Diskoseismology: Probing accretion disks. I. Trapped adiabatic oscillations

We analyze the normal modes of acoustic oscillations within thin accretion disks which are terminated by an innermost stable orbit around a slowly rotating black hole or weakly magnetized compact neutron star. The dominant relativistic effects, which allow modes to be trapped within the inner region of the disk, are approximated via a modified Newtonian potential. A general formalism is developed for investigating the adiabatic oscillations of arbitary unperturbed disk models. We explore the generic behavior via an expansion of the Lagrangian displacement about the plane of sysmmetry and by assuming separable solutions with the same radial wavelength for the horizontal and vertical perturbations. We obtain the lowest eigenfrequencies and eigenfunctions of a particular set of radial and quadrupole modes which have minimum motion normal to the plane. These modes correspond to the standard dispersion relation of disk theory. In future papers we will study a wide variety of modes and compute the rates of growth or damping of these modes due to gravitational radiation and various models of viscosity.