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Transcript
Kepler’s Laws
Testing Models


Geocentric (or Ptolemaic) means the Earth is at the
center and motionless.
Heliocentric (or Copernican) means the Sun is at the
center and motionless.

Scholars wanted to differentiate models by
comparing the predictions with precise observations.

This originated the modern scientific method.
Kepler’s Work


Tycho Brahe led a team
which collected data on the
position of the planets (15801600 with no telescopes).
Mathematician Johannes
Kepler was hired by Brahe to
analyze the data.




He took 20 years of data on
position and relative
distance.
No calculus, no graph paper,
no log tables.
Both Ptolemy and
Copernicus were wrong.
He determined 3 laws of
planetary motion (16001630).
Kepler’s First Law

The orbit of a planet is an ellipse with the sun at one
focus.
A path connecting the two foci to the
ellipse always has the same length.
Orbital Description

An ellipse is described by two axes.
• Long – semimajor (a)
• Short – semiminor (b)

The area is pab (becomes pr2 for a circle).
b
a
Orbital Speed

The centripetal force is due to gravity.
• GMm/r2 = mv2/r
• v2 = GM/r

Larger radius orbit means slower speed.

Within an ellipse larger distance also gives slower
speed.
Kepler’s Second Law

The line joining a planet and the sun sweeps equal
areas in equal time.
Dt
The planet moves
slowly here.
Dt
The planet moves
quickly here.
Orbital Period

The speed is related to the period in a circular orbit.
• v2 = GM/r
• (2pr/T)2 = GM/r
• T2 = 4p2r3/GM

Larger radius orbit means longer period.

Within an ellipse, a larger semimajor axis also gives a
longer period.
Kepler’s Third Law

The square of a planet’s period is proportional to the
cube of the length of the orbit’s semimajor axis.
• T2/a3 = constant
• The constant is the same for all objects orbiting the Sun
direction of orbit
semimajor axis: a
The time for one orbit
is one period: T
Hyperbolic Orbits

Some satellites have so much speed that gravity
can’t hold them in an orbit.

These objects take a hyperbolic orbit that never
returns.