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kepler’s laws of planetary motion with equations

How did Johannes Kepler come up with his three laws of planetary motion?

Using newton’s laws of motion and law of universal gravitation,we can understand and analyze the behavior of all the bodies in the solar system:the orbits of the planets and comets about the sun and of natural and artificial satellites about their planets.We make two assumption that simplify the analysis:

  1. We consider the gravitational force only between the orbiting body and the central body (the sun) ,ignoring the perturbing effect of the gravitational force of other bodies (such as other planets).
  2. We assume that the central body is so much more massive than the orbiting body that we can ignore its motion under their mutual interaction.In reality,both objects orbit around their common center of mass,but if one object is very much more massive than the other,the center of mass is approximately at the center of the more massive body.Exceptions to this second assumption will be noted.

The empirical basis for understanding the motions of the planets is kepler’s three laws,and we now show how these laws are related to the analytical results of newton’s laws.

What is the Kepler’s first law (the law of orbits)?

“All planets move in elliptical orbits,with the sun at one focus.”


kepler 1st law

Consider a planet of mass ‘m’ moving in such an orbit around the sun,whose mass is M.We assume that M>>m,so that the center of mass of the planet sun system is approximately at the center of mass of the planet sun system is approximately at the center of the sun.The orbit is described by two parameters:the semi major axis ‘a’ and the eccentricity ‘e’.The distance from the center of the ellipse to either focus is ‘ea’.The maximum distance Ra of the planet from the sun is called aphelion,similarly,the closest distance RP is called perihelion.

Fro the figure we see that:

Ra=a +ea =a(1+e) ———-(1)

Rp=a-ea=a(1-e) ————–(2)

A circular orbit is a special case of an elliptical orbit with e=0.

For circular orbit,from equation (1) and (2) we get,


What does Kepler’s second law (The law of areas) state?

“A line joining any planet to the sun sweeps out equal areas in equal times.”


“Areal velocity of the planet around the sun is constant.”


kepler 2nd law

Consider a planet of mass is moving in elliptical orbit around the sun.The sun and the planet are separated by distance r.Consider the small area ∆A covered in a time interval ∆t,as shown in figure.The area of the wedge is approximately the area of triangle with base ‘rΔθ‘ and height ‘r’.Since the area of a triangle is one half of the base times the height.

kepler ist law

kepler ist law


Where dA/dt is called areal velocity.Since angular momentum ‘L’ and mass of the planet is constant.

What does Kepler’s third law (The law of periods) state?

“The square of the period of any planet about the sun is proportional to the cube of the planet’s mean distance from the sun.”


kepler 3rd law

Let us prove this result fro circular orbits.Consider a planet of mass ‘m’ is moving around the sun of mass ‘M’ in circular orbit of radius ‘r’ as shown in figure.The gravitational force provides the necessary centripetal force to the planet for circular motion.Hence

kepler's 3rd law

T2    r3



Similar result is obtained for elliptical orbits with radius ‘r’ replaced by semi major axis ‘a’ given by the relation:


T2    a3

watch also video about Kepler’s law of planetary motion:

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