Transcript PHY101_Lec_Sept12 - MSU Physics and Astronomy Department

The solar system
Pluto
Neptune
Uranus
Saturn
Jupiter
Mars
Earth
Venus
Mercury
Sun
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Sun
Planets
Asteroids
Comets
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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 Tycho’s 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
1. Each planet moves on an elliptical orbit.
2. The radial vector sweeps out equal areas in equal
times.
3. 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?
Mars
Compare the heliocentric model to
naked-eye astronomy
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 Kepler’s
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)
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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:
1. The laws of motion
2. The theory of universal gravitation
3. Calculus, a new branch of mathematics
F=
F
a=
m
Gm1m2
r2
“If I have been able to see farther than others it is
because I stood on the shoulders of giants.”
--- Newton’s letter to Robert Hooke,
probably referring to Galileo and Kepler
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Newton’s Theory of Universal Gravitation
Newton and the Apple
Newton asked good questions  the key to his success.
Observing Earth’s gravity
acting on an apple, and seeing
the moon, Newton asked
whether the Earth’s 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
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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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Circular Orbits
(a pretty good approximation for all the
planets because the eccentricities are
much less than 1.)
(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
T =
2 r
v
r
4 r
2
= 2 r
, or T =
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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FIN
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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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The Law
of
Equal
Areas: fastest
Perihelion
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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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).
4 a
4 a
T =

G (M  m ) GM
2
2
3
2
3
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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