Math Makes the World(s) Go ‘Round
A Mathematical Derivation of Kepler’s
Laws of Planetary Motion
by Dr. Mark Faucette
Department of Mathematics
University of West Georgia
A Little History
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.
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.
A Little History
In the sublunary region, substances were made up of the four elements, earth, water, air, and fire.
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.
A Little HistoryHeavenly 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.
A Little HistoryIn 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.
A Little History
Ptolemy used three basic construc- tions, the eccentric, the epicycle, and the equant.
A Little HistoryWith such
combinations of constructions, Ptolemy was able to account for the motions of heavenly bodies within the standards of observational accuracy of his day.
A Little History
However, the Earth was still at the center of the cosmos.
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.
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.
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.
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.
A Little HistoryHe 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.
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).
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.
A Little History
Unlike Brahe, Johannes Kepler (1571-1630) believed firmly in the Copernican system.
A Little HistoryKepler 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.
A Little History
Kepler formulated three laws which today bear his name: Kepler’s Laws of Planetary Motion
Kepler’s Laws
Kepler’s Laws
Kepler’s
First
Law
The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
Kepler’s Laws
Kepler’s
Second
Law
The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
Kepler’s Laws
Kepler’s
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:
Mathematical Derivation of Kepler’s Laws
Mathematical Derivation of Kepler’s Law
Kepler’s Laws can be derived using the calculus from two fundamental laws of physics:
• Newton’s Second Law of Motion
• Newton’s Law of Universal Gravitation
Newton’s Second Law of Motion
The relationship between an object’s 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.
Newton’s 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.
Choosing the Right Coordinate System
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 uθ corresponding to the polar coordinate system:
Choosing the Right Coordinate System
Taking derivatives, notice that
Choosing the Right Coordinate System
Now, suppose θ is a function of t, so θ= θ(t). By the Chain Rule,
Choosing the Right Coordinate System
For any point r(t) on a curve, let r(t)=||r(t)||, then
Choosing the Right Coordinate System
Now, add in a third vector, k, to give a right-handed set of orthogonal unit vectors in space:
Position, Velocity, and Acceleration
Position, Velocity, and Acceleration
Recall
Also recall the relationship between position, velocity, and acceleration:
Position, Velocity, and Acceleration
Taking the derivative with respect to t, we get the velocity:
Position, Velocity, and Acceleration
Taking the derivative with respect to t again, we get the acceleration:
Position, Velocity, and Acceleration
We summarize the position, velocity, and acceleration:
Planets Move in Planes
Planets Move in Planes
Recall Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion (in vector form):
Planets Move in Planes
Setting the forces equal and dividing by m,
In particular, r and d2r/dt2 are parallel, so
Planets Move in Planes
Now consider the vector valued function
Differentiating this function with respect to t gives
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.
Boundary Values
Boundary Values
We will set up our coordinates so that at time t=0, the planet is at its perihelion, i.e. the planet is closest to the sun.
Boundary Values
By rotating the plane around the sun, we can choose our θ coordinate so that the perihelion corresponds to θ=0. So, θ(0)=0.
Boundary Values
We position the plane so that the planet rotates counterclockwise around the sun, so that dθ/dt>0.
Let r(0)=||r(0)||=r0 and let v(0)=||v(0)||=v0. Since r(t) has a minimum at t=0, we have dr/dt(0)=0.
Boundary Values
Notice that
Kepler’s Second Law
Kepler’s Second Law
Recall that
we have
Kepler’s Second Law
Setting t=0, we get
Kepler’s Second Law
Since C is a constant vector, taking lengths, we get
Recalling area differential in polar coordinates and abusing the notation,
Kepler’s 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.
Kepler’s First Law
Kepler’s First Law
Recall
Dividing the first equation by m and equating the radial components, we get
Kepler’s First Law
Recalling that
Substituting, we get
Kepler’s First Law
So, we have a second order differential equation:
We can get a first order differential equation by substituting
Kepler’s First Law
So, we now have a first order differential equation:
Multiplying by 2 and integrating, we get
Kepler’s First Law
From our initial conditions r(0)=r0 and dr/dt(0)=0, we get
Kepler’s First Law
This gives us the value of the constant, so
Kepler’s First Law
Recall that
Dividing the top equation by the bottom equation squared, we get
Kepler’s First Law
Simplifying, we get
Kepler’s First Law
To simplify further, substitute
and get
Kepler’s First Law
Which sign do we take? Well, we know that dθ/dt=r0v0/r2 > 0, and, since r is a minimum at t=0, we must have dr/dt > 0, at least for small values of t. So, we get
Hence, we must take the negative sign:
Kepler’s First Law
Integrating with respect to q, we get
Kepler’s First Law
When t=0, θ=0 and u=u0, so we have
Hence,
Kepler’s First Law
Now it all boils down to algebra:
Kepler’s 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.
Kepler’s Third Law
Kepler’s Third Law
The time T is takes a planet to go around its sun once is the planet’s orbital period. Kepler’s Third Law says that T and the orbit’s semimajor axis a are related by the equation
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 b2+c2=a2. The eccentricity of the ellipse is defined to be e=c/a. Hence
Kepler’s Third Law
On one hand, the area of an ellipse is πab. On the other hand, the area of an ellipse is
Kepler’s Third Law
Equating these gives
Kepler’s Third Law
Setting θ=π in the equation of motion for the planet yields
Kepler’s Third Law
So,
This gives the length of the major axis:
Kepler’s Third Law
Now we’re ready to kill this one off. Recalling that
we have
Kepler’s Third Law
Kepler’s Third Law
since
Kepler’s Third Law
This is Kepler’s Third Law.
Now for the Kicker
Now for the Kicker
What is truly fascinating is that Kepler (1571-1630) formulated his laws solely by analyzing the data provided by Brahe.
Now for the Kicker
Kepler (1571-1630) derived his laws without the calculus, without Newton’s Second Law of Motion, and without Newton’s Law of Universal Gravitation.
Now for the Kicker
In fact, Kepler (1571-1630) formulated his laws before Sir Isaac Newton (1643-1727) was even born!
References
References
History:http://es.rice.edu/ES/humsoc/Galileo/Things/
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/Copernicus.
http://csep10.phys.utk.edu/astr161/lect/history/brahe.html
http://es.rice.edu/ES/humsoc/Galileo/People/kepler.html
http://csep10.phys.utk.edu/astr161/lect/history/newton3laws.html
http://www.marsacademy.com/orbmect/orbles1.htm
References
Mathematics:Calculus, Sixth Edition, by Edwards & Penney, Prentice-Hall,
2002
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