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A perturber may excite a coherent mode in a star cluster or galaxy. If thestellar system is stable, it is commonly assumed that such a mode will bestrongly damped and therefore of little practical consequence other thanredistributing momentum and energy deposited by the perturber. This paperdemonstrates that this assumption is false; weakly damped modes exist and maypersist long enough to have observable consequences. To do this, a method forinvestigating the dispersion relation for spherical stellar systems and forlocating weakly damped modes in particular is developed and applied to Kingmodels of varying concentration. This leads to the following remarkable result:King models exhibit {\it very} weakly damped $m=1$ modes over a wide range ofconcentration ($0.67\le c\le1.5$ have been examined). The predicted dampingtime is tens to hundreds of crossing times. This mode causes the peak densityto shift from and slowly revolve about the initial center. The existence of themode is supported by n-body simulation. Higher order modes and possibleastronomical consequences are discussed. Weakly damped modes, for example, mayprovide a natural explanation for observed discrepancies between density andkinematic centers in galaxies, the location of velocity cusps due to massiveblack holes, and $m=1$ disturbances of disks embedded in massive halos.Gravitational shocking may excite the $m=1$ mode in globular clusters, whichcould modify their subsequent evolution and displace the positions of exoticremnants.Comment: 19 pages, AAS LaTeX macros v3.0, 10 figures, UMASS-APT-00

Weakly damped modes in star clusters and galaxies