OUR UNIVERSE

1
OUR UNIVERSE
Lectures 4 - 6
WEEK 2
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Astronomy seems to have been practised by most
ancient civilisations. Many ideas, myths and
misconceptions have occurred over and over. We
follow a Western history from the Ancient Greeks
(400 BC to the present)
Astronomy seems to have been practised by most
ancient civilisations. Many ideas, myths and
misconceptions have occurred over and over. We
follow a Western history from the Ancient Greeks
(400 BC to the present)
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Gravitation Planetary Motion or The Copernican
Revolution Geocentric versus Heliocentric cosmogo
ny
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The Geocentric cosmogony Def. A theory of the
Earth's place in the Universe
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Merry-go-round analogy
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Gravitation Planetary Motion The key
difficulty is the retrograde motion of the
planets (wanderers) In the geocentric view this
required epicycles
MARS
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July 2005 to February 2006 Mars retrograde
motion
MARS
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• Aristotle (384-322 BC)
• Earth does not feel
• as if its moving
• Natural state for any body
• is to be stationary
• The circle the perfect form
• Cycles epicycles required

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Geocentric explanation of retrograde
motion Ptolemy (140 AD) in Alexandrias
Library set up precise epicycles to fit the
observed planetary motions.
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Geocentric explanation of retrograde
motion Ptolemy (140 AD) in Alexandrias
Library set up precise epicycles to fit the
observed planetary motions.
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• Ptolemy (140 AD)
• Refined the geocentric model to a high degree
• Very accurate, but also very complicated - 80
  circles!
• Refinements kept being added to account for data.
• No coherent theory behind it.

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Ptolemeys 13 -Volume Almagest covered elements
of spherical astronomy, solar, lunar, and
planetary theory, eclipses, and the fixed
stars. It remained the definitive authority on
its subject for nearly 1500 years.
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Nicolaus Copernicus (1473 - 1543) Polish
Polymath Lawyer, physician, economist, canon of
the church, and artist. Gifted in Mathematics
and influenced by the ideas of Aristarchus, he
turned to Astronomy in the early 1500s.
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Nicolaus Copernicus (1473 - 1543) The
heliocentric model explains retrograde motion
easily.
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• Nicolaus Copernicus (1473 - 1543)
• Worked out many details
• Ordering of planetary orbits.
• Mercury Venus, Inferior planets,
• always seen near Sun.
• Mars, Jupiter, Saturn, Superior planets,
• sometimes seen on opposite side of the
• celestial sphere to Sun, high
• above horizon - Earth between Sun and
• these planets.

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• Nicolaus Copernicus (1473 - 1543)
• Explained why planets appear in
• different parts of the sky on different
• dates
• Mercury Venus, Inferior planets,
• seen in west near Sunset, then in east
• just before sunrise - elongation.
• Mars, Jupiter, Saturn, Superior planets,
• best seen at night in opposition.

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• Conjunction
• The Earth, Sun and a Planet form a straight line
  in the direction of the Sun (as seen from the
  Earth)
• Opposition
• The Earth, Sun and a Planet form a straight line
  in the direction away from the Sun (as seen from
  the Earth,

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• Inferior Planets
• Inferior planets can never be in opposition
  (they are cannot be away from the sun as seen
  from the earth).
• Two Types of Conjunction
• Inferior conjunction (same side as the earth)
• Superior conjunction (opposite side)

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• Elongation of a Planet
• Elongation is the angular distance of an
  inferior planet from the Sun as seen from the
  earth.

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• Elongation of Inferior Planets
• Greatest Elongation is the maximum angular
  distance of an inferior planet from the Sun.
• Mercury 18o 28o
• Venus 45o 47o (eliptical orbits)
• If visible in the morning (Eastern Elongation)
• If visible in the evening (Western Elongation)
• Minimum Elongation occurs at .?

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• Elongation of Inferior Planets
• Greatest Elongation is the maximum angular
  distance of an inferior planet from the Sun.
• Mercury 18o 28o
• Venus 45o 47o (eliptical orbits)
• If visible in the morning (Eastern Elongation)
• If visible in the evening (Western Elongation)
• Minimum Elongation occurs at conjunction (0o
  either inferior or superior)

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• Elongation of Superior Planets
• The minimum elongation of a superior planet
  occurs at conjunction ( zero degrees)
• The greatest elongation of a superior planet
  occurs at opposition ( 180o)

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Elongation Period

• Greatest elongations of a planet happen
  periodically, with a eastern followed by western,
  and vice versa.
• The period depends on the relative angular
  velocity of Earth and the planet, as seen from
  the Sun.
• The time it takes to complete this period is the
  synodic period of the planet.

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Elongation Period

• Let
• T be the period between successive greatest
  elongations,
• ? be the relative angular velocity,
• ?e Earth's angular velocity and
• ?p the planet's angular velocity.
• Then

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Elongation Period
Hence
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Elongation Period

• But ? ?p ?e

Hence
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Elongation Period

• But ? ?p ?e

Hence
Hence
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Elongation Period

• Since

Hence
Then
Tp/e are the siderial periods
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Elongation Period

• Since

Hence
Then
Tp/e are the siderial periods
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Elongation Period

• Since

Hence
Then
Tearth 365 days Tvenus 225 days T
584 days
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Relationship between synodic and siderial periods

• Copernicus devised a mathematical formula to
  calculate a planet's sidereal period from its
  synodic period.

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Relationship between synodic and siderial periods

• Copernicus devised a mathematical formula to
  calculate a planet's sidereal period from its
  synodic period.
• E siderial period of the Earth
• P siderial period of the Planet
• S the synodic period.

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Relationship between synodic and siderial periods

• During the time S,
• the Earth moves over an angle of (360/E)S
  (assuming a circular orbit)
• and the planet moves (360/P)S.

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Relationship between synodic and siderial periods

• Let us consider an inferior planet.
• which will complete one revolution before the
  earth by the time the two return to the same
  position relative to the sun.

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Relationship between synodic and siderial periods
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Relationship between synodic and siderial periods
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Relationship between synodic and siderial periods
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for inferior planet
S is observed as time interval between
successive overtakings of one planet by the other.
for superior planet just swap
for superior planet
Box 4-1
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Nicolaus Copernicus (1473 - 1543) Determined
planetary distances from Sun by geometry in
terms 1 AU
Planet------Copernicus---Modern Mercury
0.38 AU 0.39 AU Venus 0.72 AU
0.72 AU Mars 1.52 AU
1.52.AU Jupiter 5.22 AU 5.20
AU Saturn 9.07 AU 9.54 AU
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• Nicolaus Copernicus (1473 - 1543)
• His results showed that the larger the
• orbit, the longer the period the smaller
• the speed.
• Noticed variable speed on orbits and so
• included epicycles to keep using circular
• motion!
• This made his model no better than
• Ptolemys geocentric one to astronomers
• at the time. MORE EVIDENCE NEEDED

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Tycho Brahe (1546 - 1601) Danish Astronomer
Observed Supernova Nov. 11, 1572 Danish king
financed observatory Uraniborg (sky castle) on
Hven Island. Made measurements of stars and
planets with unprecedented accuracy. Repeated
measurements with different instruments to assess
errors - pioneer of our modern practices.
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Tycho Brahe (1546 - 1601) Danish Astronomer
Observed Supernova Nov. 11, 1572 Danish king
financed observatory Uraniborg (sky castle) on
Hven Island. Made measurements of stars and
planets with unprecedented accuracy. Repeated
measurements with different instruments to assess
errors - pioneer of our modern practices.
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• Tycho Brahe (1546 - 1601)
• Attempted to test Copernicuss ideas
• about the planets orbiting the Sun.
• Failed to measure any stellar parallax
• concluded Earth was stationary and
• Copernicus wrong. (We now know the stars
• were too far away to measure parallax without a
  telescope)
• Compiled a massive data base with
• 1? 1 arcmin accuracy
• (best one can do without a telescope)

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• Tycho Brahe (1546 - 1601)
• Attempted to test Copernicuss ideas
• about the planets orbiting the Sun.
• Failed to measure any stellar parallax
• concluded Earth was stationary and
• Copernicus wrong. (We now know the stars
• were too far away to measure parallax without a
  telescope)
• Compiled a massive data base with
• 1? 1 arcmin accuracy
• (best one can do without a telescope)

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Johannes Kepler (1571 - 1630) Employed by
Tycho in 1600 in Prague. After Tychos death
Kepler inherited his data and his position as
Imperial Mathematician of the Holy Roman
Empire.
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Johannes Kepler (1571 - 1630) Employed by
Tycho in 1600 in Prague. After Tychos death
Kepler inherited his data and his position as
Imperial Mathematician of the Holy Roman
Empire.
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• Johannes Kepler (1571 - 1630)
• Kepler could be said to be the first
  astrophysicist
• He could also be said to be the last scientific
  astrologer.
• (except maybe me)

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• Johannes Kepler (1571 - 1630)
• Astrology was once kind of scientific

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• Johannes Kepler (1571 - 1630)
• Astrology was once kind of scientific
• What happened last time Venus rose in the
  constellation of the goat? Maybe something like
  it will happen again.

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• Johannes Kepler (1571 - 1630)
• Astrology
• Disaster

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• Johannes Kepler (1571 - 1630)
• Astrology
• Disaster from the Greek for bad star

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• Johannes Kepler (1571 - 1630)
• Astrology
• Disaster from the Greek for bad star
• Influenza

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• Johannes Kepler (1571 - 1630)
• Astrology
• Disaster from the Greek for bad star
• Influenza the influence of the stars

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• Johannes Kepler (1571 - 1630)
• Astrology
• Even today, how many papers have a regular
  astrology column?

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• Johannes Kepler (1571 - 1630)
• Astrology
• Even today, how many papers have a regular
  astrology column?
• But how many have a regular astronomy column?

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• Johannes Kepler (1571 - 1630)
• Astrology
• Based on the idea that the position of the
  planets in the sky fundamentally affect our
  lifes.
• But there are greater influences.

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• Johannes Kepler (1571 - 1630)
• Kepler believed in the heliocentric model.
• 29 years of struggle with the data led him to try
  elliptical orbits with dramatic success.
• He confirmed this by mapping out the shape of
  orbits by observations with Earths orbit (1 AU)
  as baseline.

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• Johannes Kepler (1571 - 1630)
• In Keplers time there were only 6 known planets
• Mercury, Venus, Earth, Mars, Jupiter and Saturn.

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• Johannes Kepler (1571 - 1630)
• In Keplers time there were only 6 known planets
• Mercury, Venus, Earth, Mars, Jupiter and Saturn.
• Why not 20, or 100?
• Why these particular spacings?
• Before Kepler no one had asked such questions.

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• Johannes Kepler (1571 - 1630)
• Consider an equilateral triangle,
• Draw a circle outside and one inside

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• Johannes Kepler (1571 - 1630)
• Consider an equilateral triangle,
• Draw one circle outside, one inside and remove
  the triangle.

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• Johannes Kepler (1571 - 1630)
• These two circles have the same ratio as did the
  orbit of Jupiter to the orbit of Saturn.

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• Johannes Kepler (1571 - 1630)
• These two circles have the same ratio as did the
  orbit of Jupiter to the orbit of Saturn.
• Spooky eh!

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• Johannes Kepler (1571 - 1630)
• These two circles have the same ratio as did the
  orbit of Jupiter to the orbit of Saturn.
• Spooky eh! But Kepler was intrigue and expanded
  on it.

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• Johannes Kepler (1571 - 1630)
• These two circles have the same ratio as did the
  orbit of Jupiter to the orbit of Saturn.
• Spooky eh! But Kepler was intrigue and expanded
  on it. A triangular prism is a tetrahedron

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• Johannes Kepler (1571 - 1630)
• These two circles have the same ratio as did the
  orbit of Jupiter to the orbit of Saturn.
• Spooky eh! But Kepler was intrigue and expanded
  on it. A triangular prism is a tetrahedron

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• Johannes Kepler (1571 - 1630)
• Could a similar geometry relate the orbits of the
  other planets?

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• Johannes Kepler (1571 - 1630)
• Could a similar geometry relate the orbits of the
  other planets?
• Kepler recalled the regular solids of Pythagoras.
• There were five.

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• Johannes Kepler (1571 - 1630)
• Could a similar geometry relate the orbits of the
  other planets?
• Kepler recalled the regular solids of Pythagoras.
• There were five.

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• Johannes Kepler (1571 - 1630)
• He believed they nested one within another.
• Hence the invisible supports of the 5 solids was
  the spheres of the 6 planets.

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Spheres enclosing solids
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Spheres enclosing solids
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Spheres enclosing solids All this, is an attempt
to fit the orbits of the planets with harmonics
in music.
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• Johannes Kepler (1571 - 1630)
• But no matter how he tried, he could not make it
  work very well.

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• Johannes Kepler (1571 - 1630)
• But no matter how he tried, he could not make it
  work very well.
• Why not?

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• Johannes Kepler (1571 - 1630)
• But no matter how he tried, he could not make it
  work very well.
• Why not?
• Because it was wrong.

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• Johannes Kepler (1571 - 1630)
• But no matter how he tried, he could not make it
  work very well.
• Why not?
• Because it was wrong.
• The later discovery of Uranus, Neptune, Pluto,
  and the others prove that

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• Johannes Kepler (1571 - 1630)
• He spent 29 years trying to make it work, but in
  the end decided that it was the observations that
  were right, not his ideas.
• Hence, he finally abandoned them.
• Astronomy wins over astrology

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• Johannes Kepler (1571 - 1630)
• In abandoning his regular solids, he was also
  able to free his mind of the perfect
  sphere/circle for orbital motion.
• Hence he considered that they may be elliptical.

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Drawing an Ellipse
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Johannes Kepler (1571 - 1630) Keplers 3
Laws of planetary motion 1) Orbital paths of
planets are ellipses, with the Sun at
one focus.(1609) 2) Line joining the planet to
the Sun sweeps out equal areas in equal
times. 3) The square of a planets orbital
period is proportional to the cube of its
semimajor axis
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Keplers 1st Law

• The orbit of every planet is an ellipse with the
  Sun at one focus.

Planet
P
Sun at a focus
Empty focus
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Keplers 1st Law
r and q are polar coordinates
e is the eccentricity of the ellipse
? is the semi-latus rectum
Planet
P
Sun at a focus
Empty focus
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Keplers 1st Law
r and q are polar coordinates
Planet
Major axis
r
r
P
q
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Keplers 1st Law
Eccentricity e
Planet
r
r
P
Semi Major axis a Semi Minor Axis b
q
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Keplers 1st Law
Eccentricity e
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Keplers 1st Law
Semi Latus Rectum ?b2/a
Planet
r
r
P
q
Note that a circle is a special type is ellipse
(one with e 0)
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Keplers 2nd Law
The line between the sun and a planet sweeps out
equal areas in equal time.
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Keplers 2nd Law
The line between the sun and a planet sweeps out
equal areas in equal time. If the planet moves
from A to B in one day. Then the Sun A and B
roughly form a triangle. The area of that
triangle is the same no matter where the planet
is on its orbit.
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Keplers 2nd Law
The orbit is an ellipse. Thus, the planet must
move faster when near perihelion than it does
near aphelion.
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Keplers 2nd Law
The orbit is an ellipse. Thus, the planet must
move faster when near perihelion than it does
near aphelion. This is because the net tangential
force involved in an elliptical orbit is zero.
As the areal velocity is proportional to angular
momentum, Kepler's second law is a statement of
the law of conservation of angular momentum..
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Keplers 2nd Law
Written symbolically,
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Keplers 3rd Law
The square of the orbital period of a planet is
proportional to the cube of its semi-major
axis. P2 ? a3
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Keplers 3rd Law
The square of the orbital period of a planet is
proportional to the cube of its semi-major
axis. P2 ? a3 Example Uranus was found to have a
period of 84 years. What is its distance from
the Sun?
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Keplers 3rd Law
The square of the orbital period of a planet is
proportional to the cube of its semi-major
axis. P2 ? a3 Example Uranus was found to have a
period of 84 years. What is its distance from
the Sun? a P2/3 842/3 19 AU
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Using his laws Kepler was the first astronomer to
predict a transit of Venus (for the year 1631)
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Galileo Galilei (1564 - 1642) One of the first to
use a telescope From 1610 onwards he saw
mountains on the Moon, sunspots on the Sun,
the rings of Saturn, Jupiters moons ( providing
a counter example to the view that Earth is the
centre of the universe)
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Galileo Galilei (1564 - 1642) One of the first to
use a telescope, His observations constitute the
beginnings of modern astronomy. His defence of
the Copernican heliocentric solar system was
published in The Starry Messenger. (Siderius
Nuncius)
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Galileo Galilei (1564 - 1642) One of the first to
use a telescope, His observations constitute the
beginnings of modern astronomy. His defence of
the Copernican heliocentric solar system was
published in The Starry Messenger. (Siderius
Nuncius)
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Galileo Galilei (1564 - 1642) He noted that as
the phases of Venus changed, so did its apparent
size. This provided decisive evidence
against Ptolemaic geocentric system.
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Venus in the Heliocentric system
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Venus in the Geocentric system
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Galileo Galilei (1564 - 1642) 1610 Using his
telescope he discovered 4 moons
orbiting Jupiter (the Galilean satellites) This
provided a counterexample to the view that Earth
is the centre of the universe
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Jupiters moons
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Jupiters moons
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1610 Galileo observed Jupiters moons.
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Isaac Newton (1642 - 1727) One of the greatest
scientists who ever lived was a great
experimentalist, mathematician, philosopher of
the scientific method .
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Isaac Newton (1642 - 1727) One of the greatest
scientists who ever lived was a great
experimentalist, mathematician, philosopher of
the scientific method .
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Isaac Newton (1642 - 1727) Principia Mathematica
1667 Newtons Laws of Motion 1) A particle will
continue moving in a straight line unless acted
on by a force. 2) Application of a force, F
causes an acceleration, a, given by ma F 3)
Action reaction are equal and opposite.
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Isaac Newton (1642 - 1727) Principia
1667 Newtons derivation of Centripetal Accelerati
on for motion in a circle using 1) A particle
will continue moving in a straight line unless
acted on by a force. 2) Application of a force, F
causes an acceleration, a, given by ma F
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Centripental
Position
Velocity
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Centripental
Position
Velocity
Draw a position vector
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Centripental
Position
Velocity
Draw a position vector
r
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Centripental
Position
Velocity
Draw a position vector
v
r
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Centripental
Position
Velocity
Draw a position vector
Draw that velocity vector
v
r
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Centripental
Position
Velocity
Draw a position vector
Draw that velocity vector
v
r
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Centripental
Position
Velocity
Draw that velocity vector
v
r
Draw a position vector some time dt later
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Centripental
Position
Velocity
Draw that velocity vector
v
r
r
Draw a radius vector some time dt later
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Centripental
Position
Velocity
v
v
r
r
Draw a position vector some time dt later
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Centripental
Position
Velocity
v
v
r
r
Draw a position vector some time dt later
Draw that new velocity vector
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Centripental
Position
Velocity
v
v
r
r
Draw a position vector some time dt later
Draw that new velocity vector
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Centripental
Position
Velocity
v
v
r
r
Now draw an acceleration vector
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Centripental
Position
Velocity
v
v
r
r
Now draw an acceleration vector
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Centripental
Position
Velocity
v
v
r
r
And here
Now draw an acceleration vector
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Centripental
Position
Velocity
v
v
r
r
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Centripental
Position
Velocity
v
v
r
r
The time taken for both the position vector and
the velocity vector to complete one cycle must be
the same.
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How long does it take the position to complete
one cycle?
v
v
r
r
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How long does it take the position to complete
one cycle? Circumference divided by the velocity.
v
v
r
r
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How long does it take the position to complete
one cycle? Circumference divided by the thing
that is changing v.
v
v
r
r
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How long does it take the velocity to complete
one cycle?
v
v
r
r
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How long does it take the velocity to complete
one cycle? The circumference divided by the thing
that is changing a
v
v
r
r
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How long does it take the velocity to complete
one cycle? The circumference divided by the thing
that is changing a
v
v
r
r
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But the periods P are the same for both.
v
v
r
r
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But the periods P are the same for both. Hence,
v
v
r
r
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But the periods P are the same for both. Hence,
v
v
r
r
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Centripetal Acceleration
v
dx
r
dq
r
v
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Centripetal Acceleration
v
v at A
dq
dv
v
v at B
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Apply Newtons 2nd Law
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Apply Newtons 2nd Law
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Isaac Newton (1642 - 1727) Principia Mathematica
1667 Newtons Laws of Motion 1) A particle will
continue moving in a straight line unless acted
on by a force. 2) Application of a force, F
causes an acceleration, a, given by maF 3)
Action reaction are equal and opposite.
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Isaac Newton (1642 - 1727) Principia
1667 Newtons Law of Universal Gravitation
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Newtons Law of Universal Gravitation
mP
r
MSun
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Newtons Law of Universal Gravitation
mP
r
MSun
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Newtons Law of Universal Gravitation
mP
r
MSun
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Newtons Law of Universal Gravitation
mP
r
How did Newton derive this law?
MSun
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Newtons Law of Universal Gravitation
mP
r
He made it up
MSun
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Newtons Law of Universal Gravitation
mP
r
Its an educated guess
MSun
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Newtons Law of Universal Gravitation
mP
r
He made a few educated guesses Until he found one
that worked.
MSun
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Isaac Newton (1642 - 1727) To keep the planet in
an orbit of radius r, requires a centripetal
force F(centripetal). This is provided by the
Suns gravitational force F(grav). F(centripetal)
F(grav)
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Earth has P 1 yr, a 1 AU
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Keplers 2nd Law
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Keplers 2nd Law
The line joining a planet to the sun sweeps out
equal areas in equal time. A consequence of the
law of conservation of momentum
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The ice skater Conserves Angular Momentum
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Angular Momentum is L Momentum ? lever
arm Illustrate for circular motion
v
Conservation is L constant
m
r
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v
r
A
r
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Area swept out on one second is
v
r
A
r
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Area swept out on one second is but P 2p/w
v
r
A
r
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Area swept out on one second is but P 2p/w
v
r
A
r
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Area swept out on one second is but P 2p/w
and v wr
v
r
A
r
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Area swept out on one second is but P 2p/w
and v wr
v
r
A
r
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Conservation of Momentum
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Conservation of Momentum
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Conservation of Momentum
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Conservation of Momentum
L, 2, and m are all constant, hence A must be a
constant.
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Real Planetary Orbits Both bodies orbit about a
common centre of mass.
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Real Planetary Orbits Both bodies orbit about a
common centre of mass.
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Real Planetary Orbits Kepler's 3rd Law (Newton's
Form)
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Example

• Jupiters moon Europa has a period of 3.55 days
  and its average distance from the planet is
  671,000 km. Determine the mass of Jupiter.

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We know 4, p, a, G, and P but neither of the two
masses, giving one equation with two unknowns.
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We know 4, p, a, G, and P but neither of the two
masses, giving one equation with two unknowns.
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We know 4, p, a, G, and P but neither of the two
masses, giving one equation with two
unknowns. Make the reasonable assumption that
the mass of Europa is zero.
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We know 4, p, a, G, and P but neither of the two
masses, giving one equation with two
unknowns. Make the reasonable assumption that
the mass of Europa is zero (i.e., that mJ mE
mJ).
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In Solar Units a in AU P in years M in
solar masses
212
THE END OF LECTURES 4-6