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The Motion of the Planets
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The Motion of the Planets
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The Solar System
Precisely how do the planets (including
Earth) move around the sun?
What are the fundamental laws of
nature that govern this motion?
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Kepler’s Laws of Planetary Motion
Johannes Kepler – a contemporary of Galileo –
studied the data on astronomical observations of
the planets, seeking to describe accurately their
motion around the sun. After 20 years of work, he
deduced three empirical laws of planetary motion.
1. A planet moves on an elliptical orbit with the
sun at one focal point of the ellipse.
2. The radial line – the line from the sun to the
planet – sweeps out equal areas in equal times.
3. The square of the period is proportional to the
cube of the semi-major axis.
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Ellipse geometry
An ellipse with semi-major
axis a and eccentricity e.
This ellipse has a = 1
and e = 0.5 .
Hint:
perihelion+aphelion = 2a
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The motion of the planets
Diagram of a planet revolving
around the sun.
The eccentricity e is grossly
exaggerated ― real orbits are
very close to circular.
In fact there are nine planets. The center of mass
of the solar system is fixed (). To a first
approximation the center of mass is at the Sun.
() actually it moves around the
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center of the galaxy
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Centripetal
acceleration
For an object in circular
motion, the centripetal
acceleration is a = v 2/r .
(Christian Huygens)
Example. Determine the string tension if a mass of 5 kg is
whirled around your head on the end of a string of length 1
m with period of revolution 0.5 s.
Answer : 790 N
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Three concepts --–
• Centripetal acceleration
• Centripetal force
• Centrifugal force
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v
ar 
r
2
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Centripetal force and “centrifugal force”
View from above
View from rear
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Isaac Newton (1642 – 1727)
Newton solved the premier scientific problem of his
day – to explain why the planets move as they do.
To solve this problem he developed …
• the three laws of motion,
• the theory of universal gravitation,
• calculus, a branch of mathematics.
Newton quote:
“If I have been able to see farther than
others it is because I stood on the shoulders
of giants.”
-- in a letter to Robert Hooke
He could be referring to Galileo and Kepler.
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Circular Orbits
(a pretty good approximation for all the
planets because the eccentricities are
small.)
(velocity)
Centripetal acceleration
v2
ar  
r
(acceleration)
Newton’s second law
mv
GMm

r
r2
2
There is a subtle approximation here: we are approximating the center of
mass position by the position of the sun. This is a good approximation.
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mv
GMm

2
r
r
2
Circular Orbits
The planetary mass m cancels out.
The speed is then
GM
v
r
Period of revolution
Time = distance / speed
i.e., Period = circumference / speed
2 r
r
4 r
2
T 
 2 r
, or T 
v
GM
GM
2
3
 Kepler’s third law: T 2  r 3
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Generalization to elliptical orbits
(and the true center of mass!)
2 3
4 a
T 
G( M  m)
2
2 3
4 a

GM
where a is the semi-major axis of the ellipse
The calculation of elliptical orbits is difficult
mathematics.
The story of Newton and Halley
Many applications ...
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Kepler’s second law -- The Law of
Equal Areas
Perihelion : fastest
Aphelion : slowest
Newton: Angular momentum is conserved—in fact
that’s true for any central force—so the areal rate
is constant.
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Westminster Abbey
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Here is buried Isaac Newton, Knight, who by a
strength of mind almost divine, and
mathematical principles peculiarly his own,
explored the course and figures of the planets,
the paths of comets, the tides of the sea, the
dissimilarities in rays of light, and, what no
other scholar has previously imagined, the
properties of the colors thus produced.
Diligent, sagacious and faithful, in his
expositions of nature, antiquity and the holy
Scriptures, he vindicated by his philosophy the
majesty of God mighty and good, and
expressed the simplicity of the Gospel in his
manners. Mortals rejoice that there has existed
such and so great an ornament of the human
race! He was born on 25th December, 1642,
and died on 20th March 1727.
Newton’s monument in
Westminster Abbey.
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Comets
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Ocean Tides -- an effect due to the
gravity of the moon and the sun
Bay of Fundy
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If you have ever lived near the ocean, you have
observed the tides at the beach.
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Newton: Ocean tides are due to the gradient of the
gravitational force
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Newton said of himself…
“ I know not what I appear to the world,
but to myself I seem to have been only like
a boy playing on the sea-shore, and
diverting myself in now and then finding a
smoother pebble or a prettier sea-shell,
whilst the great ocean of truth lay all
undiscovered before me.”
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Quiz question
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Consider the star (S),
which has two planets (P1
and P2).
P2 is twice as far from S as
P1. The period of
revolution P1 is 1 year.
What is the period of P2?
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Newton’s figure to explain
planetary orbits
Newton’s theory has stood the
test of time. We use the same
theory today for planets,
moons, satellites, etc.
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Here is the general formula. We
can always neglect m (mass of
the satellite) compared to M
(mass of the center).
4 a
4 a
T 

G (M  m )
GM
2
2
3
2
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Applications
• Earth and Sun
• Other planets
• Moon and Earth
• Artificial satellites
• Exploration of the solar system
Some of these are covered in the CAPA problems.
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Comets
Comet Hale-Bopp
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