Singular Isothermal Disks. I. Linear Stability Analysis

Abstract

As part of a larger effort to understand how binary and single stars form from the collapse of magnetized molecular cloud cores, we perform a global stability analysis of isopedically magnetized, singular isothermal disks (SIDs). The work described here has precedents in earlier studies of disturbances in power-law disks by Zang in 1976, Toomre in 1977, Lynden-Bell & Lemos in 1993, Syer & Tremaine in 1996, and Goodman & Evans in 1999. We find the analytic criteria for the bifurcation of axisymmetric disks into nonaxisymmetric forms with azimuthal periodicities m = 1 (eccentric displacements), 2 (oval distortions), 3 (triangular distortions), etc. These bifurcations, which occur at zero frequency, are the compressible and differentially rotating analogs of how the classical sequence of incompressible and uniformly rotating Maclaurin spheroids bifurcate (secularly, under dissipative forces) to become Dedekind ellipsoids with figure axes that remain fixed in space. Like Syer & Tremaine and Lynden-Bell & Lemos, we also find that zero-frequency logarithmic spirals are possible scale-free disturbances, but our interpretation of the existence of such steadily propagating wavetrains is different. We give a dynamical instability interpretation based on the onset of swing amplification by overreflection at the corotation circle of prograde spiral density waves the pattern speeds of which have nonzero and positive values. Our analysis yields identical instability criteria as the global normal-modes treatment of Goodman & Evans, and we tentatively also identify dynamical barred-spiral instabilities as the breathing mode limit of two-armed ordinary-spiral instabilities. We prove a general reciprocity theorem, which states that the overreflection factors are identical for spiral density waves launched from cavities interior or exterior to Q-barriers that straddle the corotation circle. This globally valid result supports a unifying interpretation, advocated for many years by C. C. Lin and his colleagues (see, e.g., work by Bertin & Lin): the coexistence of spiral structure in galaxies arising from the instability of internal normal modes in the combined star/gas disk or from driving by external tidal influences associated with the chance passages of companion bodies.