Final answer:
Planet Y, with a 145-day orbital period, is closer to its host star than Planet X, which has a 290-day orbital period. This conclusion is based on Kepler's third law, which relates the square of a planet's orbital period to the cube of its semi-major axis distance from the star.
Step-by-step explanation:
The question involves understanding Kepler's third law of planetary motion, which states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit (P² ≈ a³). Given two exoplanets with different orbital periods around a star with the same mass as the Sun, Planet Y, with an orbital period of 145 days, would be closer to the star compared to Planet X, with an orbital period of 290 days. According to Kepler's third law, the shorter the orbital period, the smaller the mean distance from the star, making Planet Y closer to the star than Planet X.
To determine the semi-major axis of the orbits for Planet X and Planet Y, one would apply the formula a = (P²/GM)^(1/3), where P is the orbital period, G is the gravitational constant, and M is the mass of the star. The semi-major axis would be measured in Astronomical Units (AU), where 1 AU is the average distance from the Earth to the Sun.