Hamiltonian hydrodynamics of eccentric discs. Issue 4 (29th December 2018) - Ogilvie, Gordon I; Lynch, Elliot M
Abstract: We show that the ideal hydrodynamics of an eccentric astrophysical disc can be derived from a variational principle. The non-linear secular theory describes the slow evolution of a continuous set of nested elliptical orbits as a result of the pressure in a thin disc. In the artificial but widely considered case of a 2D disc, the hydrodynamic Hamiltonian is just the orbit-averaged internal energy of the disc, which can be determined from its eccentricity distribution using the geometry of the elliptical orbits. In the realistic case of a 3D disc, the Hamiltonian needs to be modified to take into account the dynamical vertical structure of the disc. The simplest solutions of the theory are uniformly precessing non-linear eccentric modes, which make the energy stationary subject to the angular momentum being fixed. We present numerical examples of non-linear eccentric modes up to their limiting amplitudes. Although it lacks dissipation, which is important in many astrophysical contexts, this formalism allows a simpler theoretical approach to the non-linear dynamics of eccentric discs than that derived from stress integrals, and also connects better with established methods of celestial mechanics for cases in which the disc interacts gravitationally with one or more orbital companion.