Mars has a moon (Phobos) with an orbital period of 7.67 hours and an orbital radius of
9.4x106 m. Use Kepler’s 3rdLaw to estimate the mass of Mars.
To estimate the mass of Mars using Kepler's 3rd Law, we need to use the formula:
T² = (4π² / GM) * r³
Where:
T is the orbital period of Phobos (in seconds)
G is the gravitational constant (approximately 6.67430 × 10^-11 m³ kg⁻¹ s⁻²)
M is the mass of Mars (in kilograms)
r is the orbital radius of Phobos (in meters)
First, let's convert the orbital period to seconds. We are given that the orbital period of Phobos is 7.67 hours, which is equal to 7.67 * 60 * 60 seconds.
T = 7.67 * 60 * 60 = 27612 seconds
Now we can plug the values into the formula and solve for the mass of Mars:
T² = (4π² / GM) * r³
(27612)² = (4π² / G) * (9.4 × 10⁶)³
Simplifying further:
G = 6.67430 × 10^-11 m³ kg⁻¹ s⁻²
π = 3.14159265359
(27612)² = (4 * 3.14159265359² / (6.67430 × 10^-11)) * (9.4 × 10⁶)³
Solving this equation will give us the value of the mass of Mars.