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Kepler’s Laws
Kepler and the
Physics of Planetary Motion
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Laws of Planetary Motion
Law 1 - Law of Ellipses
 Law 2 - Law of Equal Areas
3 2
 Law 3 - Harmonic Law (r /T = C)
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Kepler’s laws provide a concise and
simple description of the motions of the
planets
Kepler’s First Law
The Law of Ellipses:
The planets move in
elliptical orbits with the
Sun at one focus.
eccentrici ty 
distance between focus points
length of major axis
The Ellipse
Semi-major Axis = ½ Major Axis
aphelion
90°
Focus Points
Center
eccentrici ty 
distance between focus points
length of major axis
e=0 perfect circle
e=1 flat line
perihelion
Verifying Kepler’s 1st
aphelion
L1 + L2= L3 + L4 ??
P2
L3
P1
L1
L4
Center
L2
perihelion
Kepler’s Second Law
As a planet orbits the Sun, it
moves in such a way that a line
drawn from the Sun to the
planet sweeps out equal areas in
equal time intervals.
Lunar Orbit of
Explorer 35
apoluna
Points represent satellite
positions separated by
equal time intervals.
Moon
periluna
Verifying Kepler’s 2nd
Area = ½ base X height
Equal area in equal time.
A1 = A2 ??
A2
A1
Kepler’s Third Law
The ratio of the average
distance* from the Sun cubed to
the period squared is the same
constant value for all planets.
* Semimajor axis
r3 = C
T2
Summarizing Kepler’s Laws
Kepler's First Law:
Each planet’s orbit around the Sun is
an ellipse, with the Sun at one focus.
Kepler's Second Law: Line
joining planet and the Sun sweeps out
equal areas in equal times
Kepler's Third Law: The squares of
the periods of the planets are proportional to
the cubes of their semi-major axes or:
r3 = C
T2
5. Universal Laws of Motion
“If I have seen farther than others, it
is because I have stood on the
shoulders of giants.”
Sir Isaac Newton (1642 – 1727)
Physicist
Newton’s Universal Law of
Gravitation
Isaac Newton discovered that it is gravity that plays
the vital role of determining the motion of the
planets - concept of action at a distance.
Gravity is the force that results in centripetal
acceleration of the planets.
Orbital Paths
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Extending Kepler’s Law
#1, Newton found that
ellipses were not the only
orbital paths.
possible orbital paths
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ellipse (bound)
parabola (unbound)
hyperbola (unbound)
Newton’s Universal Law of
Gravitation
Between every two objects there is an attractive
force, the magnitude of which is directly
proportional to the mass of each object and
inversely proportional to the square of the
distance between the centers of the objects.
Newton’s Universal Law of
Gravitation
G=6.67 x 10-11 m3/(kg s2)
Newton’s Version of Kepler’s
Third Law
Using calculus, Newton was able to derive
Kepler’s Third Law from his own Law of Gravity.
In its most general form:
2
2
3
T = 4 r / G M
If you can measure the orbital period of two
objects (T) and the distance between them (r),
then you can calculate the mass of the central
object, M.
What have we learned?
•
What is the universal law of gravitation?
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The force of gravity is directly proportional to the product
of the objects’ masses and declines with the square of the
distance between their centers (Inverse Square Law).
What types of orbits are possible according to the
law of gravitation?
•
Objects may follow bound orbits in the shape of ellipses
(or circles) and unbound orbits in the shape of parabolas
or hyperbolas.
What have we learned?
•
How can we determine the mass of distant
objects?
•
Newton’s version of Kepler’s third law allows us
to calculate the mass of a distant object if it is
orbited by another object, and we can measure
the orbital distance and period.
Combining Newton’s and
Kepler’s Laws, we can . . . .
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Determine the mass of an unknown
planet.
Determine the escape and orbiting
velocities for a satelite.
Determine the acceleration due to
gravity on a planet.
You should be able to derive equations
for the above determinations.
Derivations
Escape velocity
Orbiting velocity
“g”
Kepler’s “C”
G and g

Geosynchronous satellites orbit the
Earth at an altitude of about 3.58 x 107
meters. Given that the Earth’s radius is
6.38 x 106 meters and its mass is
5.97 x 1024 kg, what is the magnitude of
the gravitational acceleration at the
altitude of one of these satellites?
Orbiting velocity

The International Space Station orbits
the Earth at an average altitude of 362
kilometers. Assume that its orbit is
circular, and calculate its orbital speed.
The Earth’s mass is 5.97 x 1024 kg and
its radius is 6.38 x 106 meters.
Gravity is a source of energy

Because gravity is a force, it can be
associated with potential energy:
Recall:
dU
F 

dx
Solving, the formula for gravitational PE is:
GmM
Ug  
r
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The minus sign indicates that PE decreases
as the masses get closer together.
Gravitational Potential Energy
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Gravitational PE is
negative.
PE increases as r
decreases.
Potential energy vs. separation distance
Sample problem
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What is the minimum escape speed
from Earth?
KEat Earth’s surface = PEouter space
½ mvesc2 = GmM/r
vesc =
2GM
r
Sample problem.

Calculate the total energy of a satellite
in circular orbit about Earth with a
separation distance of r?
The change in a system’s
energy equals work.

How much work is required to move the
satellite to an orbit with a separation
distance of 2r?