Question

Use the ratio form of Kepler's third law, ((t_a/t_b)² = ?), and the data provided to determine the time it takes Mars to orbit the Sun. Round your answer to the nearest tenth. Earth's orbital period = 1.0 Earth year, Earth's distance from the Sun = 1.0 AU, Mars's distance from the Sun = 1.5 AU. It takes Mars about _______ Earth years to orbit the Sun.

Answer 1

Using the ratio form of Kepler's third law and the given data, it takes Mars approximately 1.9 Earth years to complete one orbit around the Sun.

Step-by-step explanation:

To calculate the time it takes Mars to orbit the Sun using the ratio form of Kepler's third law, we will use the ratio of the orbital periods of Earth and Mars, given that the distance of Mars from the Sun is 1.5 AU compared to Earth's 1.0 AU.

Kepler's third law states that the square of the orbital period of a planet is proportional to the cube of its semimajor axis in astronomical units (AU). So in the ratio form, this law can be written as:

(ta/tb)² = (aa/ab)³

Given that tb (Earth's orbital period) is 1.0 Earth year, and aa (Mars's distance from the Sun) is 1.5 AU, and ab is 1.0 AU, we can rearrange the equation to solve for Mars's orbital period:

ta² = (1.5 / 1.0)³ = 3.375

Taking the square root of both sides, we find:

ta = √3.375 ≈ 1.84 Earth years

Therefore, it takes Mars about 1.9 Earth years to orbit the Sun, when rounded to the nearest tenth.