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Astron. Astrophys. 351, 192-194 (1999)

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2. Ambipolar diffusion induced ionisation

[FORMULA] is used to signify the magnitude of the magnetic force per unit volume. In steady state,

[EQUATION]

where inertial terms have been neglected and [FORMULA] is the force transferred to ions per unit volume per unit time due to elastic collisions with neutrals, and is given by Eq. (3.1) in Draine (1986) if the [FORMULA] fluid and the [FORMULA] fluid consist of ions and neutrals respectively. We consider a nonmolecular pure hydrogen medium. For similar assumptions the ion temperature is given by

[EQUATION]

where [FORMULA] is the energy transferred to the ion fluid per unit volume per unit time due to elastic collisions and is given by Eq. (3.7) in Draine (1986).

In ionisation equilibrium

[EQUATION]

where [FORMULA] is a background ionisation rate assumed so that for low values of [FORMULA] some ionisation occurs, [FORMULA] and [FORMULA] are the number densities of [FORMULA] and [FORMULA] respectively, [FORMULA] is the recombination rate coefficient, and [FORMULA] is the ionisation rate coefficient due to ambipolar diffusion given by Eqs. (25) through (27) and (29) of Draine et al. (1983). We assumed [FORMULA], where [FORMULA] and [FORMULA] are the neutral and electron temperatures. In the evaluation of [FORMULA] and [FORMULA], we used expressions for the elastic scattering cross section for low and high ion-neutral relative speeds, as recommended by Draine (1986). In fact, over the range of calculations for which we present results, errors of only a few per cent were found when we used the high-speed form of the elastic cross sections at all relative speeds.

Fig. 1 presents results for [FORMULA], the magnitude of the mean relative velocity between ions and neutrals, as a function of [FORMULA]. Fig. 2 contains plots of [FORMULA], the fractional ionisation, as a function of [FORMULA]. In each figure, the curves are labelled with the number density of neutrals, [FORMULA]. As mentioned above, we assumed values for [FORMULA] and [FORMULA]. They are probably higher than would be appropriate where [FORMULA] is very low; then at most they should most likely be several thousand degrees, as for low [FORMULA] the material in a disk is probably mostly molecular and cooling due to dissociation will limit the neutral temperature to about [FORMULA]. For low values of [FORMULA], we have consequently somewhat overestimated [FORMULA]. When [FORMULA], the neutral temperature is close to the ion temperature we have calculated, whereas in reality the ion temperature is near the electron temperature which cooling by line emission keeps near [FORMULA]. Thus, because of the symmetry of [FORMULA] and [FORMULA] for ions and neutrals of the same mass, we have probably made good estimates of [FORMULA] for very high values of [FORMULA]. In any case, the point that [FORMULA] goes up by many orders of magnitude but [FORMULA] only increases roughly from [FORMULA] to [FORMULA] as [FORMULA] changes by many orders of magnitude is secure.

[FIGURE] Fig. 1. Ion-Neutral Relative Speed

[FIGURE] Fig. 2. Fractional Ionisation

To get a reasonable approximation for the value of [FORMULA] required for ambipolar diffusion to induce sufficient ionisation that [FORMULA], one may take [FORMULA] where

[EQUATION]

The magnetic force will have a magnitude given roughly by twice the magnetic pressure divided by the length scale associated with changes in the field. This yields

[EQUATION]

The value of [FORMULA] required to induce sufficient ionisation that [FORMULA] would be lowered by an order of magnitude if the critical velocity ionisation phenomena proposed by Alfvén (1954, 1960) were to operate efficiently. This effect has been studied in the laboratory (e.g. Piel et al. 1980) and observed in near-Earth space experiments (e.g. Deehr et al. 1982). However, Formisano et al. (1982) have suggested that in cases in which the ion gyrofrequency is large compared to the ionisation frequency, given by their Eq. (4), the fraction of an ion's kinetic energy that goes into the production of ions via the critical velocity ionisation mechanism is small. In a region where [FORMULA] and [FORMULA], the critical velocity ionisation mechanism can be neglected if Formisano et al. (1982) are correct.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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