The shearing sheet and swing amplification revisited

Abstract

ABSTRACT The principal results of the classic analysis of the shearing sheet and swing amplification by Julian and Toomre (JT) are re-derived in a more accessible way and used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs’ 1965 dispersion relation and Toomre’s 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomre’s important 1969 paper on the group velocity implied by the Lin–Shu–Kalnajs (LSK) dispersion relation, we discover that two wavepackets emerge inside corotation: one each side of the inner Lindblad resonance. An extended form of the JT equation is used to investigate the impact of there being a deficit or surplus of stars in a narrow range of angular momenta. Swing amplification of leading waves introduced by such a groove gives rise to transient trailing spirals that extend further in radius and live longer at smaller azimuthal wavenumbers. Although the LSK dispersion relation provides useful interpretations of wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances do not avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. c++ code is provided to facilitate applications of winding spiral waves.