Transcript Two-Body Central-Force Problems

Kepler Orbits
Last time we saw that this equation
describes an ellipse.
rmin = perihelion (for solar orbits) or perigee
rmax = aphelion (for solar orbits) or apogee
Kepler Orbits
• We can rewrite this in a the
more familiar equation for
an ellipse:
Kepler’s First Law: the planets follow elliptical orbits with the Sun at one focus
Brief History of Astronomy
Copernicus (1473-1543):
• proposed Sun-centered model
(published 1543)
• used model to determine layout of
solar system (planetary distances
in AU)
But . . .
• model was no more accurate than
Ptolemaic model in predicting
planetary positions, because still used
perfect circles.
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Tycho Brahe (1546-1601)
• Compiled the most accurate (one arcminute)
naked eye measurements ever made of planetary
positions.
• Still could not detect stellar parallax, and thus
still thought Earth must be at center of solar
system (but recognized that other planets go
around Sun)
• Hired Kepler, who used his observations to
discover the truth about planetary motion.
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• Kepler first tried to match Tycho’s observations
with circular orbits
• But an 8 arcminute discrepancy led him
eventually to ellipses…
Johannes Kepler
(1571-1630)
If I had believed that we could ignore these eight
minutes [of arc], I would
have patched up my hypothesis accordingly. But,
since it was not permissible to ignore, those eight
minutes pointed the road to a complete
reformation in astronomy.
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What is an Ellipse?
An ellipse looks like an elongated circle
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Kepler’s First Law: The orbit of each planet
around the Sun is an ellipse with the Sun at one
focus.
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Kepler’s Second Law: As a planet moves
around its orbit, it sweeps out equal areas in
equal times.
 means that a planet travels faster when it is nearer to the Sun and
slower when it is farther from the Sun.
Whiteboards: Derive Kepler’s 2nd Law from conservation of angular momentum.
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Kepler’s Third Law
More distant planets orbit the Sun at slower average speeds,
obeying the relationship
p2 = a3
p = orbital period in years
a = avg. distance from Sun in AU
Whiteboards: Derive Kepler’s 3rd Law starting with
F = m a.
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Graphical version of Kepler’s Third Law
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Question
An asteroid orbits the Sun at an average distance a = 4 AU.
How long does it take to orbit the Sun?
A. 4 years
B.
8 years
C. 16 years
D. 64 years
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Galileo Galilei
Galileo (1564-1642) overcame major
objections to Copernican view. Three
key objections rooted in Aristotelian
view were:
1.
2.
3.
Earth could not be moving because
objects in air would be left behind.
Non-circular orbits are not “perfect”
as heavens should be.
If Earth were really orbiting Sun,
we’d detect stellar parallax.
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Overcoming the first objection (nature of motion):
Galileo’s experiments showed that objects in air would stay with a
moving Earth.
• Aristotle thought that all objects naturally come to rest.
• Galileo showed that objects will stay in motion unless
a force acts to slow them down (Newton’s first law of motion).
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Overcoming the second objection (heavenly perfection):
• Tycho’s observations of comet and
supernova already challenged this
idea.
• Using his telescope, Galileo saw:
• sunspots on Sun
(“imperfections”)
• mountains and valleys on the
Moon (proving it is not a perfect
sphere)
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Overcoming the third objection (parallax):
• Tycho thought he had measured stellar distances, so lack of
parallax seemed to rule out an orbiting Earth.
• Galileo showed stars must be much farther than Tycho thought
— in part by using his telescope to see the Milky Way is countless
individual stars.
• If stars were much farther away, then lack of detectable parallax
was no longer so troubling.
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Galileo also saw four moons
orbiting Jupiter, proving that not
all objects orbit the Earth…
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… and his observations of phases of Venus proved that it orbits the
Sun and not Earth.
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The Catholic Church ordered
Galileo to recant his claim that
Earth orbits the Sun in 1633
His book on the subject was
removed from the Church’s index of
banned books in 1824
Galileo was formally vindicated by
the Church in 1992
Galileo Galilei
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Historical Overview
Isaac Newton (1643 - 1727)
• Building on the results of Galileo and Kepler
•
Adding physics interpretations to the mathematical descriptions of astronomy by
Copernicus, Galileo and Kepler
Major achievements:
1. Invented calculus as a necessary tool to solve mathematical
problems related to motion
2. Discovered the three laws of motion
3. Discovered the universal law of mutual gravitation
The Universal Law of Gravity
• Any two bodies are attracting each other through
gravitation, with a force proportional to the product of
their masses and inversely proportional to the square of
their distance:
F=-G
Mm
r2
(G is the gravitational constant.)
Orbital Motion (II)
In order to stay on a closed orbit, an object
has to be within a certain range of
velocities:
Too slow : Object falls back down to
Earth
Too fast : Object escapes the Earth’s
gravity
Relating Energy to Eccentricity
• Use the relation that E = Ueff(rmin) and
rmin = c/(1+ε), you can show
Summary of Orbits
eccentricity
energy
orbit
e=0
E<0
circle
0<e<1
E<0
ellipse
e=1
E=0
parabola
e>1
E>0
hyperbola
Changes in Orbit
• One way to change an objects orbit is using a
tangential thrust at perigee.
• Let λ= thrust factor
• (ratio of speed after/ speed before)
• Angular momentum changes by the same factor
• The eccentricity of the new orbit is:
Changes in Orbit
The Slingshot Effect
•
•
A close encounter with a planet can
dramatically increase a spacecraft’s
speed
Elastic Collision
•
•
Momentum is conserved
Kinetic energy is conserved
The Slingshot Effect
 0  mB v p 0  mA v mB v px 

 
   

m
v

0

0

m
v
 A 0 
    B py 

 0 
 
 0 
 
 0 
 
 0 

1
1
1
1
mA v 02  mB v P2 0  mA v 2  mB v P2
2
2
2
2
The Slingshot Effect
 0  mB v p 0  mA v mB v px 

 
   

m
v

0

0

m
v
 A 0 
    B py 

 0 
 
 0 
 
 0 
 
 0 

1
1
1
1
mA v 02  mB v P2 0  mA v 2  mB v P2
2
2
2
2
The probe’s speed after the encounter
is exceeds 2x the planet’s initial speed!
Eyes on the Solar System
• Go to http://solarsystem.nasa.gov/eyes/
• Watch the 4 tutorials
• Follow the Voyager 2 spacecraft on its journey from
Earth to the outer solar system. Identify significant
energy boosts that the probe received and explain
qualitatively how it received them (through what
mechanism).
• Repeat for a mission to Mars (you decide on the
mission).