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Kepler's laws of planetary motion: The law of orbits, areas, periods

Kepler's laws of planetary motion: The law of orbits, areas, periods
The ancient astronomers contributed a great deal by identifying the planets in the solar system and carefully plotting the variations in their positions of the sky over the periods of many years. These data eventually led to models and theories of the solar system.

Planetary motion

 

The ancient astronomers contributed a great deal by identifying the planets in the solar system and carefully plotting the variations in their positions of the sky over the periods of many years. These data eventually led to models and theories of the solar system.

 

The first major theory, called the Geo-centric theory was developed by a Greek astronomer, Ptolemy. The Earth is considered to be the centre of the universe, around which all the planets, the moons and the stars revolve in various orbits. The great Indian Mathematician and astronomer Aryabhat of the 5th century AD stated that the Earth rotates about its axis. Due to lack of communication between the scientists of the East and those of West, his observations did not reach the philosophers of the West.

Nicolaus Copernicus, a Polish astronomer proposed a new theory called Helio-centric theory. According to this theory, the Sun is at rest and all the planets move around the Sun in circular orbits. A Danish astronomer Tycho Brahe made very accurate observations of the motion of planets and a German astronomer Johannes Kepler analysed Brahe's observations carefully and proposed the empirical laws of planetary motion.

 

Kepler's laws of planetary motion


(i) The law of orbits

Each planet moves in an elliptical orbit with the Sun at one focus. A is a planet revolving round the Sun. The position P of the planet where it is very close to the Sun is known as perigee and the position Q of the planet where it is farthest from the Sun is known as apogee.


(ii) The law of areas

The line joining the Sun and the planet (i.e radius vector) sweeps out equal areas in equal interval of times.


The orbit of the planet around the Sun is as shown in Fig.. The areas A1 and A2 are swept by the radius vector in equal times. The planet covers unequal distances S1 and S2 in equal time. This is due to  the variable speed of the planet. When the planet is closest to the Sun, it covers greater distance in a given time. Hence, the speed is maximum at the closest position. When the planet is far away from the Sun, it covers lesser distance in the same time. Hence the speed is minimum at the farthest position.

Proof for the law of areas

Consider a planet moving from A to B. The radius vector OA sweeps a small angle dθ at the centre in a small interval of time dt.


From the Fig., AB = rd θ. The small area dA swept by the radius is,

dA = x r x rdθ

Dividing by dt on both sides

dA/dt = x r2 x dθ/dt

dA/dt = r2 ω

where ω is the angular velocity.

The angular momentum is given by L = mr2ω

r2ω = L/m

dA/dt = L/M

Since the line of action of gravitational force passes through the axis, the external torque is zero. Hence, the angular momentum isconserved

dA/dt = constant.

(iii) The law of periods

The square of the period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance between the

planet and the Sun.

(i.e) T 2 α r3

T 2 / r3 = constant

Proof for the law of periods

Let us consider a planet P of mass m moving with the velocity v around the Sun of mass M in a circular orbit of radius r.


The gravitational force of attraction of the Sun on the planet is,

F = GMm / r2

The centripetal force is, F = mv2/r

Equating the two forces

Mv2/r  = GMm/r2

v2 = GM/r   .(1)

If T be the period of revolution of the planet around the Sun, then

v = 2πr /T

Substituting (2) in (1)

R3/T2 = GM / 2

GM is a constant for any planet

T2 α r3

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