ASTR 111

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Lecture 6

• ASTR 111 Section 002

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Ptolmey
Copernicus
Brahe
Kepler
Galileo (Galilei)
Newton
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Outline

• Exam Results
• Finish Chapter 4
• Keplers Laws
• Newtons Laws

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Exam Results - Average 89
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Percent error in students guess of their score
out of 100 Average -0.88.
Error (out of 100)
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Kepler proposed elliptical paths for the planets
about the Sun

• Using data collected by Brahe, Kepler deduced
  three laws of planetary motion
• the orbits are ellipses
• a planets speed varies as it moves around its
  elliptical orbit
• the orbital period of a planet is related to the
  size of its orbit

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Text these numbers
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Abbreviation
Circle with radius 1. x from -1.0 to 1.0 in
steps of 0.1. Compute y using
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Equation for a circle
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Equation for an ellipse
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Keplers First Law
Planets orbit the Sun in an ellipse
b
a
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Keplers Second Law
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Keplers Third Law
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Keplers Laws

• Planet orbit is ellipse
• Equal area in equal time
• Farther away planets orbit slower

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Is this an ellipse?
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• Suppose that you are looking down on a solar
  system with one planet orbiting a star. You take
  a picture every 10 days.
• Does this planet obey Keplers laws? How do you
  know?
• How would the speed of this planet change? How
  would you measure the change in speed?

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8
6
5
9
4
10
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3
2
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Based on Lecture-Tutorials for Introductory
Astronomy 2nd ed., Prather et. al, page 21
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• The following planet obeys Keplers second law.
• Draw two lines one connecting the planet at
  Position A to the star and a second line
  connecting the planet at Position B to the star.
  Shade in the triangular area swept out by the
  planet when traveling from A to B.
• Which other two planet positions, out of C-I,
  could be used together to construct a second
  swept-out triangular area that would have
  approximately the same area as the one you shaded
  in for Question 3? Shade in the second swept-out
  area using the planet positions that you chose.
  Note Your triangular area needs to be only
  roughly the same size no calculations are
  required.
• How would the time it takes the planet to travel
  from A to B compare to the time it takes to
  travel between the two positions you selected in
  the previous questions? Explain your reasoning!
• During which of the two time intervals for which
  you sketched the triangular areas in questions 3
  and 4 is the distance traveled by the planet
  greater?
• During which of the tow time intervals for which
  you sketched the triangular areas in Questions 3
  and 4 would the planet be traveling faster?
  Explain your reasoning!

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C
D
B
E
A
F
G
H
I
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1. The drawing on the following slide shows another
   planet. In this case, the twelve positions are
   exactly one month apart. As before, the plane
   obeys Keplers second law.
2. Does the planet appear to be traveling the same
   distance each month?
3. At which position would the planet have been
   traveling the fastest? The slowest? Explain
   your reasoning.
4. At position D, is the speed of the planet
   increasing or decreasing? Explain.
5. Provide a concise statement that describes the
   relationship that exists between a planets
   orbital speed and the planets distance from its
   companion star.

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E
D
C
F
B
G
A
L
H
K
J
I
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Lingering questions

• Keplers laws are not so clean
• Need to explain
• Why orbits of planets are elliptical
• Why distance from Sun is related to orbital
  period
• Why planet velocity changes during orbit
• Also want a recipe that gives good predictions of
  when eclipses will occur, where the planets will
  be in the future.

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Lingering questions

• Keplers laws are not so clean
• Need to explain
• Why orbits of planets are elliptical
• Why distance from Sun is related to orbital
  period
• Why planet velocity changes during orbit
• Why people on the south pole dont fall into
  space
• Also want a recipe that gives good predictions of
  when eclipses will occur, where the planets will
  be in the future.

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Isaac Newton

• Isaac developed three principles, called the
  laws of motion, that apply to the motions of
  objects on Earth as well as in space

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Newts Principles (Laws of Motion)

1. The law of inertia a body remains at rest, or
   moves in a straight line at a constant speed,
   unless acted upon by a net outside force
2. F m x a the force on an object is directly
   proportional to its mass and acceleration,
   provided the mass does not change
3. The principle of action and reaction 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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Group Question

• An object at rest tends to stay at rest. An
  object in motion tends to stay in motion.
• What is wrong with this statement?
• Why dont we observe objects in motion tending
  to stay in motion more often?

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Newtons Law of Universal Gravitation
A number (T.B.D.)
Mass m2
Mass m1
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• Mass and Weight are not the same
• Mass refers to how much stuff is in an object
  (atoms, molecules, etc).
• Weight refers to how much that stuff will push
  down on a scale. This depends on what planet you
  are on.

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Newtons Law of Universal Gravitation
Mass m1
A spring
Weight is a number that tells you about how much
this spring will compress. It depends on m1 and
r.
r
Mass m2
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How to get Weight mass x gravity
Mass of Earth
m/s2
Radius of Earth
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What about Bob Beamon?
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• The law of universal gravitation accounts for
  planets not falling into the Sun nor the Moon
  crashing into the Earth

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v
v
m2
m2
(You will need to take my word on this equation)
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Now suppose Earth provides pull instead of
string and arm
v
v
m2
m1
m2
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(Force needed to keep it in orbit)
(Force that can be provided)
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Is this right?

• G 6.7 x 10-11 N.m2/kg2
• m1 2 x 1030 kg
• Mars
• Orbital velocity 24 km/s
• Distance from Sun 228 x 109 km
• Earth
• Orbital velocity 30 km/s
• Distance from Sun 150 x 109 km

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Compare

• Keplers 3rd law relates orbital speed and radius
• Newtons law of gravitation was used to derive a
  relationship between orbital speed and radius
• Both will give the same answer. Which is
  better?

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To get something in orbit, you need a special
horizontal velocity

• 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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Question

• How far would you have to go from Earth to be
  completely beyond the pull of gravity?
• Suppose the Earth was 2x its current radius (with
  the same mass). How would your mass change? How
  would your weight change?

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• Given that Earth is much larger and more massive
  than the Moon, how does the strength of the
  gravitational force that the Moon exerts on Earth
  compare to the gravitational force that Earth
  exerts on the Moon? Explain your reasoning.
• Consider the following debate between two
  students about their answer to the previous
  question.
• Student 1 I thought that whenever one object
  exerts a force on the second object, the second
  object also exerts a force that is equal in
  strength, but in the other direction. So even if
  Earth is bigger and more massive than the Moon,
  they still pull on each other with a
  gravitational force of the same strength, just in
  different directions.
• Student 2 I disagree. I said that Earth exerts
  the stronger force because it is way bigger than
  the Moon. Because its mass is bigger, the
  gravitational force Earth exerts has to be bigger
  too. I think that you are confusing Newtons
  third law with the law of gravity.
• Do you agree or disagree with either or both
  students? Explain.
• How would the strength of the force between the
  Moon and Earth change if the mass of the Moon
  were somehow made two times greater than its
  actual mass?

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Earth
Mars

• In the picture, a spaceprobe traveling from Earth
  to Mars is shown at the halfway point between the
  two (not to scale).
• On the diagram, clearly label the location where
  the spaceprobe would be when the gravitational
  force by Earth on the spacecraft is strongest.
  Explain.
• On the diagram, clearly label the location where
  the spaceprobe would be when the gravitational
  force by Mars on the spacecraft is strongest.
  Explain your reasoning.
• Where would the spaceprobe experience the
  strongest net (or total) gravitational force
  exerted on it by Earth and Mars? Explain your
  reasoning.
• When the spacecraft is at the halfway point, how
  does the strength and direction of the
  gravitational force on the spaceprobe by Earth
  compare with the strength and direction of the
  gravitational force on the spaceprobe by Mars.
  Explain your reasoning.

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Earth
Mars

• If the spaceprobe had lost all ability to control
  its motion and was sitting at rest at the
  midpoint between Earth and Mars, would the
  spacecraft stay at the midpoint or would it start
  to move.
• If you think it stays at the midpoint, explain
  why it would not move.
• If you think it would move, then (a) Describe the
  direction it would move (b) describe if it would
  speed up or slow down (c) describe how the net
  (or total) force on the spaceprobe would change
  during this motion and (d) identify when/where
  the spaceprobe would experience the greatest
  acceleration.
• Imagine that you need to completely stop the
  motion of the spaceprobe and have it remain at
  rest while you perform a shutdown and restart
  procedure. You have decided that the best place
  to carry out this procedure would be at the
  postion where the net (or total) gravitational
  force on the spaceprobe by Mars and Earth would
  be zero. On the diagram, label the location
  where you would perform this procedure. (Make
  your best guess there is no need to perform any
  calculations here.) Explain the reasoning behind
  your choice.
• Your weight on Earth is simply the gravitational
  force that Earth exerts on you. Would your
  weight be more, less, or the same on the Mars.
  Explain your reasoning.

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