Kepler’s Laws

 ▪ Kepler’s Laws:

1. First law :  The Law of Orbits

- All planets move around the sun in elliptical orbites with the sat
  one of the foci

2. Second law : The Law of Equal Areas

- A planet sweeps out equal areas in equal intervals of time

3. Third law : The Law of Periods

- The square of the orbital period of a planet is proportional to
   the cube of the orbit’s semi major axis.

The square of the time period of the planet is directly proportional to the cube of the semimajor axis of its orbit.

                                                    T²∝ a³

That means the time ‘ T ‘ is directly proportional to the cube of the semi major axis i.e. ‘a’. Let us derive the equation of Kepler’s 3rd law. Let us suppose,

• m = mass of the planet
• M = mass of the Sun
• v = velocity in the orbit

So, there has to be a force of gravitation between the Sun and the planet.

                                                       F =  GmMr²

Since it is moving in an elliptical orbit, there has to be a centripetal force.

Fc =  mv²r²

mv²r²mv²

mv²r²Fcr

Now, F = Fc

⇒ GMr = v²

Also, v = circumferencetime = 2πrt

Combining the above equations, we get

⇒ GMr =  4π²r²T²

T² = 4π2r3)GM

⇒ T² ∝ r³

 r²