The semi-major axis of the orbit of Saturn is nine times that of Earth. The time period of the revolution of Saturn is approximately equal to?

• (A) 81 years
• (B) 27 years
• (C) 729 years
• (D) 81 years
• (E) 9 years

The time period of a planet's revolution around the Sun can be estimated using Kepler's Third Law of Planetary Motion. This law states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit. The formula can be expressed as:

T² ∝ a³

For Earth, the semi-major axis is defined as 1 astronomical unit (AU), and its orbital period is 1 year. If Saturn's semi-major axis is nine times that of Earth, we can set up the equation:

Calculating Saturn's Orbital Period

• Let a = 9 AU (for Saturn)
• Using Kepler's law: T² = k * a³
• For Earth: T² = 1² = 1 (where k is a constant)
• For Saturn: T² = k * (9)³ = k * 729

Since the constant k remains the same for both planets, we can relate their periods:

Finding the Value of T

From the relationship, we know:

• 1 = k * 1³ (for Earth)
• Thus, k = 1
• Now, for Saturn: T² = 729

Taking the square root gives us:

Final Calculation

T = √729 = 27 years.

Therefore, the time period of the revolution of Saturn is approximately equal to 27 years. The correct answer is (B) 27 years.