Final answer:
The factor by which the orbital period is increased between two planets with different mean distances from the Sun can be found by dividing their orbital period equations. In this case, planet Y is twice the mean distance away from the Sun as planet X. Therefore, the orbital period of planet Y is increased by a factor of 2^(1/2), which is option B: 2₁₂.
Step-by-step explanation:
The equation T squared = A cubed represents Kepler's third law, which relates a planet's orbital period (T) to its mean distance from the Sun (A) in astronomical units (AU). Given that planet Y is twice the mean distance away from the Sun compared to planet X, we can determine the factor by which the orbital period is increased.
Let's assume that the mean distance of planet X from the Sun is A. Therefore, the mean distance of planet Y from the Sun is 2A. Using Kepler's third law, we can write the equation:
Tx² = Ax³
Ty² = (2A)³
Ty² = 8A³
Divide the second equation by the first equation to find the factor by which the orbital period is increased:
Ty² / Tx² = 8A³ / Ax³
Ty / Tx = √(8A³ / Ax³)
Ty / Tx = √(8 / 1)
Ty / Tx = √8
Ty / Tx = 2^(1/2) = 2^(1/2) * 1
The factor by which the orbital period is increased is 2^(1/2), which is option B: 2₁₂.