Kepler’s Law of Planetary Motion – The Heliocentric Model

Last updated on October 10th, 2020 at 05:57 pm

kepler's-law-of-motion

The centuries-old dispute between the Geocentric Model and the Heliocentric Model Of The Solar System was finally put to the rest by the German astronomer Johannes Kepler. In fact, with Kepler’s law of planetary motion; he solved the riddle that we are living in the Heliocentric Model of the Solar System.

Meaning, the sun is at the center of our solar system, not The Earth. Well, Copernicus Heliocentrism also played its crucial role.

But, the role of Kepler’s law overshadowed that of Nicolaus Copernicus. Johannes Kepler Heliocentric Model showed that the orbits of planets are ellipses rather than circular.

 

Kepler’s Law Of Planetary Motion – The Heliocentric Model

Here is the list of things I will be discussing in this article related to Johannes Kepler Heliocentric Model. So, just buckle up.

  1. Historical View And Development Of Kepler’s Law Of Motion
  2. Comparison b/w Kepler Universe Model To Copernican Model
  3. Kepler’s Three Laws Of Planetary Motion
  4. Eccentricity Of The Planets

 

Historical View And Development Of Kepler’s Law Of Motion

Well, before the emergence of the Scientific Revolution or Copernican Revolution, the Aristotelian-Ptolemaic Universe was widely accepted as the working model of the Universe. Just for simplicity, let’s say it as Aristotelian Universe.

According to Aristotelian Universe or Earth-Centered Universe, (stationary) earth is the center of the universe; and each planet within our solar system included the sun revolves around the earth.

copernican-celiocentrism-similar-to-kepler's-law-of-motion
Copernican model/Credit: Wikimedia Commons

This geocentric model of the solar system was prevailing until the arrival of Copernican Heliocentrism.

Well, for those who don’t know, Nicolaus Copernicus was not the first person to proclaim that the sun is the center of the solar system, not earth. Therefore, every planet including earth revolves around the sun.

Related, Top 6 Obsolete cosmological Theories Of The Universe

It was the Greek astronomer Aristarchus of Samos who proposed the heliocentric model. Though there are no written works of him left now.

aristarchus-diagram-for-calculating-size-of-earth-moon-and-sun-similar-to-kepler's-laws
Aristarchus’s 3rd-century BC calculations on the relative sizes of (from left) the Sun, Earth, and Moon, from a 10th-century AD Greek copy/Credit: Wikimedia Commons

The heliocentric model was not originally accepted when it was first proposed by Aristarchus in the second-century bc. Simply, Because of the prevailing Aristotelian Universe.

In fact, as a contemporary of Aristotle, he always criticized Aristotle beliefs of Earth-Centered Universe.

 

Problems In the Copernican Model

There were some notable problems in Nicolaus Copernicus heliocentric model. In the Copernican model, all the planets revolve around the sun in a circular motion which was enhanced by Johannes Kepler.

Of course, with the help of his mentor Tycho Brahe’s orbital calculation. In order to know more about Tycho Brahe’s works, You can check this article.

Later, Sir Issac Newton showed that Johannes Kepler laws of planetary motion are the direct consequence of Newton’s Law Of Gravitation which governs the forces between two massive objects.

 

Comparison b/w Kepler Universe Model To Copernican Model

Although, both the Johannes Kepler Heliocentric model and the Copernican model contradicted the classical view of the Aristotelian-Ptolemaic Universe. Yet, there were some ideological differences between them; such as:

  1. In the Copernican Model, all planet revolves around the sun in a circular orbit. While on the other hand; in Kepler Universe Model, all planet revolves around The Sun but in a flattened circular orbit that means the elliptical orbit.
  2. Copernican Model states that the sun is located at the center of the orbit. While on the other hand; Johannes Kepler Heliocentric model states that the sun is located at one of the foci of an elliptical orbit.
  3. In the Copernican Model, the speed of the planet in an orbit remains constant. While on the other hand; in Johannes Kepler Heliocentric Model, the speed of the planet in an orbit is not constant but the area speed remains constant.

 

Kepler’s Three Laws Of Planetary Motion

kepler's-law-of-planetary-motion
Figure 1: Illustration of Kepler’s 3 laws with two planetary orbits. The orbits are ellipses, with focal points F1 & F2 for the first planet and F1 & F3 for the second planet. The Sun is placed in focal point F1. The two shaded sectors A1 & A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2. The total orbit times for planet 1 and planet 2 have a ratio/Credit: Wikimedia Commons

The model of the solar system that Johannes Kepler proposed has three 3 important laws. In fact, the importance of the sun in keplers laws of motion can be seen in these three laws. Kepler’s Three Laws Of Planetary Motion can be described as follow:

 

Kepler’s First Law Of Planetary Motion

When planets travel in an elliptical orbit around the sun with sun located at one of the foci is known as Kepler’s First Law Of Planetary Motion. That’s why Kepler’s first law is also known as the law of ellipses.

Mathematically, according to the Kepler’s first law, an ellipse can be represented by:

keplers first law images

where,

r = distance from the sun to the planet
p = semi-latus rectum
ε = eccentricity of an ellipse
θ = angle of the planet’s current position from it’s the closest approach, as seen from the sun.

 

Kepler’s Second Law Of Planetary Motion

kepler's-second-law-of-planetary-motion-giff
The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity/Credit: Wikimedia Commons

In planetary motion, the line joining the planet and the Sun sweep out equal areas in equal intervals of time. This is a direct consequence of Kepler’s second law of planetary motion.

That’s why Kepler’s second law of planetary motion is also known as the law of equal areas.

 

Kepler’s Third Law Of Planetary Motion

According to Kepler’s third law, the square of the time period of a planet is proportional to the cube of the semi-major axis of its orbit. That’s why Kepler’s third law of planetary motion is also known as the law of harmonies.

Well, mathematically, Kepler’s third law can be represented by the formula:

{displaystyle {frac {P^{2}}{a^{3}}}={frac {4pi ^{2}}{G(M+m)}}approx {frac {4pi ^{2}}{GM}}=mathrm {constant} }

where,
M = mass of the sun
m = mass of the earth
G = gravitational constant
P = time taken by a planet to complete an orbit around the sun
a = mean value between the maximum and minimum distances between the planet and the sun.

 

Eccentricity Of The Planets

The Eccentricity Of The Planet is the ratio between the distance from the center to the focus divided by the semi-major axis. ε is the symbol of Eccencrity.

Well, it can be divided into two groups. Perihelion i.e the closest point of a planet to the sun. On the other hand,  Aphelion i.e the farthest point of a planet to the sun.

 

Conditions For Eccentricity

  1. If ε = 0, then the planet’s orbit will be circular; same as Copernican Heliocentrism
  2. Or, 0 < ε > 1, then planet’s orbit will be elliptical
  3. If ε = 1, then the orbit will be parabolic
  4. Or, if ε > 1, then the orbit will be hyperbolic

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About Atul Sinha

I am a mechanical engineer by profession. Just because of my love for fundamental physics, I switched my carrier, and therefore I just completed my postgraduate degree in physics. Right now I am a loner (as like ever) and a Physics blogger too. My sole future goal is to do a Ph.D. in theoretical physics, especially in the field of cosmology. Because in my view, every part of physics comes within the range of cosmology. And I love traveling, especially the Sole one.

2 Replies to “Kepler’s Law of Planetary Motion – The Heliocentric Model”

  1. There are a lot of ellipticals on the market these days. But if you understand how to compare them effectively, you will be able to sift through the junk to find the right elliptical for your home gym in no time.

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