A MODEL FOR TIDALLY DRIVEN ECCENTRIC INSTABILITIES IN FLUID DISKS

Observations and simulations of superhump phenomena in close binary star systems and observations and previous studies of eccentric planetary rings, such as the rings of Uranus, indicate that eccentric instabilities occur in disks and rings. We investigate the eccentric stability of a fluid disk or ring in the presence of the tidal field of a companion object which is in circular orbit. The disk can have pressure, vicosity, and self-gravity. Eccentric Lindblad resonances that lie within a disk cause eccentricity growth by the following process. For a particular perturbing potential component phi-m(r) cos [m(theta - OMEGA-p(t))] a simple tidal response is felt by the disk that scales with phi-m. This response couples with the imposed eccentricity to launch density waves at each eccentric Lindblad resonance, producing a response that scales with e-phi-m. The coupling of these two responses produces a stress that increases eccentricity with growth rate that scales as phi-m2. We consider an eccentric disk with slowly varying orbital elements. An upper limit to the growth rate is obtained in the ideal case that eccentricity growth proceeds with minimal dissipation. In that case, for a narrow ring, mechanical energy changes E due to an eccentric Lindblad resonance for potential phi-m are related to angular momentum changes J by E = OMEGA-e(J), where OMEGA-e = m-OMEGA-p/(m +/- 1) is the pattern speed of the resonance, with the minus (plus) sign applying to an eccentric inner (outer) Lindblad resonance. For m > 1, the ideal eccentricity growth rate lambda is related to resonant angular momentum change as lambda = [1/(m +/- 1)](\J\/e2J). For high m, these results apply to the eccentric planetary rings. For application to superhump phenomena, the m = 3 eccentric inner Lindblad resonance likely dominates over other eccentric Lindblad and eccentric corotational resonances. This resonance produces an ideal eccentricity growth rate in the disk lambda = 2.08C-OMEGA-pq2r(res)/max (r(o) - r(i), w), where C is a function of the disk structure that is equal to unity for a narrow ring, q << 1 is the binary mass ratio, w is the resonance width, and r(res), r(i), and r(o) are the resonant, inner, and outer radii of the disk. A tightly wrapped, trailing, inwardly propagating acoustic wave is launched at this resonance with periodicity in 2-theta - 3-OMEGA-p(t).