Question

Kepler law of periods .....................?????

Answer 1

Law of periods :-

The square of time taken by a planet to complete a revolution around the sun is directly proportional to the cube of semi - major axis of the elliptical orbit.

$\boxed{\rm T^2 \propto r^3}$

Answer 2

Answer:

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 ellipsethat 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’s Second Law – The Law of Equal Areas

Kepler’s second law states ” The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time”

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)

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 can also be stated as “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

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

Shorter the orbit of the planet around the sun, shorter the time taken to complete one revolution. Using the equations of Newton’s law of gravitation and laws of motion, Kepler’s third law takes a more general form:

P2 = 4π2 /[G(M1+ M2)] × a3

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