Planetary Rotation

Suppose that the planet is rotating rigidly about the axis $\theta=0$ at the angular velocity ${\mit\Omega}$ (where ${\mit\Omega}\gg \omega_1$ ). The planet's rotational angular velocity vector is thus

$$\mbox{\boldmath$\Omega$} = {\mit\Omega}\,\cos\theta\,{\bf e}_r - {\mit\Omega}\,\sin\theta\,{\bf e}_\theta.$$ (12.43)

Let

$$\phi = \varphi - {\mit\Omega}\,t.$$ (12.44)

It follows that $r$ , $\theta$ , $\phi$ are spherical coordinates in a frame of reference that co-rotates with the planet.