Suppose that, in the future, observations with some new telescope reveal a planet about 16 AU from a star whose mass is the same as our sun's mass. How long does it take the planet to orbit the star?
Use Keplers law. You know at 1AU the period is 365 days...
To find the orbital period of the planet located 16 AU away from its star, we can use Kepler's Third Law of Planetary Motion. This law relates the orbital period of a planet (T) to its average distance from the star (r).
Kepler's Third Law states that the ratio of the square of the orbital period (T^2) to the cube of the average distance (r^3) is constant for all planets orbiting the same star. Mathematically, this can be written as:
(T1^2 / r1^3) = (T2^2 / r2^3)
Given that we know the orbital period at 1 AU is 365 days, we can use this information to solve for the orbital period at 16 AU.
Let's assign the values:
T1 = 365 days
r1 = 1 AU
r2 = 16 AU
Substituting these values in the formula, we get:
(T1^2 / r1^3) = (T2^2 / r2^3)
(365^2 / 1^3) = (T2^2 / 16^3)
Simplifying further:
365^2 / 1 = T2^2 / (16^3)
365^2 * (16^3) = T2^2
Solving for T2, take the square root of both sides:
T2 = √(365^2 * (16^3))
Calculating this value, we find that the orbital period of the planet located at 16 AU from its star is approximately 24,612 days.
So, it takes the planet approximately 24,612 days to orbit the star.