= Planet-induced debris-disk warp
An inclined inner <planet> causes exterior <debris disk> rings to undergo <nodal precession> at different rates. In the <quadrupole approximation>, $\nu\simeq(3/4)n(M_p/M_\star)(a_p/a)^2\propto a^{-7/2}$. Rings that have precessed appreciably tilt away from their original <orbital plane>, whereas distant rings remain near it, making an evolving warp. Conservative <secular perturbations> alone do not damp every ring into the planet's plane; unresolved <phase mixing> can instead produce a thick inner disk whose mean normal is near the forced plane.
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