Tuesday, December 5, 2017

Kepler's Law Of Planetary Motion-- The Heliocentric Model

planets revolving around the sun in an elliptical orbit
Credit: MathWorks

The centuries-old dispute between The Geocentric Model Of The Universe and The Heliocentric Model Of The Universe was finally put to the rest by the German Mathematician, Astronomer, and Astrologer Johannes Kepler. He solved the riddle that we are living in the Heliocentric Model of the universe i.e the sun is at the center of our solar system-- not the earth.

Historical View And Development

Before the emergence of Scientific Revolution or Copernican Revolution, Aristotelian-Ptolemaic Universe was widely accepted as the working model of the Universe. According to Aristotelian-Ptolemaic Universe or the Geocentric Model Of The Universe, the earth is stationary and regarded as The Center Of The Universe; and each planet within our solar system included the sun revolves around the earth. This Geocentric Model was prevailing until the arrival of Copernican Heliocentrism.

see also, History Of Classical Physics

Copernicus heliocentric model
Credit: Wikimedia Commons

Well for those who don't know that Nicolaus Copernicus was not the first person to proclaim that the sun is the center of the universe, not earth; and every planet including earth revolves around the sun. It was the Greek astronomer Aristarchus of Samos who proposed the heliocentric model; though there are no written works of him left until now.

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

There was some problem in the Heliocentric Theory Of Nicolaus Copernicus such as the planets revolve around the sun in circular motion etc (which we will discuss in the later section) which were enhanced by the German Mathematician, Astronomer, and Astrologer Johannes Kepler (with help of his mentor Tycho Brahe's orbital calculation). Later Sir Issac Newton showed that Kepler's Law Of Planetary Motion is a direct consequence of Newton's Law Of Gravitation which governs the forces between two massive objects. 

kepler's heliocentric model
Credit: The Scientific Revolution

Comparison Of Kepler's Model To Copernican Model 

Although both Johannes Kepler and Nicolaus Copernicus contradicted the classical view of Aristotelian-Ptolemaic Universe yet there was some ideological difference between them; such as:

  1. In Copernican Model-- all planet revolves around the sun in a circular orbit; while on the other hand, in Kepler's Model-- all planet revolves around the sun but in a flattened circular orbit that means the elliptical orbit.
  2. In Copernican Model-- the sun is located at the center of the orbit; while on the other hand, in Kepler's Model-- the sun is located at one of the focal of the elliptical orbits.
  3. In Copernican Model-- the speed of the planet in an orbit remains constant; while on the other hand in Kepler's Model-- the speed of the planet in an orbit is not constant but the area speed remains constant.
See also, 8 Scientific Laws And Theories You Really Should Know

Kepler's Three Laws Of Planetary Motion

Diagram of Kepler's three laws with two planetary orbits
Credit: Wikimedia Commons

Kepler's Three Laws Of Planetary Motion can be described as follows:

1. Kepler's First Law-- The Law Of Ellipse

All planets revolve in an elliptical orbit with the sun at one of the two foci.
Mathematically, an ellipse can be represented by the formula

r=\frac{p}{1+\varepsilon\, \cos\theta},

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 closest approach, as seen from the sun.

2. Kepler's Second Law-- The Law Of Equal Areas

A line segment joining a planet and the sun sweeps out equal areas in equal intervals of time

3. Kepler's Third Law-- The Law Of Harmonies

The square of the time period of a planet is proportional to the cube of the semi-major axis of its orbit.

Mathematically, it can be represented by the formula

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

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 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. Eccentricity is represented by ε. eccentricity can be divided into two groups such that;
Perihelion-- is the closest point of a planet to the sun
Aphelion-- is the farthest point of a planet to the sun 

Four conditions for eccentricity
  1. If ε = 0 then planet's orbit will be circular; same as Copernican Heliocentrism
  2. If 0 < ε > 1 then planet's orbit will be elliptical
  3. If ε = 1 then orbit will be parabolic
  4. If ε > 1 then orbit will be hyperbolic

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