Johannes Kepler: The Heavens

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Johannes Kepler: The Heavens

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Johannes Kepler was an accomplished mathematician and astronomer. We see Kepler’s work in modern science in the form of Kepler’s laws of planetary motion. The works that Kepler did to produce the three laws of planetary motion are far too numerous and complex to cover in this small project. Instead I intend to give a brief overview of his three laws of planetary motion, and the modifications of the three laws needed for justification of the laws.

 

 

Kepler believed in a sun-centered universe, “From the heavenly music to the hearer; from the Muses to Apollo the choirmaster; from the six planets which go around and make harmonies, to the Sun at the center of all the orbits, motionless in his place, but revolving on his own axis” (1, P. 492). In believing in a centered universe Kepler was one step closer to understanding the motion of the planets.

 

Kepler began to attempt to make a geometric model of the universe based on Tycho Brahe’s observational data. Tycho did not use telescopes for his data. Kepler first began by using a series of offset circles and equants, and reached agreement with the data for all except for two points. The two points were off by two times the accuracy of Tycho’s observations. Upon reaching this result Kepler decided that Tycho did not make a mistake, instead Kepler decided that using equants and offset circles were not accurate. As a result Kepler abandoned the idea of circular planetary motion (3, P.25-6).

 

Because of the disagreement with the two data points using offset circle and equant models, Kepler rejected the assumption that planets move with circular motion. Kepler now assumed that planets moved with elliptical motion. An easy way to describe an ellipse is an elongated circle, as shown below. 

 

A general shape of an ellipse. (elongated circle)
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If you will an ellipse has two “center” points, each called foci. The so called “center” points are not precisely located at the center of the ellipse.

 


A description of an ellipse

 

“Kepler’s First Law A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse” (3, P.27).

 

Keplers first law
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Kepler’s Second Law A line connecting a planet to the Sun sweeps out equal areas in equal time intervals” (3, P.27).

Kepler's second law
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Kepler's second law says that a planet in the perihelion moves faster than when it is at aphelion. Another way to think of this is that a planet moves faster as it gets closer to the sun and slower as it gets further away from the sun.

The first two laws were published in a different book called Astronomica Nova (The New Astronomy), ten years before the third law was published. The third law was published in the book Harmonica Mundi (The Harmony of the World) (3, P.27).

 

Kepler’s Third Law P2 = a3” (3, P. 27).

 

So what does this third law say? First I have to define “P” and “a.” “P” stands for the period of the planet in question. Period is the term that stands for the time for one revolution around the sun to occur or the time of one rotation around an object. For example the period of the earth’s revolution around the sun is 1 year or approximately 365 days. “a” stands for the average distance between the planet and the sun.


Kepler did not understand why his model worked in a physical sense. He only discovered the mathematical device that models planetary motion.

 

What else is needed to explain planetary motion?

 

The answer to this question seems simple to the modern scientist, but in Kepler’s time, the theory of gravity was incomplete. Sir Isaac Newton’s theory of Universal Gravitation fit beautifully with Kepler’s laws of planetary motion. In fact Newton was able to improve the accuracy of Kepler’s third law.

 

So what is the theory Universal Gravitation?

 

Simply put, gravity is an attractive force between two objects which contain matter. That means that picking any two objects and placing them on a table that there is an attractive force between them. The attractive force is extremely small. Since the earth is extremely massive compared to any object that is on the earth, the attractive force due to the earth cancels out any visible gravitational force between any two objects.

 

Newton’s First Law The Law of Inertia. An object at rest will remain at rest and an object in motion will remain in motion in a straight line at a constant speed unless acted upon by an unbalanced force” (3, P. 34).

 

Newton’s Second Law The net force (the sum of all forces) acting on an object is proportional to the objects mass and its resultant acceleration” (3, P. 34).

 

So what does the second law mean?

 

Think of Newton's second law as the law that says force equals mass multiplied by acceleration. F = ma (F “Force acting on the object,” m “Mass of the object,” and a “The acceleration due to gravity”)

 

Newton’s Third Law For every action there is an equal and opposite reaction” (2, P. 35).

 

Now, since we are armed with the laws of Newtonian mechanics we can explain in greater detail Kepler’s laws of planetary motion.

 

I must cover a term that is commonly used in physics that is essential for both theories. This term is called the center of mass. The center of mass is a distance between two or more objects with respect with there masses. Often the center of mass is not located on one of the objects. Think of the center of mass in terms of a seesaw. The center of mass on a seesaw is in the center. When two people of equal weight get on both sides of the seesaw, then the seesaw will be perfectly balanced, but when the two weights are unequal then the heavier person will need to move closer to the center to balance the two objects. The center of mass is the balancing point of a seesaw for any collection of objects. In this case it will only contain two objects.

 

A little bit more of an explanation of the center of mass
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Kepler’s First Law (revisited) Both objects in a binary orbit move about the center of mass in ellipses, with the center of mass occupying one focus of each ellipse” (3, P. 45).

Instead of the center of the sun occupying the one focus of the ellipse, the "seesaw" point between the planet and the sun occupies that point.


Kepler’s second law stays intact conceptually, but the math just gets more complicated.

 
Kepler’s Third Law, however, is changed substantially. Instead of the simple equation,   P2 = a3, P = (4π2 a3)/(G(m1 + m2)). “G” is the gravitational constant which was not known until after Newton died, but “m1 + m2” was the major contribution of Newton. What this means is that the center of mass was taken into account in the laws of planetary motion. The reason Kepler did not notice this was because the data he used was obtained by the naked eye, which had less accuracy than the data used by Newton which was obtained by telescopic observations (3, P.52).

Interpretation

Kepler was a man of great importance to astronomy and physics. Without Kepler’s laws of planetary motion we would not understand much of anything surrounding the earth. Newton’s theory of Universal Gravitation would not be complete, because it would be without a model to describe planetary motion. Without Universal Gravitation we would not be anywhere near to the level of understanding that we now have reached in physics or astronomy. 

 
Sources

1.) Kepler, Johannes, The Harmony of the World. Translated by E.J. Aiton, A.M. Duncan, J.V. Field. The American Philosophical Society: c 1997.

Notes:  The Harmony of the Worlds was very confusing for me to read. It was written as a scientific book with theological justifications. This work is presented in a much different from what we now refer to as "Kepler's third law." For me this book was extremely confusing.

2.) Livio, Mario, The Golden Ratio: The Story of Phi The World’s Most Astonishing Number. New York : Broadway, c2003. 

Notes:  Mario Livio is the head of the Science Division at the Space Telescope Science Institute. He obviously has knowledge in the area of math, physics and astronomy. In this book he not only describes the golden ratio in context to all three areas. He also approaches this topic with respect to many historical accounts of figures in math, physics and astronomy. In this book he devotes a section for Kepler. In this section he describes an overview of Kepler's life, and his achievements involving the golden ratio and not involving the golden ratio. This book helped me understand the information contained in Kepler's works involving his three laws.

3.) Carroll, Bradley W., Dale A. Ostilie, An Introduction to Modern Astrophysics. Weber State University: ADDISON-WESLEY PUBLISHING COMPANY, INC, c1996.

Notes: Bradley W. Carroll and Dale A. Ostlie are both members of the Department of Physics at Weber State University at Ogden, Utah. This book was essential for my project because it presented Kepler's Three laws and Newton's revisions to Kepler's Three laws in a modern approach to Physics.

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