Question

What is Kepler’s constant?

Answer 1

Hint: There are three Kepler’s law of planetary motion. The third law of Kepler’s planetary motion should be used to obtain Kepler’s constant. The third law gives the relation between the radius of the orbit and the period of the orbit, thus, using this, we will obtain the expression for Kepler’s constant.

Complete answer:
Among the three laws of planetary motion of Kepler, the third law of planetary motion defines the Kepler’s constant as the relation between the radius of the orbit and the period of the orbit, that is, the time taken by the planet to complete one orbit.
Every planet has its own Kepler’s constant value, as all the planets have different values of the radius and the time period of their orbit.
In the case of orbit, the gravitational force is the cause for the centripetal force. Thus, equating both the equations.
$${\displaystyle \frac{m{v}^2}{r}}=G{\displaystyle \frac{Mm}{r^2}}$$
Here, the velocity of the planet orbiting is given by the formula as follows.
$$v={\displaystyle \frac{2\pi r}{T}}$$
Substitute this expression of the velocity in the equation of Kepler’s third law.
$${\displaystyle \frac{m}{r}}{\left({\displaystyle \frac{2\pi r}{T}}\right)}^2=G{\displaystyle \frac{Mm}{r^2}}$$
Rearrange the terms.
$${\displaystyle \frac{T^2}{r^3}}={\displaystyle \frac{4{\pi }^2}{GM}}$$
If we remove the constant value, then, we will get a proportional equation.
$$\therefore T^2\propto r^3$$
Thus, we can say that the square of the orbital period of the planet is directly proportional to the cube of the semi-major axis of its orbit.
Thus, Kepler’s constant is the square of the orbital period of the planet by the cube of the semi-major axis of its orbit.
$$K={\displaystyle \frac{T^2}{r^3}}$$
The numerical value of Kepler’s constant is, $$K={\displaystyle \frac{4{\pi }^2}{GM}}$$.
Where G is the gravitational constant and M is the mass of the object being orbited.
 $$\therefore$$ Kepler’s constant is the square of the orbital period of the planet by the cube of the semi-major axis of its orbit.

Note:
Using the formula of Kepler’s constant, we can compute the value of Kepler’s constant of any object, such as the Sun, the planets as, the Earth, Mars, etc. While computing the constant, Kepler considered the orbit to be circular and the radius to be the average radius of the orbit.