The Development of Kepler’s Law of Planetary Motion: An Overview

Kepler’s law of planetary motion is of great contribution to the field of celestial mechanics. In his work, Kepler considered that the Sun is firmly at the center of the solar system. He is the first kind who introduced celestial motion which is not circular, but elliptical.

About Kepler

Johannes Kepler, a German Astronomer, Mathematician and Physicist who is very well-known for his three laws of planetary motion which revolutionized the fields of Astronomy and classical mechanics. He is known to be the founder of Celestial Mechanics. Kepler was an influenced follower of Copernicus, who stated himself being Copernican as “physical or if you prefer, metaphysical reasons”. In 1597, he published “The Mysteriumcosmographicum” (The Sacred Mystery of the Cosmos), which is a supportive work defending the Helio-centric theory (Copernican system).

He stated in his work “I wanted to become a theologian, and for long I was uneasy, but now, see how through my efforts God is being celebrated in Astronomy.” In this work, he considered Sun being the center of the system and planets revolving around it in circles in their orbits. Though, he gave the wrong description of planetary orbits. (Read more on Parts of The Sun)

Later, he worked as an assistant under Tycho Brahe in fitting Brahe’s astronomical observations on Mars with Tycho Brahe’s cosmological model. In 1609, he published his work “Astronomia Nova” (New Astronomy) which constituted the first two laws, i.e., The Law of Ellipses and The Law of Equal Areas.

He took nearly six years to derive the first law of planetary motion, where he observed Mars moves in an elliptical orbit around the Sun. However, Astronomers did not readily accept Kepler’s first two laws as there was no proper explanation of orbits being elliptical and the second law was ignored for nearly eight decades.

Kepler's Law of Planetary Motion

After ten years, in 1619, he published “Harmonice Mundi” (The Harmony of the World) in which he gave his third law of planetary motion: The Law of Harmonies, wherein the derived the heliocentric distances of the planets and their periods from considerations of musical harmony.

In the Harmony of the World, he stated …if you want the exact moment in time, it [the correct form of the law] was conceived mentally on the 8th of March in this year one thousand six hundred and eighteen but submitted to the calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind.

So strong was the support from the combination of my labor of seventeen years on the observations of Brahe and the present study, which conspired together, that at first, I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances…[Harmony of the World, Linz, 1619]

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  • Kepler’s First Law of Planetary Motion: The Law of Ellipses

According to this law, The planets revolve around the Sun in elliptical orbits, with the Sun at one of the two foci of the elliptical orbit.”

Kepler’s Law of Planetary Motion

Though, this law is not widely accepted by scientists at that time as it went against the major assumption of bodies move in circular orbits at that time. But later Newton showed that his law of gravitation gave the same results as that of Kepler’s laws and also showed that the shape of the orbits were conic sections.

  • Kepler’s Second Law of Planetary Motion: The Law of Equal Areas

According to this law, An imaginary line from the Sun to the planet revolving around it sweeps out equal areas in equal intervals of time.”

kepler's law of planetary motion

This law is based on the speed of the planet as it orbits. Though Kepler did not speak about speeds when he devised this law.

  • Kepler’s Third Law of Planetary Motion: The Law of Harmonies

According to this law, The square of the period of orbit divided by the cube of the mean distance of the planet from the sun is equal to Kepler’s constant for that planet being orbited.

This law is a great mathematical contribution from Kepler where he related the mean distance of the planet from the Sun to its period of the orbit, i.e., the time of revolution.

Mathematically, Kepler's Law of Planetary Motion

Where,

T is the period of revolution,

a is the mean distance of the planet around the Sun

K is Kepler’s constant, which is a constant remains the same for any planet revolving the Sun.

If we consider a different object revolving around the Sun, or say, moon or satellites around the Earth then Kepler’s constant changes, i.e., Earth’s Kepler’s constant is different from Sun’s Kepler’s constant. Thus, Kepler’s laws of planetary motion play a vital role in understanding the motions of natural and for placing artificial satellites as well as unpowered spacecraft in orbit in near planets.

Determining stellar masses using Kepler’s third law:

The mass of celestial bodies like stars is called stellar mass. The Kepler’s constant considered in Kepler’s third law of Planetary Motion depends on the total mass of the celestial bodies involved which helps in determining the mass of the system which moves around each other.

Mathematically,

Kepler's Law of Planetary Motion

Where,

T is the period of revolution,

a is the mean distance of the planet to the Sun,

M1 is the mass of the Sun,

M2 is the mass of the Earth,

k is the Gaussian Gravitational Constant, which is defined in terms of Earth’s orbit around the Sun and its value is 0.01720209895 (10 significant digits).

What according to Kepler made planets to move around the Sun in such a way?

Kepler was unable to give the correct explanation of why planets had to move in such a manner. From his laws, he realized that the planets move slowly when they are further from the Sun. He guessed that the planets are revolved in some way because of Sun and change in orbital speeds is due to some kind of Sun’s magnetic force.

He did not make any attempt to devise or define actual physical processes governing the planetary motion, which was made by Isaac Newton in the late 17th century.

Influential Laws of any time and Usefulness

Kepler’s three laws of planetary motion are of great contribution to the field of celestial mechanics. In his work, Kepler considered that the Sun is firmly at the center of the solar system. He is the first kind who introduced a celestial motion which is not circular, but elliptical.

Though he discarded Aristotle’s theory of motion model, he understood that his model was lacking in motion dynamics and was unable to explain why planets had to move elliptically. Also, he doubted the planetary motion is caused due to some magnetic force originating from the Sun, which became an enthralling intellectual problem in the following century.

These laws can be applied to any celestial body revolving around any star. Like, an exoplanet orbiting a star. With some necessary modifications, these laws can be used to describe binary systems, or how a star orbit around a compact object, or any compact object orbiting another compact object.

( Read More- Binary Star or Double Star | Why are Binaries Important? | ScienceMonk )


 

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