Final answer:
By applying Kepler's Third Law to the given satellite orbit details, such as its altitude and period, we can calculate the mass of planet Zeeman-474.
Step-by-step explanation:
To find the mass of planet Zeeman-474, we can use Kepler's Third Law and the provided values: A satellite is in a circular orbit 230 km (or 230,000 m) above the surface of Zeeman-474, the orbital period is 89 minutes (or 5340 seconds), the radius of Zeeman-474 is 6.38 × 10⁶ m, and the gravitational constant is 6.67 × 10⁻¹¹ N ⋅ m²/kg².
The total radius of the orbit r is the radius of Zeeman-474 plus the altitude of the satellite:
r = 6.38 × 10⁶ m + 230,000 m = 6.61 × 10⁶ m
Using the circular orbit version of Kepler's Third Law, we calculate the mass (M) of the planet:
M = ⅔ ∙ (⅔ ∙ r³) / (G ∙ T²)
Where T is the orbital period of the satellite and G is the gravitational constant.
Substituting the given values:
M = ⅔ ∙ (⅔ ∙ (6.61 × 10⁶ m)³) / (6.67 × 10⁻¹¹ N⋅m²/kg² ∙ 5340 s²)
This calculation gives us the mass of Zeeman-474, which we can calculate using a calculator or appropriate software.