Question

Use the power function to calculate the half major axis of the asteroid Vesta, which has an orbital period around the sun of 3.63 years.

Answer 1

To calculate the semi-major axis of asteroid Vesta with an orbital period of 3.63 years, use Kepler's Third Law by taking the orbital period P, squaring it, and finding the cube root to get the semi-major axis, which is approximately 2.35 AU.

Step-by-step explanation:

To calculate the half major axis (or semi-major axis) of the asteroid Vesta with an orbital period around the sun of 3.63 years, we will use Kepler's Third Law of planetary motion. This law states that the square of the orbital period (P) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit around the sun. Kepler's Third Law can be expressed as P² = a³, where P is in years and a is in Astronomical Units (AU).

To find the semi-major axis of Vesta, we rearrange the formula to solve for a:

1. 
   
3. Take the orbital period P, which is 3.63 years.
4. 
   
6. Calculate P²: (3.63² = 13.1769).
7. 
   
9. Find the cube root of P² to get a: a³ = P², so a = ∛(P²) = ∛(13.1769) ≈ 2.35 AU.

Therefore, the semi-major axis of asteroid Vesta is approximately 2.35 AU.