Math Makes the World(s) Go

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Math Makes the World(s) Go Round

• A Mathematical Derivation of Keplers Laws of
  Planetary Motion

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by Dr. Mark Faucette

• Department of Mathematics
• University of West Georgia

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A Little History
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A Little History

• Modern astronomy is built on the interplay
  between quantitative observations and testable
  theories that attempt to account for those
  observations in a logical and mathematical way.

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A Little History

• In his books On the Heavens, and Physics,
  Aristotle (384-322 BCE) put forward his notion of
  an ordered universe or cosmos.

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A Little History

• In the sublunary region, substances were made up
  of the four elements, earth, water, air, and
  fire.

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A Little History

• Earth was the heaviest, and its natural place was
  the center of the cosmos for that reason the
  Earth was situated in the center of the cosmos.

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A Little History

• Heavenly bodies were part of spherical shells of
  aether. These spherical shells fit tightly around
  each other in the following order Moon, Mercury,
  Venus, Sun, Mars, Jupiter, Saturn, fixed stars.

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A Little History

• In his great astronomical work, Almagest, Ptolemy
  (circa 200) presented a complete system of
  mathematical constructions that accounted
  successfully for the observed motion of each
  heavenly body.

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A Little History

• Ptolemy used three basic construc- tions, the
  eccentric, the epicycle, and the equant.

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A Little History

• With such combinations of constructions, Ptolemy
  was able to account for the motions of heavenly
  bodies within the standards of observational
  accuracy of his day.

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A Little History

• However, the Earth was still at the center of the
  cosmos.

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A Little History

• About 1514, Nicolaus Copernicus (1473-1543)
  distributed a small book, the Little Commentary,
  in which he stated
• The apparent annual cycle of movements of the sun
  is caused by the Earth revolving round it.

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A Little History

• A crucial ingredient in the Copernican revolution
  was the acquisition of more precise data on the
  motions of objects on the celestial sphere.

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A Little History

• A Danish nobleman, Tycho Brahe (1546-1601), made
  im-portant contribu- tions by devising the most
  precise instruments available before the
  invention of the telescope for observing the
  heavens.

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A Little History

• The instruments of Brahe allowed him to determine
  more precisely than had been possible the
  detailed motions of the planets. In particular,
  Brahe compiled extensive data on the planet Mars.

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A Little History

• He made the best measurements that had yet been
  made in the search for stellar parallax. Upon
  finding no parallax for the stars, he (correctly)
  concluded that either

• the earth was motionless at the center of the
  Universe, or
• the stars were so far away that their parallax
  was too small to measure.

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A Little History

• Brahe proposed a model of the Solar System that
  was intermediate between the Ptolemaic and
  Copernican models (it had the Earth at the
  center).

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A Little History

• Thus, Brahe's ideas about his data were not
  always correct, but the quality of the
  observations themselves was central to the
  development of modern astronomy.

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A Little History

• Unlike Brahe, Johannes Kepler (1571-1630)
  believed firmly in the Copernican system.

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A Little History

• Kepler was forced finally to the realization that
  the orbits of the planets were not the circles
  demanded by Aristotle and assumed implicitly by
  Copernicus, but were instead ellipses.

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A Little History

• Kepler formulated three laws which today bear his
  name Keplers Laws of Planetary Motion

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Keplers Laws
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Keplers Laws

• Keplers
• First
• Law

• The orbits of the planets are ellipses, with the
  Sun at one focus of the ellipse.

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Keplers Laws

• Keplers
• Second
• Law

• The line joining the planet to the Sun sweeps out
  equal areas in equal times as the planet travels
  around the ellipse.

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Keplers Laws

• Keplers
• Third
• Law

• The ratio of the squares of the revolutionary
  periods for two planets is equal to the ratio of
  the cubes of their semimajor axes

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Mathematical Derivation of Keplers Laws
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Mathematical Derivation of Keplers Law
Keplers Laws can be derived using the calculus
from two fundamental laws of physics

• Newtons Second Law of Motion
• Newtons Law of Universal Gravitation

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Newtons Second Law of Motion
The relationship between an objects mass m, its
acceleration a, and the applied force F is F
ma. Acceleration and force are vectors (as
indicated by their symbols being displayed in
bold font) in this law the direction of the
force vector is the same as the direction of the
acceleration vector.
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Newtons Law of Universal Gravitation

• For any two bodies of masses m1 and m2, the force
  of gravity between the two bodies can be given by
  the equation
• where d is the distance between the two objects
  and G is the constant of universal gravitation.

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Choosing the Right Coordinate System
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Choosing the Right Coordinate System
Just as we have two distinguished unit vectors i
and j corresponding to the Cartesian coordinate
system, we can likewise define two distinguished
unit vectors ur and u? corresponding to the polar
coordinate system
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Choosing the Right Coordinate System
Taking derivatives, notice that
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Choosing the Right Coordinate System
Now, suppose ? is a function of t, so ?
?(t). By the Chain Rule,
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Choosing the Right Coordinate System
For any point r(t) on a curve, let r(t)r(t),
then
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Choosing the Right Coordinate System
Now, add in a third vector, k, to give a
right-handed set of orthogonal unit vectors in
space
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Position, Velocity, and Acceleration
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Position, Velocity, and Acceleration
Recall
Also recall the relationship between position,
velocity, and acceleration
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Position, Velocity, and Acceleration
Taking the derivative with respect to t, we get
the velocity
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Position, Velocity, and Acceleration
Taking the derivative with respect to t again, we
get the acceleration
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Position, Velocity, and Acceleration
We summarize the position, velocity, and
acceleration
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Planets Move in Planes
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Planets Move in Planes
Recall Newtons Law of Universal Gravitation and
Newtons Second Law of Motion (in vector form)
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Planets Move in Planes
Setting the forces equal and dividing by m,
In particular, r and d2r/dt2 are parallel, so
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Planets Move in Planes
Now consider the vector valued function
Differentiating this function with respect to t
gives
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Planets Move in Planes
Integrating, we get
This equation says that the position vector of
the planet and the velocity vector of the planet
always lie in the same plane, the plane
perpendicular to the constant vector C. Hence,
planets move in planes.
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Boundary Values
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Boundary Values
We will set up our coordinates so that at time
t0, the planet is at its perihelion, i.e. the
planet is closest to the sun.
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Boundary Values
By rotating the plane around the sun, we can
choose our ? coordinate so that the perihelion
corresponds to ?0. So, ?(0)0.
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Boundary Values
We position the plane so that the planet rotates
counterclockwise around the sun, so that
d?/dtgt0. Let r(0)r(0)r0 and let
v(0)v(0)v0. Since r(t) has a minimum at
t0, we have dr/dt(0)0.
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Boundary Values
Notice that
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Keplers Second Law
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Keplers Second Law
Recall that
we have
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Keplers Second Law
Setting t0, we get
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Keplers Second Law
Since C is a constant vector, taking lengths, we
get
Recalling area differential in polar coordinates
and abusing the notation,
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Keplers Second Law
This says the rate at which the segment from the
Sun to a planet sweeps out area in space is a
constant. That is, The line joining the planet
to the Sun sweeps out equal areas in equal times
as the planet travels around the ellipse.
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Keplers First Law
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Keplers First Law

• Recall

Dividing the first equation by m and equating the
radial components, we get
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Keplers First Law

• Recalling that

Substituting, we get
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Keplers First Law

• So, we have a second order differential equation

We can get a first order differential equation by
substituting
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Keplers First Law

• So, we now have a first order differential
  equation

Multiplying by 2 and integrating, we get
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Keplers First Law

• From our initial conditions r(0)r0 and
  dr/dt(0)0, we get

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Keplers First Law

• This gives us the value of the constant, so

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Keplers First Law

• Recall that

Dividing the top equation by the bottom equation
squared, we get
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Keplers First Law

• Simplifying, we get

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Keplers First Law

• To simplify further, substitute

and get
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Keplers First Law

• Which sign do we take? Well, we know that
  d?/dtr0v0/r2 gt 0, and, since r is a minimum at
  t0, we must have dr/dt gt 0, at least for small
  values of t. So, we get

Hence, we must take the negative sign
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Keplers First Law

• Integrating with respect to q, we get

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Keplers First Law

• When t0, ?0 and uu0, so we have

Hence,
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Keplers First Law

• Now it all boils down to algebra

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Keplers First Law

• This is the polar form of the equation of an
  ellipse, so the planets move in elliptical orbits
  given by this formula. This is Kepler's First
  Law.

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Keplers Third Law
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Keplers Third Law

• The time T is takes a planet to go around its sun
  once is the planets orbital period. Keplers
  Third Law says that T and the orbits semimajor
  axis a are related by the equation

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Anatomy of an Ellipse

• An ellipse has a semi-major axis a, a semi-minor
  axis b, and a semi-focal length c. These are
  related by the equation b2c2a2. The
  eccentricity of the ellipse is defined to be
  ec/a. Hence

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Keplers Third Law

• On one hand, the area of an ellipse is pab. On
  the other hand, the area of an ellipse is

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Keplers Third Law

• Equating these gives

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Keplers Third Law

• Setting ?p in the equation of motion for the
  planet yields

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Keplers Third Law

• So,

This gives the length of the major axis
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Keplers Third Law

• Now were ready to kill this one off. Recalling
  that

we have
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Keplers Third Law
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Keplers Third Law
since
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Keplers Third Law
This is Keplers Third Law.
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Now for the Kicker
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Now for the Kicker
What is truly fascinating is that Kepler
(1571-1630) formulated his laws solely by
analyzing the data provided by Brahe.
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Now for the Kicker
Kepler (1571-1630) derived his laws without the
calculus, without Newtons Second Law of Motion,
and without Newtons Law of Universal Gravitation.
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Now for the Kicker
In fact, Kepler (1571-1630) formulated his laws
before Sir Isaac Newton (1643-1727) was even born!
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References
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References
History http//es.rice.edu/ES/humsoc/Galileo/Thin
gs/ptolemaic_system.html http//www-gap.dcs.st-and
.ac.uk/history/Mathematicians/Newton.html http//
www-gap.dcs.st-and.ac.uk/history/Mathematicians/C
opernicus. http//csep10.phys.utk.edu/astr161/lect
/history/brahe.html http//es.rice.edu/ES/humsoc/G
alileo/People/kepler.html http//csep10.phys.utk.e
du/astr161/lect/history/newton3laws.html http//ww
w.marsacademy.com/orbmect/orbles1.htm
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References
Mathematics Calculus, Sixth Edition, by Edwards
Penney, Prentice-Hall, 2002
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