On the cavity of a debris disc carved by a giant planet

One possible explanation of the cavity in debris discs is the gravitational perturbation of an embedded giant planet. Planetesimals passing close to a massive body are dynamically stirred resulting in a cleared region known as the chaotic zone. Theory of overlapping mean-motion resonances predicts the width of this cavity. To test whether this cavity is identical to the chaotic zone, we investigate the formation of cavities by means of collisionless N-body simulations assuming a 1.25-10 Jupiter mass planet with eccentricities of 0-0.9. Synthetic images at millimetre wavelengths are calculated to determine the cavity properties by fitting an ellipse to 14 per cent contour level. Depending on the planetary eccentricity, e(pl), the elliptic cavity wall rotates as the planet orbits with the same (e(pl) < 0.2) or half (e(pl) > 0.2) period that of the planet. The cavity centre is offset from the star along the semimajor axis of the planet with a distance of d = 0.1q(-0.17) e(pl)(0.5) in units of cavity size towards the planet's orbital apocentre, where q is the planet-to-star mass ratio. Pericentre (apocentre) glow develops for e(pl) < 0.05 (e(pl) > 0.1), while both are present for 0.05 <= e(pl) <= 0.1. Empirical formulae are derived for the sizes of the cavities: delta a(cav) = 2.35q(0.36) and delta a(cav) = 7.87q(0.37) e(pl)(0.38) for e(pl) <= 0.05 and e(pl)> 0.05, respectively. The cavity eccentricity, e(cav), equals to that of the planet only for 0.3 <= e(pl) <= 0.6. A new method based on Atacama Large Millimeter/submillimeter Array observations for estimating the orbital parameters and mass of the planet carving the cavity is also given.