Final answer:
To find the mass of Neptune from the orbital parameters of its moon Nereid, we can use Kepler's third law of planetary motion. Using the period and radius of Nereid's orbit, we can calculate the mass of Neptune and then determine how many Earth masses it is.
Step-by-step explanation:
To find the mass of Neptune, we can use Kepler's third law of planetary motion, which states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis. First, we need to convert the period from days to seconds. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, so 360 days is equal to 360 * 24 * 60 * 60 seconds. Next, we can use the formula:
T² = (4π²/GM)r³
Where T is the period, G is the gravitational constant (6.67 × 10^-11 Nm²/kg²), M is the mass of the planet, and r is the radius of the moon's orbit. Rearranging the formula, we can solve for M:
M = (4π²/G)r³/T²
Substituting the given values:
M = (4π²/(6.67 × 10^-11 Nm²/kg²))(5.59 × 10^9 m)³/((360 * 24 * 60 * 60 s)²)
Calculating the mass, we find:
M ≈ 1.86 × 10^19 kg
To determine how many Earth masses this is, we can divide the mass of Neptune by the mass of Earth, which is approximately 5.97 × 10^24 kg:
E = (1.86 × 10^19 kg)/(5.97 × 10^24 kg)
Calculating this ratio, we find:
E ≈ 3.11 × 10^-6 Earth masses