Question

derive Kepler's third law​

Answer 1

Answer:

This is known as Kepler's Third Law or 3rd Law of Kepler. This law states that square of the Orbital Period of Revolution is directly proportional to the cube of the radius of the orbit. We will derive the equation for Kepler's 3rd Law using the concept of Period of Revolution and the equation of orbital velocity

Answer 2

1619, Johannes Kepler published a relationship between how long a planet takes to orbit the Sun and the size of that orbit, something we now call his 3rd law of planetary motion, or just “Kepler’s 3rd law”. It states that

T^{2} \propto a^{3}

where

T

is the period of the orbit and

a

is the size of the orbit. Kepler also found that the planets orbit the Sun in elliptical orbits (his 1st law), and so the size of the orbit

a

that we refer to is actually something called the “semi-major axis”, half the length of the long axis of an ellipse.

Any proportionality can be written as an equality if we introduce a constant, so we can write

T^{2} = k a^{3} \text{ (equation 1)}

where

k

is our constant of proportionality.

Kepler found that the planets orbit the Sun in ellipses, with the Sun at one of the foci. The long axis of an ellipse is called its major axis.

Kepler found that the planets orbit the Sun in ellipses, with the Sun at one of the foci. The long axis of an ellipse is called its major axis. The

a

in Kepler’s 3rd law refers to the length of the semi-major axis of a planet’s ellipse.