# Kepler’s Laws of Planetary Motion

In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the sun.

• Kepler first law: The law of orbits
• Kepler’s second law: The law of equal areas
• Kepler’s third law: The law of periods

## Introduction to Kepler’s Laws

Motion is always relative. Based on the energy of the particle under motion, the motions are classified into two types:

• Bounded Motion
• Unbounded Motion

In bounded motion, the particle has negative total energy (E<0) and has two or more extreme points where the total energy is always equal to the potential energy of the particle i.e the kinetic energy of the particle becomes zero.

For eccentricity 0≤ e <1, E<0 implies the body has bounded motion. A circular orbit has eccentricity e = 0 and elliptical orbit has eccentricity e < 1.

In unbounded motion, the particle has positive total energy (E>0) and has a single extreme point where the total energy is always equal to the potential energy of the particle i.e the kinetic energy of the particle becomes zero.

For eccentricity e ≥ 1, E > 0 implies the body has unbounded motion. Parabolic orbit has eccentricity e = 1 and Hyperbolic path has eccentricity e>1.

Kepler’s laws of planetary motion can be stated as follows:

## Kepler First law – The Law of Orbits

According to Kepler’s first law, all the planets revolve around the sun in elliptical orbits having the sun at one of the foci. The point at which the planet is close to the sun is known as perihelion and the point at which the planet is farther from the sun is known as aphelion.

It is the characteristics of an ellipse that the sum of the distances of any planet from two foci is constant. The elliptical orbit of a planet is responsible for the occurrence of seasons.

Kepler First Law – The Law of Orbits

## Kepler’s Second Law – The Law of Equal Areas

As the orbit is not circular, the planet’s kinetic energy is not constant in its path. It has more kinetic energy near perihelion and less kinetic energy near aphelion implies more speed at perihelion and less speed (vmin) at aphelion. If r is the distance of planet from sun, at perihelion (rmin) and at aphelion (rmax), then,

rmin + rmax = 2a × (length of major axis of an ellipse) . . . . . . . (1)

Kepler’s Second Law – The law of Equal Areas

For an infinitesimal movement of the planet in a time interval in an elliptical orbit, the area swept by the planet in time is given by;

dA/dt = d/dt [ 1/2 × r × (v dt)]= 1/2 × rv . . . . . (2)

At perihelion r = rmin, v = vmax then from Equation 2;

dA/dt = 1/2 × rmin × vmax) = [m × vmax × rmin]/2m = L/2m;

At aphelion r = rmax, v = vmin then from Equation 2;

dA/dt = 1/2 × vmin × rmax = [m × vmin × rmax]/2m = L/2m

Kepler’s second law states the areal velocity of a planet revolving around the sun in elliptical orbit remains constant which implies the angular momentum of a planet remains constant. As the angular momentum is constant all planetary motions are planar motions, which is a direct consequence of central force.

⇒ Check: Acceleration due to Gravity

## Kepler’s Third Law – The Law of Periods

Shorter the orbit of the planet around the sun, shorter the time taken to complete one revolution. According to Kepler’s law of periods, the square of the time period of revolution (of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis).

T2 ∝ a3

Using the equations of Newton’s law of gravitation and laws of motion, Kepler’s third law takes a more general form:

P= 4π2 /[G(M1+ M2)] × a3

where M1 and M2 are the masses of the two orbiting objects in solar masses.

#### Practise This Question

The largest and the shortest distance of the earth from the sun are r1 and r2, its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun