To find the percentage increase in the radius of the orbit of revolution, we need to understand the relationship between the time period of revolution and the radius of the orbit.


According to Kepler's third law of planetary motion, the square of the time period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit.

Mathematically, this can be represented as:
T^2 ∝ r^3 (Equation 1)

Where T is the time period of revolution and r is the radius of the orbit.


Let's assume the initial time period of revolution as T1 and the initial radius of the orbit as r1.
The increased time period of revolution is 3√3 times the present value, which can be represented as:
T2 = 3√3 * T1

To find the percentage increase in the radius of the orbit, we need to find the ratio of the new radius (r2) to the initial radius (r1), and then express it as a percentage.


Using Equation 1, we can write the relationship between T1 and r1 as:
T1^2 ∝ r1^3 (Equation 2)

Similarly, for the increased time period T2 and new radius r2, we can write:
T2^2 ∝ r2^3 (Equation 3)

Dividing Equation 3 by Equation 2, we get:
(T2^2 / T1^2) = (r2^3 / r1^3)

Substituting the values of T2 and T1, we get:
[(3√3 * T1)^2 / T1^2] = (r2^3 / r1^3)

Simplifying the equation, we get:
27 * T1^2 / T1^2 = r2^3 / r1^3

Canceling out T1^2, we get:
27 = r2^3 / r1^3

Taking the cube root of both sides, we get:
3 = r2 / r1

Therefore, the new radius (r2) is 3 times the initial radius (r1).


To find the percentage increase, we can use the formula:
Percentage Increase = (New Value - Initial Value) / Initial Value * 100

Substituting the values, we get:
Percentage Increase = (3r1 - r1) / r1 * 100
Percentage Increase = 2r1 / r1 * 100
Percentage Increase = 200%

Therefore, the percentage increase in the radius of the orbit of revolution is 200%.