if a planet has a period of 1 century, what is its average distance from the Sun in AU?
To determine the average distance of a planet from the Sun, we can use Kepler's Third Law of Planetary Motion. The formula for Kepler's Third Law is:
T^2 = k * R^3
Where T is the period of the planet in years, R is the average distance from the Sun in AU (Astronomical Units), and k is a constant.
In this case, the period of the planet is 1 century, which is equivalent to 100 years. So we have:
T = 100 years
To find the average distance (R), we need to rearrange the formula to solve for R:
R = (T^2 / k)^(1/3)
Now, what we need is the value of the constant k. Kepler's Third Law states that k is the same for all planets in our solar system. The value of k depends on the units used for T (period) and R (average distance) in the equation. In this case, the values of T and R should be in years and AU, respectively.
By using the known values of other planets, we can find the value of k. The average distance of Earth from the Sun is approximately 1 AU, and its period is 1 year. Plugging these values into the equation, we get:
1^2 = k * 1^3
Simplifying, we find that k = 1.
Now we can substitute the values of T = 100 years and k = 1 into the rearranged formula and calculate the average distance (R):
R = (100^2 / 1)^(1/3) = 100^(2/3)
Calculating this expression gives us the average distance from the Sun in AU for a planet with a period of 1 century.