# 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:

The tidal gravitational potential \(\PhiT\) is expressed as a superposition

of partial tidal potentials defined by

(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,

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).