THE MOTION OF CELESTIAL BODIES

Ptolemaic or geocentric theory

The first theory explaining the structure of the Universe and the motion of celestial bodies was proposed by Aristotle, in the IV century B.C. According to this theory, all celestial bodies (the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn and the so-called "fixed stars") were inserted in concentric rigid spheres, which uniformly rotate around the Earth. The several peculiarities of the planetary motions, were explained by means of complicated motions along circumferences centered on theses spheres.
Celestial spheres were perfect and unchanging.
The geocentric theory was commonly accepted until the XVI century, when the Polish astronomer Nicolaus Copernicus (1473-1543) conceived the hypothesis that the Sun, not the Earth, is the center of the Universe.
 

Copernican or heliocentric theory

Copernicus said that the Earth is a simple planet orbiting the Sun, just like the other ones. This theory is then called heliocentric. Copernicus' hypothesis was supported by an accurate study which explained the motion of planets. However, the scientific community strongly opposed to the theory. Its definitive affirmation was due to the studies by Galileo Galilei (1564-1624), and to the demonstration that the orbits of all planets are ellipses, where the Sun occupies on the the two foci. Johannes Kepler (1571-1630) provided this demonstration, using the observations carried out by the Danish astronomer Thyco Brahe. We know today that the Sun is not the center of the Universe. It is just one of the many stars in our Galaxy, and the latter in just one of the galaxies populating the Universe.

A sketch illustrating the heliocentric theory, from Copernicus' "De rivolutionis".

Kepler enunciated three laws that rule the motion of the planets around the Sun. This motion is called "revolution". The time a planet takes to go back to the same point of its orbit, is called "period" of the revolution. Kepler's three laws were deduced from the observations without any theoretical basis. Isaac Newton (1642-1727) later revealed that these laws are just special cases of the universal gravitational law, which describes the interactions between any physical bodies.

KEPLER'S FIRST LAW
 

All planets move around the Sun on elliptical orbits. The Sun occupies one of the two foci, the same focus for all ellipses.

The ellipse is a plane figure, obtained by cutting a cone with a plane not orthogonal to its axis. The sum of the distances from two points, called foci, to any of of its points is constant. Since planets describe elliptical orbits, the Sun-planet distance varies as a function of time, and it has a maximum and a minimum values. The former occurs when the planet is in a point called "aphelion", while the latter occurs when it is in the "perihelion". The ratio of the distance from one focus to the center, over the length of the semimajor axis, is called "eccentricity" of the ellipse. A circumference can be imagined as a special ellipse, whose eccentricity is zero.

KEPLER'S SECOND LAW
 

The vector radius covers equal areas in equal times.
 

The vector radius connects the center of the Sun to that of the planet. Its length changes along the orbit. If we take two equal areas defined by the vector radius, the second law implies that the planet's revolution is not constant in speed. It is faster at the perihelion, and slower at the aphelion.

KEPLER'S THIRD LAW
 

The revolution periods of planets, squared, is proportional to the major semiaxis of their orbits, raised to the third power.
 

This law implies that the larger is the distance from the Sun, the slower is a planet's revolution. In fact, the closer to the Sun it is, the more the planet is affected by its attraction. Therefore, it must move faster in order not to fall upon it. Actually, both the Sun and the planet rotate around the common baricenter, but the Sun is much more massive than the planet. The baricenter then almost coincides with the Sun's center, so the only apparent revolution is that of the planet. This behavior is true each time a body rotates around a more massive one. Indeed, these laws are not only valid for the planets in the Solar System, but also for any celestial body.

If two bodies have comparable masses, then their baricenter does not coincide with any of them, and their orbit around this point become visible. This is the case, for example, of binary stars.

If instead there are three or more bodies with comparable masses, then their relative orbits cannot be predicted by any laws of Mechanics, since their description becomes too complex.

UNIVERSAL GRAVITATIONAL LAW
 

Kepler's three laws are just the consequence of Newton's universal gravitation law. It was enunciated in 1688:

Each body exerts an attractive force on any other body. The force is directed along the direction joining the two bodies. Its intensity is directly proportional to the product of their masses, and inversely proportional to the square of their distances.

If we have two bodies, whose masses are M₁   and M₂  , and whose distance is r, the attractive force is equal to

F = K (M₁ M₂)/r²
 

where K is called universal gravitation constant. It does not depend on the shape, the size or the chemical composition of the bodies. This law tells that each planet attracts the other ones just like the Sun, but with a much smaller strength. The result is that the planets' orbits are actually not perfect ellipses, since they are affected by gravitational perturbations caused by the rest of the planets.
 

ROTATION
 

Revolution is not the only motion of planets around the Sun. The other main motion is that of rotation around one axis. The time interval taken ot complete one turn is called "rotation period", or "day". A result of the rotation is the alternation of "day" and "night", just like a result of the revolution is the alternation of the seasons.