An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that T = k/R √r3/g where k is a dimensionless constant and g is acceleration due to gravity.
An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that T = k/R √r3/g where k is a dimensionless constant and g is acceleration due to gravity.

Kepler’s third law states that,

i.e., square of time period of a satellite revolving around a planet, is proportional to the cube of the radius of the orbit .

On applying Kepler’s third law,

Also, depends on and .

Let
…..(i)

where,
dimensionless constant.

On writing the dimensions of various quantities on both the sides, we get

Comparing the dimensions of both sides,

…..(ii)

…..(iii)

From Eq. (ii), we get

Putting the values of and in Eq. (i), we get