# Kepler's law

Kepler's laws of planetary motion can be expressed as follows:

The First law of Kepler - the law of orbit

According to Kepler's first law: "All the planets revolve around the Sun in an elliptical orbit, the Sun is at one focus". The point where the planet is closest to the Sun is called perihelion (about 17 million kilometers from the Sun), and the point where the planet is farthest from the Sun is called aphelion (152 million kilometers from the Sun). An ellipse is characterized by the fact that the sum of the distances of any planet from two foci is constant.

Kepler's laws of planetary motion

Kepler's Second Law - The Law of Equal Areas

Kepler's second law states: "A radiation vector drawn from the sun to a planet sweeps out equal areas at equal intervals of time."

Since the orbit is not a circle, the kinetic energy of the planet is not constant. It has more kinetic energy near perihelion and less kinetic energy near Apel means higher velocity at perihelion and lower velocity (vmin) at aphelion. If r is the distance of the planet from the Sun at perihelion (rmin) and  aphelion (rmax), then

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

Kepler's second law

Kepler's second law - the plane law

Using the law of conservation of momentum, the law can be verified. The angular momentum at any time can be represented as L = mr2ω.

Now consider a small area ΔA described by a short time interval Δt and subtended by an angle Δθ. Let the radius of curvature of the road be r, then

then the length of the arc covered = r Δ θ

ΔA = [r. (r. Δ θ)] = r2Δθ

Therefore, $\frac{∆A}{∆t}$$\left[\frac{1}{2}{r}^{2}\right]$$\frac{∆\theta }{∆t}$

Taking limits on both sides as, Δt→0, we get,

Now, due to the conservation of momentum, L is constant.

So $\frac{dA}{dt}$ = constant

The area swept at regular intervals is constant.

Kepler's second law can also be stated as follows: "The regional speed of a planet in an elliptical orbit around the Sun remains constant, which means that the angular momentum of the planet remains unchanged". Since momentum is constant, all planetary motions are planar motions,  a direct result of centripetal force.

Kepler's third law - The Law of Periods:

According to Kepler's periodic law: "The Square of the time it takes a planet to revolve around the Sun in an elliptical orbit is proportional to the cube of its semi major axis".

The shorter the planet's orbit around the sun, the shorter the time it takes to make one revolution. Using the equations of the law of gravity of Newton and the laws of motion, the third law of Kepler takes a more general form.

P=

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