 Resonant torque from a rotating perturbation

The leading second-order specific torque on a collisionless distribution from a weak steadily rotating perturbation with ${\Phi }_{\mathbf{k}}={\Psi }_{\mathbf{k}}{e}^{-im{\Omega }_{p}t}$ and $m=k_{\varphi }$. The sum includes conjugate harmonics. Linear response in action-angle variables gives the finite-time kernel $\sin (\Delta t)/\Delta$; its integrated long-time resonance kernel selects orbital resonances. This limit assumes a smooth continuum, regular resonant surfaces and negligible secular redistribution during the response calculation.