Model and Wave Equations

As in Pilipp et al. (1987) we consider small-amplitude waves in a static uniform molecular cloud consisting of neutral atomic and molecular species, the ionized atomic and molecular species, the electrons, neutral dust grains, and negatively charged dust grains. The cloud is assumed cold, so that the gas pressures of all the species may be neglected. A 4-fluid model of the plasma is used, which employs the linearized fluid momentum equations for plasma ions (singly charged), electrons, neutral molecules and charged dust grains: (We neglect the motion of neutral dust grains that was included by Pilipp et al. (1987))
    
where is the wave electric field, is the species mass and is the species velocity in the wave. is the collision frequency of a particle of species s with the particles of species t. We have neglected electron inertia and momentum exchange between ions and electrons, but have included ion and neutral molecule inertia terms because we are mainly interested in the frequency regime above the dust cyclotron frequency, where the ion and neutral molecule dynamics are important.

To complete the system of equations, Maxwell's equations ignoring the displacement current are used, with the conduction current density given by

where equilibrium charge neutrality is expressed by (1).

The background magnetic field is assumed to be in the z-direction, and the steady electron, ion and dust densities are , and . The parameter measures the charge imbalance in the plasma, with the remainder of the charge residing on the dust particles, so that the total system is charge neutral. The equations are linearized, so that since we can define the transverse direction of wave field variation to be the x-axis, without loss of generality the wave fields are assumed to vary as . Thus our analysis differs from that of Pilipp et al. (1987) in that oblique propagation (i.e. non-zero ) is allowed for. We assume for simplicity that the charge on the dust particles is not affected by the wave, i.e. we neglect the dust charging effects discussed by Vladimirov (1994a,b).

Define the Alfvén speed , where , the ion-cyclotron frequency and the dust-cyclotron frequency . Eliminating and , and using the assumed time dependence, we obtain, for ,
    
where is the wave magnetic field, and . The neglect of gas pressure in all species implies that .

© Copyright Astronomical Society of Australia 1997