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We investigate the evolution on secular time-scales of a radially extendeddebris disc under the influence of a system of two coplanar planets interior tothe disc, showing that the secular resonances of the system can produce adepleted region in the disc by exciting the eccentricities of planetesimals.Using Laplace-Lagrange theory, we consider how the two exterior secularresonance locations, time-scales and widths depend on the masses, semi-majoraxes and eccentricities of the planets. In particular, we find that unless theresonances are very close to each other, one of them is very narrow andtherefore unimportant for determining the observable structure of the disc. Weapply these considerations to the debris disc of HD 107146, identifyingcombinations of the parameters of a possible unobserved two-planet system thatcould configure the secular resonances appropriately to reproduce the depletionobserved in the disc. By performing N-body simulations of such systems, we findthat planetesimal eccentricities do indeed become large near the theoreticalsecular resonance locations. The N-body output is post-processed to set theinitial surface density profile of the disc, and to include the possibleeffects of collisional depletion. We find that it is possible to obtain adouble-ringed disc in these simulations but not an axisymmetric one, with theinner ring having an offset whose magnitude depends on the eccentricities ofthe planets, and the outer ring showing spiral structure.

Empty gaps? Depleting annular regions in debris discs by secular resonance with a two-planet system