. Moon’s phases come out fine.
e. But there are small, nagging problems.
1
Outline of Lecture on Copernican Revolution:
4. Problems with the geocentric model:
a. Retrograde motion of the planets.
Particularly, Mars, Jupiter, and Saturn occasionally slow
down relative to the stars and actually go backwards for a
time.
b. Phases of Venus (an objection more notable after the
invention of the telescope).
Full cycle of phases, correlated with Venus’ apparent size.
5. Retrograde motion “explained.”
a Explained by epicyclic motion.
a.
motion
b. Imperfect, but still circles, just more of them.
c. This idea came from the Greek, Hipparchus, and was
elaborated by Ptolemy, who lived in Alexandria in the
second century AD.
The problem with the retrograde motion was:
• Hard to see why a god would have established so complex a
motion.
• This going forward and then occasionally backward, and then
forward again would seem to demand constant attention from a
god, who should more properly have better things to be
attending to.
Ptolemy’s epicycles derived the back and forth motion of the
planets from circles turning on circles.
• This motion might seem possible for a god to set going and
then have it run on forever, since uniform circular motion
seemed like something that could go on forever without
attention. People had been able to build devices (wheels) that
acted this way, although our human imperfection resulted in
the wheels eventually slowing down and coming to rest.
However divine wheels would be perfect and keep on turning.
2
But Ptolemy’s epicyclic motions were actually not built out of
uniform circular motions.
• The motion of the planet around the epicycle was uniform.
• But the motion of the guiding center of the epicycle about its
orbit circle was non-uniform.
• But it was only slightly non-uniform.
• This imperfection of Ptolemy’s model no doubt bothered the
experts who thought about it, but then there weren’t very many
experts around.
There was another problem with Ptolemy’s epicycles, they were
uncomfortably large.
large
• This made the motions of the planets quite bizarre.
• Ptolemy’s model might never have caught on had he used the
epicycles for the outer planets that describe their actual motion
about the sun. More on this in a moment.
The epicycles in this
diagram from
your textbook
are not drawn
correctly.
In Ptolemy’s
model,
for most
of these
planets the
epicycles
are much
larger than
these in relation
to the radii of the
planets’ orbits about the earth.
For the curious: In fact,
to get the motion of
a planet correct,
the ratio of the
radius of its
i l to
epicycle
that of its
orbit must
equal the
ratio of the
radius of its
orbit about the
sun to that of the
earth, or the inverse
of that ratio,
whichever is
smaller.
3
Through a telescope, Mars actually
appears larger during its retrograde
motion, because it is closer to the
earth at that time.
The line of dots in the
b k
background
d are
the distant
planet
Uranus.
4
With a telescope, we can measure the
variation in size of the visible disk of
Mars, and hence deduce the ratio of its
orbital radius to that of the earth.
The distance of Mars from the earth was
arbitrary in Ptolemy’s model. Its value could not be determined by observation
with the technology of Ptolemy’s time.
It was not until the invention of the telescope and Galileo’s training of
that new instrument upon the sun, moon, and planets that it became clear
that Venus exhibits a complete cycle of phases, like the Moon.
The pphases of the pplanet Venus.
Note that the disk of Venus becomes smaller as it nears its “full” phase.
Without a telescope, one could see that Venus must be closer to the earth
than the sun is, at least some of the time, because it has a “new” phase
when it is clearly between us and the sun. A “full” Venus is only
geometrically possible if Venus is further away from us than the sun.
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Outline of Lecture on Copernican Revolution:
6. Phases of Venus.
a. Hard to see phases of Venus without a telescope.
b. But variations in brightness large and obvious.
c Phases possible if Venus orbits Earth closer than the sun,
c.
sun at
a very different speed.
d. But Venus is never very far from the sun, so:
1) it must orbit earth at same average speed as the sun
2) epicycle must make it go back and forth from one side
to the other of the sun in the sky.
e. In this case, Venus can never be “full.” Its disk can at most
be only half illuminated (and actually less than this).
The epicycles in this
diagram from
your textbook
are not drawn
correctly.
If we take
the orbit
×
of the
guiding
center of
the epicycle
off Venus
V
as
shown, then this
epicycle must have
its center precisely on the line
joining the earth and the sun, and its
radius should be about as indicated.
The correct ratio of the
radius of a planet’s
epicycle to that
of its guiding
center’s orbit
is the ratio
of the radius
of the planet’s
actual orbit
about the sun
to that of the
earth’ss actual
earth
orbit. For an outer
planet like Mars,
it is the inverse
of this ratio.
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The size of Venus’
epicycle on this
diagram is
fixed by the
requirement
that the
greatest
×
observed
angle on
the sky
between
V
Venus
and
d th
the
sun come out
correctly. Thus if
we make Venus’ orbit
twice as far from the earth, we must
double the radius of its epicycle.
With the correct size
of Venus’ epicycle,
we would be
advised to
make the
radius of
the sun’s
orbit on
this diagram a bit
bigger, so
th t no one
that
might conclude
that Venus was
orbiting the sun and
not orbiting us.
×
The guiding center for the
epicycle of Venus
must lie precisely
on the line joining the earth
and the sun.
sun
This was
known to be
true, because
over thousands
of years, Venus
had never been
seen more than
about 40° away
from the
sun.
The epicycles for Jupiter
and Saturn, when
correctly drawn,
are indeed quite
small on this
diagram but
diagram,
the epicycle
for Mars
must be 66%
×
of the radius
of the orbit of
its guiding center
as shown. To get
Mars out of the
way of Venus,
we can move
it outward.
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In the Ptolemaic system, Venus orbits the earth on a circle at a
rate that keeps it close to the sun in the sky. The phases of Venus
therefore must be restricted. In particular, Venus can never be
“full.” At the very most, only half the disc of Venus could appear
illuminated by the sun.
In the Copernican system, where both the earth and Venus orbit
the sun, a full range of phases of Venus should be visible from
the earth if the radius of Venus’ orbit is smaller than that of the
earth (i.e. the orbit of Venus is inside that of the earth).
Galileo observed a full set of phases for Venus.
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Role of Ptolemy:
7. Ptolemy built a geocentric model that:
a) used centuries of observations in library at Alexandria,
b) used circular orbits and epicycles, with non-uniform
motion of the centers of the epicycles along their circular
orbits (known as the “deferent circles”),
c) was set out in 13-volume Almagest,
d) fit all observations for hundreds of years.
8. If it works, why change it?
99. By the fifteenth century,
century clear that was something wrong
wrong.
10. Aristarchus, an ancient Greek astronomer, had suggested that if
the earth orbited the sun with the other planets, retrograde
motion of the other planets was a natural consequence.
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Illustration
of how
parallax can
result in
apparent
retrograde
motion of an
outer planet.
Modern Explanation of Retrograde Motion:
11. The apparent retrograde motion of the planets is an especially
obvious example of parallax.
a. Parallax causes a nearby object to appear to move relative
to distant objects when an observer moves.
b You
b.
Yo see this effect eevery
er da
day. JJust
st look ooutt the window
indo of
a moving car.
c. Your brain uses parallax to construct a stereo view of the
world.
d. If planets appear to move “backwards” because of parallax,
then stars should exhibit this same effect.
e. Ancient
A i t people
l were able
bl to
t supportt their
th i view
i off the
th earth
th
at the center of the universe by pointing out that stars
showed no parallax.
f. Of course, stars do show parallax, but stars are so distant
that this effect was unmeasurable before the invention of
the telescope (the nearest star is only about 1 parsec away).
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This change
g
in the
apparent
position of
the star
relative to
more distant
stars is too
small to see
without a
telescope.
The largest such motion is by an amount equal to about 1/1800 of
the angular diameter of the observed disk of the Moon.
As educated, modern people, we know that in fact both the earth and Mars
orbit the sun. Because we are stuck on the earth, it is useful
to define a coordinate system with the earth fixed and
at the center. These coordinates must describe our
perception of reality. They may not, however, provide
the most useful description of reality, since fate may
not have placed us at a particularly good spot from which
to observe reality. In the coordinates in which we stay in one
central location and everything else may move, it is clear that the sun will
appear to orbit us in a circle. Because Mars is in turn orbiting the sun in something
pretty close to a circle, the orbit of Mars about us must therefore appear to be a circle
riding on a circle – an epicyclic orbit, just as Ptolemy said it was in his model of the
solar system. But just a minute, the radius of the epicycle for Mars is larger than the
radius of the guiding center of the epicycle (the radius of the orbit of the sun).
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Our modern knowledge of the motions of the planets therefore tells us that
the “actual” epicyclic orbit that Mars appears to follow about
us is quite weird. It is a big cicle riding along a little
circle, which would have seemed very strange to a
classically trained guy like Ptolemy. The idea of
epicyclic motion is fine, but the idea of a big epicycle
riding on a little guiding center circle is not fine at all.
It is simply implausible. Who would build a solar system
in such a bizarre fashion?
The model is saved, however, by a trick of geometry. It is a fact, and those who do
not believe it can come talk with me in my office hours, that precisely the same
apparent motion for Mars is generated if we switch the guiding center circle radius
and the epicycle radius. Now everything is just fine, and the epicycle looks more like
a detail, although a big detail.
Here I have fit two circles over the orbits in the diagram
from your textbook, and I have put red and blue dots over
the positions of the two planets marked in the diagram.
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We obtain the same apparent motion by letting the small circle
move along the large one as we do by letting the large circle
move along the small one.
Here the motion along each circle is by the same amount from dot to dot as in
the original diagram from your textbook.
In both cases, we move half way around the small circle in 6 equal steps, and about a
quarter of the way around the large circle in 6 equal steps.
The radius of the small circle is the radius of the orbit of the earth about the sun.
The radius of the large circle is that of the orbit of Mars about the sun.
In both diagrams, the epicycle moves counterclockwise along the deferent circle, and
the planet moves counterclockwise around the epicycle.
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We get precisely the same motion, because it does not matter whether, as on the left, we
go first along the vector with length equal to the radius of Mars’ orbit and then along the
one with length equal to the radius of Earth’s orbit, or if, as on the right, we travel along
these vectors in the opposite order.
For the case of perfectly circular orbits and perfectly uniform motion along the circles,
the paths traced out by the planet’s motion relative to us are identical.
The two constructions are different, but the resulting paths of the planet are the same.
Kepler discovered that the orbits of the planets are actually ellipses, and this
modification of the planetary motion was approximated in Ptolemy’s model by a nonuniform motion of the epicycle around the deferent circle. Copernicus made an identical
correction in his model by using a very small epicycle, which was truly a detail.
Here we see the same point once again, perhaps more clearly.
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15
Here we see how the two different constructions do indeed
produce the identical result.
In Ptolemy’s model, on the left, the guiding center of Mars’ epicycle orbits the
earth on the “deferent” circle. We need to specify the radii of the deferent and
of the epicycle, which must have a special ratio but which can both be scaled
together arbitrarily.
arbitrarily We must also specify the orbital periods for both the
deferent and the epicycle.
In the Copernican model, we must specify only the ratio of the orbital radii of
Mars and the earth as well as the ratio of their orbital periods. Two arbitrary
constants are eliminated from Ptolemy’s model, the radius of the deferent for
Mars and one period (which is replaced with the period of the earth’s orbit). An
epicycle disappears in the process as well.
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In Ptolemy’s model, on the left, the guiding center of Mars’ epicycle orbits the
earth on the “deferent” circle precisely once every year.
Surely experts noted this. It is very hard to believe that this fact is an accident.
It results naturally if the earth in fact orbits the sun, but in Ptolemy’s model it
emerges as a completely unmotivated result.
In fact, in Ptolemy’s model the periods in which also Jupiter and Saturn go
around their individual epicycles are also, astonishingly, precisely one year.
One would imagine that this accidental result would have caught someone’s
attention. Ptolemy offers absolutely no reason why this is so.
We can use this same diagram to understand the motion of an inner planet
like Venus as it appears from the point of view of the earth.
If we now ride on Mars and observe the apparent motion of Earth, we see
that the sun will orbit us (Mars), and the Earth will travel on this deferent
circle executing an epicyclic motion with an epicycle radius equal to that of
the Earth’s orbit about the sun. It will exhibit a brief retrograde motion
between points 3 and 5 on its orbit.
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These are the actual “orbits” of Venus
and Mercury as observed regarding the
Earth as fixed. Hopefully, you can see from these last few
slides why the idea of epicyclic motion was quite fitting as
a description of the observed motions of the planets.
At the left is Kepler’s
p
rendering
g of the orbit of Mars as seen from an imagined
g
point above the earth. At the right is a modern rendering. As Mars appears to
come close to the earth, the loops caused by its apparent epicyclic motion are
not all the same. This is caused by the nonuniform motion of the guiding center
along the deferent circle in Ptolemy’s model. In Kepler’s model, this feature
results because the orbit of Mars is an ellipse and not a circle. The apparent
motion of Mercury on the previous slide shows a similar nonuniform character,
but to a lesser degree.
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Both Ptolemy and Copernicus had to explain the nonuniform
motion of the planets in their orbits.
Planets move faster when they are closer to the sun.
Ptolemy “explained” this by a nonuniform motion of the guiding
center of the epicycle about the guiding center circle.
Copernicus, perhaps rediscovering something that muslim
mathematicians had realized two centuries before him, used a
small epicycle that executes precisely 2 revolutions for every
single revolution of its guiding center.
The two constructions yield exactly the same result.
Hence Copernicus built a theory that produced precisely the same
motions as that of Ptolemy.
Its results were not better, but its construction was far less
arbitrary, and it proved far more conducive to productive
thought. This is one sign in science of being on the right track.
The angular motion of the epicycle about the deferent circle is uniform only
when viewed from the equant point in Ptolemy’s model. This makes the
epicycle move faster when it is closer to the earth. This is turn gives rise to the
narrower and broader loops in the apparent motion of Mars shown on the
previous slide.
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A series of 13th and 14th century Muslim astronomers in Persia
and in Damascus had discovered that Ptolemy’s device of the
equant could be replaced by either a circle with two small
epicycles or a circle with a single eccentric (i.e. off-center)
epicycle.
Copernicus may have rediscovered this fact independently, or he
may have heard about it somehow (no one knows how).
This device has the advantage, for an Aristotelian, of replacing a
non-uniform motion about a circle with uniform motion about
multiple circles.
Copernicus used two epicycles at first, and in his final model, laid
out in his landmark book, he switched to the single, off-center
epicycle.
Copernicus’ model therefore exactly reproduces Ptolemy’s model,
and therefore it was no more accurate.
Nevertheless, Copernicus was able to eliminate the large (single)
epicycles in Ptolemy’s model (although replacing them with
much smaller ones) by using instead a single orbit for the earth
about the sun to explain all of these in one fell swoop.
Copernicus thus showed that the epicycles in Ptolemy’s model
were not in fact arbitrary or the result of accidents, but were in
fact related to one another by the single orbit of the earth.
For one thing,
g, this explained
p
whyy the epicycles
p y
for the inner
planets moved along their guiding (or “deferent”) circles at
precisely the same speed as the sun moved around the earth.
These planets were known to be closer than the sun, because they
were sometimes observed in the “new” (dark) phase.
20
Ignore Ptolemy’s “equant” on the left. It is too complicated and will not be on the
test. This diagram, however, serves to show that the epicycles in Copernicus’ model
really are very small and really make very little impact on the apparent motion of the
planets. They simply serve to generate a faster motion when the planets are closer to
the sun. Today we understand that this is necessary because the force of the sun’s
gravity is more strongly felt along this portion of the orbit. But it is a small effect.
We can be sure that Copernicus used only very small epicycles
compared with the ones used by Ptolemy, because otherwise
Copernicus would have created orbits that did not simply
slow down a little when the planet was farthest from the sun
but would have actually produced real, not just apparent,
retrograde motion.
To see this, we build orbits using Copernicus’ construction with
different sizes for the epicycles.
We MUST have the period of the epicyclic motion precisely half
the period of the orbit of the guiding center.
For too large an epicycle radius,
radius we get real retrograde motion
(shown on the next slide).
For a smaller epicycle, the effect distorts the orbits shape
considerably (second following slide).
A still smaller orbit looks just right.
21
22
This is the kind of orbit
we are after. The planet
will orbit faster at the
right
i ht andd slower
l
att the
th
left, while the orbit
remains roughly circular.
To get the right motion,
we need to place the sun
nearer to the right than to
the left portions of the
orbit.
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5a: So, what was wrong with Ptolemy’s model to a contemporary mind?
First, everything was an accident.
That is to say that each planet had its own special “deferent” circle (i.e.
guiding center circle) and its own special epicycle radius and period.
For each planet, you needed to come up with the following completely
unrelated and arbitrary parameters:
1. The radius of its deferent circle, that is, of the orbit of its guiding center
about the earth.
2. The displacement of the center of the deferent circle from the earth.
3. The period of its orbit about the deferent circle.
4. The radius of its epicycle.
5. The period of its orbit about the epicycle.
Th was no way tto establish,
There
t bli h within
ithi the
th model,
d l a necessary ordering
d i off
the sun, moon, and planets in terms of their distances from the earth.
The same observed positions on the sky could be explained by
multiplying both the radius of the deferent and the radius of the
epicycle by a common factor. The distances from the earth cannot
therefore be determined in this model and are hence arbitrary. The
work of an arbitrary-minded god?
5b: So, what was wrong with Ptolemy’s model to a contemporary mind?
One also had to explain the following amazing coincidences:
1. The guiding centers of the epicycles for both Mercury and Venus
revolve about the earth at precisely the speed of the sun’s orbit.
2. These guiding centers are both always located on the line joining the
earth and the sun.
3. The periods of the motions about the earth of ALL the guiding centers
of the planets are identical to the period of the motion of the sun about
the earth if we choose for the outer planets to generate their motion
using big epicycles traveling on small deferent circles rather than the
other way around. (That is, we must explain why it is possible to build
a model like Tycho Brahe built, that produces the same motions but in
which all the epicyclic guiding center periods are identical.)
4 If we generate
4.
t motions
ti
ffor th
the outer
t planets
l t using
i small
ll epicycles,
i l as
Ptolemy did, we need to explain why all the periods of the motions of
the outer planets about their epicycles are identical and equal to the
period of the sun’s motion about the earth. (That is, we need to explain
why the sun’s period should be involved at all here, since the motions
of the planets around the earth have nothing to do with the sun in
Ptolemy’s model.)
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Outline of Lecture on Copernican Revolution:
5b: So, what was wrong with Ptolemy’s model to a contemporary
mind?
In short, Ptolemy had an awful lot of explaining to do.
Usually, when a scientist is looking for a mechanism that can
explain a collection of observations, coming upon such an
apparently accidental equality of seemingly unrelated numbers
is an important clue to the solution to the problem.
It is often a sign that these numbers come out to be the same for a
reason that is fundamental to the mechanism that creates the
observed phenomena.
Modern scientists are now familiar with this, but in Ptolemy’s time
the clue might have been more easily missed.
But the fact that the Greek’s were aware that a heliocentric model
could naturally produce the observations suggests that the
coincidences had indeed caused people to question and to think.
In Ptolemy’s model, on the left, the guiding center of Mars’ epicycle orbits the
earth on the “deferent” circle. We need to specify the radii of the deferent and
of the epicycle, which must have a special ratio but which can both be scaled
together arbitrarily.
arbitrarily We must also specify the orbital periods for both the
deferent and the epicycle.
In the Copernican model, we must specify only the ratio of the orbital radii of
Mars and the earth as well as the ratio of their orbital periods. Two arbitrary
constants are eliminated from Ptolemy’s model, the radius of the deferent for
Mars and one period (which is replaced with the period of the earth’s orbit). An
epicycle disappears in the process as well.
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If we consider the motion of only a single planet, like Mars, which
is shown here, Ptolemy’s model does not seem much more
complicated than Copernicus’.
However, when we take the motions of all the 5 other known
pplanets at the time,, Copernicus
p
had a veryy much simpler
p
construction.
He eliminated all of Ptolemy’s large epicycles and explained the
equality of all their periods, for the outer planets, and the
periods of their guiding centers, for the inner planets.
Details remained in both models. These result because the actual planetary
orbits are elliptical and not circular.
Ptolemy dealt with this using uniform motion about the equant point for an
eccentric (displaced) deferent circle, and Copernicus dealt with it by
introducing a small epicycle for each planet.
As a result
result, the 2 models were equally inaccurate (since both were wrong),
wrong) but
Copernicus’ model was much more appealing. This appeal was not purely
aesthetic; its improved construction led the way to revisions that later
produced a correct model and suggested the mechanism of gravity.
Thus Ptolemy’s model, in which so many things appeared arbitrary or
accidental, led nowhere and proved a scientific dead end.
But Copernicus’ model produced a framework for thought in which it was soon
possible to correct the inaccuracies through a slight modification (circles
became ellipses).
The model also exposed tantalizing trends in the radii and periods of planetary
orbits, and these helped to lead later scientists to uncover the underlying
mechanism driving the motions, the force of gravity.
Thus Copernicus started out along an extremely fertile path of discovery.
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