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 third law, we need to relate the orbital period of Phobos to the orbital radius of Phobos.

Kepler's third law states that the square of the orbital period (T) of a celestial object is directly proportional to the cube of its average distance (r) from the central object. Mathematically, this can be written as:

T^2 = (4π^2/G)(r^3/M),

where T is the orbital period, r is the orbital radius, G is the gravitational constant, and M is the mass of the central object.

To estimate the mass of Mars, we need to rearrange the formula to solve for M:

M = (4π^2/G)(r^3/T^2).

Now, let's plug in the given values:

T = 7.67 hours = 7.67 * 3600 seconds (convert to seconds)
r = 9.4 × 10^6 m
G = 6.674 × 10^-11 m^3/(kg s^2) (gravitational constant)

First, we need to convert the time from hours to seconds:

T = 7.67 hours * 3600 seconds/hour = 27582 seconds.

Now, we can calculate the mass of Mars:

M = (4π^2/G)(r^3/T^2)
M = (4π^2/(6.674 × 10^-11))(9.4 × 10^6)^3/(27582)^2

Using these values in a calculator, we find the mass of Mars to be approximately 6.42 × 10^23 kg.

So, the estimated mass of Mars is 6.42 × 10^23 kg.