planet is in circular orbit around the Sun. Its distance from the Sun is 2.9 times the average distance of Earth from the Sun. What is the period of this planet, in Earth years?
period^2/r^3=1year^2/re^3
so period= sqrt (2.9^3)
To find the period of the planet in Earth years, we need to use Kepler's Third Law of Planetary Motion, which states that the square of a planet's orbital period is directly proportional to the cube of its average distance from the Sun.
Let's break down the problem step by step:
Step 1: Determine the average distance of Earth from the Sun.
The average distance of Earth from the Sun is known as an Astronomical Unit (AU), which is approximately 149.6 million kilometers (93 million miles).
Step 2: Find the distance of the planet from the Sun.
Using the given information, we know that the distance of the planet from the Sun is 2.9 times the average distance of Earth from the Sun. Therefore, the distance of the planet from the Sun is:
2.9 x 149.6 million kilometers ≈ 434.84 million kilometers.
Step 3: Calculate the period of the planet.
Applying Kepler's Third Law, we set up a proportion by comparing the orbital periods and the average distances.
(Period of the planet / Period of Earth) = (Distance of the planet / Distance of Earth)^(3/2)
We can rewrite this as:
Period of the planet = (Period of Earth) x (Distance of the planet / Distance of Earth)^(3/2)
Substituting the values into the equation:
Period of the planet = 1 year (since we want the answer in Earth years) x (434.84 million kilometers / 149.6 million kilometers)^(3/2)
Calculating this expression gives us:
Period of the planet ≈ 1 year x (2.9053)^(3/2)
Finally, we evaluate the expression:
Period of the planet ≈ 1 year x 5.815
Therefore, the period of the planet, in Earth years, is approximately:
Period of the planet ≈ 5.815 Earth years.