Nonaxisymmetric secular instabilities driven by star/disk coupling

Abstract

We determine conditions for the onset of nonaxisymmetric secular instabilities in polytropes with a wide range of angular momentum distributions using Lagrangian techniques, and then calculate the growth rate of such instabilities when driven by the coupling of the perturbed star to a circumstellar disk. We use Langrangian displacement vectors with azimuthal dependence proportional to exp (im phi), where m is an integer and phi is the azimuthal coordinate. The onset of secular instability in terms of the quantity T/absolute value of W, the ratio of rotational kinetic energy to gravitational potential energy, is affected by both the compressibility and the angular momentum distribution of the polytrope. The largest effects occur when the angular momentum distribution is varied. For polytropic index n = 3/2, the onset of secular instability for the m = 2 mode (the bar mode), as determined by its neutral point, shifts from T/absolute value of W = 0.141 to 0.093, while the m = 5 mode neutral point shifts from T/absolute value of W = 0.088 to 0.031 over the range of angular momentum distributions we consider. The smallest critical T/absolute value of W-values occur for the angular momentum distributions which are the most peaked toward the equator. For the angular momentum distribution of a Maclaurin spheroid, as the polytropic index n is increased from 3/2 to 5/2, the neutral point for m = 2 shifts from T/absolute value of W = 0.141 to 0.144 and the netural point for m = 5 shifts from T/absolute value of W = 0.069 to 0.078. The netural points for m = 2 and 5 for the Maclaurin sequence (n = 0) are 0.137 and 0.0629, respectively. As the angular momentum distribution becomes more peaked toward the equatorial radius of the polytropes, the critical T/absolute value of W-values generally become less sensitive to the compressibility of the polytrope. Star/disk coupling can drive the secular instability in systems where the star is surrounded by a massive disk and, if the instability can grow to moderate amplitude, then the coupling can transport significant amounts of angular momentum from the star into the circumstellar disk. We find, for the particular case of rotating protostars during the accretion phase, that the instability growth time can be shorter than the accretion time. Further, if the instability can grow to amplitudes on the order of several percent, the star/disk coupling can remove angular momentum from the forming star faster than it is added by accretion.