Kepler's Laws: how to understand the motion of the planets

planets

One of the questions that haunted the great thinkers of mankind was how the planets moved and they wanted to find an effective way to predict their behavior. However, from the raw data of thousands of observations they managed to obtain the famous Kepler's laws, an easy way to understand the behavior of the planets around the sun and here I tell you about it.

Solar system distances
Artistic representation of the scale of distances between the orbit of each planet of the solar system. Click on the image to explore it.

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Creating a star map

Tycho Brahe was a very important character for history, taking advantage of the 4% of the gross domestic product of Denmark; granted by the king of that time, built a complex whose objective was to produce the most complete ephemeris possible.

Employing dozens of different instruments, and at a time when the telescope was not yet used to observe the sky, possible. Employing dozens of different instruments, he is considered to be the greatest observer of the sky before the use of the telescope in astronomical fields. He and his team created a continuous record of the position of stars, planets and other objects in the celestial vault.

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Drawing of Uraniborg, Tycho Brahe's complex. It was capable of manufacturing scientific instruments, analyzing results and even publishing and printing them.

Shortly after his death, Johannes Kepler gained access to all this data. At the time there was still the mystery of why Mars went through a phenomenon known as retrograde motion; in which the planet appears to stop, and backward and forward again, which was not satisfactorily explained by any model of the solar system. Thus, from Brahe's records, he proposed a model and three laws that governed the movement of the stars.

Kepler's three laws of planetary motion

  • Planets have elliptical trajectories with the sun at one of their foci.

    To begin with, we already obtain enough information about the shape of the trajectory traced by the stars around the Sun, through flattened circles or ellipses. In this case two new terms appear, perihelion and aphelion, that is, the closest and farthest point of the orbit, respectively.

    Once the general shape of the trajectory is known, different mathematical tools can be used to better understand the observed phenomena. And, in fact, it is from this special geometry of ellipses that the following laws are constructed.

  • A planet travels equal areas in equal times

    Together with the observations made by Tycho Brahe and Johannes Kepler's analysis of these, different changes in the linear velocities of the planets were observed depending on their distance from the sun. But at the same time those changes involved a conserved quantity; something very important in physics, and very useful to better understand gravity.

    No matter in which direction to the Sun a planet was observed, by letting it move for a fixed time should always cover the same area between the area described by its trajectory and the focus of the ellipse, This is due to the change in velocity depending on the distance to the star.

1200px Kepler second law.svg
Sample of Newton's first two laws, where the orbit of the planet is an ellipse with the star at one of its foci. The shaded areas correspond to two different areas but obtained at equal times.
  • The square of the period is directly proportional to the cube of the semi-major axis.

    In order to characterize an ellipse, two fundamental numbers are required, the eccentricity; a number that takes values greater than zero and less than one that describes how flattened it is, and the semi-major axis; the length between the center and the farthest point. Then, to understand the orbits we use the period, which refers to the time it takes for a planet to complete one revolution around the Sun.

    Together, these two values maintain an intrinsic and very useful relationship when studying the movement of the planets, in such a way that by knowing one you can know the other without any problem. However, the constant that served as a transformation between the period and the semi-major axis was not known and had to be estimated from direct observations of the then known six planets.

Newton and the law of universal gravitation

Inspired by the work of Johannes Kepler on the study of the motion of the planets around the Sun, Newton wanted to unveil not only the dynamics that governs them; that is, to study exclusively the motion, but to go a step further and unravel the mechanics that made it possible; that is, the forces that are responsible for inducing those peculiar trajectories.

Omega Centauri or NGC 5139 is a globular cluster of stars seen in the constellation Q90.jpeg Q90.jpeg
Photograph of the Omega Centauri cluster in the direction of the constellation Centaurus. The photograph shows hundreds of stars bound together by the force of gravity.

From the analysis of the Moon and the planets, he found that an attractive force whose magnitude was directly proportional to the product of the masses and inversely proportional to the square of the distance separating the two objects worked incredibly well for predicting and justifying planetary orbits and, but with less precision, that of the Moon around the Earth.

This is how Newton's law of universal gravitation was born, a simple but incredibly powerful way to explain not only the motion of the stars, but it was also applicable to our daily lives. For example, gravity can be used to correctly understand the period of oscillation of a pendulum, the fall time of an object at a given distance, the approximate distance a cannonball would travel, among many other things.

               The two-body problem

In physics and mathematics, "solving the equations of motion" is understood as being able to know all the dynamics; motion, of a system under certain forces. In this case; Sun and planet, it is called the two-body problem and since our star has a mass several orders of magnitude more than the bodies orbiting it, it becomes a one-body problem. With this simplification and through mathematical tools, Kepler's laws can be obtained without major problems.

movluna
3D artistic recreation of the orbital plane of the Moon around the Earth, which approximately satisfies Kepler's laws. At the same time, the trajectory of the Earth around the Sun.

Kepler not only did a fascinating job in deriving his laws solely and exclusively from observations, but his findings proved to be correct for the system under study. Kepler's laws about the motion of the planets can also be extended to moons and satellites that are not far from the Earth.

Francisco Andrés Forero Daza