**Kepler’s laws of planetary motion**

Hello dear students in this article we are going to discuss about the motion of planets around sun, path that is followed by planets to revolve around sun, the time required to do the same and all other details. Sir Johannes Kepler one of the greatest astrophysicists, mathematician who was an assistant to Brahe studied the motion of planets and proposed three laws. These laws were further used by Sir Isaac Newton to describe the concept of gravitation in more detail.

Before proceeding with the above laws let’s discuss some important properties of ellipse.

In above figure, we can see that two points marked as S_{1} and S_{2}, known as focus of ellipse. If we join any point on circumference of ellipse to focus S_{1} and S_{2}, the sum of segments formed by addition is always same.

i.e. S_{1}A+S_{2}A= S_{1}B+S_{2}B

This property is known as focus directrix property which is further applied for study of motion of planets.

Now let’s discuss the laws stated by Sir J. Kepler.

**1 ^{st} law:**

This law describes the path followed by planets around the sun.

Kepler’s first law stated as, “Every planet revolves around the sun in an elliptical orbit with sun acts as one the foci of ellipse.”

This law is also known as law of orbit.

**2 ^{nd} law:**

Kepler’s second law helps us to understand the variation of speed, momentum f planets when the moves around the sun.

It’s stated as, ‘The line joining centres of sun and planets sweeps equal areas in equal intervals of time’. Or ‘Areal velocity of planets is always constant’.

Consider a planet of mass ‘m’ revolves around sun situated at origin such that it moves from point A to B in small interval of time ‘△t’. If ‘p’ is momentum of planet, ‘r’ is the distance from centre of sun and ‘v’ be the velocity of planet then the distance travelled by planet is

Distance = v. △t

Area traced by planet in time △t will be given as,

∆ A = 1/2 (r × length of arc AB)

∆A=1/2 (r × v.∆t)

Momentum of planet can be written as

p = m.v

v = p/m

∴ ∆ A = 1/2 (r × p/m)∆t

∴∆A/∆t=1/2(r×p/m)

∴∆A/∆t=1/2(L×1/m)

(since, r × p = angular momentum of body

By law of conservation of angular momentum, the angular momentum of rotating body about a centre is always constant. Thus in above equation L and m are constant,

∴ ∆A/∆t = Constant

The same can be proved for position of planets from C to D and E to F.

i.e. the area sweeps by line joining planet and sun sweeps equal areas in equal intervals of time.

**3 ^{rd} law:**

Kepler’s third law helps us to describe the relation between periodic time of planets and mean distance of planets from the sun.

It stated as, ‘The square of period of planet around the sun is directly proportional to cube of semi-major axis of orbit of ellipse’.

∴T2 α r3

∴T2 = k r3 where ‘k’ is constant of proportionality.

This law is also known as law of period.

1600,00,00,0000

Consider the following examples for more detail explanation of the above law.

Eg: Find the period of planet at distance of 10^{11} km from the sun, if the distance between earth and sun is 16×10^{10} km.

Ans: Here, T_{E}= 1 year, r_{E}= 16×10^{10} km, r_{P}= 10^{11} km, T_{2}= ? yr

By Kepler’s 3^{rd} law, we can write,