Question

Use Kepler's Law, which states that the square of the time, T, required for a planet to orbit the Sun varies directly with the cube of the mean distance, a, that the planet is from the Sun. Using the Earth's distance of 1 astronomical unit (A.U.), determine the time, in Earth years, for a planet to orbit the Sun if its mean distance is 2.59 A.U. (Round your answer to two decimal places.)

Answer 1

The time for a planet to orbit the Sun can be found using Kepler's Law. For a planet with a mean distance of 2.59 A.U., the time is approximately 91.64 Earth years.

Step-by-step explanation:

To solve this problem, we can use the equation based on Kepler's Law: T2 = ka3. We know that the mean distance of Earth from the Sun is 1 A.U., and the mean distance of the other planet is 2.59 A.U. We can substitute these values into the equation and solve for T.

T2 = k(2.59)3
T2 = k(17.691)
T2 = 17.691k

Since we are comparing the time for the other planet to orbit the Sun with the time for Earth to orbit the Sun, we can set up a ratio:

Tother2/TEarth2 = 17.691k/1
Tother2/12 = 17.691k/1
Tother2 = 17.691k

We know that TEarth = 1, so we can substitute Tother2 = 17.691k and solve for Tother:

1 = 17.691k
k = 1/17.691

Substituting this value of k back into the equation:

Tother2 = 17.691(2.59)3
Tother2 = 17.691(17.691)(2.59)2
Tother2 = 17.691(474.667)
Tother2 = 8391.82
Tother = √8391.82

Using a calculator, we find that Tother ≈ 91.64 Earth years.

Learn more about Kepler's Law