Final answer:
Using Kepler's Third Law and converting Planet X's orbital period from days to years, we calculate the semi-major axis to be approximately 0.856 AU, indicating that Planet X is farther from its host star compared to Planet Y, which has a shorter orbital period.
Step-by-step explanation:
To calculate the approximate length of Planet X's orbital period, we need to use Kepler's Third Law, which states that the square of a planet's orbital period (P²) is directly proportional to the cube of the semi-major axis of its orbit (a³). This law applies to planets orbiting our Sun, and it can also be used for planets orbiting other stars, assuming the star has a similar mass to the Sun.
If Planet X has an orbital period of 290 days, we can convert this to years since Kepler's Third Law is usually expressed in astronomical units (AU) and years. To convert days to years, we divide by 365.25 (the average number of days in a year, considering leap year): 290 days / 365.25 days/year = 0.794 years (approximately).
Now, we use the proportionality constant from Kepler's Third Law, which is approximately 1 when we measure the period in years and the semi-major axis in AU. Therefore, for a star with mass similar to the Sun, this constant can be omitted, giving us:
P² = a³
0.794² = a³
0.63 (approx.) = a³
Now we solve for a by taking the cube root:
a = √(0.63)
a = 0.856 AU (approximately)
Thus, the semi-major axis of Planet X is approximately 0.856 AU.
If we apply the same method to Planet Y which orbits in 145 days, we can see that Y is closer to the host star than X, thus verifying that Planet Y is closest to its host star, and we could calculate its semi-major axis similarly.