On the gravitational stability of a disk of stars

This paper considers the question of the large-scale gravitational stability of an arbitrary, highly flattened stellar system, which is assumed initially to rotate in approximate equilibrium between its self-gravitation and the centrifugal forces. It is concluded that no such disk, if fairly smooth or uniform, can be entirely stable against a tendency to form massive condensations within its own plane, unless the root-mean-square random velocities of its constituents, in the directions parallel to that plane, are everywhere sufficiently large. Lacking such random motions, it is shown that the system must be vulnerable to numerous unstable disturbances, the dimensions of which may approach its over-all radius, and whose times of growth are to be reckoned in fractions of the typical periods of revolution. The minimum root-mean-square radial velocity dispersion required in any one vicinity for the complete suppression of all axisymmetric instabilities is calculated (in collaboration with A. Kalnajs) as 3.36 G /K, where G is the gravitational constant, and and K are the local values of the projected stellar density and the epicyclic frequency, respectively. From that, and the observed j# and K, together with their uncertainties, this minimum for the solar neighborhood of our Galaxy is estimated to fall between 20 and 35 k /sec, a range which indeed encompasses the actual radial velocity dispersions of the most predominant types of stars in our vicinity. It is pointed out that both this curious agreement, and also the well-known discrepancy between the z- and r-velocity dispersions at least of the older disk stars, may be explainable in terms of past instabilities of this galactic disk.