Planetary Motion
by Nick D’Anna
Earth Science Teacher
Plainedge Middle School
Planet Names
Tuesday  Martedì (Italian)  Mars’ day
Wednesday  Mercoledì (Italian) 
Mercury’s day
Thursday  Giovedì (Italian)  Jupiter’s day
Friday  Venerdì (Italian)  Venus’ day
Saturday  Saturn’s day
Sunday  Sun’s day (not a planet but still
important)
Monday  Lunedì (Italian)  Moon’s day
(also not planet, but also important because it
moves differently than the other things in the
sky.
The Greeks believed that the
planets traveled in circular paths.
Since the
acceleration
(force of
gravity) is
perpendicular
to the
velocity of
the body, the
torque on the
body is zero.
Thus, the
velocity of
the body
remains
constant.
However, the planets did not move
with constant speed.
Planet
comes from
the Greek
word
Planētēs:
WANDERER
The planets move differently than
all the other celestial objects.
Retrograde motion explained by
Hipparchos & Ptolemy
• Ptolemy believed in a
Geocentric (Earth
Centered) model of the
Solar System
• Ptolemy explained
retrograde motion with
DEFERENTS &
EPICYCLES.
• The math involved for
Ptolemy’s model with
epicycles became
extraordinarily
complicated.
Copernicus & the Heliocentric
model (Sun-centered) Solar System
Copernicus was able
to explain the
retrograde motion of
the planets just as
well as Ptolemy.
However,
Copernicus’ model
still had it’s
problems.
Copernicus used perfect circular motion, unlike
Ptolemy, who had the Earth offset as an equant (not
centered in circular orbits
Ockham’s Razor
Cited from http://sbast3.ess.sunysb.edu/fwalter/AST101/occam.html
The most useful statement of the principle for
scientists is: "when you have two competing
theories which make exactly the same
predictions, the one that is simpler is the
better.“
The Copernican system was more elegant and
more aesthetic than Ptolemy’s system. Hence, it
had favor.
Johannes Kepler (1571 – 1630)
• Believed the Universe was driven by
mathematical principals
• There must be a force, propelling
planets to move. The force was
something like magnetism between the
Sun and the planets.
• Devised Three Laws of Planetary Motion
Kepler’s Laws
• Law of Ellipses (1609)
• Law of Equal Areas (1609)
• Harmonic Law (1618)
Kepler’s First Law
An ellipse is a geometric
shape somewhere
between a circle and a
parabola.
ECCENTRICITY
measures how round or
flattened an ellipse.
Ellipses
Eccentricity
E = distance between the foci ÷ Length of major axis
Effects of elliptical orbits
• Changes in gravitational pull between
planet and Sun
• Changes in orbital velocity
• Changes in apparent angular diameter
Kepler’s 2nd Law
If the net torque on a body is zero, then the
angular momentum will be conserved
Kepler’s 2nd law can be equated to the
conservation of angular momentum.
A of ∆AoB closely
approximates the
area swept out in
time (dt) by a line
connecting the Sun
and the planet
dѲ
The base of
∆AoB = rdѲ and
Area of triangle = ½(base x height)
the height is r.
Area = ½(r)(rdѲ) = ½r2dѲ
dA/dt = ½(r2)(dѲ/dt)
dA/dt = ½r2ω or r2ω/2
dѲ/dt = ω, where
ω is the angular
velocity
The angular momentum (L) of a
planet around the sun is the product
of the r and the component of the
momentum perpendicular to r.
L = rp┴ = (r)(mv┴) = (r)(mωr) = mr2ω
Bringing it all together:
dA/dt = (r2/2)(dѲ/dt) = r2ω/2
L = rp┴ = (r)(mv┴) = mr2ω r2ω = L/m
dA/dt = r2ω /2
dA/dt = L/2m
If angular momentum is
conserved, L is constant, then
dA/dt must also be constant.
Kepler’s 3rd Law: Harmonic Motion
Galileo
• Lived at the same time as Kepler.
• Studied falling bodies and the way they
accelerate toward Earth
• Introduced the Law of inertia
• Made crucial astronomical observations:
– Moon’s orbiting Jupiter.
– The surface of the Moon looks like the surface
of Earth. It has mountains and craters, etc…
It is not perfect.
• Dealt the final blow to the Ptolemic system of
the Solar system. And also a major problem for
the Roman Catholic Church
Isaac Newton (1643 – 1727)
Unified Kepler’s and Galileo’s work.
Ode to Newton
“Once in a great while, a few times in
history, a human mind produces an
observation so acute and unexpected that
people can’t quite decide which is the
more amazing – the fact or the thinking of
it. Principia was one of those moments”
Bill Bryson, A Short History of Nearly Everything.
Newton’s 1st Law:
Inertia and Momentum
Inertia: A moving body tends to keep moving,
and a stationary body tends to remain at rest.
Momentum: The product of mass and velocity ρ = mv
Newton’s 2nd Law: Force
ƒ = ma
Newton’s 3rd Law: Reaction
For every applied force
there is an equal and
opposite reaction force
Derivation of the Universal Law of
Gravity from Newton’s Laws of
motion and Kepler’s Laws.
ƒ = ma
For circular motion
ƒ=m
a = v2/r
2
v /r
Centripetal Force on a planet
From F = mv2/r, let’s look at v
Velocity is distance over time. For simplicity
we’ll use a circular path, so the distance is
2πr
(the circumference of a circle)
And the time for a planet to travel in its
orbit is called the Period
(P)
Therefore,
V = 2πr/P
In the centripetal force equation, F
2
= mv /r , the velocity is squared
Recall v = 2πr/P
Square it
V2 =
4π2r2/P2
Substituting everything into the
centripetal force equation, F = mv2/r
F= m•
2
2
2
2
4π
v r /P r
Recall Kepler’s 3rd Law of
Harmonic Motion P2 = Ar3
Apply this law to the
centripetal force
equation:
F = m4π2r2/ Ar
P2 3r
Simplify the equation to:
F=
2
2
m4π /Ar
F = m4π2/Ar2
Remove the constant value from the above equation
F ά m/r2
The mass of the planet (m) is
also constant, therefore,
Fά
2
1/r
Where, ά means proportional to
We’re not done.
According to Newton’s 3rd law
(reaction), if the sun exerts a force
on the planet, the planet must exert
a force on the Sun.
F ά m/r2 is the force on the planet by the sun, then F
ά M/r2 is the force on the Sun exerted by the planet,
where M is the mass of the Sun.
Which produces a net force of :
Fά
2
mM/r
Satellite in Motion
Bibliography
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Universe, 6th Edition. Saunders College Publishing, 1991.
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Physics, Volume 1, 5th Edition. John Wiley & Sons Inc., 1997.
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1983.
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Physics of Satellite Motion. New York: Holt, Rinehart and Winston, Inc.,
1964.
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