Two-fluid gravitational instabilities in a galactic disk

Abstract

We forlulate and solve the hydrodynamic equations describing an azimuthally symmetric galactic disk as a. two-fluid system. The stars and the gas are treated as two different isothermal fluids of different velocity dispersions (C_s >> C_g ), which interact gravitationally with each other. The disk is supported by rotation and random motion. The formulation of the equations closely follows the one-fluid treatment by Toomre. We solve the lineraized perturbation equations by the method of modes, and study the stability of the galactic disk agaInst the growth of axisymmetric two-fluid gravitational instabilities. We find that even when both the fluids in a two-fluid system are separately stable, the joint two-fluid system, because of the gravitational interaction between the two fluids, may be unstable. The ratio of the gas contnbution to the stellar contribution toward the formation of two-fluid instabilities is substantially greater than μ_g/μ_s, the ratio of their respective surface densities; this is due to the lower gas velocity dispersion as compared to the stellar velocity dispersion (C_g << C_s). The two contributions are comparable when the gas fraction (μ_g/μ_s) IS only ~ 0.10-0.25. Therefore, a galactic disk is a meaningful two-fluid system even when the gas constitutes only 1O %-20 % of the total surface density. The ratio of the amplitude in the gas to the amplItude In the stars is an increasIng function of the wavenumber of the two-fluid perturbation. The wavelength and the time of growth of a typical two-fluid instability in the inner galaxy, for μ_g/μ_s = 0.1-0.2, and ~2-3 kpc and ~2-4 × 10⁷ years, respectively, and each of these contains gas of mass 4 × 10⁷-10⁸ M. The two-fluid analysis presented here is applocable to any general disk galaxy consisting of stars and gas.