If a planet's closest distance to the Sun is 1.00 AU and its furthest is 9.00 AU, how long is its orbital period in years?

If a planet's closest distance to the Sun is 1.00 AU and its furthest is 9.00 AU, how long is its orbital period in years?

• A. 5.5 years
• B. 11.2 years
• C. 9.6 years
• D. 15.0 years

Answer: (B)

To determine the orbital period of the planet, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet (in years) is proportional to the cube of the semi-major axis of its orbit (in astronomical units, AU). In this case, the closest distance to the Sun (perihelion) is 1.00 AU, and the furthest distance (aphelion) is 9.00 AU. The semi-major axis is the average of these two distances, calculated as follows: \[ \text{Semi-major Axis} = \frac{\text{perihelion} + \text{aphelion}}{2} = \frac{1.00 \, \text{AU} + 9.00 \, \text{AU}}{2} = 5.00 \, \text{AU} \] According to Kepler’s Third Law, we can express this as: \[ T^2 = a^3 \] where \( T \) is the orbital period in years and \( a \) is the semi-major axis in AU. Substituting \( a = 5.00 \): \

To determine the orbital period of the planet, we can use Kepler's Third Law of Planetary Motion, which states that the square of the orbital period of a planet (in years) is proportional to the cube of the semi-major axis of its orbit (in astronomical units, AU).

In this case, the closest distance to the Sun (perihelion) is 1.00 AU, and the furthest distance (aphelion) is 9.00 AU. The semi-major axis is the average of these two distances, calculated as follows:

[

\text{Semi-major Axis} = \frac{\text{perihelion} + \text{aphelion}}{2} = \frac{1.00 , \text{AU} + 9.00 , \text{AU}}{2} = 5.00 , \text{AU}

]

According to Kepler’s Third Law, we can express this as:

[

T^2 = a^3

]

where ( T ) is the orbital period in years and ( a ) is the semi-major axis in AU.

Substituting ( a = 5.00 ):

\