On Types of Velocity Distribution and Stability of Disc Galaxies

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Abstract
The local stability analysis of rotating sheets due to Toomre is generalized for all elliptically-symmetric velocity distributions. The velocity distribution of least energy for which a disc is stable to axisymmetric disturbances of all wavelengths is a δ -function, i.e. one in which all Lindblad epicycles are equal; it has about 20 per cent less energy than the Schwarschild distribution which has just sufficient energy for stability. It is suggested that an instability in the galaxy produces a more stable velocity distribution, typically one having fewer stars with small peculiar velocities. This should be observable in classes of old stars. If a large proportion of mass has small peculiar velocities, then the total dispersive energy required to produce stability is large, especially at short wavelengths. A velocity distribution consisting of three components, and which approximates to that of our galaxy, could explain the spiral arm spacing. The components are (i) 75 per cent by mass of old stars with high velocity dispersion, (ii) 20 per cent by mass of stars and gas clouds with low-medium velocity dispersion, and (iii) 5 per cent by mass of clouds with very low velocity dispersion. A velocity distribution consisting of four components, which might be considered to be an evolution of that with three, due to the decaying dispersion of the gas clouds, has a most unstable wavelength which could explain the spiral arm spacing in Andromeda.

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