The solar system

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The solar system
Sun Planets Asteroids Comets
Pluto Neptune Uranus Saturn Jupiter Mars Earth Ven
us Mercury Sun
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Historical figures in the Copernican Revolution
Ptolemy the geocentric model, that the Earth is
at rest at the center of the Universe.
Copernicus published the heliocentric model.
Galileo his observations by telescope verified
the heliocentric model.
Kepler deduced empirical laws of planetary
motion from Tychos observations of planetary
positions.
Newton developed the full theory of planetary
orbits.
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The Copernican Revolution
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Nicolaus Copernicus

• The Earth moves, in two ways.
• It rotates on an axis (period 1 day).
• It revolves around the sun (period 1 year).

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Johannes Kepler (1571 1630)
discovered three empirical laws of planetary
motion in the heliocentric solar system

• Each planet moves on an elliptical orbit.
• The radial vector sweeps out equal areas in equal
  times.
• The square of the period is proportional to the
  cube of the radius.

(needed for the CAPA)
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How did Kepler determine the planetary orbits?
Compare the heliocentric model to naked-eye
astronomy
Mars
The inner planet is Earth the outer one is Mars.
Plot their positions every month. Mars lags
behind the Earth so its appearance with respect
to the Zodiac is shifting.
Earth
The most complete data had been collected over a
period of many years by Keplers predecessor,
Tycho Brahe of Denmark.
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Ellipse Geometry
To draw an ellipse Take a string. Tack down the
two ends. Put a pencil in the string and pull the
string taut. Move the pencil around keeping the
string taut.
An ellipse is the locus of points for which the
sum of the distances to two fixed points is fixed.
The two fixed points are called the focal points
of the ellipse.
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Parameters of an elliptical orbit (a,e)
? Semi-major axis a one half the largest
diameter
? Eccentricity e ratio of the distance
between the focal points to the major diameter
For example, this ellipse has a 1 and e 0.5.
? Perihelion and aphelion
Perihelion r2 0.5 Aphelion r1 1.5
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Isaac Newton
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The observed solar system at the time of Newton
Sun Mercury Venus Earth Mars Jupiter Saturn
(all except Earth are named after Roman gods,
because astrology was practiced in ancient Rome)
Three outer planets discovered later Uranus
(1781, Wm Herschel) Neptune (1846 Adams
LeVerrier) Pluto (1930, Tombaugh)
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Isaac Newton
Newton solved the premier scientific problem of
his time --- to explain the motion of the planets.
To explain the motion of the planets, Newton
developed three ideas

• The laws of motion
• The theory of universal gravitation
• Calculus, a new branch of mathematics

If I have been able to see farther than others
it is because I stood on the shoulders of
giants. --- Newtons letter to Robert
Hooke, probably referring to Galileo and Kepler
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Newtons Theory of Universal Gravitation
Newton and the Apple
Newton asked good questions ? the key to his
success.
Observing Earths gravity acting on an apple, and
seeing the moon, Newton asked whether the Earths
gravity extends as far as the moon.
(The apple never fell on his head, but sometimes
stupid people say that, trying to be funny.)
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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 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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(a pretty good approximation for all the planets
because the eccentricities are much less than 1.)
Circular Orbits
(velocity)
(acceleration)
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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Circular Orbits
The planetary mass m cancels out. The speed is
then
Period of revolution
Time distance / speed i.e., Period
circumference / speed
? Keplers third law T 2 ? r 3
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Generalization to elliptical orbits
(and the true center of mass!)
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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FIN
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Newtons figure to explain planetary orbits
Newtons theory has stood the test of time. We
use the same theory today for planets, moons,
satellites, etc.
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The Law of Equal Areas
Perihelion fastest Aphelion slowest
Newton Angular momentum is conservedin fact
thats true for any central forceso the areal
rate is constant.
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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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Here is the general formula. We can always
neglect m (mass of the satellite) compared to M
(mass of the center).