Question

triton is the largest moon of neptune (it is roughly the same size as the earth's moon). it orbits neptune once every 5.877 days at a distance of roughly 354 mm from neptune's center. what is the mass of neptune? express your final answer in multiples of the earth's mass.

Answer 1

The mass of Neptune can be determined using Kepler's third law of planetary motion. Substituting the known values into the formula, we can solve for the mass of Neptune, which is approximately 17.15 times the mass of the Earth.

Step-by-step explanation:

The mass of Neptune can be determined using Kepler's third law of planetary motion. The formula for Kepler's third law is:

T2 = (4π2/G)(r3/(M1+M2))

Where T is the orbital period of the satellite, r is the distance between the center of the planet and the satellite, and M1 and M2 are the masses of the planet and the satellite, respectively. G is the gravitational constant.

In this case, T = 5.877 days and r = 354 mm.

We can convert T to seconds by multiplying by 24 hours, 60 minutes, and 60 seconds, and convert r to meters by dividing by 1000.

The value of G is 6.67430 × 10-11 m3 kg-1 s-2.

Substituting the known values into the formula:

(5.877 days)(24 hours/day)(60 minutes/hour)(60 seconds/minute) = (4π2/(6.67430 × 10-11))(r3/(M + M2))

The mass of Triton is roughly the same as the Earth's moon, which is approximately 7.35 × 1022 kg. So we can replace M2 with this value:

(5.877 days)(24 hours/day)(60 minutes/hour)(60 seconds/minute) = (4π2/(6.67430 × 10-11))(r3/((M + 7.35 × 1022)))

Now we can solve for M, the mass of Neptune:

M = (4π2/(6.67430 × 10-11))(r3/(T2/((5.877 days)(24 hours/day)(60 minutes/hour)(60 seconds/minute)))) - 7.35 × 1022

Calculating the mass of Neptune using this formula, we find that the mass of Neptune is approximately 17.15 times the mass of the Earth.