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For a thin spherical shell, the radial pressure force per unit volume is $-dP/dr$. By the shell theorem, the inward gravitational acceleration is $GM(r)/r^2$, because exterior spherical shells contribute no force. Balancing force densities in hydrostatic equilibrium givesThe second equation is mass conservation; it closes the pressure-support equation once an equation of state is supplied.

The corotation radius is where the stellar spin matches Keplerian rotation:A parcel forced into corotation at $r_m$ has outward centrifugal acceleration ${\Omega }_m^2{r}_m$. This exceeds the inward gravitational acceleration whenThis is the onset condition for a centrifugal propeller effect. It does not by itself guarantee escape to infinity. For example, a parcel released with only azimuthal speed ${\Omega }_m{r}_m$ has nonnegative specific orbital energy only ifAdditional magnetic work, radial motion or pressure can alter that escape condition.

The constant $f=2\Omega \sin \phi$ is the positive Coriolis parameter at northern latitude $\phi$, $g$ is the gravitational acceleration, and $h$ is the undisturbed mean depth of the layer.

The linearized shallow water equations assume a homogeneous, incompressible, inviscid layer over a flat bed; a horizontal scale much larger than the depth, so the shallow-water approximation and hydrostatic approximation apply; depth-independent horizontal velocity; a free-surface displacement $|\eta |\ll h$ and sufficiently small velocity to neglect nonlinear products; and an f-plane on which $f$ is constant. The equations also omit viscosity, forcing, and bottom topography.

The acceleration tidal tensor is $T_{jk}={\partial }_k{g}_j=-{\partial }_j{\partial }_k\Phi$ for gravitational acceleration $\mathbf{g}=-\nabla \Phi$. It describes differential acceleration $\Delta \mathbf{g}\simeq T\Delta \mathbf{r}$. Some authors instead call the positive Hessian matrix ${\partial }_j{\partial }_k\Phi$ the tidal tensor, reversing this sign convention. For a point mass and the positive potential convention $\mathbf{g}=\nabla \psi$, the tensor is $Gm(3\mathbf{R}{\mathbf{R}}^{\mathsf{T}}/R^5-1/R^3)$.