# Linearized Equations

The linearized tidal equations are similar to the linearized oscillation equations, but include an extra term in the momentum equation (6) representing the tidal force exerted by the companion:

$\rho \pderiv{\vv'}{t} = - \nabla P' - \rho' \nabla P - \rho \nabla \Phi' - \rho \nabla \PhiT.$

The tidal gravitational potential $$\PhiT$$ is expressed as a superposition

(14)$\PhiT = \sum_{\ell=2}^{\infty} \sum_{m=-\ell}^{\ell} \sum_{k=-\infty}^{\infty} \PhiTlmk,$

of partial tidal potentials defined by

$\PhiTlmk = - \epsT \, \frac{GM}{R} \, \cbar_{\ell,m,k} \left( \frac{r}{R} \right)^{\ell} Y^{m}_{\ell}(\theta, \phi) \, \exp(- \ii k \Oorb t).$

(the summation over $$\ell$$ and $$m$$ comes from a multipolar space expansion of the potential, and the summation over $$k$$ from a Fourier time expansion). Here,

$\epsT = \left( \frac{R}{a} \right)^{3} = \frac{\Oorb R^{3}}{GM} \frac{q}{1+q}$

quantifies the overall strength of the tidal forcing, in terms of the companion’s mass $$q M$$, semi-major axis $$a$$ and orbital angular frequency $$\Oorb$$. These expressions, and the definition of the tidal expansion coefficients $$\cbar_{\ell,m,k}$$, are presented in greater detail in Sun et al. (2023).