math makes the world(s) go ‘ round

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 constructions, 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


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:


Taking derivatives, notice that


Now, suppose θ is a function of t, so θ= θ(t). By the Chain Rule,


For any point r(t) on a curve, let r(t)=||r(t)||, then


Now, add in a third vector, k, to give a right-handed set of orthogonal unit vectors in space:

Position, Velocity, and Acceleration


Recall

Also recall the relationship between position, velocity, and acceleration:


Taking the derivative with respect to t, we get the velocity:


Taking the derivative with respect to t again, we get the acceleration:


We summarize the position, velocity, and acceleration:

Planets Move in Planes


Recall Newton’s Law of Universal Gravitation and Newton’s Second Law of Motion (in vector form):


Setting the forces equal and dividing by m,

In particular, r and d2r/dt2 are parallel, so


Now consider the vector valued function

Differentiating this function with respect to t gives


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


Recall that

we have


Setting t=0, we get


Since C is a constant vector, taking lengths, we get

Recalling area differential in polar coordinates and abusing the notation,


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


Recall

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


Recalling that

Substituting, we get


So, we have a second order differential equation:

We can get a first order differential equation by substituting


So, we now have a first order differential equation:

Multiplying by 2 and integrating, we get


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


This gives us the value of the constant, so


Recall that

Dividing the top equation by the bottom equation squared, we get


Simplifying, we get


To simplify further, substitute

and get


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:


Integrating with respect to q, we get


When t=0, θ=0 and u=u0, so we have

Hence,


Now it all boils down to algebra:


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


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


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


Equating these gives


Setting θ=π in the equation of motion for the planet yields


So,

This gives the length of the major axis:


Now we’re ready to kill this one off. Recalling that

we have


since


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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