Gravitation

Kepler’s Law of Planetary Motions – Orbits, Areas, Periods

Kepler’s Law states that the planets move around the sun in elliptical orbits with the sun at one focus. There are three different Kepler’s Laws. Law of Orbits, Areas, and Periods. Let us know about them one by one.

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Kepler’s Three Law:

  1. Kepler’s Law of Orbits – The Planets move around the sun in elliptical orbits with the sun at one of the  focii.
  2. Kepler’s Law of Areas – The line joining a planet to the Sun sweeps out equal areas in equal interval of time.
  3. Kepler’s Law of Periods – The square of the time period of the planet is directly proportional to the cube of the semimajor axis of its orbit.

Kepler’s 1st Law of Orbits:

This law is popularly known as the law of orbits. The orbit of any planet is an ellipse around the Sun with Sun at one of the two foci of an ellipse. We know that planets revolve around the Sun in a circular orbit. But according to Kepler, he said that it is true that planets revolve around the Sun, but not in a circular orbit but it revolves around an ellipse. In an ellipse, we have two focus. Sun is located at one of the foci of the ellipse.

Kepler’s 2nd Law of Areas:

This law is known as the law of areas. The line joining a planet to the Sun sweeps out equal areas in equal interval of time. The rate of change of area with time will be constant.  We can see in the above figure, the Sun is located at the focus and the planets revolve around the Sun.

Assume that the planet starts revolving from point P1 and travels to P2 in a clockwise direction. So it revolves from point P1 to P2, as it moves the area swept from P1 to P2 is Δt. Now the planet moves future from P3 to P4 and the area covered is Δt.

As the area traveled by the planet from P1 to P2 and  P3 to P4 is equal, therefore this law is known as the Law of Area. That is the aerial velocity of the planets remains constant. When a planet is nearer to the Sun it moves fastest as compared to the planet far away from the Sun.

Kepler’s 3rd Law of Periods:

This law is known as the law of Periods.  The square of the time period of the planet is directly proportional to the cube of the semimajor axis of its orbit.

T² \( \propto\) 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 =  \( \frac{GmM}{r²} \)

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

F=  \( \frac{mv²}{r²} \)

Now, F = Fc

⇒ \( \frac{GM}{r} \) = v²

Also, v = \( \frac{circumference}{time} \) = \( \frac{2πr}{t} \)

Combining the above equations, we get

⇒ \( \frac{GM}{r} \) =  \( \frac{4π²r²}{T²} \)

T² = \( \frac{4π^2r^3)}{GM} \)

⇒ T² \(\propto\) r³

Solved Questions For You

Q1. A planet moves around the sun in an elliptical orbit with the sun at one of its foci. The physical quantity associated with the motion of the planet that remains constant with time is:

  1. velocity
  2. Centripetal force
  3. Linear momentum
  4. Angular momentum

Answer: D. Angular momentum is conserved ( constant) because of the force of gravitational attraction between the planets and the sun exerts zero torque on the planet.

Q2. Kepler’s second law states that the radius vector to a planet from the sun sweeps out equal areas in equal intervals of time. This law is a consequence of the conservation of:

  1. Time
  2. Mass
  3. Angular momentum
  4. Linear momentum

Answer: C. Area velocity = \( \frac{ΔA}{Δt } \)  = \( \frac{L}{2m } \). Since the radius vector of planet sweeps out equal area in equal interval of time, thus, \( \frac{ΔA}{Δt } \)  = constant

⇒ L = Constant

Thus Kepler’s second law is a consequence of the conservation of angular momentum.

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