Kepler's Law

 Kepler’s Laws

Kepler’s Laws of Planetary Motion.The motion of planets in the solar system follows specific rules known as Kepler’s Laws of Planetary Motion. These laws were formulated by Johannes Kepler based on observational data collected by Tycho Brahe. Kepler published his first two laws in 1609 and the third law in 1619.

Kepler’s First Law – The Law of Orbits

"All planets move in elliptical orbits around the Sun, with the Sun at one of the foci of the ellipse."

The orbit of a planet around the Sun is not a perfect circle but an ellipse.
The Sun is not at the center but at one of the two foci of the elliptical orbit.
The major axis of the ellipse is the longest diameter, while the minor axis is the shortest.
At Perihelion, the planet is closest to the Sun.
At Aphelion, the planet is farthest from the Sun.

---

Kepler’s Second Law – The Law of Areas

"The line joining a planet and the Sun sweeps out equal areas in equal intervals of time."

This law implies that planets move faster when they are closer to the Sun and slower when they are farther.
The areal velocity remains constant for a planet in orbit.

Mathematical derivation
This is derived from the conservation of angular momentum, as the gravitational force acts as a central force.
Consider a planet of mass m moving around the Sun in an elliptical orbit under the influence of gravitational force. At any instant, let:

The infinitesimally small area dA swept in dt can be approximated as the area of the triangular sector formed by r and dr.

We recall that angular momentum Lis given by: L=m(r×v)

Since there is no external torque acting on the system (only the gravitational force, which is central), the angular momentum L is conserved. Thus,

Physical Significance of Kepler’s Second Law 

Planets move faster when closer to the Sun (perihelion) and slower when farther (aphelion). This law is a direct consequence of the conservation of angular momentum. It explains why planets do not move at a uniform speed but instead adjust their velocity to maintain constant areal velocity.

---

Kepler’s Third Law – The Law of Periods

Kepler’s Third Law – The Law of Periods The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. 

 Mathematical Form

If T is the time period of revolution and r is the semi-major axis, then:

---

<<<Introduction│ │ Universal law of Gravitation>>> 

---