Kepler's Laws On The Movement Of The Planets

Astronomy was of interest to Kepler from an early age, but smallpox transferred in childhood hit the future scientist’s eyes hard and seriously limited him in his ability to independently conduct observations. However, interest in science has not been lost. While studying at the university in Tübingen, Kepler got acquainted with the heliocentric system of Nicolaus Copernicus and immediately supported it. The great astronomer published his first book “The Mystery of the Universe” in 1596 while working at the University of Graz – this was his first attempt to find the mathematical harmony of the Universe, later the work was republished with numerous corrections.

At the beginning of the century, Kepler moved to Prague, where he worked for about a year as an assistant to Tycho Brahe, and after the death of the latter, he became his successor as a royal astronomer and astrologer. Kepler got his hands on a huge, perfectly systematized and extremely accurate, by the standards of that time, Brahe archive with data on planetary observations.

Processing and searching for patterns in such a volume of information is a difficult task even for modern computers. But already in 1609, the first two Kepler’s laws, obtained by intuitive analysis and manual calculations, were published in the book “New Astronomy”. Probably in order to avoid unnecessary conflicts with the church, the new laws were applied exclusively to describe the movement of Mars.

He lived in an era when there was still no confidence in the existence of some general pattern for all natural phenomena. What a deep faith he had in such a pattern, if, working alone, not supported and understood by anyone, for many decades he drew strength from it for a difficult and painstaking empirical study of the motion of planets and the mathematical laws of this motion!
Albert Einstein

The third law was discovered in 1618, after Kepler’s move to Linz, and was first published in the book “The Harmony of the World”. This time, the scientist applies the discovered patterns to all the planets and satellites of the solar system. All of Kepler’s works, including those unpublished at the time of his death in 1630, were published in a collection in 22 volumes, four of which are considered lost, and the rest are kept in the St. Petersburg branch of the RAS archive.

Kepler's Laws — Kepler’s Laws on the Movement of the Planets

Contents

Kepler’s first law (law of ellipses)

The orbit of each planet in the solar system is an ellipse with the sun at one of its focal points.

It should be noted right away that the shape of the ellipse is characterized by the ratio e = c / a , where c is the distance between the focus and the center, a is the semi-major axis, i.e. half of the largest possible diameter. A circle is a special case of an ellipse in which c = 0 .

The ratio mentioned above is called eccentricity and, with its help, characterizes the degree of elongation of the planet’s orbit. As you can see in the figure, the planet, as it moves, can approach the Sun at a minimum distance (perihelion) or, on the contrary, move away as much as possible (aphelion). The greatest eccentricity of Mercury’s orbit is 0.205; then comes Mars with a value of 0.094.

Before the discovery of Kepler’s first law, the orbits of the planets were considered circular, and if this did not agree with the observational data, then additional small circles called epicycles were used. Philosophers of that time postulated the perfection and harmony of the heavenly structure, and the circles and spheres were considered ideal figures. After analyzing the data of Tycho Brahe and discarding prejudices, Kepler came to the conclusion that the orbits of the planets are ellipses nested into each other.

Kepler’s second law (area law)

The radius vector connecting the planet and the Sun describes equal areas at equal intervals.

Applying this law to our planet, we can conclude that the speed of its movement in orbit is uneven. Passing through perihelion (early January), the Earth moves as fast as possible, and at aphelion (early July) its speed is minimal. You can see this effect by observing the movement of the Sun along the ecliptic. The planet’s speed at perihelion is only 1.0339 times faster than at aphelion, which gives an absolute value of about 1 kilometer per second. Kepler’s second law is a consequence of the angular momentum conservation law and tells us that the force acting on the planets is directed towards the sun.

Kepler’s third law (harmonic law)

The squares of the orbital periods of the planets are referred to as cubes of their major semiaxes.

In this equation, T 1 and T 2 are the orbital periods of the planets, and a 1 and a 2 are the lengths of the semi-major axes. If in the previous two laws we are talking about the orbits of individual objects, then using the third law we can compare the orbital parameters of different planets in relative units. The distance from the Earth to the Sun is equal to one astronomical unit (~ 150,000,000 km), and it takes one year to complete a revolution around the star. Therefore, using these units of measurement, we can simplify the formula by removing the orbital parameters of our planet from it:

(T ₁ ) ² = (a ₁ ) ³

Observing the motion of Mars, it is not difficult to establish that its orbital period is 687 days or 1.88 years. Substituting this value in the formula, we get 1.524 astronomical units. It was not possible to calculate the value of the radius of the Mars orbit in absolute units during Kepler’s lifetime since at that time the distance from the Earth to the Sun had not yet been established with any acceptable accuracy. However, even the relative measurement of the solar system was a giant breakthrough for astronomy.

From Newton to the present day

Thanks to Kepler’s discoveries, the heliocentric system of the world reached a qualitatively new level of development: the uneven motion of the planets was explained; The earth lost its special status, which was attributed to it by Copernicus; managed to get rid of the pile of epicycles. In his works, Newton confirmed the correctness of Kepler’s laws, the scientist deduced them mathematically from the laws of mechanics, the law of conservation of angular momentum and the law of universal gravitation. He also managed to find an inaccuracy in the third law and refine it by adding the mass of the star and the masses of the planets to the formula:

The theory outlined by Kepler is true not only for any planetary system in the Universe but can also be used to calculate the orbits of artificial and natural satellites – in this case, the planet will be considered the centre of gravity. At the moment, astronomers have discovered almost four thousand exoplanets and, although they are not available for direct observation, Kepler’s equations are actively used to calculate the parameters of their orbits.

A fairly large number of stars are found in binary systems, and the study of such systems is extremely important for understanding the processes of star formation. Even if one of the companion stars is too dim to observe directly, by applying Kepler’s third law, astronomers can calculate the characteristics of the system and find out its total mass.

And I would like to end the material with a quote from the great astronomer regarding astrology:

Of course, this astrology is a stupid daughter, but, my God, where would her mother, highly wise astronomy, if she did not have a stupid daughter! After all, the world is much more stupid and so stupid that for the good of this wise old mother, the stupid daughter must talk and lie. And the salaries of mathematicians are so negligible that the mother would probably starve if her daughter did not earn anything.
Johannes Kepler

Recommended Books:

1. Kepler

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