Kepler and the planetary harmony – Part II

The three laws that Kepler published in his book “Astronomy Nova“, is a legacy that demonstrates the elegance and precision of the planetary movement; forming the foundation that describes the interaction of different celestial bodies, fundamental principle to understand how the cosmos works.

As mentioned in the previous article, even with a complicated life, Kepler’s perseverance and temperance allowed him to finally find himself at the right time and place to finally have the means to work on his approach to explain the planetary movement. Still unresolved at the end of the sixteenth century.

 

Planetary mechanics

In the sixteenth century, although the planets’ movement had been measured with good precision, was still debated how it really was the mechanics of their movement, particularly given the backward motion or retrograde movement that was identified with them. In some periods the planet gives the appearance of temporarily moving in the opposite direction of its original trajectory, to finally return again in the same direction it had originally.

Mars’ retrograde orbit

Mars’ retrograde orbit (Wikipedia.org)

Theories

To explain this phenomenon, different astronomers from that time proposed different mechanisms of planetary movement to explain these trajectories, including the configuration of concentric circles of Plato, the geocentric model of Ptolemy, where the earth was located at the center of the solar system, a system that was accepted and endorsed as officially by the church. And the proposal of Tycho Brahe with his Tychonic System, with a complex configuration where the Earth was maintained in the center of the solar system, and the rest of the planets revolved around the Sun; while this, in turn, revolved around the Earth.

Tycho worked for many years compiling detailed records of the planets and other celestial bodies’ movement. With the intention of proving his theory about the Tychonic system he integrated Kepler as part of his team; recognizing his analytical capabilities; Tycho asked for his collaboration on very specific problems, but never giving him the opportunity to have full access to his data, to avoid any opportunity for Kepler to make some discovery by himself.

Kepler himself also defined a model of the solar system, considering the radii of the orbits for the six planets known at that time; this by identifying a correspondence between the proportions of the planet’s orbits and those of the Platonic Solids, this discovery caused a big impression to Kepler, therefore calling it the “Misterium comsographicum“.

Kepler platonic solids (famousscientists.org)

Kepler’s laws

The sudden death of Tycho Brahe, and subsequent appointment of Kepler as Imperial Mathematician for the Emperor Rudolf second, was what gave him the opportunity to analyze the data from observations made by Tycho. With this information he defined a new model of the solar system, the heliocentric model that we all now know, with the Sun at the center of the solar system and all the planets, including the Earth, spinning it around.
As mentioned, Kepler was finally able to publish this new model in 1609 in his book “Astronomía Nova“, where he presented his first two laws; now known as Kepler’s laws, which establish that.

I. Planets move in elliptic orbits, with the Sun located in one the foci.

The first law is simple to comprehend; in an ellipse there are two points called foci. In the case of the orbit of the planets, the sun is in one of these foci.

II. The radius vector describes equal areas in equal times.

This law describes the difference in speed with which the planets move around the Sun, which traverse a bigger distance from their orbit when they are closer to the Sun, or basically, closer to the body that they’re orbiting (this also happens in the movement of satellites orbiting planets, like the case of the Moon). This law describes that the relative area covered is constant.

Second law

Second law (Britannica.com)

III. For any planet, the square of its orbital period is directly proportional to its to the cube of the major semi-axe of its elliptic orbit.

His third law, published in 1619, ten years after the first two laws in his book “Armonices Mundi“, or harmony of the worlds, presents a new relationship in the planetary movement. It states that the time it takes for a planet to describe an orbit around the Sun (what we consider to be the year), is proportional to how far the planet is from the sun. And, with this in mind, we can deduce that the time it takes for planets to complete an orbit is greater the more distant from the sun the planet is.

For example, considering the case of Mars; it is at a distance from the Sun 1.524 times greater than that of the Earth, this means that the orbital period or “year” of Mars is equivalent to the square root of the resulting value of the distance to the sun raised to the cube; calculating the cube of this distance 1.524 equals 3.54; and the square root value of it is in turn 1.88. This means that the year of Mars is in proportion 1.88 times the year of the Earth. This relationship can be proven for any other planet, and shows how, given this law, the more distant a planet is from the sun, the longer is the time to complete its orbit, or the duration of its year.

 

Relationship and proportion

But a much more interesting fact that can be deduced with the third law of Kepler is when we add the constant value that regulates this relationship. It should be remembered that in the world of mathematics, a proportion and a relation are different, a relation compares two quantities, while a proportion is an equation that proves that two quantities are equivalent.

When this principle is used, Kepler’s third law which is a relationship, can be expressed as a proportion or an equation that regulates the relationship between the Sun and the planets, and is based on the constant 4π2 / G (M + m) where G is the gravitational constant and M is the mass of the greater body, in this case the Sun, and m is the minor body, or the planet, all these terms derived from the discoveries of Newton, who used Kepler’s laws as a reference to formulate his laws.

This proportion allows us not only to determine the duration of the orbit of a planet around the Sun, but, using the principles of Newton, allows us to calculate the mass of the Sun, or considering other orbital systems such as the Moon orbiting the Earth, with these proportions to know the Earth’s mass. So, in this way we can get the mass of any planet that has a satellite.

The three laws of Kepler that describe the planetary movement were a prelude that allowed to better understand the functioning of our cosmos; and were a fundamental element in the later works of Newton with his laws of universal gravitation and, centuries later, of the works of Einstein with his theory of relativity. Being one of the pioneers of modern astronomy.

Regards
Alex, ScienceKindle

 

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