Kepler law of planatery
Answers
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's Laws of Planetary Motion
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 Laws of Planetary Motion
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:
P2 = 4π2 /[G(M1+ M2)] × a3
where M1 and M2 are the masses of the two orbiting objects in solar masses.