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Astron. Astrophys. 330, 1070-1076 (1998)

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3. Governing equations

In the context of the magnetospheres of YSO as protostellar class I objects and T-Tauri stars one has to deal with partially ionized dusty magnetoplasmas characterized by a dynamical rather than static dust component that plays an outstanding role for the overall dynamics. We consider a frame comoving with the neutral component ([FORMULA]). Thus, the relevant fluid balance equations (cf. Birk et al. 1996, Shukla et al. 1997 for slightly different approximations) that describe the low-frequency (with respect to the dust gyro-frequency) dynamics of such multispecies quasineutral plasmas in ionization equilibrium are the continuity equations of the charged components (electrons, ions and dust):

[EQUATION]

where [FORMULA] and [FORMULA] are the mass density and the fluid velocity and the indices e, i and d denote the electron, ion and dust component, the total momentum balance equation of the charged fluids (electrons and ions are assumed to be inertialess as compared to the dust component):

[EQUATION]

where p, [FORMULA], [FORMULA] and c denote the thermal pressure, the magnetic field, the collision frequencies of collisions between species [FORMULA] and [FORMULA] (elastic collisions as well as charge exchange) and the velocity of light, and the energy equations as, e.g., the dust energy equation (the other energy equations can be derived accordingly):

[EQUATION]

where [FORMULA], m, [FORMULA], [FORMULA], and [FORMULA] denote the ratio of specific heats, the particle mass, the collisional resistivity, the current density and the Boltzmann constant, respectively. These equations have to be completed by Ohm's law or an induction equation that governs the dynamical evolution of the magnetic field. It can be derived (cf. Birk et al. 1996) from the inertialess ion momentum equation

[EQUATION]

[EQUATION]

where [FORMULA] is the electric field and e the elementary charge. We neglect the Hall-like as well as the pressure term and obtain

[EQUATION]

with the constant collisional resistivity [FORMULA] which may alternatively be replaced by a turbulent resistivity caused by microinstabilities.

In deriving Eq. (5) we have used Faraday's law

[EQUATION]

and Amperère's law

[EQUATION]

where [FORMULA] is the dust charge number. Additionally, we assumed that the dynamical friction is dominated by dust-neutral collisions, i.e. we are dealing with a plasma where the dust component is important for the overall dynamics rather than only an impurity effect.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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