Satellite Motion
Low Orbit

A container falls off the space station while in low
earth orbit. It will move
A) straight down toward Earth.
B) curving slowly down toward Earth.
C) in the same orbit as the space station.
D) ever farther away due to lower mass.
E) rapidly away into space.
Short Period


An object in space would go
in a straight line without
another force.
Gravity supplies a force to
hold objects in circular orbits.

In low orbit the period is
related to the gravitational
acceleration.
mv 2
mg 
RE
 2RE 
v 
  gRE
 T 
2
4

RE
T2 
g
2
2
no gravity
gravity
Low Earth orbit period: T < 90 min.
Geosynchronous Orbit

In higher orbits, the
gravitational force is
significantly less than on the
surface.
• Use the force of universal
gravitation.
• Fgrav = G M m / r2

The height for a satellite with
a 24 hr period can be found.
GMm mv 2

2
r
r
 2r  GM
v2  
 
r
 T 
2
GMT
r3 
4 2
2
radius: r = 4.22 x 107 m
altitude is r - 6400 km = 36,000 km
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 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

An ellipse is described by
two axes.

• Long – semimajor (a)
• Short – semiminor (b)

• v2 = GM/r
• (2r/T)2 = GM/r
• T2 = 42r3/GM
The area is ab (becomes
r2 for a circle).
b
a
The speed is related to the
period in a circular orbit.

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
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