Geoff Schwindt

Footnot 18 Project

2:40 MWF

Kepler's Laws of Planetary Motion

In 1605, Johannes Kepler who had been studying the precise observations of the planets as calculated by Tycho Brahe discovered that the orbits of the planets followed three mathematic rules. Though he was unable to show why these rules behaved as they did, he was able to extrapolate them from Brahe's data. When Isaac Newton invented calculus, he was able to derive the formulas regulating the motion of the planets from his own laws of motion and gravitation.

Kepler's three laws are:

1. The orbit of every planet is an ellipse with the sun at one foci;

2. A line joining a planet and the sun sweeps out in equal areas in equal times;

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

Proof 2:

The angular momentum equation is:

so

A vector crossed with itself is 0, and the book establishes that rXa is also equal to 0

Therefore, by integrating, L is equal to a constant.

The change in area of the triangular area between two vectors r (radius) and v (angular momentum) region is: where r goes from time t=t₁ to t=t₂

However,

Substituting into the area function, we find that

Since we have established that L is a constant and m is also a constant, is constant, so with an equal change in time the area swept out by the position vector of the planet will be constant, proving Kepler's second law.

Kepler's Third Law

Following from Kepler's Second Law, multiplying both sides by dt gives us

Integrating both sides with respect to their variables gives us

Squaring both sides we find that

Solving for t^2 we find that

The area of an ellipse is equal to where a is the length of the semimajor axis and b is the length of the semiminor axis

So

And where c is the distance between a foci and the center of the ellipse

So

where is the eccentricity of the ellipse.

So

So

The equation for the angular momentum can be rewritten as

From the book we know that , a constant vector. So Solving for m and squaring both sides we find that

Substituting this in we find that

This reduces to

Using a property of ellipses we can say that where p is the directrix

So

so

From the book we know that where G is the universal gravitational constant and M is the mass of the sun.

So

Canceling the h's and solving for c leaves us with

Substituting that into the equation gives us

Simplifying gives us

Since G and M are constants, this proves Kepler's third law that the square of the period of revolution is proportional to the cube of its major axis.

3. Using Kepler's 3^rd Law and inputting the given data we get that

Using Maple's solve command for a, we find that that a is equal to

4. To find this, the satellite must have a period of revolution equal to one day, therefore

Solving for a in Maple gives us but we must subtract the radius of the Earth, This leaves us with a final altitude of above the earth.

Reference - MAT294, Calculus III, Fall 2006 - Henry Braun, "Kepler's Laws"