PowerPoint Presentation Astronomy 112

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Astronomy 112
Dr. Steve Desch Arizona State University Spring
2009
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• RecapWe discussed Kepler's new model for the
  Solar System
• Earth is a sphere floating in space. Consistent
  with observations that of shadow lengths in
  different locations, with fact that some stars
  invisible from Phoenix, and the shape of Earth's
  shadow during lunar eclipses.
• The Moon orbits around the Earth. Consistent
  with eclipses consistent with phases of the
  Moon.
• The Earth/Moon and all planets orbit the Sun.
  Consistent with the positions of the planets in
  the sky Mercury and Venus near the Sun Mars,
  Jupiter and Saturn exhibit retrograde motion.
• So far that's not an improvement on the Ptolemaic
  model...

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RecapWe discussed Kepler's new model for the
Solar System Planet orbits obey Three Laws (1)
Planets orbit the Sun in ellipses, with the Sun
at a focus of the ellipse.
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a semi-major axis
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• RecapWe discussed Kepler's new model for the
  Solar System
• Planet orbits obey Three Laws
• 1. Planets orbit the Sun in ellipses, with the
  Sun at a focus of the ellipse.
• 2. Period of a planet around the Sun, P, is
  related to semi-major axis of the orbit, a
• P2 a3
• P period in years, a orbit size in AU

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Planet a (AU) a3/2 P (yr) Mercury
0.38 0.24 0.24 Venus 0.72
0.61 0.61 Earth 1.00 1.00
1.00 Mars 1.52 1.88 1.88
Jupiter 5.2 11.8 11.8 Saturn
9.6 29.5 29.5
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• RecapWe discussed Kepler's new model for the
  Solar System
• Planet orbits obey Three Laws
• 1. Planets orbit the Sun in ellipses, with the
  Sun at a focus of the ellipse.
• 2. Period of a planet around the Sun, P, is
  related to semi-major axis of the orbit, a
• P2 a3
• P period in years, a orbit size in AU
• 3. A planet's speed varies along its orbit such
  that it sweeps out equal areas in equal times.

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larger distance from star means planets moves
less along orbit, i.e., slower
1 week
1 week
If these areas are equal, the planet must take
equal times to cross them.
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• RecapWe discussed Kepler's new model for the
  Solar System
• Significant improvement over Ptolemaic model.

Mars predicted to be here in Ptolemaic model
Mars actually is here.
Mars predicted to be here in Kepler's model.
8 minutes of arc 8/60 of a degree, 1/2700 of
a circle.
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• RecapWe discussed Kepler's new model for the
  Solar System
• Consistent with lots of new observations, many of
  them made by Galileo
• Venus has phases goes around the Sun
• Jupiter has moons that go around it and have
  orbits that obey Kepler's laws, too!

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• RecapWe also discussed observations of stars.
• Parallax is when a star appears to move in the
  sky because of our changing vantage point.
• Tycho Brahe could not detect any parallax. He
  thought this meant the Earth stood still, but
  really (as we'll see) the stars are just very,
  very far away.

Earth, December
new star
Earth, June
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• RecapWe also discussed observations of stars.
• Stellar aberration is when the Earth's motion
  affects the direction we perceive light to be
  coming from.

Star might really be directly above Earth's
orbit...
But we have to tilt the telescope to get light to
hit the back of it.
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c speed of light
V speed of telescope / Earth
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c
V
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c
V
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Star expected here
Star actually seen here
V
? ? 20 1/180 degree
c
tan ? V / c
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• RecapWe also discussed observations of stars.
• Finally we discussed how eclipses of Jupiter's
  moons were used by Roemer to determine the size
  of the Solar System.

eclipses on time
eclipses 8 minutes early
eclipses 8 minutes late
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Recap The heliocentric model of Kepler makes
predictions about the size of the Solar
System. Earth's radius 6380 km (Eratosthenes's
experiment) Distance to Moon 385,000 km
(parallax of Moon) Radius of Moon 1750 km (from
its apparent size) Distance to Sun 1 AU
?? Aristarchus's experiment said 1 AU 20 x
Earth-Moon distance (actually 390 x, we know
now). Aberration of starlight timing of
Jupiter's eclipses say 1 AU distance light
travels in 8 minutes ??
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Speed of light wasn't known for sure in the 1600s
and 1700s. Distance from Earth to Sun 1 AU was
determined by observing transits of Venus
Very rarely, Venus goes between the Earth and
Sun, in 1639, 1761, 1769, 1874, 1882, 2004, 2012
US Naval observatory photo of 1882 Venus transit
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Captain Cook 1769
From South Pacific
From England
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D
D d (Rv)/(Re-Rv) d(0.72)/(1-0.72) d2.5
From this, the radius of the Sun is found to be
696,000 km Then the distance to the Sun is 1 AU
149,000,000 km
Multiple methods are giving the same answers now
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Gravity
Why do the planets go in ellipses around the
Sun? Why do all the planets obey P2 Const x
a3? Can we figure out what the Constant should
be, and then calculate the value of 1 AU?
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Isaac Newton (1642 - 1727) was the first person
to figure out the whole thing both how things
move, and how gravity makes them
accelerate. (Building on work by Galileo)
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• Newtons Three Laws
• A body (e.g., hockey puck) at rest stays at rest.
• A body in motion (e.g., a hockey puck) will
  continue in a straight line at constant speed
  until it feels a force (a push or pull)
• For every force there is an counter-force, equal
  in strength and opposite in direction. You push
  on the hockey puck, it pushes back.

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t 10 s
t 0 s
position
x 0 m
x 100 m
Force
velocity
v 30 m/s
v 0 m/s
acceleration
a (30 m /s) (10 s) 3 m/s2
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a acceleration how quickly velocity is
changing F force causing the acceleration m
mass the heaviness of the object a F / m
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Y velocity doesnt change (A body at rest remains
at rest...)
y
x
X velocity doesnt change (A body in motion
continues in a straight line at constant speed...)
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Y velocity changes because there is a force in
that direction (Newtons 3rd law)
y
Force
x
X velocity doesnt change (A body in motion
continues in a straight line at constant speed...
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String with tension, pulling the weight toward
the center
velocity after acceleration 2
Force
2
1
velocity after acceleration 1
2
velocity after acceleration 1
1
original velocity
accelerated this way
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Some other force is pulling objects toward the
center
velocity after acceleration 2
Force
2
1
velocity after acceleration 1
2
velocity after acceleration 1
1
original velocity
accelerated this way
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This force is called centripetal force F m V2 /
r
velocity after acceleration 2
Force
2
1
velocity after acceleration 1
2
velocity after acceleration 1
1
original velocity
accelerated this way
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Lets estimate the strength of that centripetal
force. How much pull is needed to keep a mass
moving in a circle?
Average acceleration a (2 V) / ( ? R / V) a
V2 / R
Change in velocity 2 V
Change in time ? R / V
Centripetal acceleration
t 0
t ? R / V
t 2 ? R / V
Centripetal force F m V2 / r
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The apple does not remain at rest. It is
accelerated downward. It feels a force.
a F / m
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The force is due to gravity. The Earth is
pulling ALL objects down. The strength of the
pull depends on the mass of the object (apple,
etc.). F m g g 9.8 m/s2
a F / m
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In the absence of other forces, all objects are
accelerated toward the Earth equally. a F / m
(m g) / m g 9.8 m/s2
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In the absence of other forces, all objects are
accelerated toward the Earth equally. a F / m
(m g) / m g 9.8 m/s2
t 1 s V 9.8 m/s
t 2 s V 19.6 m/s
t 3 s V 29.4 m/s
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In the absence of other forces, ALL objects are
accelerated toward the Earth equally... including
the Moon! a F / m (m g) / m g 9.8 m/s2
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The same force of gravity is pulling the Moon
toward the center of the Earth
velocity after acceleration 2
Force
2
1
velocity after acceleration 1
2
velocity after acceleration 1
1
original velocity
accelerated this way
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Actually, the Moon is not pulled as strongly as
the apple is, because it is farther away. g
9.8 m/s2 on the surface of the Earth, but the
pull gets smaller as one goes farther away. g G
M r2 F G M m r2 Newtons
Law of Gravity
r distance to center of the Earth M mass of
the Earth m mass of the Moon or apple
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Newton invented vector calculus and used it to
show that this formula for gravity predicts
planets should move in ellipses, obeying all of
Kepler's laws. Other formulas for gravity do
not predict elliptical orbits.
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F m v2 / r G M m / r2
centripetal force needed to keep the Moon moving
around the Earth
Gravitational force of the Earth pulling on the
Moon

Remember v (2 ? r) / P (distance traveled
divided by time) m (2 ? r)2 G M m
r P2 r2
Holy cow! Thats Keplers law!
P2 (4 ?2 / GM) r3
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But whats G?
Henry Cavendish (1731 - 1810) experiment of 1783
found G. G 6.67 x 10-8 m3 / kg / s2
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This was the missing piece of information that
allowed people to determine the mass of the Earth
RE 6380 km g G ME 9.8 m/s2
RE2 ME 5.98 x 1024 kg
g 2.7 x 10-6 m/s2 at the distance of the
Moon (R 385,000 km 3.85 x 108 m) ... weak,
but enough to keep the Moon orbiting the Earth!
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V 7.8 km/s 17,500 mph is needed for
orbit. Too fast and gravity cannot keep the
object moving in a circle. Too slow and gravity
totally wins.
A circular orbit happens only if the supplied
gravitational force G ME m / r2 exactly equals
the centripetal force m V2 / r needed to keep the
thing moving in a circle
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Example the Earth
P2 (4 ?2 / G ME) r3
The Moon P 27.3 days 2.36 x 106 s
r 385,000 km 3.85 x 108 m We can
solve for G ME 4.04 x 1014 m3 / s2 A
satellite in orbit has r 6600 km. Knowing r
and G ME, we can solve for P 88 minutes Note V
(2 ? r) / P (2 ? 6600 km) / (88 min) 7.9
km/s 28,000 km/hr 17,500 mph
orbital velocity
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Example Jupiter
P2 (4 ?2 / G MJ) r3
Io P 1.77 days 1.53 x 105 s
r 425,000 km 4.25 x 108
m Knowing G, too, we can solve for the mass of
Jupiter MJ 1.90 x 1027 kg 318 ME
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Example Sun
P2 (4 ?2 / G MJ) r3
Earth P 365.25 days 3.155 x 107 s
r 149,600,000 km 1.496 x 1011
m Knowing G, too, we can solve for the mass of
the Sun MS 1.99 x 1030 kg 333,000 ME
!!!
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Summary By the end of the 1700s, we know the
followingPlanets move in orbits around the Sun,
described by Kepler's laws 1.) elliptical
orbits, Sun at one focus 2.) period
related to the size of the orbit 3.) equal areas
swept out in equal times. Size of Earth's orbit
1 AU determined from transits of Venus, 1 AU
149,000,000 km. Aberration of starlight, and
timing of Jupiter's moons' eclipses says speed of
light is 1 AU / 8 minutes 300,000 km/s. Newton
figured out formula for pull of gravity
F G M m / r2 . Constant G measured,
masses of Earth and Sun determined.