The orbit of the comet Encke has a perihelion distance of 0.331 AU and an eccentricity of 0.850. Suppose that the arrival of the comet at its perihelion distance happens to occur at a point directly in line between the Sun and the Earth. Roughly how many years will it be before such an event happens again?
To determine how many years it will be before the comet Encke arrives at its perihelion distance at a point directly in line between the Sun and the Earth again, we need to calculate the period of the comet's orbit.
The period can be calculated using Kepler's Third Law of Planetary Motion, which states that the square of a planet's orbital period is proportional to the cube of its average distance from the Sun.
The average distance of the comet Encke from the Sun can be calculated by adding the perihelion distance and the aphelion distance and dividing by 2. However, we are given only the perihelion distance (0.331 AU) and eccentricity (0.850).
To find the aphelion distance, we can use the formula:
aphelion distance = perihelion distance / (1 - eccentricity)
Plugging in the given values, we can calculate the aphelion distance:
aphelion distance = 0.331 AU / (1 - 0.850) = 0.331 AU / 0.150 = 2.207 AU
Now, we can calculate the average distance:
average distance = (perihelion distance + aphelion distance) / 2 = (0.331 AU + 2.207 AU) / 2 = 1.269 AU
Next, we can use Kepler's Third Law to find the period of the comet's orbit. The time period (T) in years can be calculated using the following formula:
T^2 = (average distance)^3
Squaring both sides of the equation:
T^2 = (1.269 AU)^3
Taking the square root of both sides:
T = sqrt((1.269 AU)^3)
Calculating this square root, we find:
T ≈ 3.812 years
Therefore, it will take roughly 3.812 years before the comet Encke arrives at its perihelion distance at a point directly in line between the Sun and the Earth again.