Final answer:
The time it takes for a comet to complete one orbit around the Sun can be calculated using Kepler's third law of planetary motion. We get au^(3/2) years.
Step-by-step explanation:
Answer:
The time it takes for a comet to complete one orbit around the Sun can be calculated using Kepler's third law of planetary motion.
Kepler's third law states that the square of the period of an orbit is equal to the cube of the average distance from the Sun.
In this case, since the comet has an elliptical orbit, its average distance from the Sun would be the semi-major axis of its orbit.
Let's represent the average distance from the Sun as 'a' and the period as 'T'.
According to the problem statement, the comet's closest distance from the Sun is 'au' and its greatest distance from the Sun is also 'au'.
Since the comet's orbit is elliptical, the average distance 'a' would be equal to half the sum of the closest distance and the greatest distance.
So, 'a' = (au + au)/2 = au.
Now we can use Kepler's third law to find the period 'T' in years.
The equation is T^2 = a^3, where 'T' is measured in years and 'a' is measured in astronomical units (AU).
Rearranging the equation, we have T = √a^3.
Substituting the value of 'a' as 'au', we get T = √(au)^3.
Evaluating this expression, we find that T ≈ au^(3/2) years.