Kepler’s Laws of Planetary Motion

Introduction

In 1609, Johannes Kepler gave his first two laws of planetary motion after examining Tycho Brahe’s astronomical data. After a gap of 10 years, he reported his third law in 1619. 

Before starting his research on planetary motion, he believed in the Copernican model of the solar system like many of his time. The Copernican model stated that planets had circular orbits. However, comparing another scientist’s observations of the planet of Mars helped him derive his first law of planetary motion.

Along with Newton’s Principles, Kepler’s work on planetary motion contributed immensely to our understanding of motion in physics. 

Kepler’s Laws

Planets revolve around the sun in a counterclockwise direction when observed from the sun’s north pole. The planets’ orbits are also aligned on an ecliptic plane.

We owe our understanding of the ways in which planets move entirely to Johannes Kepler.

Tycho Brahe, a renowned scientist Kepler was interning under, gave Kepler an assignment to study the orbital movement of Mars. The orbit of Mars did not conform to the theories proposed by Aristotle and Ptolemy. As he studied the orbit of Mars, Kepler observed that the shape of the orbit was not circular, as was popularly believed at the time, but instead was elliptical.

The Ellipse

An ellipse is defined by two points, each of which is referred to as a focus and which together are referred to as foci. The eccentricity of an ellipse is directly proportional to how flat the ellipse is. Each ellipse has an eccentricity that ranges from zero, which is a circle, to one, which is a flat line called a parabola. 

The ellipse is also defined by two axes: the major axis, which is the ellipse’s longest axis, and the minor axis, which is the ellipse’s shortest axis.

The Law of Orbits, or Kepler’s First Law of Planetary Motion

The orbit of every planet around the sun is an ellipse. The centre of the sun is found at one of the foci in the planet’s elliptical orbit. The sun is in a single point of focus, and all the planets travel in an elliptical orbit.

This means that the distance between the sun and the planet changes as the planets move.

The point at which a planet is closest to the sun is called the perihelion. The point at which the planet is found at its furthest distance from the sun is called the aphelion.

The elliptical shape of orbit allows us to find the distance between two focal points in which the planet travels. This is also why planets have different seasons.

The Law of Equal Areas, or Kepler’s Second Law of Planetary Motion

The imaginary line that joins both the planet and sun sweeps an equal area of space in equal intervals of time. Eventually, the planets do not move at a constant speed along the orbits as believed to be. Instead, their speed varies with the equal area of space swept by the planet in a particular time limit.

The highest speed is observed at the perihelion, while the planet moves slower at the aphelion.

If r = the distance from the sun, then perihelion is rmax, and aphelion is rmin.

rmin + rmax =  2 a  (length of major axis of ellipse)

The confirmation of this law can be obtained by using the formula of angular momentum, 

L =  mr2ω.

We will arrive at: dA / dt = constant.

From the above equation, it can be deduced that the area swept in an equal interval of time is a constant value.

The Law of Periods, or Kepler’s Third Law of Planetary Motion

The cube of a planet’s semi-major axis is directly proportional to the square of its time period of revolution around the sun in an elliptical orbit. 

This may be represented by the formula:

T2 ∝ a3

According to Kepler’s third law, the period of a planet’s orbit around the sun increases when the radius of the orbit is expanded. Thus, the shorter the planet’s orbit around the sun, the faster it takes to complete one revolution. 

Kepler’s third law is more generalised using the equations of Newton’s law of gravitation and laws of motion:

T2 = 42G(M1+M2)a3

As a result, it can be deduced that Mercury, the planet closest to the sun, orbits it in only 88 earth days. We all know that the earth takes 365 days to complete one revolution around the sun, while Saturn takes 10,759 days.

Conclusion

Though Kepler had no knowledge of gravitation when he devised his three laws, they were essential in Isaac Newton’s development of his theory of universal gravitation, which explains Kepler’s Third Law’s unknown force. 

Kepler’s theories were crucial in gaining a better grasp of the dynamics of our solar system, as well as serving as a springboard for newer models that more properly approximate our planetary orbits.

Kepler’s laws are applied in understanding and predicting the motions of natural and human satellites, along with those of star systems and extrasolar planets. 

It is worth noting that Kepler’s laws apply not only to gravitational forces but also to electromagnetic forces within the atom as long as relativistic and quantum effects are taken into account.