# Kepler's Laws Of Planetary Motion

## Introduction

The German astronomer, Johannes Kepler was born in the year 1571. When he was training to become a minister, he came across the work of Nicolaus Copernicus. At that time it was believed that the sun and other planets orbit the earth and not how as we know it now (everything orbits the sun!). Copernicus however, hypothesized that the earth and other planets orbit the sun. He had no proof of this though. All these theories were derided by the church in any case and as everyone knows Galileo was even thrown in jail.

With his continued work in astronomy, he impressed a wealthy Danish astronomer, Tycho Brahe, who then invited him to Prague to work with him. Brahe had an observatory where he kept detailed information about planetary motions. But Brahe didn’t trust Kepler. He thought that his much smarter assistant may surpass him and become the astronomer of the age. Because of this, he didn’t let Kepler access all his data. When Brahe passed away, Kepler finally had access to his data using which he solved the very confusing Martian problem (varying speeds and distances of the planet observed at different times) and through this formulated the Kepler’s Laws of Planetary Motion that he is famous for.

## Kepler’s Laws Of Planetary Motion

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler’s laws of planetary motion can be stated as follows:

### Kepler’s First Law: Law of Orbits

Kepler’s 1st law means that all planets move around the Sun in elliptical orbits. The orbit of a planet about its star (the sun for our solar system), follows an elliptical path with the star occupying the position of one of the foci of the ellipse formed.

An ellipse traced out by a planet around the sun. The closest point is P and the farthest point is A, P is called the perihelion and A the aphelion. The semimajor axis is half the distance AP.

### Kepler’s Second Law: Law of Equal Areas

If we join a planet to its star with an imaginary line, then equal areas are swept by this line in equal intervals of time.

The planet P moves around the sun in an elliptical orbit. The shaded area in the above image is the area ∆A swept out in a small interval of time ∆t.

### Kepler’s Third Law: Law of Harmonics or Law of Periods

According to this, the square of the time period (revolution time) is proportional to the cube of the semi-major axis of the elliptical orbit formed.

So, Kepler’s Third Law says

$P^{2}\alpha a^{3}$

where ‘a’ is expressed in astronomical units and ‘P’ is the period.

After applying Newton’s Newton’s Law of Gravity and Laws of Motion, Kepler’s Third Law takes a more general form:

$P^{2}=\frac{4\pi^{2}}{G(M1+M2)}a^{3}$

where M1 and M2 are the masses of the two orbiting objects in solar masses.

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#### Practise This Question

The period of revolution of an earth satellite close to surface of earth is 90 min. The time period of another satellite in an orbit at a distance of three times the radius of earth from its surface will be