Final answer:
To find the semi-major axis of Pallas's orbit, we apply Kepler's third law which relates the square of the orbital period to the cube of the semi-major axis. Using the given orbital period of 4.62 years, we calculate that the semi-major axis of Pallas's orbit is approximately 2.77 AU.
Step-by-step explanation:
To find the semi-major axis of the asteroid Pallas's orbit, we can apply Kepler's third law, which states that the square of the orbital period (P^2) is directly proportional to the cube of the semi-major axis (a^3) of the orbit. Since Pallas has an orbital period of 4.62 years, we can calculate P^2 as 4.62^2.
However, to find the semi-major axis (a), we need the constant of proportionality which is determined by the mass of the Sun and the gravitational constant.
For our solar system, when period (P) is given in Earth years and semi-major axis (a) is given in Astronomical Units (AU), the proportionality constant is 1 due to Kepler's third law being empirically determined for our solar system's scale. Therefore, P^2 = a^3, and we solve for 'a'.
Using the equation:
a^3 = P^2
a = ∛(P^2)
Calculating:
a^3 = 4.62^2
a^3 = 21.3444
a = ∛(21.3444)
a = 2.77 AU (approximately)
Therefore, the semi-major axis of Pallas's orbit is approximately 2.77 AU.