# Kepler’s law

Do you still remember the memories of first riding a car? When in a moving car, you see as if a tree or building is moving. At that time you might think the trees or buildings are moving, while you and the car are rest. In fact, you and the car move, while the trees or buildings are rest. This experience of fake motion is actually experienced every day. Every morning “sunrises” on the eastern horizon then move west and “sets” on the western horizon in the afternoon.

Likewise, at night, you often see the moon moving from east to west. Have you ever thought or guessed that the sun and moon moved around the earth, while the earth was rest?

**History of the development of astronomy**

Humans who lived in antiquity (early BC) also considered the sun, moon, and other celestial bodies moving around the earth, while the earth was rest. In other words, the earth is considered the center of the universe (geocentric). This assumption is based on the limited sensory experience of humans, who daily observe the sun, moon, and stars moving, while the earth is felt to be rest. Similar to being in a moving car, you see as if trees or buildings are moving. The assumption that the earth is the center of the universe was researched and developed by the Greek astronomer Claudius Ptolemaeus (100-170 AD) in the second century AD and believed for the next 1400 years.

According to Ptolemy, the earth was at the center of the solar system. The sun and planets surround a circle (rotational motion) where the center of this circle surrounds the earth in a circular path (revolutionary motion).

In 1543, Polish astronomer Nicolaus Copernicus (1473-1543) proposed a heliocentric model, in which the sun was at the center of the solar system. The planets include the earth around a circle (rotational motion) where the center of this circle surrounds the sun in a circular path (revolutionary motion). Copernicus had a more advanced understanding of Ptolemy because he placed the sun at the center of the solar system. Nevertheless, Copernicus still uses circles as a form of planetary motion.

Exciting debates about geocentric and heliocentric models encourage astronomers to make more careful observations. Astronomers at that time observed celestial bodies using only the eyes, did not use instruments like telescopes or star binoculars. At that time the telescope has not been made. The telescope that can be used to observe celestial bodies was first made by an Italian scientist, Galileo Galilei, in 1609. Galileo used his artificial telescope to observe celestial bodies and used his observational data to argue with supporters of the geocentric model.

A famous Danish astronomer named Tycho Brahe (1546-1601) was the last astronomer to observe celestial bodies using only the eyes. After observing from 1576 to 1599, then Tycho Brahe collaborated with a German astronomer, Johannes Kepler (1571-1630), who was also a mathematician. Kepler is Tycho Brahe’s assistant. The collaboration between Tycho Brahe and Kepler lasted not long because Tycho Brahe died. After Tycho Brahe died, Kepler used astronomical data obtained by his teacher and spent about twenty years of his life creating mathematical models to explain the motions of the planets.

Kepler’s first work in the field of astronomy entitled The Mystery of the Universe was published in 1596. In the book, he sought to find an alignment between planetary orbits according to Copernicus and Tycho Brahe’s observations. But Kepler failed to find harmony between the models developed by Copernicus and Ptolemy with the results of Tycho Brahe’s observations. Therefore he abandoned the Ptolemaic and Copernicus models and then sought new models. In 1609, Kepler found that ellipses were very suitable with the results of Tycho Brahe’s observations. Kepler no longer uses a circle as a form of the trajectory of celestial bodies but ellipses.

**Kepler’s law**

This law was put forward by Kepler half a century before Isaac Newton proposed his three laws of motion and the law of universal gravitation.

**Kepler’s first law**

*The path of each planet when it surrounds the sun is an ellipse, where the sun at one focus.*

F_{1 }and F_{2 }are elliptical focal points. The sun is at one of the focal points (for example in the image selected F_{2}), the planet is at a distance of r_{2} from F_{2} or r_{1} from F_{1}. If the planet’s position changes, r_{2} and r_{1} also change.

Even so, r_{1} + r_{2} is always the same. The distance a is called the semimajor axis and 2a is called the major axis. The distance b is called the semiminor axis and 2b is called the minor axis. The ellipse’s focal point is located at distance c from the center of the ellipse, where c^{2} = a^{2} + b^{2}.

The ellipse shape is determined by the eccentricity (e) of the ellipse, where e = c / a. The eccentricity of an ellipse ranges from 0 to 1 (0 <e <1). For a circle, c = 0 therefore e = 0. The smaller the eccentricity, the ellipse is closer to the circle. Conversely, the greater the eccentricity, the ellipse is longer and thinner.

If the planet is at the left end of the ellipse (to the left of F_{1}) then the planet’s distance to the sun is a + c. This point is called aphelion. When the planet is at aphelion, the planet is at its farthest distance from the sun. If the planet is at the right end of the ellipse (right of F_{2}) then the planet’s distance to the sun is a-c. This point is called perihelion. When the planet is at the point of perihelion, the planet is at the closest distance from the sun.

**Kepler’s second law**

The imaginary line that drawn from the sun to the planet, sweeps out equal areas, for the same time interval.

In the figure below, there are only two examples of areas, namely the abc area and the ade area. These two areas have the same size. During the same time interval, the imaginary line connecting the planet and the sun sweep the area that has the same magnitude, so when moving from b to c (the planet is at aphelion), the speed of the smaller planet or planet moves more slowly, on the contrary when moving from d to e (the planet is at perihelion), the speed of the larger planet or planet moves faster. So the maximum planetary rate when it is at the perihelion point and the minimum planetary rate when it is at the aphelion point.

**Kepler’s third law**

The ratio of the squares of the orbital period of a planet to the cube of the average distances of the planet with the sun (T^{2}/r^{3}) is constant and the value is the same for all planets. If T_{1} and T_{2} state the period of two planets, r_{1 }and r_{2} represent the average distance of each planet from the sun: