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Astron. Astrophys. 331, 121-146 (1998)

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1. Introduction

Chondrules, a type of meteoritic inclusion, are small ([FORMULA]) beads of glassy rock having structures that imply that they cooled from temperatures of 1700 K on timescales of tens of minutes to hours (Fujii & Miyamoto 1983; Hewins 1983). (For a review of models proposed for chondrule formation and an evaluation of these models see Grossman, 1988; Levy, 1988). The various flash heating mechanisms that have been invoked to account for chondrule structures include: impacts (Wasson 1972; Kieffer 1975); aerodynamic drag on dust accreting onto the protosolar nebula (Wood 1984; Hood & Horanyi 1991, 1993; Ruzmaikina & Ip 1994); dissipation of shear in the nebula (Wood 1986); magnetic flares (Sonett 1979, Levy & Araki 1989); absorption of pulsed radiation of unspecified origin (Eisenhour & Buseck 1995); absorption of solar radiation as material is dragged out of the disk by a magneto-centrifugally driven wind and before it falls back to the disk (Shu et al. 1996).

The possibility that lightning occurred in the protosolar nebula and was the source of chondrule heating has also been considered (Whipple 1966; Cameron 1966; Levy 1988; Morfill & Sterzik 1990; Pilipp et al. 1992; Morfill et al. 1993; Horanyi et al. 1995; Love et al. 1995). However, the only detailed calculations of the electric field for relevant gas and dust grain number densities and specified fluid dynamics and gas phase charge and grain charge conditions were restricted to small amplitude acoustic waves (Pilipp et al. 1992). The volume associated with an acoustic wave having an electric field strong enough to trigger discharge was found to be too small to contain sufficient energy to ionize a discharge channel. According to estimates by Pilipp et al. lightning in the protosolar nebula could give rise to significant heating and ionization in the discharge channels only if it were generated on a more global scale, i.e. if the spatial extent of a region in which the electric field is strong enough to induce discharges is larger than [FORMULA] where [FORMULA] is the number density of neutral molecules of the [FORMULA] gas. Morfill & Sterzik (1990) and Morfill et al. (1993) have suggested that global electric fields of sufficient strengths could have been established by gravitational sedimentation of dust particles and size sorting due to gas drag on these particles.

Gibbard et al. (1997) have recently performed detailed calculations for the growth and charging of dust grains (i.e. ice particles) due to collisions between the grains in the presence of turbulent gas motion in the chondrule - formation region of the protoplanetary nebula. The electrical conductivity of the nebula was estimated self-consistently under the assumption that radioactive decay of [FORMULA] induced ionization in the gas. In their model grain - grain collisions give rise to the growth of grains up to a maximum size limited by assumption to [FORMULA], and most of the mass ends up in mm - sized grains with the mass in smaller sized grains being small. Gas phase ions and electrons are removed mainly in collisions with mm - sized grains. The grain - grain charge transfer rate was calculated from a formula based on experimental results and the time scale for the evolution of the electric field is determined by the currents due to movements of charged grains as they descend under the force of gravity and by the currents of gas phase ions and electrons driven by the electric field. Whereas a similar model applied to terrestrial thunderstorm conditions results in the prediction of the growth of the terrestrial electric field to its breakdown value for lightning discharge on a realistic time scale, the model indicates that precipitation - induced lightning cannot occur in the protosolar nebula.

Gibbard et al. construct a self-consistent model, incorporating particle growth by grain-grain collisions in turbulent gas. The resulting particle-size spectrum turns out to be deficient in small grains as compared with what is observed in chondritic meteorites, where the mass of fine-grained matrix is at most a factor of a few smaller than the mass of the chondrules themselves (Scott et al. 1988). The origin of this discrepancy is not clear. On the one hand, the actual coagulation process in the nebula may have left behind a large quantity of fine-grained material for reasons not captured in the coagulation model, or collisionally induced fracturing may have returned a large quantity of fine particles to the mix. On the other hand, it may be that meteorite-matrix material had indeed been coagulated into larger - perhaps chondrule sized - accumulations before being incorporated into the meteorite parents, in agreement with the coagulation-model results, and simply avoided being melted in a chondrule-producing heating event. In any case, the particle-size distribution is a matter of some significance inasmuch as it is collisions between large and small particles that produces electric charge separation, and it is collisions on grains - mainly the smallest grains, which present the greatest surface area per unit mass - that removes free electrons and ions from the gas. Thus a larger abundance of small grains produces conditions more favorable for the production of strong electric fields.

In this paper we explore the possibility that lightning occurred in the protosolar nebula on the assumption that in many cases (but not all) the micron and/or submicron sized particles contained about half or a fifth of the solid mass. In addition, for the grain - grain charge transfer rate we do not use an expression based on experimental results as did Gibbard et al. Instead we adopt an expression showing clearly the relative importance of the Elster-Geitel mechanism (a particular charge transfer process occurring when the grains' surface charge distributions are polarized by a large scale electric field) and non-inductive charge transfer (e.g. that due to different work functions for the material of the colliding grains). The amount of charge transferred per collision by the non-inductive process depends on two free parameters. In addition, we assume in many cases that cosmic ray induced ionization was negligible and take the decay of [FORMULA] as the only other source of ionization.

We present the results of calculations for the spatial dependence of the steady electric field arising in a medium with gas and dust number densities that may have obtained in the protosolar nebula at positions between Earth and Jupiter and experiencing largescale neutral gas motions antiparallel to an effective gravitational field.

Three sets of models were considered. One set is appropriate for steady convective flow perpendicular to the disk midplane at roughly the Earth's present orbit while another set is applicable to such flow at roughly Jupiter's present orbit. A third set of models is for radial flows in a subdisk with a dust - to - gas mass ratio that is enhanced by a factor of 100 over that of a typical interstellar cloud which we adopted for the other models; in such a subdisk the grains are subject (in the frame corotating with the neutral gas) to a radially inwardly directed effective gravity comparable to the radial component of the gradient of the gas thermal pressure divided by the gas mass density. Ideally, we would like to construct multidimensional time-dependent models of the electric field evolution; however, in order to maintain computational simplicity we have restricted ourselves to one independent variable which we have taken to be a spatial variable (rather than time as it would be in a one point calculation) because, as mentioned above, the spatial extent of the region in which the electric field is strong enough to induce discharge determines whether a discharge channel is heated significantly.

Each model is a five-fluid model with the fluids consisting of neutrals (the dynamics of which are specified), gas phase ions, gas phase electrons, "big" grains identical to one another with each carrying the mean charge of a big grain, and "small" grains identical to one another except for the charge carried; the small grains are divided into subfluids in which each grain carries the same charge so that the charge distribution on the small grains could be calculated. While big grain - small grain collisions are assumed to transfer charge we have assumed that no coagulation or fracturing of grains occurs. The gas phase and grain charge conditions were calculated along with the largescale electric field and fluid densities and velocities in a self-consistent manner for various assumed gas phase ionization rates and grain charging rates.

In Sect. 2 we give the fluid and electrostatic equations on which the model is based. A consideration of the rates at which momentum transfer and charge transfer between species occurs is presented in Sect. 3. Sect. 4 contains numerical results of our model calculations. In Sect. 5 we discuss results, and in Sect. 6 we give conclusions on whether lightning may have been the source of chondrule heating.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998