# Kepler’s Laws of Planetary Motion

###### Stuid
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According to Kepler’s Law, planets move around the sun in elliptical orbits with the sun at the focus. The three different Kepler’s laws are namely the laws of orbits, areas, and periods.

Kepler’s Three Laws:

1. Kepler’s Law of Orbits: Planets will move around the sun in elliptical orbits with the sun placed at one of the foci.
2. Kepler’s Law of Areas: The line which joins a planet to the sun will sweep out equal areas in equal intervals of time.
3. Kepler’s Law of Periods: The square of the time period of the planet will be directly proportional to the cube of the semimajor axis of its orbit.

## Kepler’s 1st Law of Orbits

It is commonly known as the law of orbits. The orbit of a planet is basically an ellipse around the sun, with the sun placed at one of its two foci. Kepler had accepted that planets move around the sun, but not in circular orbits! ‘There are two foci for an ellipse’. Sun will be located at one of the foci of the ellipse.

## Kepler’s 2nd Law of Areas

The law is called the law of areas. As per the law, a line joining the planet will sweep out equal areas in equal intervals of time. However, the rate of change of area with time will remain constant. We know the sun is located at the focus and the planets revolve around it.

• Let’s assume that the planet starts revolving from point P1 and moves to P2 in a clockwise direction.
• The area swept from P1 to P2 is Δt.
• The area swept as it moves from P3 to P4 will also be Δt.
• Since the area covered by the planet from P1 to P2 and P3 to P4 is equal, the law is called the Law of Areas.
• The areal velocity of the planet will remain constant.
• A planet moves fastest when it is nearer to the sun, as compared to when it is away from the sun.

## Kepler’s 3rd Law of Periods

The third law is widely known as the law of periods. As per the law, the square of the time period of the planet will be directly proportional to the cube of the semimajor axis of its orbit.

T² ∝ a³

Here, time ‘T’ is directly proportional to the cube of the semi major axis ‘a’.

## Derivation

• ‘m’ indicates the mass of the planet.
•  ‘M’ denotes the mass of the sun.
• ‘v’ stands for velocity in the orbit.

Hence, there will be a force of gravitation between the Sun and the planet.

F = GmM/r²

As it is moving in an elliptical orbit, there should be a centripetal force.

Fc= mv²/r²

Therefore, F = Fc

⇒ GM/r = v²

And also,

v = circumference/time2πr/t

When we combine the above equations, we get:

⇒ GM/r = 4π²r²/T²

T² = 4π²r³)/GM

⇒ T² ∝ r³