4' How Orbits Work

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Astronomy 1001, Sept 2007 Prof. K. Davidson
4. How Orbits Work
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KEPLERS LAWS (c. 1610)
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1. Each planet follows an elliptical orbit,
with the Sun at one focus 2.
Equal areas in equal times 3. Period squared
radius cubed (Note the orbits arent
aligned)
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KEPLERS FIRST LAW
(except that planets orbits are
usually more circular than this)
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KEPLERS SECOND LAW EQUAL AREAS IN EQUAL
TIMES
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KEPLERS THIRD LAW
relates the speeds of different planets
those farther out move slower. If a
average distance from the Sun and P
orbital period, then P
2 (constant) x a 3 . (Distance is
reckoned from the center of the Sun.)
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Keplers third law -- the numbers
a (AU) P (yr) a 3
P 2 MERCURY 0.387 0.241 0.058
0.058 VENUS 0.723 0.615
0.378 0.378 EARTH 1.000 1.000
1.00 1.00 MARS 1.524 1.881
3.54 3.54 JUPITER 5.203 11.86
141 141 SATURN 9.54 29.46
868 868
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Isaac Newton (1642 1727)
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Newtons theory of dynamics,
also called mechanics Motions of
objects impelled by forces
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Brief digression ... A VECTOR IS A
QUANTITY THAT HAS A DIRECTION IN
SPACE. examples POSITION
VELOCITY
ACCELERATION
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vector addition
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VELOCITY tells us how POSITION changes
with time (in all directions)

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VELOCITY tells us how POSITION changes
with time (in all directions) ACCELERATION
tells us how VELOCITY changes with time
(in all directions).

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VELOCITY tells us how POSITION changes
with time (in all directions) ACCELERATION
tells us how VELOCITY changes with time
(in all directions). So If we know an
objects three-dimensional acceleration
at all times, then we
can predict its motion.
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If we know an objects acceleration at all
times, then we can predict its motion.
( Newton invented a new kind of math
specifically for this purpose
differential calculus, which concerns
rates of change. )

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Newton also invented the concept of
force in the modern sense. FORCE
MASS x ACCELERATION, or
F m a , or
a F / m . ( F and a are
vectors they have directions.)

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Newtons crucial thought experiment (1665)
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The Moons orbit around Earth r 384000
km about 60 x (radius of Earth), v
3700 km/hr 1.02 km/s. So the required
acceleration toward earth is a v 2 /
r 0.27 cm / s / s.
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Moons acceleration toward Earth is about
0.27 cm / s / s. So what? -- Our
acceleration toward Earth is g 980 cm / s
/ s. Newton noticed that these have the
ratio 3600. 60 x farther makes gravity
weaker by a factor of 3600 x.
This is obviously 60 x 60 60 2 !
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Newtons law of gravity ( attractive force
between M and m ) G x M x m /
(distance) 2 . For instance, 3 x farther
makes it 9 x weaker 10 x farther makes
it 100 x weaker.
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THE INVERSE SQUARE LAW OF GRAVITY WAS
ENOUGH TO EXPLAIN KEPLERS
LAWS! 1. Orbits are ellipses, Sun at one
focus 2. Equal areas in equal times 3.
Period squared orbit radius cubed
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NEWTONS REVOLUTION WAS AS IMPORTANT
AS COPERNICUS. There are laws of physics
that apply everywhere, from this room to
the edge of the universe. In 1680 this
was a breathtaking new idea! It led
to modern physical science.
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Newtons influence on the following century
-- the age of reason, at least for
philosophers. His historical importance
was recognized in his own time -- arguably
the most important man in the world,
outranking even Louis XIV.
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ORBIT SHAPES ELLIPSE, PARABOLA, HYPERBOLA
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CONIC SECTIONS CIRCLE ELLIPSE
PARABOLA HYPERBOLA
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ENERGY IN ORBITS -- KINETIC
ENERGY -- POTENTIAL ENERGY K.E.
P.E. constant The Roller Coaster
analogy
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CELESTIAL MECHANICS (ORBITAL
DYNAMICS) Keplers laws are not
exactly true ...
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