** Kepler’s laws of planetary motion**, are the laws that describe the motions of the planets in the solar system. These laws were derived by

**, a German astronomer, in the early 1600s. And hence these laws are named after him.**

*Johannes Kepler*These laws are mainly used in ** Astronomy** and

*Classical Physics**.*

Index

## History

Kepler was able to summarize the carefully collected data of his mentor –** Tycho Brahe**– a Danish astronomer. Kepler published his first two laws in 1609, in the book

*. The third law was published nearly a decade later, in 1619, in the book*

**Astronomia Nova***.*

**Harmonices Mundi**## Kepler’s First Law

The path of the planets about the sun is elliptical, with the sun at one of the two foci. This is known as the *law of Ellipses*.

## Kepler’s Second Law

An** **imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. This is known as the *Law of Equal areas.*

## Kepler’s third law

The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit( i.e T^{2 }α R^{3} ). This is known as the *Law of Harmonies.*

Which means..

(T_{1}/T_{2})^{2} = (R_{1}/R_{2})^{3}

## Applications

- They have been used to find the orbits of many objects such as
*Comets*and*Asteroids.* - These laws played a key role in the discovery of dark matter in the Milky Way.
- Although Kepler originally developed his laws in the context of planetary orbits, the results hold valid for any system with radial force obeying the inverse square laws
- The 1
^{rst}and the 2^{nd}laws help us determine the path of the objects in space. - Whereas the third law is used to determine the period and radius of the elliptical paths.

## Example Problems

**Question 1:** A satellite on a circular orbit of radius R around Mars has an orbital period of 10.0 hr. A second satellite is to be placed in a circular orbit so that its orbital period will be 5.00 hr. The radius of the orbit of the second satellite should be closest to?

**Ans:** For Satellite-1 with circular orbit about a planet:

The radius of the orbit: *R*

Time, T_{1} = 10.0 hr

For Satellite-2 with circular orbit about the same planet:

The radius of the orbit: *R’ *(To be found)

Time, T_{2} = 5.00 hr

** **T_{2}^{2} ∕ T_{1}^{2} =^{ }*R’* ^{3} /* R*^{3}

(5.00 hr)^{2} ∕ ( 10.0 hr)^{2} =^{ }*R’ ^{3 }/ R^{3 }*

We get *R’ = 0.630 × R*

**Question 2:** A satellite A of mass M is at a distance of r from the center of the earth. Another satellite B of mass 2m is at a distance of 2r from the earth’s center. Their time periods are in the ratio of ..?

**Ans:** Here we should note that Mass of the satellite does not affect the time period.

We know that

(T_{1}/T_{2})^{2} = (R_{1}/R_{2})^{3}

R_{1}=r;

R_{2}=2r;

So we get the ratio of the time period = 1/2√ 2 *sec*

**FAQ**s

**What is AU (or) Astronomical unit ?**

It is considered as the distance between the center of the Sun and the center of the Earth.

**What does Kepler’s 2nd law deal with ?**

It deals with speed/area the planet travels.

**Kepler’s first law is also known as ?**

Law of Ellipses.

So, that is all about Kepler’s laws of planetary motion.

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