Download Lecture #5 Copernicus, Kepler, Galileo, and Newton 11 June 2012

Lecture #5
Copernicus, Kepler, Galileo, and Newton
11 June 2012
Eratosthenes of Cyrene (c. 276 BCE – c. 195 BCE)
June 19, 240 BCE
Hipparchus of Nicaea (c. 190 BCE – c. 120 BCE)
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Developed trigonometry.
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Probably the best astronomical observer of antiquity.
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He constructed a very large, reasonably accurate star catalog,
and devised the magnitude system for stellar brightnesses.
He measured the longitude of Spica (the brightest star in the
constellation Virgo) sometime in the period 147 to 127 BCE.
Comparing that to the results from Timocharis and Aristillus,
roughly a century earlier, he determined that it had moved by 2o.
Well, not moved itself; rather the longitude system had moved
with respect to Spica.
From this, he determined that the Earth's axis of rotation moves
slowly, tracing a circle on the sky.
The Political Situation
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30 BCE – the Romans under Octavian (later Augustus) annexed
Egypt and the Ptolemaic empire.
Claudius Ptolemy (c. 90 CE – c. 168 CE)
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Another academic at the Library of Alexandria
Wrote the Almagest (as it was later called in Arabic), the only
surviving major astronomical treatise from antiquity.
In Planetary Hypotheses, he laid out a physical structure of
nested spheres to explain planetary motions. This introduced
the epicycle-deferent-equant concept.
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Concept of “saving the appearances”.
His Geographia was the great summation of all that was known
about the world at the time. His maps were scientific, in that
they attempted to handle the projection of the sphere onto a flat
map correctly.
His Harmonics revived the Pythagorean ratio concept, and
emphasized how such ratios related to mathematical equations.
Claudius Ptolemy (c. 90 CE – c. 168 CE)
Ptolemy's
particular
contribution
Skipping over much history
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Ptolemy's system was picked up by Islamic scholars who, in the 8th to 10th centuries
were avid translators of Greek texts and aggressive pursuers of the ideas therein.
There was a significant argument between those who felt the Greek knowledge was
irrelevant to Islam and those who did not.
Islamic follow-through on Greek science declined from the 11th century on, as one
side took ascendance.
At about that time, though, a number of Western European scholars became
interested in both the Greek and Arabic work, and began an active translation
movement, with a special interest in the works of Aristotle.
Again, there were pro- and anti-Aristotle factions, as well as pro- and anti-Averroes
(Ibn Rushd) factions. (Ibn Rushd had promoted a reconciliation of faith and reason in
order to pursue science,)
From the 13th century on, much work on natural philosophy was carried out in the
universities, which spread rapidly from Bologna, Paris, and Oxford to the rest of
Europe.
Mikołaj Kopernik (Nicolaus Copernicus; 1473 – 1543)
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Born in Torun (Thorn), part of Poland under the Kingdom of Prussia.
Studied from 1491 to 1495 at the University of Krakow (now Jagiellonian University),
leaving without a degree.
Studied canon law and humanities from 1496 to 1501 at the University of Bologna.
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Studied medicine from 1501 to 1503 at the University of Padua.
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Became more focused on astronomy, reading complaints by Puerbach and
Regiomontanus about the Ptolomaic system, and making observations of
Aldeberan that seemed to confirm their concerns.
Began studying Greek to be able to work with the original texts that related
both to astronomy and to general humanist ideas.
Eventually settled as an administrator (and eventually chancellor) in the city of
Frauenberg, in the region of Warmia, in the NE part of Poland (but full of Prussians;
the original Warmians were a Prussian tribe – and Warmia was eventually annexed by
Prussia in the First Partition of Poland, 1772).
The Political Situation
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Martin Luther posted his Disputation of Martin Luther on the Power and Efficacy of
Indulgences (known later as the Ninety-Five Theses) in 1517. He was
excommunicated, and declared outlaw in the Holy Roman Empire, in 1521.
The various parts of the Holy Roman Empire began to identify themselves either with
the Roman Catholic church or the Lutheran church.
Conflict among them triggered the Peasant Revolts of 1524-1526.
Political and social tensions continued to increase as what we now call the
Reformation began to work itself out.
Eventually, Pope Julius III convened the Council of Trent in 1551-1552, condemning
Protestantism and initiating what we now call the Catholic Counter-reformation
The Emperor, Charles V, convened a Diet in Augsberg in 1555, and established the
principle of Cuius regio, eius religio, religious uniformity in each state according to the
faith of the ruler, so long as it was either Catholic or Lutheran.
Mikołaj Kopernik (Nicolaus Copernicus; 1473 – 1543)
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Having thought through the matter, and having made relevant observations over the
course of his travels for his work, Copernicus wrote up a short treatise on his version
of the heliocentric theory: Nicolai Copernici de hypothesibus motuum coelestium a se
constitutis commentariolus, now known as the Commentariolus.
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It was just argument without mathematical or detailed observational
support, intended only for distribution to his colleagues, in order to get
feedback.
In his role as chancellor (economist), he wrote Monetae cudendae ratio, which
expressed the first version of Gresham's law (bad money drives out good money).
In 1532, he completed De revolutionibus orbium coelestium, but withheld it from
publication. People were already aware of what he was doing, and not liking it.
In 1539, Georg Joachim Rheticus came to be his student. In the course of that work,
he published a short description of the theory, the Narratio prima. Rheticus pressed
Copernicus to publish De revolutionibus, and was eventually given the manuscript to
get it published in Nurenberg.
Rheticus had to leave Nurenberg, so Andreas Osiander oversaw completion of the
printing, adding an infamous preface.
The legend is that Copernicus was given a galley copy on his death bed in 1543.
What Copernicus Didn't Like About the Ptolemaic Theory
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Why do Mercury and Venus only appear close to the
Sun?
Why do the retrograde motions of the outer planets
only appear at opposition?
There is no particular reason for ordering the planets
the way we do.
Why should all the various epicycles be such as to
exactly correct for the Earth-Sun motion? A
coincidence?
Why should the vast shell of stars be moving, rather
than the little Earth?
What Copernicus Liked About His Theory
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He explained why all the various epicycles were such as to 'exactly' correct
for the Earth-Sun motion.
The reflection of the Earth-Sun motion explains all the retrograde motions
with one concept; all Ptolemy's epicycles are ad hoc.
The order of the planets is fixed by the theory, and explains the maximum
elongations of Mercury and Venus, and the opposition retrograde motions of
the superior planets.
The relative sizes of the planetary orbits can be determined, rather than just
posited.
He got rid of the equant (which violated the spirit of uniform circular motion)
by adding additional epicycles.
Didn't look so complicated (from the above-ecliptic view).
It used a physical theory (the 'true causes') to explain observations, rather
than just an explanatory structure ( a basic philosophical disagreement with
his contemporaries).
When Copernicus Says That His System is 'Simpler'...
PTOLEMY
COPERNICUS
He Means Particularly That the Motions of the Planets,
as Seen from Above the Plane, Don't Look Like This.
Complaints to Copernicus (well, not directly to
him, because he was already dead)
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Why don't we have any sensation of the diurnal motion of the Earth? Why is it that, when you
throw something over your head, it doesn't fall behind you (because the Earth has moved under
it)?
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And by the way, if the other planets aren't naturally falling toward the Earth, then
why does stuff on the Earth still fall toward it?
And by the way, why is the Moon still orbiting the Earth, not the Sun?
Copernicus says that this means the stars are quite far away.
All these circles on circles... you have more epicycles than Ptolemy!
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The counter argument to that is: where does this 'natural' motion come from. You
need a new physics to explain this.
If the Earth is going around the Sun, we should see a parallax shift of the nearby stars. But we
don't.
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Copernicus argued that the rotation of the Earth is 'natural', not 'violent', so all
objects on it naturally partake in its rotation.
Copernicus had to do that to get rid of the equants.
Speaking of the principle of economy, why is there so much space between the orbit of Saturn
and the fixed stars? It seems a waste...
Bottom Line
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Simplicity really isn't there in the details, to a large part because
Copernicus was still wedded to the idea of circular motion.
If you only think of a theory as a method for calculating the
appearances of the phenomena (which was the approach put
forward by Osiander, totally contrary to what Copernicus would
have wanted), then either Ptolemy's theory or Copernicus'
theory was good enough for most predictions.
The big difference was that Copernicus had created a physical
theory, which claimed that it wasn't just useful, but actually
represented what the real universe looked like, and how the real
universe acted.
And as a physical theory, it could give answers for questions
that hadn't been asked, like “What is the order of the planets?”,
and “How far away are the stars?”
Clicker Question
What part of Ptolemy's theory of the heavens did
Copernicus agree with?
A – The planets revolve about the Earth.
B – The orbits of the planets are circular.
C – The fixed stars are close to the most distant
planet (Saturn).
D – The most important criterion for a theory is
that it fits the data, “saves the appearances”.
Clicker Answer: B
What part of Ptolemy's theory of the heavens did
Copernicus agree with?
A – The planets revolve about the Earth. (Copernicus
said they revolve about the Sun.)
B – The orbits of the planets are circular. (Yes, but it meant he
needed more epicycles.)
C – The fixed stars are close to the most distant planet
(Saturn). (Copernicus put the stars far away.)
D – The most important criterion for a theory is that it fits the
data, “saves the appearances”. (Copernicus wanted his
theory to be physical, to correspond to the physical truth.)
Tyge Ottesen Brahe (Tycho Brahe; 1546 – 1601)
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An exceptional astronomer because he was good at getting funds, from the Kings of
Denmark and Bohemia.
Established observatories in Hven (Uraniborg, in Denmark) and Benátky nad Jizerou
(in the part of Bohemia that is now the Czech Republic)
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These were true research institutes
Reported the November 11, 1572 “new” star (now known as SN 1572) in Cassiopeia,
in De nova stella, coining the word 'nova'.
Its lack of parallax indicated it must be much farther away than the moon. At the
time, it was thought that changes only occurred in the sublunar regions; the idea that
there could be a change among the fixed stars was revolutionary.
His observations of the Great Comet of 1577 showed that it was farther away than
the orbit of the Moon. At the time, comets were thought to be atmospheric
phenomena.
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Kepler later showed from these data that the comet was crossing the
“spheres” of the planets, which meant that they could not be physical.
SN 1572
Great
Comet
of
1577
Tycho's Instrumentation enabled the best measurements visual
astronomy could make, with accuracies ~ one arc minute.
Equatorial armillary
Sextant
Tycho's Theory
The superior planets
orbit the Earth.
Venus and Mercury orbit
the Sun, as it orbits the Earth
The Earth is still in the center
Johannes Kepler (1571 – 1630)
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In 1594, at 23, began teaching mathematics and astronomy at
the Protestant School of Graz, Austria (which became the
University of Graz).
While talking about the occurrences of the conjunctions of
Jupiter and Saturn, he realized that their orbits defined circles
with ratios that match those of the circles circumscribed and
inscribed by a triangle.
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He soon extended this to having the spheres of the
orbits be spaced by regular polygons, and published the
result in the Mysterium cosmographicum.
This began a long fascination with searching out geometric and
harmonic structure in the universe.
1570-1770
1300-1490
Johannes Kepler (1571 – 1630)
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In 1600, Kepler began working with Tycho, and continued to analyze Tycho's data
after his death in 1601.
He was expelled from Graz because he refused to become a Catholic. Fortunately,
he was hired by Emperor Rudolf II to be the imperial mathematician.
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In 1603, Kepler completed Astronomiae Pars Optica (The Optical Part of
Astronomy), which described the inverse square law for brightness versus distance,
reflections from mirrors (both flat and curved), the red color of the Moon when it is
eclipsed, and even the optical performance of the human eye.
*
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Imperial mathematician meant court astrologer.
Kepler was the first to show that the image produced by the lens on the
retina was upside-down.
He also worked out significant elements of projective geometry, including the idea
that, as the foci of a conic section shift, the figure changes continuously from one form
to another.
*The full title: Ad Vitellionem paralipomena, quibus Astronomiae pars optica traditur
Conic Sections and the Ellipse in Particular Are
Very Important to Kepler's Work
The ellipse is defined as the set
of points with the same sum of
distances from the two foci.
FOCI
Kepler showed how the conics smoothly
change from one to another.
Astronomia Nova
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Kepler put most of his effort at calculations in understanding the orbit of
Mars, which was the most difficult to fit to existing theories.
Part of what helped was thinking that there might be some motive force
directing the planets towards the Sun, similar to the magnetic force he had
read about in William Gilbert's De Magnete (1600).
In 1604, he was able to establish two laws of planetary motion:
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All planets move on ellipses, not circles, with the Sun at one focus.
As they move along their orbits, they sweep out equal areas in
equal times.
Area of section a =
Area of section b =
Area of section c
Galileo Galilei (1564 – 1642)
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Born in Pisa, Galileo trained in medicine at the University of
Pisa, but was distracted by doing experiments in physics.
Became chair of mathematics first at Pisa, and then at the
University of Padua.
In 1608, various Dutch spectacle makers, including Hans
Lipperhey and Zacharias Janssen, in Middelburg, and Jacob
Metius in Alkmaar, began to produce refracting telescopes.
Galileo acquired one, and made some modifications of his own,
and turned it to the heavens.
He published the results in 1610, in the Sidereus Nuncius (The
Starry Messenger).
This began a series of important celestial observations of
surprising accuracy.
Galileo's Observations of Jupiter
If there are bodies that go around
Jupiter, then there can be other
centers of motion In the universe
than just the Earth or the Sun!
JAN 7
JAN 8
JAN 10
JAN 11
JAN 13
JAN 15
JAN 17
JAN 17
4h later
JAN 18
JAN 19
Clear evidence that the
Moon has mountains and
other surface structure,
and so is like the Earth.
Thomas Hariot (1610)
Moon as seen through telescope
Motion of Jupiter's moons
The Telescope Shows Many Faint Stars Not
Visible to the Naked Eye
Neptune
Clicker Question
Which of these was not one of Galileo's
discoveries?
A – Sunspots
B – All revolutions don't have to be around the
same center.
C – Annual parallax of the nearest stars.
D – Venus has phases like the Moon.
Clicker Answer: C
Which of these was not one of Galileo's
discoveries?
A – Sunspots (Yes, it's in the book, not in the lecture.)
B – All revolutions don't have to be around the same center. (Yes,
that's what the moons of Jupiter prove.)
C – Annual parallax of the nearest stars. (No, his telescope wasn't
strong enough.)
D – Venus has phases like the Moon. (Yes, it's in the book, not in
the lecture.)
More Kepler
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Responding to the Sidereus Nuncius, Kepler began further work on optics,
leading to the Dioptrice (1611), on the optics of the telescope.
At the time, the primary optical theory presumed that light originated in the
eye, and illuminated the objects we see (the emission theory).
Although there were a number of contrary works, especially the Kitab alManazir (Book of Optics; 1011-1021) by Abū Alī al-Ḥasan ibn al-Ḥasan ibn
al-Haytham (Alhazen; 965 – 1040), it was Kepler's work that finally settled the
understanding of light as an external phenomenon.
It's a good thing that he did this, because without a theory of how telescopes
form images, people were not easily convinced about what Galileo was
seeing.
Over the course of 1617–1623, Kepler published Epitome astronomiae
Copernicanae (Epitome of Copernican Astronomy), which extended his
analysis of observations, and introduced his third law, the harmonic law.
Kepler's Third Law
log (T2a-3) = log k
2logT – 3loga = log k
logT = (3/2)loga + log k
So, on a log-log plot,
we have a straight line.
But the constant, log k, is
different for Jupiter's moons!
The Galileo Affair
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Galileo's Dialogue Concerning the Two
Chief World Systems, was published in
1632, as the summation of the
arguments for the Copernican theory.
There were issues political, personal,
and religious in the subsequent
controversy. But the end result was
that, in 1633, Galileo was tried, found
"vehemently suspect of heresy", and
sentenced to house arrest; and the
Dialogue was put on the Index of
Forbidden Books, until 1835.
Isaac Newton (1642 – 1727)
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In 1661, started as a sizar (a work-study student) at Trinity
College, Cambridge.
Took degree in 1665, and went home to avoid the plague.
Returned as a fellow in 1667.
After a conversation with Edmund Halley about the orbits of
comets, he worked up a paper for the Royal Society in 1684, De
motu corporum in gyrum (On the motion of bodies in an orbit),
which showed that Kepler's Laws were a consequence of
motion under an inverse square-law force.
He then wrote up his years of work on the mathematics of the
motions of bodies, Philosophiæ Naturalis Principia Mathematica
(Mathematical Principles of Natural Philosophy; 1687), in which
he presents his Three Laws of Motion.
Newton's First Two Laws
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The velocity of a body is constant if there are no
forces on it.
If there is a force (F), then the change of velocity with
time (the acceleration, a) is proportional to the force,
and inversely proportional to the mass, m.
VELOCITY is
the PAIR of
SPEED and
DIRECTION
ACCELERATION is
the CHANGE IN VELOCITY –
either the
CHANGE IN SPEED
or the
CHANGE IN DIRECTION
or the
CHANGE IN BOTH.
a=
F
m
Newton's Third Law
Newton's Third Law
To hold the rock up,
Sisyphus pushes on it,
exerting a force.
Sisyphus
pushing on
the rock
Newton's Third Law
By Newton's third law,
the rock exerts an equal
and opposite force.
Sisyphus
pushing on
the rock
The rock
pushing on
Sisyphus
Newton's Third Law
In order to counter that,
Sisyphus exerts a force
on the ground.
Sisyphus
pushing on
the rock
The rock
pushing on
Sisyphus
Sisyphus
pushing on
the ground
Newton's Third Law
So that by Newton's third
law, the ground pushes
back.
Sisyphus
pushing on
the rock
The rock
pushing on
Sisyphus
The ground
pushing on
Sisyphus
Sisyphus
pushing on
the ground
Newton's Third Law
The forces of the
rock material add
up to balance this
push.
Everything balances.
It's an equilibrium.
Sisyphus
pushing on
the rock
The rock
pushing on
Sisyphus
The ground
pushing on
Sisyphus
These two forces
balance each other.
Sisyphus
pushing on
the ground
The forces of the
ground material
add up to balance
this push.
Newton's Law of Gravitation
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G is the constant
of proportionality.
Suppose we have two masses,
m1 and m2.
Each exerts a gravitational
force, F, on the other.
The force is proportional to m1
and is proportional to m2.
The force is inversely
proportional to the square of the
distance between them, r.
The distance, r, is measured
between the centers of the two
masses.
Newton's Law of Gravitation
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G is the constant
of proportionality.
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If you increase m1 to twice its mass,
then the force is increased by a factor
of 2.
If you increase the size of m1 without
increasing its mass, then the force
stays the same.
If you increase the distance, r,
between the two masses, the force
drops to one-quarter of the original.
We adopt for our units:
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Kilograms for mass
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Meters for distance
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Newtons for force.
Then G = 6.67300 × 10-11 m3 kg-1 s-2