# Kepler's Laws of Planetary motion, Dedication

## Kepler's Laws of Planetary motion

In astronomy, * Kepler's laws* of planetary motion, proposed by Johannes Kepler during the period of 1609 to 1619, describe the orbits of planets around the Sun. The motion of planets has been a subject of much interest for astronomers from very early times.

Kepler's Laws of Planetary motion |

In the above figure, the orbits are ellipses, with focal points F_{1} and F_{2} for the first planet and F_{1} and F_{3} for the second planet. The Sun is placed in focal point F_{1}.

The two shaded sectors A_{1} and A_{2} have the same surface area and the time for planet 1 to cover segment A_{1} is equal to the time to cover segment A_{2}.* Kepler's laws* of planetary motion are as follows :

### (i) The Law of Elliptical Orbits (First Law)

According to this law, "* Every planet moves in an elliptical orbit around the sun, the sun being at one of the foci*."

The point at which the planet is close to the sun is known as * perihelion* and the point at which the planet is farther from the sun is known as

*.*

**aphelion**#### Dedication of Kepler's First Law

Let S be the sun (mass M) and a planet of mass m, revolving around the sun in elliptical orbit. The force acting between planet and sun is gravitational and given by

`F=-\frac{GMm}{r^{2}}\hat{r}`

**r** being position vector of planet relative to sun

`F=-\frac{GMm}{r^{3}}r=f(r)r`

`f(r)r=-\frac{GMm}{r^{3}}`

Torque acting on planet.

`Ï„=r\times F=r\times f(r)r`

`Ï„=f(r)r\times r=0`

As torque acting on planet is zero, the angular momentum of the planet relative to sun is conserved.

The angular momentum of planet relative to sun is

`J=r\times p=r\times mv=constant`

It follows that **J**, being constant, must be perpendicular to the plane containing position vector **r** and velocity **v** that is the orbit of the plane must lie in a plane.

### (ii) The Law of Areas ( Second Law)

According to this law, "* The radius vector, drawn from the sun to a planet, sweeps out equal areas in equal times i.e., the areal velocity of the radius vector is constant*."

As the orbit is not circular, the planet’s kinetic energy is not constant in its path. It has more kinetic energy near perihelion and less kinetic energy near aphelion implies more speed at perihelion and less speed at aphelion.

#### Deduction of Kepler's Second Law

Let P be the instantaneous position of planet relative to sun. The position vector of P relative to sun is **r**. After time Î”t, the planet is at Q, having position vector r+Î”r such let Î”**r** = PQ

The area swept out by radius SP in time Î”t.

Î”A = Area of triangle PSP'

`Î”A=\frac{1}{2} r × Î”r`

Area swept by radius vector per second or Areal velocity,

`\frac{dA}{dt}=\lim_{Î”t \rightarrow 0}\frac{Î”A}{Î”t}`

`\frac{dA}{dt}=\lim_{Î”t \rightarrow 0}\frac{\frac{1}{2}r\timesÎ”r}{Î”t}`

`\frac{dA}{dt}=\frac{1}{2}r\times\frac{dr}{dt}`

`\frac{dA}{dt}=\frac{1}{2}r\times v`

`r\times v=\frac{J}{m}=constant`

Areal velocity,

`\frac{dA}{dt}=\frac{1}{2}\cdot\frac{J}{m}=constant`

It follows that the are velocity of the radius vector of planet relative to sun remains constant. This is * Kepler's Second Law*.

### (iii) The Harmonic Law (Third Law)

This Law states that, "* The square of the period of revolution of the planet around the sun is proportional to the cube of the semi-major axis of the ellipse*."

`T^{2}∝a^{3}`

It is clear from the above equation that shorter the orbit of the planet around the sun, shorter the time taken to complete one revolution.

#### Dedication of Kepler's Third Law

Let T be period of revolution of the planet in elliptical orbit, area of ellipse = Ï€ab a and b being semi-major and semi-minor axes of the ellipse.

Areal velocity,

`\frac{dA}{dt}=\frac{J}{2m}`

∴ Period of revolution

`T=\frac{Area-of-ellipse}{Areal-velocity}`

`T=\frac{Ï€ab}{\frac{J}{2m}}`

`T^2=\frac{4Ï€^{2}m^{2}a^{2}b^{2}}{J^2}`

If *l* is the semi-latus rectum of ellipse, then

`l=\frac{b^{2}}{a}`

`T^2=\frac{4Ï€^{2}m^{2}a^{2}la}{J^2}`

i.e.,

`T^{2}∝a^{3}`

Since all other quantities are constant. This is * Kepler's third law*.