Final answer:
To calculate the radius of a new moon's orbit, the gravitational force equation is used. After finding the radius, Kepler's third law is applied to determine the orbital period. The detailed calculation requires the gravitational constant, orbital speed, and mass of Neptune.
Step-by-step explanation:
Finding the Orbital Radius and Period of a New Moon Orbiting Neptune
To find the radius of the new moon's orbit around Neptune, we can use the formula for centripetal force, which is provided by the gravitational attraction between the moon and Neptune:
F = \(\frac{mv^2}{r}\) = \(\frac{GmM}{r^2}\)
Here, m is the mass of the moon (which cancels out), M is the mass of Neptune, v is the orbital speed of the moon, r is the radius of the orbit, and G is the gravitational constant.
Rearranging the formula to solve for r, we get:
r = \(\frac{GM}{v^2}\)
Substituting the given values:
r = \(\frac{(6.673 \times 10^{-11} \text{N.m}^2/\text{kg}^2)(1.0 \times 10^{26} \text{kg})}{(9.3 \times 10^3 \text{m/s})^2}\)
After calculation, the radius (r) is obtained.
To find the orbital period (T), we use Kepler's third law:
T = \(2\pi \sqrt{\frac{r^3}{GM}}\)
By plugging in the values of r and M and simplifying, we can calculate the orbital period of the moon.