Abstract

In young extrasolar systems, planets embedded within gaseous protoplanetary discs experience strong gravitational interactions with the disc, causing the planet to migrate inwards or outwards on a timescale far shorter than the disc’s lifetime. Understanding the strength and sign of the torque exerted by the disc on the planet as a function of the disc’s parameters is therefore essential for predicting and understanding extrasolar system architectures.

The flow induced by an Earth-mass planet admits a particularly elegant linearisation and solution. This involves large-scale excited spiral density waves which transport angular momentum radially away from the planet, and ‘horseshoe orbits’ within a critical layer which allow fluid elements on similar orbital radii to the planet (that is, corotating fluid elements) to periodically exchange angular momentum with the planet.

The total torque on the planet may be expressed in terms of these angular momentum dynamics, and depends at leading order on the local surface density and temperature gradients of the disc, as well as a factor arising from the cylindrical geometry. We derive novel equations which capture uniformly the dynamics of potential vorticity and entropy within the critical layer, as well as the larger scale excited density wave structure. Importantly, this resolves the previously overlooked significant density wave excitation induced by entropic effects at corotation. The result is a more robust calculation of the torque on an Earth-mass planet.