Question

A comet orbits the Sun with a period of 89.17 yr. At perihelion, the comet is 1.331 AU from the Sun. How far from the Sun (in AU) is the comet at aphelion?

Short answer

Answer: The distance of the comet from the Sun at aphelion is approximately 10.6504 AU.

Step-by-step solution

Recall Kepler's Third Law of Planetary Motion

Kepler's Third Law states that the square of the period of any celestial body's orbit is proportional to the cube of the semi-major axis (average distance from the Sun) of its orbit. Mathematically, this can be written as: P^2 = a^3 where P is the period in Earth years, and a is the semi-major axis in astronomical units (AU).

Identify the given values in the exercise

We are given the following information: Period (P) = 89.17 years Distance at perihelion (closest point of the orbit) = 1.331 AU Our goal is to find the distance at aphelion (farthest point of the orbit).

Calculate the semi-major axis using Kepler's Third Law

Using Kepler's Third Law, we can write the equation as: (89.17)^2 = a^3 To determine the semi-major axis (a), we compute the cube root of both sides of the equation: a = (89.17^2)^(1/3) a ≈ 5.9907 AU The semi-major axis is roughly 5.9907 AU.

Calculate the distance at aphelion

To find the distance at aphelion, we can use the property that the sum of the distances at perihelion and aphelion is equal to twice the semi-major axis (because the semi-major axis is the average of the perihelion and aphelion distances). Mathematically, this is represented as: (perihelion distance) + (aphelion distance) = 2 * semi-major axis We can then solve for the aphelion distance: aphelion distance = (2 * semi-major axis) - perihelion distance aphelion distance = (2 * 5.9907) - 1.331 aphelion distance ≈ 10.6504 AU Therefore, the comet is approximately 10.6504 AU away from the Sun at aphelion.