Dissipative spreading of a planetary ring ($E_J=E_{\mathrm{osc}}-\frac{3}{8}{\Omega }^2{\sum }_i{m}_i{X}_i^2$)

In a Keplerian shearing sheet, a particle's radial epicyclic guiding center $X$ contributes $-3{\Omega }^2{X}^2/8$ to its specific Jacobi energy in a shearing sheet. Radial and vertical oscillations contribute nonnegative energies. For a collection of point masses, writeInternal collisions preserve the mass-weighted mean guiding centre $\bar{X}$ through azimuthal momentum conservation. If $V_X={\sum }_i{m}_i(X_i-\bar{X}{)}^2$, thenThus loss of Jacobi energy in a shearing sheet requires guiding-centre spreading when the oscillation energy is unchanged between the compared states, for example in an approximately steady random-motion budget. Outward angular momentum transport then accompanies inward motion of some particles and outward motion of others.

Dissipation alone is insufficient: it can instead reduce $E_{\mathrm{osc}}$. For example, a purely radial inelastic collision between equal particles with equal azimuthal velocities reduces their radial relative motion while leaving both guiding centres unchanged. A general statement that every inelastic energy loss must increase ring width would miss this possibility. Predicting a secular spreading rate requires the collision and stress model, including how shear replenishes random motion.