### Question

What are Kepler's laws of motion and what exactly do they mean?

Born in Germany in 1571, Johannes Kepler lived in the wake of the Copernican revolution.

It was a popular belief of all peoples of the known world, inherited from the Peripatetic thinkers of the ancient Greek world, that the Earth was at the centre of the universe (all objects fell toward the centre of the universe, after all...).

This so-called anthropocentric system was fully supported by the Catholic Church, that punished severely those who tried to spread ideas that differed from their dogmas.

Copernicus was the first to assert that it was not the earth to be at the centre of the universe, i.e. the solar system but, indeed, the sun.

Kepler went further ahead, and, basing on the results gathered by astronomer Tycho Brahe whom he assisted in Prague, Czech Republic, starting in the year 1600, he formulated the laws of planetary motion. In other words, he was the first to realise that the motion of all planets is ruled by the same laws.

Kepler formulated three laws: the first and second law to start with, and the third law seven years later.

Before recalling the three laws, someone might find them rather simple, but when assessing their significance, it would be apt to put them in context, in other words it should never be neglected that at the time the peoples of the world had not got a clue about what goes on in the heavens.

Moreover, at the time methodology in science was at the dawn, therefore drawing scientifically sound conclusions was a very hard task indeed.

That's why Kepler is by all means among the key players of one of the most significant scientific revolutions in the history of humanity.

The first law states that each planet moves around the sun in an ellipse, with the sun at one focus.

An ellipse has two foci, i.e. two centres, unlike the circle that has just one. Therefore an ellipse is a foreshortened circle.

According to the second law, planets do not revolve around the sun at a uniform speed, but they are faster when closer to the sun, and slower when further from it. In the second law, Kepler used the idea of radius vector, which is the line drawn from the sun to any point in a planet's orbit, and stated that the radius vector from the sun to a planet sweeps out equal areas in equal intervals of time.

This law is significant because it correlates the portions of orbit when a planet is close to the sun with those when it is further away and moves more slowly, and really means that the velocity at which a planet moves is part of the grander plan of the harmony of nature.

The third law states that the squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits the period being the time a planet takes to cover a whole orbital ellipse. The semimajor axes of an ellipse are the longest lines crossing the area covered by the ellipse, and including the two foci.
Answered by: Roberto Ruggiu, M.S., Free-lance scientific consultant

Kepler's Laws of Planetary Motion were based upon observational data from Tycho Brahae, a Dane, and a very careful astronomical observer.

Based on these data, Kepler was able to formulate some general rules that gave a "best fit" to the data, although he was somewhat obsessed with making the planetary motions fit somehow into the Pythagorean ideal of the Five Perfect Solids. Kepler labored intensely to find a way to use the five solids to describe planetary motion, for he believed that some nested arrangement of the solids would show how the Creator meant Nature to be simple and elegant and geometrically symmetric.

Well, obviously, the five solids have nothing to do with the motions of the planets! So, Kepler eventually gave up on this crusade and admitted to himself that he would have to build a system that was BASED on the evidence of observational data. In this sense, Kepler was the father of experimental science! He abandoned his cherished idea of a universe based on geometric solids when it was obvious that it simply did not fit the evidence.

Back to Kepler's Laws.

Kepler's First Law states that planetary orbits follow elliptical paths.

Kepler's Second Law states that the areas swept out by a planet during equal time intervals must be identical. Imagine a pie chart, but with an ellipse rather than a circle. A slice of the pie for the Earth, representing one day's worth of motion when the Earth is near the sun will have the same AREA as a slice of the pie representing one day's worth of motion when the Earth is far from the sun. Even though the "near" pie slice will be thicker, it will also be more squat. Likewise, even though the "far" pie slice will be thinner, it will also be more tall. So, these thing compensate and when you calculate the area, you find they are identical for equal time intervals.

Kepler's Third Law states the relationship between the period of motion for a planet and the semi-major axis of the planet's elliptical orbit. The relationship is that the square of the period is proportional to the cube of the semi-major axis.

The really interesting thing is that Kepler's Laws "fall out" of Newton's Laws! For example, when you solve for the orbit of a planet using Newton's Laws, you get the equation for an ellipse. When you solve the general problem of motion under a central force (which gravity is) that varies as the inverse square of the distance (which gravity also is) using Newtonian analysis, Kepler's second and third Laws are self-evident. So, while Kepler used purely computational techniques to discover his Laws (meaning, he did a lot of number crunching), his Laws are obvious consequences of the analytical approach of Newton! This is an example of what is called the "correspondence principle," where newer theories must encompass any correct observations of past theories and/or laws.
Answered by: Andrew Paradis, M.S., High School Physics and Calculus teacher, CT USA

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### Science Quote

'Natural science does not simply describe and explain nature, it is part of the interplay between nature and ourselves.'

Werner Heisenberg
(1901-1976)