Chapter 4 Gravitation and the Waltz of the Planets

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Chapter 4 Gravitation and the Waltz of the
Planets
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Announcements

• Exam 1
• Friday (Feb 14), not Wednesday.
• In class, closed book.
• 25 multiple choice questions
• Covers Chapters 1-4
• Review session Thursday evening 630-730 in 70
  VAN.
• Practice exam will be on web site Wednesday
  afternoon.

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Ancients knew that five wandering stars seemed
to slowly move among the constellations.
These wandering stars, commonly known as
planetes, typically move from west to east,
except during brief periods where they move
backwards or retrograde. The early Greek model of
a celestial sphere did not adequately account for
these retrograde loops.
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Ancient astronomers invented geocentric models to
explain complex planetary motions
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Summary of Important Historical Benchmarks in
development of heliocentric model

• Aristarchus (c. 280 BCE) determines relative size
  of Earth, Moon, Sun, argues for heliocentric
  model.
• Eratosthenes (c.200 BCE) determines size of
  earth
• C. Ptolemy (c.150).Geocentric model (retrograde
  motion caused by epicycles)
• N. Copernicus (c.1580) Heliocentric model but
  circular orbits. This model is not in any better
  agreement w. obs. than Ptolemy
• Tycho Brahe (c.1600) Most accurate observations
  Use parallax to that some objects (comets,) must
  be much farther away than previously thought.
  Observations dont agree with either model!
• J. Kepler (c.1610) heliocentric model with
  elliptical orbits provide much better agreement
  w. obs. Three laws of planetary motion
• G. Galileo (c.1630) Observations of Jupiters
  moons, Sun, Venus all support heliocentric
  hypothesis.
• Isaac Newton (c.1665) Law of gravity explains
  Keplers laws, all observations. Heliocentric
  theory becomes widely accepted.

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Claudius Ptolemy (c. 85-165 AD)

• Greatest ancient astronomer
• Worked in Alexandria at the great library
• Ptolemy published the Almagest, an encyclopedia
  in which he used centuries of Babylonian
  observations of the motions of the planets to
  justify a geocentric (Earth centered) model of
  the universe.
• The ideas of the Ptolemaic system ruled the
  world of astronomy for 1,500 years, until
  Copernicus and Galileo (c. 1600)

9th cent. Greek MSS of the Almagest (Vatican
Library)
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Ptolemy devised the longest used geocentric model
to explain retrograde loops by putting planets on
epicycles and deferents.
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Epicycles explain retrograde motion in geocentric
models
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Nicolaus Copernicus (c. 1580) devised the first
comprehensive heliocentric (Sun-centered) model

• Copernicus imagined a universe where the Sun was
  at the center instead of Earth.
• He suggested that Earths motion around the Sun
  provided a more natural explanation for
  retrograde loops as Earth passed the other
  planets.

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Heliocentric planetary position terminology is
stated relative to Earth
Opposition Inferior conjunction Superior
conjunction Greatest eastern elongation (appears
east of the Sun in the sky) Greatest western
elongation (appears west of the Sun in the sky)
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In this heliocentric model, the planets just
appear to move backwards as the faster moving
Earth laps the more distant planet once each
year when it is at opposition.
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Tycho Brahes astronomical observations disproved
ancient ideas about the heavens.

• Brahe constructed enormous instruments to
  meticulously record the precise positions of the
  planets in the sky to an accuracy never
  previously obtained.

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Tycho Brahes astronomical observations disproved
ancient ideas about the heavens.

• Using PARALLAX, Brahe was able to demonstrate
  that the comet of 1577 was beyond the Moons
  orbit and that the supernova of 1572 was in the
  distant realm of the stars.

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Johannes Kepler (1571- 1630)
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Keplers model for the Solar System A Concentric
Series of Crystalline Spheres
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Keplers Three laws

• 1st Law All orbits are Elliptical with Sun at
  one focus (circular orbits are special case of
  elliptical with e 0)
• 2nd Law Planets sweep out equal areas in equal
  time
• 3rd Law P2 a3 (P in yrs, a in AU)

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1st Law Kepler proposed elliptical paths for the
planets about the Sun.

• Elliptical Eccentricity (e) a number ranging
  between zero (for a flat line) and one (for a
  perfectly round circle).

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2nd Law Kepler proposed speed of planets in
orbit varied, fastest at perihelion, slowest at
aphelion

• A line joining a planet and the Sun sweeps out
  equal areas in equal intervals of time.

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3rd Law Kepler found an algebraic relationship
between a planets orbital period and its average
distance from the Sun

• Keplers Third Law of Planetary Motion
• The square of the sidereal period of a planet is
  directly proportional to the cube of the
  semi-major axis of the orbit.
• PYr2 aAU3

Example If a 10 AU, P 103/2 yrs 33 yrs.
Keplers laws explain how the universe works, but
they do not explain why. The why explanation
was given by Isaac Newton.
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Keplers laws PRS quiz

• A planet moves fastest in its orbit at
• Perihelion
• Aphelion
• Along ecliptic
• At one focus
• If the distance of a planet from the Sun is 2 AU,
  its orbital period is closest to
• 1.2 yrs
• 2.0 yrs
• 2.8 yrs
• 4.0 yrs
• Which of the following is true?
• All planetary orbits are ellipses with the Sun
  at one focus
• All planetary orbits are circles except Pluto
• All planetary orbits are centered on the Sun
• All planetary orbits are parabolas with the Sun
  at the focus

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Galileos discoveries of Jupiters moons with his
telescope showed that Earth was not the center of
all orbits strongly supported a heliocentric
model even though Copernicus model was no more
accurate than Ptolemys.
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Galileo also observed blemishes on the Sun
(sunspots) This contradicted perfect spheres
idea of Aristotle. This was an important example
of scientific method Observations can test (and
disprove) hypotheses
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Galileos discoveries of Venus phases with his
telescope showed that Venus must orbit the Sun
strongly supported a heliocentric model
even though Copernicus model was no more
accurate than Ptolemys.

• Venus is clearly smallest when it is at superior
  conjunction and largest when it is close to
  inferior conjunction.

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Galileos discoveries of Venus phases with his
telescope showed that Venus must orbit the Sun
strongly supported a heliocentric model
even though Copernicus model was no more
accurate than Ptolemys.

• Venus can only go through phases if it orbits the
  Sun.

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Galileo and the Roman Inquisition

• 1610-1616 First telescopic observations support
  heliocentric hypothesis. Galileo publishes
  results in the book De Revolutionibus Orbium
  Coelestium and several letters.
• 1616 After formal complaints from friars to
  Rome, Galileo first reports to Office of
  Inquisition. De Revolutionibus put on Index of
  Forbidden Books, Galileo ordered silent about
  heliocentric theory.
• 1630. After receiving permission tfrom the Pope
  (Urban II) to discuss heliocentric hypothesis as
  a theory, not a fact, he publishes Dialogue
  Concerning the Two Chief World Systems, or
  Diologio.
• 1633 Charged with heresy, retried by Office of
  Inquisitor. Galileo, age 68, formally recanted,
  was sentenced to house arrest in his home near
  Florence (Arcetri), where he died in 1642.
• 1966 Roman Church abolishes Index of Banned
  Books, admits Galileo trial was a mistake.

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The Unfortunate Case of Giordano Bruno (1548-1600)

• 1548 Born near Naples. Italian polymath
  (theologian, mathematician, astronomer,
  Dominican friar)
• 1576-1590 Flees Italy after threat of
  Inquisition. Lives in France, England, Germany,
  Hungary
• 1584 pubishes De l'Infinito, Universo e Mondi
  ("On the Infinite Universe and Worlds") in
  London.
• 1591 arrested in Venice, recanted under threat of
  torture.
• 1591-1600 imprisoned in Rome, frequently
  questioned about beliefs
• 1600 refuses to recant again, burned at the
  stake.
• Statue of Bruno is presently at site of his
  execution in Rome (Plaza Navona)

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Recommended book about Galileos life Galileos
Daughter (Sobel)
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Sir Isaac Newton (1642 1727) the greatest
physicist who ever lived
As a young faculty member at Cambridge University
(c.1665)
As Warden of the Royal Mint (1705)
Newton was knighted by Queen Anne. However, the
act was "an honor bestowed not for his
contributions to science, nor for his service at
the Mint, but for the greater glory of party
politics in the election of 1705"
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Newtons thought experiment (gedanken) Note
ignores air friction

• Cannonball speed must be 7.9 km/s to achieve
  orbit (17,700 MPH, 26,000 fps). This is much
  higher than muzzle velocities from guns
• Note that cannonball is always in free fall, but
  does not hit Earth if it is in orbit.
• Equation for V

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The law of universal gravitation accounts for
planets not falling into the Sun nor the Moon
crashing into the Earth Paths A, B, and C do not
have enough horizontal velocity to escape
Earths surface whereas Paths D, E, and F
do. Path E is where the horizontal velocity is
exactly what is needed so its orbit matches the
circular curve of the Earth.
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Isaac Newton formulated three laws that describe
fundamental properties of physical reality.

• A body remains at rest, or moves in a straight
  line at a constant speed, unless acted upon by a
  net outside force.
• F ma (the force on an object is directly
  proportional to its mass and acceleration).
• Whenever one body exerts a force on a second
  body, the second body exerts an equal and
  opposite force on the first body.

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Newtons description of gravity accounts for
Keplers laws and explains the motions of the
planets.

• Newtons Law of Universal Gravitation
• Two bodies attract each other with a force that
  is proportional to the mass of each body and
  inversely proportional to the square of the
  distance between them.

Where the gravitational constant G 6.67 10-11
N m2/kg2 and r is the distance between the
objects with masses m1 and m2.
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Newtons law of gravitation
r
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Newtons Law of Gravity PRS Quiz

• If the distance between 2 equal masses is
  doubled, the gravitational force between the
  masses
• Doubles
• Halves
• Is 4x stronger
• Is ¼ as strong
• If one mass is doubled and the other is halved,
  while the distance remains the same, the
  gravitational force
• Is double the original force
• Is ½ the original force
• Is 2x the original force
• Is equal to the original force
• An astronaut who weighs 200 lb on Earth is sent
  to Planet X, which has ¼ the mass of Earth and ½
  the radius of Earth. How much does she weight on
  Planet X?
• 50 lb
• 100 lb
• 200 lb
• 400 lb

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Mathematically speaking, Newton discovered that
orbiting bodies may follow any one of a family of
curves called conic sections.
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Comet Halley Orbit An Ellipse with Sun at one
Focus
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Elliptical Orbits How do they differ from
circular orbits?
N.B. A circle is an ellipse with 0.0 eccentricity
(e 1-b/a)
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Newtons laws also explain tidal forces which can
deform planets, reshape galaxies.
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Newtons laws also explain tidal forces which can
deform planets, reshape galaxies.
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Newtons laws also explain tidal forces which can
deform planets, reshape galaxies.
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Gravitational forces from the Sun in addition to
the Moon can create abnormally high tides, called
spring tides.
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Gravitational forces from the Sun can also
diminish each others effects. When the Sun and
Moon are at right angles, they create abnormally
low tides, called neap tides.
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Earth-Moon System Note that tides lag
Earth-Moon line (1 hr)
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Guiding Questions

• How did ancient astronomers explain the motions
  of the planets?
• Why did Copernicus think that the Earth and the
  other planets of around the Sun?
• What did Galileo see in his telescope that
  confirmed that planets orbit the Sun?
• How did Tycho Brahe attempt to test the ideas of
  Copernicus?
• What paths do the planets follows as they move
  around the Sun?
• What fundamental laws of nature explain the
  motions of objects on Earth as well as the
  motions of the planets?
• Why dont the planets fall into the Sun?
• What keeps the same face of the Moon always
  pointed toward the Earth?