Chapter 5, Problem 56P

Neptune is an average distance of 4.5 x 10⁹ km from the Sun. Estimate the length of the Neptunian year using the fact that the Earth is 1.50 x 10⁸ km from the Sun on average.

To determine

The length of the Neptunian year using the given data.

Answer to Problem 56P

Solution:

  $T_N=164.32\text{years}$ .

Explanation of Solution

Given:

The average distance between the Sun and Neptune, $x_N=4.5\times {10}^9\text{km}$

The average distance between the Sun and Earth, $x_E=1.5\times {10}^8\text{km}$

Formula Used:

From the Kepler’s third law, for the masses orbiting the Sun

  ${\left(\frac{T_N}{T_E}\right)}^2={\left(\frac{x_N}{x_E}\right)}^3$

Where, $T_E$ is the period of the Earth.

  $T_N$ is the period of the Neptune.

Calculation:

Considering the period of the Earth as one year, the period of the Neptune ca be calculated as follows:

  $T_N=T_E{\left(\frac{x_N}{x_E}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

  $T_N=1{\left(\frac{4.5\times {10}^9}{1.5\times {10}^8}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

  $T_N=164.32\text{years}$

Conclusion:

Hence, the length of the Neptunian year is obtained as 164.32 Earth years.