Download Some Calculations (cont) - Department of Physics and Astronomy

Ch 2--Gravitation/Planetary Motion
7 Sep 2000
ASTR103, Sec 3, GMU, Dr. Correll
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“Finally we shall place the Sun himself at the
center of the Universe. All this is suggested by the
systematic procession of events and the harmony
of the whole Universe, if we only face the facts, as
they say, with both eyes open.”
Nicolaus Copernicus
Greek Cosmology
• The Greek tradition of scholarship leaves a
written record of their studies
• The Greeks developed a cosmology--a theory of
the overall structure and evolution of the universe
– incorporated mathematics
– but was not truly scientific in that it was limited by an
absence of observational corroboration
• Nevertheless, their tradition became the
dominant world view in Europe and the Middle
East
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ASTR103, Sec 3, GMU, Dr. Correll
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Aristotle (384-322 BCE)
• Terrestrial and celestial realms different
– Terrestrial realm is corruptible
– Celestial realm begins at Moon’s orbit outward and is
incorruptible
• Terrestrial matter composed of four elements
–
–
–
–
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Earth
Water
Air
Fire
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Aristotelian Concepts of Motion
• Natural motion - "like seeking like”
– Earth and water go down
– Air and fire up
Fire
Air
Water
Earth
• Forced or violent motion - “forces
moving like away from like”
– Example - throwing rock upward
Forced
motion
Natural
motion
Earth
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ASTR103, Sec 3, GMU, Dr. Correll
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Celestial Motion
Prime Mover
Planet
Celestial
realm
Moon
Earth
Terrestrial
realm
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Sphere
of Stars
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Aristarchus (ca. 310 - ca. 230 B.C.)
• On the Sizes and Distances of the Sun and Moon
– Estimated Sun 20 times farther from Earth than Moon (closer to
400)
• Natural to put largest and only self-luminous body the Sun
at center--minority opinion
• Explained
– Daily movement of heavens by Earth’s rotation on its axis
– Earth and planets revolve about Sun
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Geocentric Becomes Dominant
• Geocentric became
dominant - Ptolemaic
system (2nd cent. CE)
• Geocentric cosmology
dominates until
Copernicus’ publishes
his book in 1543 CE
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ASTR103, Sec 3, GMU, Dr. Correll
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Geocentric System Order
Saturn
Jupiter
Mars
Sun
Venus
Mercury
Moon
Earth
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Accounting for Retrograde Motion
Retrograde motion--change in direction of motion of
a planet as seen against the fixed stars
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ASTR103, Sec 3, GMU, Dr. Correll
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Geocentric Cosmology
• Ptolemaic System used Cycles, Deferents and
Epicycles to account for retrograde motion
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ASTR103, Sec 3, GMU, Dr. Correll
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Heliocentric Cosmology
• Copernican model has a more “natural” explanation of
retrograde motion--especially in hindsight
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ASTR103, Sec 3, GMU, Dr. Correll
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Which is better?
• As observations of retrograde motion became
more detailed, epicycles within epicycles were
required, making a very complicated system
• Occam’s Razor (William of Occam, 14th Century)-Eliminate unnecessary hypothesis, or “keep it
simple”
• But true test is observation…so here’s how it
happened.
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ASTR103, Sec 3, GMU, Dr. Correll
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Nicolaus Copernicus (1473-1543)
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Nicolaus Copernicus
• Studied mathematics, philosophy,
astronomy, and astrology at University of
Krakow in Poland
• Studied law and medicine at Universities of
Bologna and Padua in Italy
• Elected Canon of Ecclesiastical Law by the
Church
• 1543, published, On the Revolutions of the
Heavenly Orbs
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ASTR103, Sec 3, GMU, Dr. Correll
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Nicolaus Copernicus
1543, published On the Revolutions of the Heavenly Orbs
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Copernican
System
• Orbits perfect circles
(incorrect!)
• Earth and Moon move on
circles
– Earth about Sun
– Moon about Earth
Moon
Planet
Earth
• Planets move on
epicycles, which move on
deferents (fewer
epicycles than Ptolemaic
system)
• Predictive accuracy same
as Ptolemaic system
7 Sep 2000
Epicycle
Sun
Sun
Deferent
ASTR103, Sec 3, GMU, Dr. Correll
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Arguments For Copernican System
• Quantitative material in his book
– Predictive accuracy of Ptolemaic/Copernican systems same
• Reduction of number of elements of Copernican
system compared to Ptolemaic
• Problem: Earth not at the center means you have the
possibility of seeing parallax in the position of planets
as seen at different seasons on Earth
– Parallax may not be observable because of immense
distances of sphere of stars
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ASTR103, Sec 3, GMU, Dr. Correll
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Arguments Against
• Damage to Aristotelian science, which was more than
just a physical system of world
– Aristotelian science had become Church orthodoxy - an
article of faith
• Parallax not seen (should it be?)
• Copernican system offered no observation which was
explainable only by it and not by Ptolemaic
system…at the time
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ASTR103, Sec 3, GMU, Dr. Correll
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Most Important, though...
Copernicus’ idea was the watershed event that
started a revolution in thought. It even gave a
new meaning to the word “revolution”!
The Copernican revolution forever displaced
the earth and its humble inhabitants from the
center of the universe!
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ASTR103, Sec 3, GMU, Dr. Correll
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Tycho Brahe (1546-1601)
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Tycho Brahe
• Extensive naked-eye observations which were basis
for acceptance of heliocentric cosmology
– Observations accurate to 1 minute of arc - best
naked-eye
– Published his own cosmology
• Planets orbit Sun
• Sun and Moon orbit Earth
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ASTR103, Sec 3, GMU, Dr. Correll
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Tycho Brahe
• Parallax measurements of supernova in 1572
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ASTR103, Sec 3, GMU, Dr. Correll
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Johannes Kepler (1571-1630)
…and his three laws
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Johannes Kepler
• Convinced that mathematical relations existed that could
make sense of planetary motion
– Acquired and analyzed Brahe's observations of planets
– 1609, New Astronomy, contains first two laws of planetary motion
– 1619, The Harmonies of the World, contains third law
• Extended laws derived from basically Mars to all planets,
regarded laws as universal--very imaginative step
• His big idea--orbits are ellipses, not circles!
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ASTR103, Sec 3, GMU, Dr. Correll
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Johannes Kepler
• First Law--The orbit of a planet about the Sun is an ellipse
with the Sun at one focus
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Johannes Kepler
Semi-minor axis
"a"
Semi-major axis
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Johannes Kepler
• Second Law--A line joining a planet and the Sun sweeps
out equal areas in equal intervals of time
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Johannes Kepler
• Third Law--The square of a planet’s sidereal period
around the Sun is directly proportional to the cube of the
length of its orbit’s semimajor axis
P a
2
3
» Where P is the period in years
» and a is the semimajor axis in AU of elliptical orbit
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ASTR103, Sec 3, GMU, Dr. Correll
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Johannes Kepler
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Galileo Galilei (1564-1642)
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ASTR103, Sec 3, GMU, Dr. Correll
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Galileo Galilei
• 1632, The Dialogues of Galileo Galilei on the Two
Principal Systems of the World: The Ptolemaic and
Copernican
– Powerful enemies convinced Pope that Galileo had cast
Ptolemaic system in unfavorable light
• 1633, book banned and Galileo publicly humiliated
before papal tribunal
– Forced to recant Copernican views
• Last 9 years of life spent in Arcetri under house
arrest
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ASTR103, Sec 3, GMU, Dr. Correll
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Galileo Galilei
• Galileo’s dispute with Church often recited as
classic example of “war between theology and
science”
– mostly true, but there were also conflicts of
personalities--priests and scientists are human,
too!
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ASTR103, Sec 3, GMU, Dr. Correll
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Galileo’s Astronomical Observations
• Devised telescope in 1609 having heard of its invention in
Holland
• Discoveries and observations with telescope
–
–
–
–
–
Jupiter's four large satellites
Craters and mountains on Moon
Phases of Venus
Milky Way composed of individual stars
Observes sunspots (not new)
• 1609, published discoveries in Starry Messenger
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ASTR103, Sec 3, GMU, Dr. Correll
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Galileo’s Astronomical Observations
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Method of Experimental Science
• Galileo establishes
standard for doing
experimental science
with Starry Messenger
–
–
–
–
Design experiment
Build equipment
Take and analyze data
Publish results
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ASTR103, Sec 3, GMU, Dr. Correll
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Modern Concepts of Motion
• Body in motion will continue indefinitely at same speed
and in same direction (law of inertia)
• Force is responsible for change in body’s motion
External
Force
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ASTR103, Sec 3, GMU, Dr. Correll
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Falling Bodies
• Falling body one of most common experience, but also
one of most mysterious of experiences
• In free fall in vacuum, all objects--of whatever weight,
size, or constitution--fall given distance in same interval of
time
• Free fall is uniformly accelerated motion, i.e., gains equal
increments of speed in equal times
• Earth exerts influence on body in free fall or rolling down
an inclined plane - gravity (not new to scientific
discussion)
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ASTR103, Sec 3, GMU, Dr. Correll
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Falling Bodies (cont)
Massive
Body
Less
Massive
Body
1.5 m
Time
Distance
0.1 s
0.05 m
0.2 s
0.20 m
0.3 s
0.44 m
0.4 s
0.78 m
0.5 s
1.23 m
Strike
Together
Surface of Earth
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ASTR103, Sec 3, GMU, Dr. Correll
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Galileo’s Contribution
• Galileo brings together strands of medieval
thought regarding space, time, and motion
– 1632, published Dialogues on the Two Great World
Systems
– 1638, published Discourses and Mathematical
Demonstrations Concerning Two New Sciences
Pertaining to Mechanics and Local Motion
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Isaac Newton (1642-1727)
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Isaac Newton (1642-1727)
• Educated at Trinity College, Cambridge University
– Entered in 1661 to study mathematics
• Years of 1665 and 1666 were Plague years with university
closed
– Newton spent 1665-66 on family farm
•
1687, published Mathematical Principles of Natural
Philosophy
– Instant success revealing Newton's genius
• 1704, published Opticks
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ASTR103, Sec 3, GMU, Dr. Correll
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Newton's Principia
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Newton's Principia
• A most impressive work
– Lays out essential physical concepts and laws
– Develops a new mathematical formulation of theory
– Application to many long-standing problems
• Planetary motion
• Lunar motion
• Tides
• Etc.
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ASTR103, Sec 3, GMU, Dr. Correll
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Newton's Principia
• Provides a standard for doing scientific investigations
(along with Opticks in 1704)
• Establishes for all times mathematics not only as
language of physics (and all science) but as a means
of knowing
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ASTR103, Sec 3, GMU, Dr. Correll
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Newton’s Laws
Newton’s Laws of Motion:
1 Bodies move in uniform
(straight line) motion unless
acted on by a force
2 Acceleration proportional to
force, inversely proportional
to mass
Newton’s Law of Gravity:
– Mutual gravitational
attraction is proportional to
both masses, inversely
proportional to distance
squared
F
a
m
Gm1m2
F
2
r12
F is force
a is acceleration
m is mass
3 To every action there is an
equal and opposite reaction
7 Sep 2000
ASTR103, Sec 3, GMU, Dr. Correll
•
•
•
•
F is force
G is gravitational constant
m is mass of each body
r12 is separation
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Law of Gravity
• Always attractive, never repulsive
• Based on concept of "action-at-a-distance”
– Ability of one body to influence motion of second body
without physical contact
• For two spheres, r12 is distance between centers, not
surfaces
– One can think of spherical bodies attracting each other as if
they were mathematical points
F21
F12
m2
m1
r12
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ASTR103, Sec 3, GMU, Dr. Correll
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Newtonian Mechanics
 Newton's theories of motion and gravitation reigned for
over 200 years
 accurately described all known phenomenon until late 1800s
 precession of perihelion of mercury different than predicted
 new theory of light and electomagnetism has some
philosophical problems
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ASTR103, Sec 3, GMU, Dr. Correll
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Geometries of Space
• Euclidean space - space of our experience
– Euclid, Aristotle, Galileo, Newton
• Non-Euclidean space - understood from
mathematical arguments, not practical
experience
– Einstein, and others
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ASTR103, Sec 3, GMU, Dr. Correll
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General Relativity
• 1916, general
relativity devised by
Einstein
…gravity as curvature!
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ASTR103, Sec 3, GMU, Dr. Correll
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General Relativity - Basics
• General relativity - alternative theory of gravity
– Not action-at-a-distance Newtonian force field
– Newtonian gravity may be replaced by accelerations
arising from curvature of spacetime produced by matter
– Bodies always follow shortest available path (geodesic)
in response to local structure of curved spacetime
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Two-Dimensional Spacetime Curvature
Curvature of 2-dimensional
space due to large mass star
Curvature of 2-dimensional
space due to small mass star
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Cast of Characters
Plato
Aristotle
Ptolemy
Circa 400 BC,
Greece
Circa 350 BC,
Greece
140 AD, Alexandria
Ideal form: eg—uniform
circular motion
Celestial spheres, natural
motions & violent motions
Capstone of Greek
thought—geocentric model:
Heavenly spheres with cycle
and epicycles to account for
retrograde motion of planets
Heliocentric model…no
discriminating obsevational
data, however.
N/A
N/A
Mathematical Syntaxis,
or better know as The
Almagest (thanks to the
Arabs)
De Revolutionibus
Orbium Coelestium, or
The Revolution of the
Heavenly Orbs
Galileo
Circa 1600, Italy
First telescope observations: Siderial Messenger and
Dialogue Concerning the
(make your own list)
Two Chief World
Physics of motion-Systems
acceleration of gravity
Tycho Brahe Circa 1600, Denmark 20 years of data on star and N/A
planet positions
Mystery of the Universe
Kepler
Circa 1600, Germany Keplers’ Laws
Principia
Newton
1643-1727, Great
Newtons three laws of
Britain
motion and law of gravity
Einstein
1879-1955, Germany Theory of Relativity: Gravity Journal publications
(later the USA)
as curvature of spacetime
Copernicus
7 Sep 2000
1507 AD, Poland
ASTR103, Sec 3, GMU, Dr. Correll
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Some Calculations
• Kepler’s Laws:
1 Orbits of planets are
ellipses with the sun at
one focus
2 A line from a planet to the
sun sweeps over equal
area in equal time
3 Orbital period squared is
proportional to it’s
averaged distance from
the sun cubed
• P2 = ka3
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Some Calculations
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Some Calculations
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Some Calculations
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Some Calculations (cont)
Newton’s Laws of Motion:
1 Bodies move in uniform
(straight line) motion unless
acted on by a force
2 Acceleration proportional to
force, inversely proportional
to mass
Newton’s Law of Gravity:
– Mutual gravitational
attraction is proportional to
both masses, inversely
proportional to distance
squared
F = GMm/r2
a = F/m
3 To every action there is an
equal and opposite reaction
7 Sep 2000
ASTR103, Sec 3, GMU, Dr. Correll
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Some Calculations (cont)
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Some Calculations (cont)
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Some Calculations (cont)
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Some Calculations (cont)
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Some Calculations (cont)
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…until next time
A time to embrace, and a time to
refrain from embracing;
a time to seek, and a time to
lose;
a time to keep, and a time to cast
away;
a time to rend, and a time to sew;
a time to keep silence, and a time
to speak;
a time to love, and a time to hate;
a time for war, and a time for
peace.
--Ecclesiastes 3:6-9
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