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 compare the orbital period and radius of Phobos to the mass of Mars.
Kepler's 3rd Law states that the square of the orbital period (T) of a moon is proportional to the cube of the semi-major axis (r) of its orbit. Mathematically, this can be written as:
T^2 = (4π^2 / GM) * r^3
Where:
- T is the orbital period of Phobos
- G is the gravitational constant (approximately 6.674 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of Mars
- r is the orbital radius of Phobos
We can rearrange this equation to solve for the mass of Mars (M):
M = (4π^2 / G) * (r^3 / T^2)
Substituting the given values:
T = 7.67 hours = 7.67 * 3600 seconds (converting to seconds)
r = 9.4 × 10^6 m
Calculating:
T^2 = (7.67 * 3600)^2 seconds^2
r^3 = (9.4 × 10^6)^3 m^3
Now we can use the equation to find the mass of Mars:
M = (4π^2 / G) * (r^3 / T^2)
Note that the units cancel out, leaving us with kilograms as the unit for mass.
Let's calculate it.