Question

Asynchronous satellite, which always remains above the same point on a planet's equator, is put in a circular orbit around Uranus so that scientists can study a surface feature. Uranus rotates once every 17.2 h. Use the data of this table to find the altitude of the satellite.

__km

Answer 1

We can find the altitude of the satellite from the Newton's form of Kepler's third law:

$$\dfrac{T_{U}^2}{R^3}=\dfrac{4\pi^2}{GM_U},$$ $$\dfrac{T_{U}^2}{(R_U+h)^3}=\dfrac{4\pi^2}{GM_U},$$ $$R_U+h=\sqrt[3]{\dfrac{GM_UT_U^2}{4\pi^2}},$$ $$h=\sqrt[3]{\dfrac{GM_UT_U^2}{4\pi^2}}-R_U,$$ $$h=\sqrt[3]{\dfrac{6.67\cdot10^{-11}\ \dfrac{N\cdot m^2}{kg^2}\cdot8.68\cdot10^{25}\ kg\cdot(61920\ s)^2}{4\pi^2}}-2.33\cdot10^7\ m,$$ $$h=5.9\cdot10^{7}\ m=59000\ km.$$