The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be:

Steps to Solve

1. Understanding Kepler's Third Law

Kepler's Third Law states that the square of the period of a planet (or object) is proportional to the cube of the semi-major axis (average distance) of its orbit. Mathematically, it can be expressed as: $$ T^2 \propto R^3 $$

2. Set Up the Original Orbit

Let the duration of a year when the distance is $R$ be $T_1$. Using Kepler's law, we have: $$ T_1^2 = kR^3 $$ where $k$ is the proportionality constant.

3. Set Up the New Orbit

When the distance becomes $3R$, we denote the new period as $T_2$. According to Kepler's law: $$ T_2^2 = k(3R)^3 $$

4. Calculate the New Period

Expand the expression for $T_2^2$: $$ T_2^2 = k \cdot 27R^3 $$ Now, we can relate this back to $T_1$: $$ T_2^2 = 27 \cdot kR^3 = 27T_1^2 $$

5. Find the Duration of the Year

Taking the square root, we find: $$ T_2 = \sqrt{27} T_1 = 3\sqrt{3} T_1 $$

6. Final Calculation

If the original duration $T_1$ is 1 year, then: $$ T_2 = 3\sqrt{3} \text{ years} $$