Mechanics

# Kepler’s First Law

Kepler’s first law , also known as the law of orbits, describes the shape of planetary orbits. According to this law, the planet’s orbits around the Sun are elliptical , despite having very small eccentricities. The discovery of this law represented a major advance for the heliocentric view of the Solar System .

## Introduction to Kepler’s First Law

Between 1609 and 1618, Johannes Kepler (1571-1630), a great German astronomer and mathematician, developed three laws capable of explaining the motion of the planets around the Sun. The first of his laws, the law of orbits, states that the orbit of the planets is not circular, but elliptical. Kepler was able to determine with great precision the trajectories of the planets, for this, he relied on a large amount of data carefully collected by the Danish astronomer Tycho Brahe (1546-1601).

Kepler’s laws represented a major evolution for the heliocentric model , previously developed by Nicolaus Copernicus , but also served as a basis for the formulation of Isaac Newton ‘s theory of universal gravitation .

The statement of Kepler’s first law is shown below, check it out:

“ All planets move around the Sun in elliptical orbits, with the Sun at one focus.”

Kepler deduced the shape of planetary orbits and was able to determine the eccentricity of the orbits of some of the known planets, in addition to predicting the existence of planets that had not been observed. The eccentricity of the orbit is the ratio of the distance between the two foci of the ellipse and the semi-major axis — the closer to zero, the more circular the planet’s orbit. In the Solar System, planetary orbits are practically circular, the eccentricity of the Earth’s orbit, for example, has a value equal to 0.0167, and the orbit of Mars , 0.093.

In elliptical orbits, there are positions that represent the greatest and smallest distance from the star to one of the foci of the ellipse. Take the Earth’s orbit around the Sun as an example : the point at which the Earth is closest to the Sun is called perihelion, and the point at which it is furthest from the Sun is called aphelion .

In the vicinity of perihelion, the Earth is more strongly attracted to the Sun, therefore, in this region, the Earth moves with greater speed and, therefore, with greater kinetic energy . When the Earth is at perihelion, its kinetic energy is minimum, however, its gravitational potential energy is maximum.

## Summary of Kepler’s First Law

According to Kepler’s 1st law:

• Planetary orbits are elliptical.
• The Sun occupies one of the foci of the ellipse.
• The position where the Earth is closest to the Sun is called perihelion.
• The position where the Earth is furthest from the Sun is called aphelion.

## Solved exercises on Kepler’s first law

Question 1) (Unicamp) Kepler’s first law showed that planets move in elliptical and non-circular orbits. The second law showed that planets do not move at a constant speed.

PERRY, Marvin. Western Civilization: A Concise History. São Paulo: Martins Fontes, 1999, p. 289. (Adapted)

It is correct to state that Kepler’s laws:

a) confirmed the theories defined by Copernicus and are examples of the scientific model that came into force from the High Middle Ages.

b) confirmed the theories defended by Ptolemy and allowed the production of nautical charts used in the period of discovery of America.

c) are the basis of the geocentric planetary model and have become the scientific premises that are still in force today.

d) provided subsidies to demonstrate the heliocentric planetary model and criticize the positions defended by the Church at that time.

Template: Letter D
Resolution:

Kepler’s laws represented a break with the geocentric view of the Solar System, defended by the Church.

Question 2) (UEFS) The figure represents the elliptical trajectory of a planet in translational motion around the Sun and four points on this trajectory: M, P (perihelion of the orbit), N and A (aphelion of the orbit).

The magnitude of the speed of this planet:

a) always increases in the MPN segment.

b) always decreases in the NAM stretch.

c) has the same value at point A and at point P.

d) is increasing at point M and decreasing at point N.

e) is minimum at point P and maximum at point A.

Template: Letter D

Resolution:

As the planet moves towards perihelion, its kinetic energy increases, so at point M it is gaining speed.

Question 3) (FGV) Johannes Kepler (1571-1630) was a scientist dedicated to the study of the solar system. One of his laws states that the orbits of the planets, around the Sun, are elliptical, with the Sun situated at one of the foci of these ellipses. One of the consequences of this law results in the variation:

a) the magnitude of the acceleration of gravity on the surface of the planets.

b) the amount of gaseous matter present in the atmosphere of the planets.

c) the length of day and night on each planet.

d) the length of the year of each planet.

e) the orbital speed of each planet around the Sun.

Template: Letter E

Resolution:

The orbital speed increases near perihelion, thanks to the approach between the planet and its star.