EXTENDED STELLAR HYDRODYNAMICS FOR GALACTIC DISKS

A closed system of velocity moment equations describing the three-dimensional kinematics of a cool galactic disk is developed. Truncation of the infinite hierarchy of moment equations is achieved by demanding that the kurtoses or dimensionless fourth moments of the distribution function match those of the Schwarzschild distribution function to only one higher order in a small expansion parameter (or ratio of dispersion speed to rotation speed) than is forced mathematically. It is shown that unique solutions for the velocity ellipsoid are obtainable everywhere once a potential of any kind is chosen. An essentially model-free analytic expression for the tilting of the velocity ellipsoid at points just above the plane is presented, which is in excellent agreement with all known constraints. A further analytic result is an expression for the nonisothermality of a disk population close to the plane of symmetry, which is also in detailed agreement with observations. In particular, it is shown that for all reasonable potentials the vertical velocity dispersion has a minimum on the plane. More generally, a simple expression is obtained which illustrates the close relationship between the nonisothermality and the tilting of the velocity ellipsoid, and it is also shown that the extended fluid model satisfies exactly the requirements of a diagonal velocity ellipsoid with respect to cylindrical coordinates when the potential is separable in those coordinates. Also provided is an analytical confirmation of the suggestion of van der Kruit that the answer to the long-standing question of isothermal versus exponential galactic disks is a solution intermediate between these two cases. The closed system of equations should prove useful in understanding the behavior of the velocity ellipsoid off the plane of the Milky Way, which in turn is of vital importance in determining such fundamental quantities as the column density and the amount and distribution of dark matter in the solar neighborhood.