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Gravity and the Motion of the Stars---

1. Introduction
Observing the sky, the movement of the stars, and the length of days has long been a concern of humanity for very practical reasons, such as planting and harvesting seasons, among others.
Ancient astronomers observed that the stars remained fixed in the sky while seven celestial bodies moved. There are several models that address the motion of the stars. Two of them believed that celestial bodies revolved around the Earth in circular orbits: that of Ptolemy (2nd century AD) and that of the Greeks (Aristotle – 4th century BC).
However, this model was insufficient to satisfactorily describe the motion of the planets, since it was found that their speed was not constant.
Then, around 1500 AD, Nicolaus Copernicus (1473-1543) devised the heliocentric model, which satisfactorily explained celestial phenomena.

Years after Copernicus's death, Tycho Brahe observed the motion of the stars for about 20 years. The data obtained from these observations were tabulated and formed the basis for the work of Johannes Kepler (17th century), his disciple.

2. Law of Universal Gravitation

Isaac Newton demonstrated that Kepler's three laws (which were based on observations) could be deduced from his law of gravitation.
F = G . (m² . m²)/d²
Where d is the distance between particles and G is the universal gravitational constant that applies to the id:
G = 6.67.10-¹¹ (N.m²)/kg²
Isaac Newton concluded that gravitational forces were responsible for keeping the planets in orbit around the Sun.

3. Kepler's Laws
3.1. Kepler's First Law

*In their orbits around the Sun, the planets follow elliptical orbits, with one focus occupied by the Sun.*

According to this law, the distance between the planets and the Sun varies. The closest point on the orbit to the Sun is called *perihelion*. The farthest point is *aphelion*.

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3.2. Kepler's Second Law
*The line connecting the centers of a planet and the Sun travels equal areas in equal times.*
A = K. Time variation T.

The constant K (depending on the planet) is called the *planet's areal velocity*.

The speed of a planet's orbit around the Sun is not constant, being maximum near perihelion and minimum near aphelion.

3.3. Kepler's Third Law
*The square of the period of revolution of any planet is proportional to the cube of that planet's average distance from the Sun.*
By the formula: (T²/R²) = k
where T is the planet's period, R is the planet's average distance from the Sun, and k is a constant valid for all planets orbiting the Sun.
In the elementary study of gravitation, orbits are considered circular.
Kepler's laws generally apply to any body gravitating around another of much greater mass, such as artificial satellites orbiting Earth, for example.

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