Answer to Question #164542 in Astronomy | Astrophysics for Syed Abdullah Gilani

Question

A research team has discovered that a moon is circling a planet of our solar system: The moon orbits the planet once every 7 hours on a nearly circular orbit in a distance R of 48000 km from the centre of the planet. Unfortunately, the mass m of the moon is not known. Use Newton’s law of gravitation with G = 6.67 · 10−11 m3 /(kg·s 2 ) to approach the following questions: F = G · mM R2 (a) Based on the observations, determine the total mass M of the planet. (b) Which moon and planet of our solar system is the team observing? (Use literature.)

Answer 1

(a) We can find the mass of the planet from the Newton's form of Kepler's third law:

$$\dfrac{T^2}{R^3}=\dfrac{4\pi^2}{GM},$$ $$M=\dfrac{4\pi^2R^3}{GT^2},$$ $$M=\dfrac{4\pi^2\cdot(4.8\cdot10^7\ m)^3}{6.67\cdot10^{-11}\ \dfrac{Nm^2}{kg^2}\cdot(7\ h\cdot\dfrac{3600\ s}{1\ h})^2}=1.03\cdot10^{26}\ kg.$$

(b) The research team observed the planet Neptune ($M=1.02\cdot10^{26}\ kg$) and its satellite Naiad (semi-major axis 48224 km, orbital period 7 hours).