if vortices are oscillating, or are subject to other deformations in the plane orthogonal to the rotation. One may assume, at least under stationary conditions, that $\beta''\ll \beta, \beta'$.

In the both limits of either [*strong coupling*]{} ($\eta\gg \rho_S\omega$) or [*weak coupling*]{} ($\eta\ll \rho_S\omega$) between a vortex and the normal fluid, one finds that $\beta\to 0$ as a function of $\eta$, with the maximum $\beta_{\rm max} = 0.5$ at $\eta= \rho_S\omega$. In the strong coupling limit $\beta'(\eta)\to 1$, while in the opposite weak coupling limit $\beta'(\eta)\to 0$ ( generally we assume that the quasi-particle–vortex scattering kinematics implies $\eta'\ll \eta$ and that for the relevant densities $\eta'\ll\rho_S\omega$). Note that $\beta'(\eta)$ approaches its asymptotic strong-coupling values quadratically, while $\beta(\eta)$ does so linearly; the asymptotic behavior for large $\eta$’s, therefore, is dominated by $\beta(\eta)$.

Virial equations and their perturbations
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Virial equations of various order are constructed by taking moments of the hydrodynamic equations. Since the computation of the perturbations of the these virial equations is central to the theory of superfluid ellipsoids we review this somewhat technical issue in this section. The reader who is interested only in the physics of superfluid oscillations can proceed to the next section where we discuss 