Johannes Kepler – Math Teacher's Resource Blog

As we gaze across a beautiful valley or stare in awe at a distant mountain, it is easy to forget that we are on a spinning platform that is traveling on an elliptical orbit around the sun at an average speed of 66,600 miles per hour. I find this seemly unending journey truly amazing. In this post, I would like to take a look at some of the facts that mankind has learned about this journey.

Before Nicholas Copernicus (1474 – 1543), many people thought that the Sun, planets, and stars rotated about the Earth, and each planet in turn rotates on its own private circular arc. This complicated Earth centered view of nature became so entrenched that it became an article of faith in the Catholic Church. In fact, the Catholic Inquisition threated Galileo (1564 – 1642) with torture on the rack unless he publicly retracted his belief in the Sun centered circular orbit Copernican world system. Galileo publicly retracted his belief in the Copernican world view and was spared torture on the rack, but spent the remaining years of his life under house arrest.

Johannes Kepler (1571 – 1630) discovered three laws of planetary motion which is a relatively simple description of planetary motion. (You may find it helpful to read my post Demonstrating Dynamics in a Mathematical Model.) Kepler’s first law stated that the orbit of a planet around the Sun is an ellipse where the Sun is located at one of the two foci of the ellipse. An ellipse is a very special curve where every point P on the ellipse, the distance from P to one focus point plus the distance from P to the other focus point, is a constant. The diagram below shows an ellipse with foci at F₁ and F₂, length of major axis = 10 units, length of minor axis = 6 units, and center point at (0, 0). For very point P on an ellipse, the sum of the distances from point P to the two focus points equals the length of the major axis. As indicated in the diagram below, an ellipse can be drawn by first anchoring the endpoints of a length of string on a piece of paper or cardboard. Use a pencil to make the string taunt, and then trace the curve by keeping the string taunt as you move the pencil along the elliptical curve.

To better understand planetary orbits, it’s necessary to understand what we mean by the eccentricity of an ellipse. If a = half the length of the major axis, and c = the distance from the center to a focus point, then the eccentricity e of the ellipse = c/a. Thus elliptical eccentricity e ranges from 0 to 1. If e = 0, the ellipse is a circle, and if e = 1, the ellipse degenerates to a line segment with foci at the endpoints of the major axis. (By definition, the eccentricity of a parabola equals 1, and the eccentricity of a hyperbola is greater than 1.) The two diagrams below show eccentricity values for five ellipses where the ellipse and foci have the same color. Note that eccentricity approaches 1 as the foci approach the endpoints of the major axis. The eccentricity of the Earth’s orbit = 0.0167086. This is the reason, I suspect, that Copernicus thought the Earth’s orbit was circular, not elliptical. Since half the length of the major axis of the Earth’s elliptical orbit equals 149.6 million km, it follows that the Sun is 0.0167086*149.6 million km = 2.4996 million km from the center of the Earth’s orbit.

The two diagrams below show an exaggerated oval shape of the Earth’s yearly orbit around the Sun; the purpose is to draw your attention to key time periods in a year. Orbital dates can vary slightly from year to year, and therefore the dates shown in the diagrams are approximate. The following points describe the key time periods in Earth’s orbit:

At the point of perihelion, the Earth is at its closest point of 147.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted away from the Sun, which results is less direct sunlight and cooler average temperatures.
At the point of aphelion, the Earth is at its farthest point of 152.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted towards the Sun, which results in more direct sunlight and warmer average temperatures.
The equinoxes and solstices divide a year into approximately four equal time periods or seasons. At the fall and spring equinoxes, the Earth’s polar axis is perpendicular to the plane of the Earth’s orbit which results in equal periods of daylight and darkness. At the summer and winter solstices, the Earth’s polar axis is tilted towards or away from the Sun which results the longest and shortest days of the year.
At the point of perihelion, the Earth reaches its fastest orbital speed of 109,080 km/hour.
At the point of aphelion, the Earth reaches its slowest orbital speed of 105,480 km/hour.
The average or mean orbital speed of the Earth equals 107,200 km/hour or 66,600 mph.
It takes the Earth 365.256 363 004 days to orbit the Sun. Because of the extra 0.256 363 004 days in a year, it’s necessary to add an extra day to our calendar every four years in February. To be more specific, leap years occur in years that are multiples of 4 or 400, but not multiples of 100. Hence the years 2000 and 2400 are leap years, but the years 1800, 1900, 2100, 2200 and 2300 are not leap years. All other years that are multiples of 4 such as 1868, 1936 and 2016 are leap years.
In the diagrams below, note that seasons in the northern and southern hemispheres occur at opposite times of the year.

Everyone knows that the Earth does a daily rotation about its polar axis. Here are a few facts about the Earth’s rotation.

The Earth rotates in about 24 hours with respect to the Sun and once every 23 hours, 56 minutes and 4 seconds with respect to the stars.
The Earth’s rate of rotation rate is slowing with time. Atomic clocks have demonstrated that a modern-day is about 1.7 milliseconds longer than a day in 1900. (I doubt that this fact will be reported in the national news any time soon.)
In the northern hemisphere, the Earth rotates east towards the Sun in the morning hours and away from the Sun in the west in the evening hours. This is the reason that the folks in New York see the Sun about 4 hours before the folks in California.
Technically speaking, there is no such thing as sunrise and sunset. The Sun only appears to rise and set in the sky because of the rotation of the Earth. Buckminster Fuller who was an American architect (geodesic domes) and systems theorist suggested that we should the terms sunsight and sunclipse because the terms sunrise and sunset do not accurately describe what we observe.
The Earth’s rate of rotation is not constant. The true solar day is about 10 seconds longer at the point of perihelion and 10 seconds shorter at the point of aphelion.
At the equator, the Earth’s linear speed of rotation is 465.1 m/s, 1,674.4 km/h or 1,040.4 mph. At higher latitudes, the linear rate of rotation is reduced by a factor of Cos(angle of latitude). Example: The Kennedy Space Center is located 28.59° North latitude and has a linear rotation rate of 1,674.4Cos(28.59°) = 1,470.23 km/h = 913.56 mph.

I will close this post about an epiphany I experienced many years ago. As I recall, it was about March of 1975 when my neighbor Chuck Beck invited me into his back yard to view Sun spot activity. Chuck had placed his expensive Celestron telescope with an attached power cord and lens filter on his picnic table. As I adjusted a knob on the Celestron in order to keep the Sun in view, I had the same physical sensation in my legs as if I was riding a merry-go-round. I thought to myself, “Johnson you really ARE on a moving and spinning platform in space!”

Johannes Kepler Kopie eines verlorengegangenen Originals von 1610 — Johannes Kepler
Kopie eines verlorengegangenen Originals von 1610

“In it [differential calculus], mathematics becomes a dynamic mode of thought, and that is a major step in the ascent of man,” wrote Dr. Jacob Bronowski in The Ascent of Man. In a previous post, we saw how differential calculus gives us a dynamic mode of thought. Because static graphs in a textbook fail to capture a feel for the dynamics of a model, in this post, we’ll discover how computer graphing technology can be used to create animations that demonstrate the underlying dynamics in a mathematical model of a physical system.

For the first demonstration, a polar equation of an ellipse is used to model Kepler’s first and second laws of planetary motion. The polar equation r = 0.5*20 / (1 + 0.5Cos(θ)) has eccentricity = 0.5, and foci at (0, 0) and (13.333, 180⁰). Kepler’s first and second laws of planetary motion are given below. The Earth’s elliptical orbit has eccentricity = 0.0167 which results in an almost circular orbit. This is probably why Copernicus thought that the planets traveled at a constant speed in circular orbits around the Sun. Saturn’s elliptical orbit has eccentricity = 0.0556 which results in the familiar oval shape of an ellipse.

First Law: All planets in the solar system orbit the Sun on an elliptical curve where the Sun is located at one of the focus points of the ellipse.

Second Law: The speed of a planet increases as the planet moves closer to the Sun, and decreases as the planet moves farther from the Sun. A line segment joining a planet and the Sun sweeps out equal areas during equal time intervals.

In the animation below, the Sun is located at polar point (13.333, 180⁰), and the moving trace mark represents a planet orbiting the Sun. The two sectors, marked with red segments, represent equal area sectors that were swept out in equal time intervals. Because the software uses the origin and the polar trace mark point on the curve to draw the radius of a polar trace mark, it may appear that a planet is primarily orbiting about the second focus point instead of the Sun.

The second demonstration uses the polar equation r = 20 to model a terrifying gut wrenching ride on a Ferris wheel that has a 40-foot diameter, and turns counterclockwise one revolution every 12 seconds. The moving trace mark represents a rider’s position at time t in seconds, and t = 0 seconds when the angular position of the rider = 0 degrees. Using differential calculus, the rider’s horizontal velocity and vertical velocity at time t can be deduced. Refer to the table below. Anyone who has ridden on a Ferris wheel remembers the forces acting on his/her body as the result of his/her changing horizontal and vertical velocity as the wheel turns.

Teaching Points: (Of course, what is taught depends on the mathematical level of the student.)

We live in a dynamic, changing world. Students should be exposed to the concept of variable rate of change as early as possible. Even though younger students can’t do differential calculus, a teacher-directed animation of the rate concept will help students better understand how mathematics can describe some of nature’s laws.
Students should be taught that linear relationships are characterized by a constant rate of change. The dependent variable changes at a constant rate with respect to the independent variable. Students can see the constant rate of change of the vertical velocity of the trace mark as the trace mark advances left to right on the graph of a line.
Show students the movement of a trace mark on the curve y = 8Sin(x). Because the x-variable is changing at a constant rate, the horizontal velocity of the trace mark is constant. All students can see that the vertical velocity of the trace mark changes as the trace mark advances left to right. If they imagine that they are on a roller coaster, they can feel the variable forces acting on their body as the trace mark advances left to right.
Show students the movement of a trace mark on the curve y = 3x^(1/3). Where the curve is somewhat linear, the vertical velocity of the trace mark is almost constant. As the trace mark approaches the origin, the vertical velocity of the trace mark increases. It can be explained to a calculus student why the vertical velocity of the trace mark is infinite for a fleeting instant of time x = t = 0. Because the x-variable is changing at a constant rate, the x-variable can be treated as time variable t.
The Johannes Kepler (1571-1630) and Tycho Brahe (1546-1601) relationship is an interesting story. Day and night for many years, in his observatory on the island of Hven, near Copenhagen, Tycho Brahe carefully recorded the positions of the planets and stars. In 1600 Kepler met Tycho Brahe, and gained access to Brahe’s data. In the nine-year period after Tycho Brahe’s death, Kepler used the observational data to deduce his first and second laws of planetary motion. Kepler discovered his third law of planetary motion much later. It is difficult to understand, imagine, or appreciate how Kepler was able to use inductive reasoning to discover the patterns of planetary motion. Students should be told this story because it demonstrates the monumental gift of human intelligence, and the struggle that is required to advance knowledge.

The graphics in this post were created with the program, Basic Trig Functions, which is offered by Math Teacher’s Resource. In addition to graphing x-y variable relations and polar functions, users can graph the powers or roots of a complex number, and view a list of the powers or roots, which appears to the right of the graphic output. Segments and vectors can be drawn by left-clicking and dragging the mouse. The Edit/ Edit Graphics menu provides options for setting segment color, pen width, and Head/Tail parameters.

The user interface for all program modules is simple and intuitive. When graphing equations, users can select a sample equation, which is automatically pasted into the active equation edit box. When appropriate, the program provides comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

Tag: Johannes Kepler

Orbit and Rotation of Planet Earth

Demonstrating Dynamics in a Mathematical Model