Astron. Astrophys. 331, 121-146 (1998)
4. Numerical results
In the protosolar disk the gravitational forces due to the
gravitational field of the Sun are largely balanced by the centrifugal
forces of the orbiting medium. However, in a frame comoving with the
gas of neutral molecules there is an effective gravitational field
balancing the pressure gradients in the gas, i.e. the effective
gravitational field is equal to the gas pressure gradient divided by
the gas density. In this frame the grains are subjected to this
effective gravitational field. For our model calculations we assume
that there is some turbulent motion of the gas superimposed on its
orbiting motion. We also assume that in a frame comoving with the
orbiting gas motion the turbulent gas velocity is constant, at least
within a certain distance range which is considered to be small
compared to the thickness of the protosolar disk and for which we seek
stationary solutions for the electric field. (See Weidenschilling
& Cuzzi 1993 for likely length scales of turbulent motion).
We consider three sets of models. One consists of models for the
protosolar nebula at a distance of 1 A.U. from the Sun and
another is composed of models for the protosolar nebula at a distance
of 5 A.U. (approximately the current radius of the Jovian orbit
around the Sun). For any model belonging to either of these two sets
we have assumed that the effective gravity is directed perpendicular
to and towards the midplane of the protosolar nebula and that the
turbulent neutral gas motion is away from it. The third set consists
of models for the dust enriched subdisk. A dust enriched subdisk may
be sufficiently thin that the effective gravitational force component
perpendicular to the disk is weak relative to the effective
gravitational force component in the radial direction. In models for a
dust enriched subdisk we assume that the dominant component of the
effective gravitational field is towards the Sun and the turbulent
neutral motion is away from the Sun.
In all models, at the following boundary
conditions hold:
![[EQUATION]](img155.gif)
![[EQUATION]](img156.gif)
![[EQUATION]](img157.gif)
At , is specified to be
0 except when we clearly state otherwise. The gas phase ion and
electron number densities at are given by the
solution of Eq. (1) with the derivative set to zero and Eq. (2b),
while is simultaneously determined through the
use of the condition that the total current is zero.
Table 1 gives the model parameters
adopted for models in the three different sets considered.
![[TABLE]](img161.gif)
Table 1. Standard model parameters
The ionization rates given in Table 1
are much lower than those usually assumed for molecular clouds if
ionization is mainly provided by cosmic rays. We assume that during
the formation of planets, the strong T-Tauri wind of the Sun has
prevented ionizing cosmic rays from entering the solar system and that
ionization is only due to decays of long-lived radioactive elements
trapped in the grains (see Umebayashi & Nakano 1990). Thus, the
ionization rates in Table 1 are lower limits.
The turbulent neutral velocity given in
Table 1 is about 5 per cent of the sound velocity
for the first column and about 11 per cent of
the sound velocity for the second and third columns of the table. This
is smaller than convection velocities suggested by early solar disk
models but larger by about one order of magnitude for convective
velocities derived from more recent analyses of convective instability
modes in a rotating disk (for a review see Weidenschilling &
Cuzzi, 1993). The value for the velocity of turbulent gas motion is
very uncertain but could be as high as assumed by us, at least if
extrinsic stirring mechanisms exist.
Since the grain fluxes and
are conserved (as can be derived from Eq. (3)
and according to Eq. (4)) but the bulk grain velocities as well as the
grain number densities or grain mass densities may vary with distance
z, we adopt standard parameters for the grain mass densities at
the boundary. In Table 1, standard parameters for the ratios of
grain mass densities at a distance z to neutral mass density
are expressed as the corresponding adopted ratios at the boundary
times respective ratios and
of grain bulk velocities at the boundary to
respective grain bulk velocities ocurring at distance z.
Fig. 1 gives the electric field as a function of distance for
several different models for conditions at 1 A.U. from the Sun.
The upper curve, which is not marked, is for the standard parameters
given in the first column of Table 1 and the boundary conditions
given above. The curve marked is for standard
parameters, except . The one marked
is for standard parameters except
and at . The curve
marked is for standard parameters except that
and the electron sticking coefficients for
small grains is 0.3 instead of unity. The thin horizontal lines at the
right hand side of the figure indicate the electric field as derived
numerically for the local equilibrium (i.e. all gradients in the
equations and the current J defined by Eq. (20b) are zero) for
each model. Appendix A treats the timescales for the generation of the
electric field and for the attainment of ionization equilibrium while
Appendix B concerns upper bounds and the equilibrium values of the
electric field. Results in those appendices are relevant to the
understanding of the numerical results displayed in Fig. 1 and other
figures. Timescales are converted to the appropriate lengthscales by
multiplication with . (This is exact for the
generation of electric field and is approximately true for attainment
of ionization equilibrium of dust particles where
, but does not necessarily hold for attainment
of ionization equilibrium of ions and electrons because
and deviate
drastically from for large magnitudes of the
electric field. Only in case of the three marked curves in Fig. 1
corresponding to , the ion and electron bulk
velocities stay near due to the very low
electric field strengths. Note that in case of the curve marked
the time scale as
given by (A12) must be multiplied by the factor
since in the Appendices all sticking
coefficients are assumed to be unity but lower electron sticking
factors for small grains yield lower recombination rates for electrons
increasing ).
![[FIGURE]](img35.gif) |
Fig. 1. The Electric Field as a Function of Distance for Models Appropriate for a Position at 1 A.U. from the Sun. Unless noted otherwise all parameters have the standard values given in the first column of Table 1 and the standard boundary conditions apply. The curves marked , and are for models with . For the case at . For the case the electron sticking coefficient on small grains was taken to be 0.3 rather than unity. Dashed portions of the curve indicate that the electric field is negative. . The thin horizontal lines at the right hand side of the figure indicate the electric field for the local equilibrium
|
The physical processes operating to produce the electric field
variations can be described as follows:
As argued in Appendix A the total current J can be
decomposed into a sum of several partial currents: 1) the convection
current , where is the
total space charge density (Q is defined by (A1)), 2) the
current due to the movement of the charged big
grains as seen in the rest frame of the neutrals which would occur if
the only forces to which the big grains were subjected were gravity
and friction with the neutrals, 3) the current
due to the movement of charged small grains as seen in the rest frame
of the neutrals which would occur if the only forces to which the
small grains were subjected were gravity and friction with the
neutrals, and 4) the conduction current
containing the response of all charged particles to the external
electric field E with the time scale
defined by (A4).
Right at the boundary (not shown in the
figure where results are presented only for )
we have and .
In the case of the upper curve corresponding to standard parameters
and standard boundary conditions, at the only
charged particles are ions, electrons and few charged small grains
carrying one negative elementary charge. Then, from (A3) it follows
that if then producing
according to Poisson's equation a negative electric field E as
the medium moves to distances . However, at the
same time as the medium moves upward, the big grains are charged
positively in collisions with small grains and small grains colliding
with big grains are charged negatively due to noninductive charge
transfer. This effect results in a positive total charge density for
distances z beyond but for distances
not much larger than . Here, the positive
charge density for big grains and the negative
charge density for small grains are about of
the same magnitude but since is smaller than
by 4 orders of magnitude the positive charge
density of big grains dominates in accordance with (A3). In the
distance range we have roughly a balance
between the convection current and the current
of charged big grains whereas the other
currents are of minor importance.
For distances significantly beyond the
conduction current can no longer be neglected
and eventually leads to a negative charge density at large distances.
For distances near where the electric field reaches maximum and for
distances beyond, variation of the electric field can be approximately
described by the balance between the conduction current
and the current of
charged big grains as given by (B3) (see Appendix B for more details).
In accordance with (A13) to (A16) the positive charge on big grains
increases nearly linearly with z up to a distance
and then approaches a local charge equilibrium
approximated by (A15) (i.e. the gradient in (5) vanishes) at distances
. In fact, the local equilibrium charge of big
grains is slightly lower than given by (A15) due to electron impact
not being taken into account in (A15). Most of small grains hitting
big grains are discharged in the collisions if the big grains carry a
charge near their local equilibrium charge because, in this case, the
value of the average charge carried away by
small grains is small compared to one elementary charge (see also
(18)) but grains can carry only an integral number of elementary
charges. A small fraction of small grains hitting the big grains carry
away one negative elementary charge since is
negative, and there are no small grains carrying away a charge
different from either zero or one negative elementary charge, at least
for electric field strengths not much larger than considered in this
paper for which electrostatic polarization of grain material is
negligible. In this case, small grains are charged positively by ion
impact and they are charged negatively by electron impact and by
collisions with big grains and they are mainly discharged by
collisions with big grains. Thus, the number densities of both
positively and negatively charged small grains increase significantly
with distance up to about but approach local
charge equilibrium (i.e. vanishing gradients in (3)) at distances much
larger than that distance in accordance with (A18).
The decay of the electric field beyond its maximum is caused by the
increase of the charge on big grains and the increase of the number
densities of charged small grains increasing the electrical
conductivity of the medium. Whereas for small distances (i.e. for
) the electrical conductivity is dominated by
gas phase ions and electrons, charged grains contribute the largest
share to the electrical conductivity at the maximum of the electric
field and for distances beyond of it. For distances
small charged grains dominate the electrical
conductivity of the medium and increase of their number densities with
distance dominates the decay of the electric field accordingly. For
local charge equilibrium as it should occur at distances
the electric field eventually reaches its
equilibrium value which has been numerically determined to be about
.
The total current J is negligible compared to the current
although it does not vanish for all distances
in spite of at the boundary. The main reason
for a finite total current is that for the upper curve in Fig. 1 (and
also for the curves in Figs. 3, 4a, 5, and 6) we calculate the
electron number density from (2b) rather than from (2a) as mentioned
in Sect. 2 leading to (20c) rather than to (20a). That is,
is conserved so that
with being the number density of gas phase
electrons at the boundary. For , as should be
the case for , the total current J
increases the electric field according to Eq. (A3). However, for
distances close to and beyond the maximum in E, the total
current is smaller than the sum by two orders
of magnitude or more so that the electric field is only slightly
affected by the finite total current there. Only for small distances,
i.e. , it is found that J is larger than
S, by a factor of order unity.
For the curves marked and
, the boundary conditions are the same as those
for the upper curve, again implying a negative space charge density at
. However, in contrast to the upper curve, the
big grains are charged negatively in collisions with other charged
particles as the medium moves upward resulting in a negative total
charge density for distances z beyond
but not much larger than , and consequently in
a negative electric field. Similarly as for the case of standard
parameters (upper curve), the number density of charged small grains
increases with distance due to ion and electron impact but here the
electrical conductivity is dominated by gas phase ions and electrons
for distances ranging from the boundary up to the distance where the
maximum of the electric field strength occurs and is about constant in
this distance range. Only at distances well beyond the maximum
electric field strength (i.e. for ) do the
charged small grains contribute the largest share to the electrical
conductivity decreasing significantly whereas
contribution of the charged big grains to is
always neglibible. Since here the electric field is in the negative
direction, the effect of the conduction current is to increase space
charge density according to (A3) and eventually produces a positive
space charge density at large distances. The magnitude of the electric
field shown by the curve marked is higher than
that shown by the curve marked because for the
smaller electron sticking coefficients for small grains the
recombination rate of electrons is smaller resulting in a larger
electron number density. As a consequence electron impact on grains is
increased so that big grains for which the electron sticking
coefficient is still assumed to be equal to one gain a negative charge
of larger magnitude implying a larger magnitude of negative space
charge density according to (A3). Finally, the positive electric field
at small distances for the case corresponding to the curve marked
is a consequence of positive charge on big
grains prescribed at . As can be seen from Fig.
1 both curves marked and
end up in the same negative electric field at
distances beyond so that evolution of the
electric field at large distances does not depend on the boundary
conditions.
The three marked curves had to be truncated beyond their maxima at
about since numerical solution of Poisson's
equation proved to be difficult beyond that distance. However, the
local equilibrium electric fields which are expected to occur at
sufficiently large distances were numerically derived also for these
cases. We find a local equilibrium electric field
for and
for . Also here, the
electrical conductivity is dominated by the charged small grains when
local equilibrium obtains. For each of these curves the decay of the
electric field from the respective maximum absolute values near
down to the respective absolute values for
local equilibrium is mainly due to the increase of number densities of
charged small grains as the medium moves upward.
In Fig. 2 results are given for the local equilibrium value of
E (i.e. gradients and the total current J are zero). For
each panel all input parameters but one have their standard
1 A.U. values; the value of this one input parameter varies and
the dependence of the equilibrium value of E on it is explored.
The question of whether local equilibrium might ever obtain arises.
The scaleheight of the protosolar nebula at 1 A.U. was probably
less than and results displayed in Fig. 1 (as
well as in Fig. 3 and additional results shown in Fig. 4a below) are
only meaningful for values of z somewhat less than that
scaleheight and local equilibrium was never obtained if the assumed
boundary conditions are the correct ones. However, it is possible that
a closed cyclic convective flow was maintained for many turnover
times, that the assumed boundary conditions were not the relevant
ones, and that conditions in the flow evolved to approach local
equilibrium. As described in the appendices the longest timescale in
the evolution of the electrodynamic properties of our protosolar
nebula models is the timescale for small grains to reach local charge
equilibrium, and once a particular ion, electron, big grain, and small
grain charge distribution is established the timescale for the growth
of the electric field is very short compared to the turnover
timescale. (In Sect. 5 we will comment on the question whether
electric field strengths strong enough to induce lightning can be
built up for local equilibrium.) As the numerical results given in
Fig. 1 (and in Figs. 3, 5, and 6 and additional results in Fig. 4a)
and considerations presented in the appendices show the largest
electric field strengths obtain if the big grains have had time to
charge up but the small grain charge distribution has not approached
local equilibrium; thus, a particular range of lifetimes of convective
cells would favour the generation of electric fields strong enough to
induce discharge.
![[FIGURE]](img218.gif) |
Fig. 2. Local Equilibrium Values of the Electric Field for Models Appropriate for a Position at 1 A.U. from the Sun. All independent parameters, except the one on an axis in each panel are the standard ones. In each panel, the electric field dependence on the value of the one varied independent parameter is shown. The dashed lines mark the electric field when all independent parameters have the standard values
|
![[FIGURE]](img221.gif) |
Fig. 3. The Electric Field as a Function of Distance for a Further Model Appropriate for a Position at 1 A.U. from the Sun. Results are for cm. All other parameters have the values given in the first column of Table 1, and all boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium
|
As is argued in Appendix B variation of the local equilibrium
electric field with various relevant parameters can be approximated by
(B8) if the parameters are near the standard ones. (B8) results from
the balance between the conduction current and
the current of charged big grains as given by
(B3) if can be approximated by (A15) and
can be approximated by
with given by (A21). The conditions on which
these approximations are based are detailed in Appendix C. The most
important conditions are that 1) the electrical conductivity of the
medium is dominated by charged small grains, the small grains are in
local charge equilibrium and most of the charged small grains carry
one positive or one negative elementary charge so that higher charge
states of the small grains can be neglected and that 2) the charge on
big grains is in local equilibrium and is mainly determined by
noninductive charge exchange and electrostatic relaxation from
collisions between big grains and small grains.
For Fig. 2a, (B8) holds for where the
decrease of E with increasing is due to
(see (A21)). The decrease of
results from the increase of the number
density of charged small grains with increasing
(see(A17)) which in turn results from the
increase of the number densities of gas phase ions and electrons
implying an increased production rate for charged small grains.
(B8) breaks down for and for
because in the lower range of
charged big grains and in the higher range of
it higher charge states of small grains become important for
.
For Fig. 2b, (B8) holds at least for where
the increase of E with increasing is
due to (see (A15)).
(B8) becomes questionable for because the
electric field significantly decreases and, as
a consequence, becomes smaller than given by
(A21). For (B8) breaks down because charged
big grains dominate there, i.e. the electric
field approaches as defined by (B5) where
gravitational forces and electrostatic forces on the big grains
balance.
For Fig. 2c, (B8) holds for the total parameter range considered
where E is about constant due to (see
(A15)) and so that the current of charged big
grains moving downward in the rest frame of neutrals does not vary
with .
For Fig. 2d, (B8) holds for where the
increase of E with increasing is due to
(see (A15)) and (see
(A21)). The increase of with
results from (see also
A7), i.e. from a decrease of mobility of small grains with increasing
size whereas the number density of charged small grains does not vary
with (see (A17)).
(B8) breaks down for because there higher
charge states of small grains become important for
, and for gas phase ion
and electron number densities become so high due to reduction of
recombination on small grain surfaces that gas phase ions start to
dominate the electical conductivity of the medium and, in addition,
electron impact on big grains reduce the positive charge on them
drastically so that decreases with
much faster than predicted by (A15).
As seen from Fig. 2d the local equilibrium electric field strength
is larger for than for
. Thus, Fig. 3 shows results for the dependence
of the electric field when . All other
parameters have the values given in the first column of Table 1,
and all boundary conditions are the standard ones. Also here, the thin
horizontal line at the right hand side of the figure indicates the
electric field for the local equilibrium.
It should be mentioned that for Fig. 3 the ratio of the total
current J to the sum S is much larger than for the upper
curve in Fig. 1 for distances close to and beyond the maximum electric
field, so that E is more strongly affected by the total current
there. This difference results mainly from the fact that if the small
particles are larger, the total current is
increased as a result of the increase of both
and , and that the current of the charged big
grains is lowered as a result of the decrease
of . Our numerical results show that the total
current J is never larger than half the current
and is significantly smaller than the sum
S for distances . The ratio
shows a local maximum of about 0.55 at the
distance and is about 0.3 at
, where E reaches its maximum. That is,
there is a moderate effect of J on E, increasing the
maximum electric field strength by a factor of about 1.3. In summary,
we conclude that increasing the radius of
small grains from the standard value of to
increases the maximum electric field strength
roughly by a factor 2. At large distances, i.e.
, the electric field shown in Fig. 3 approaches
its local equilibrium value as calculated by setting gradients and
J equal to zero (indicated by the thin horizontal line at the
right hand side of the figure).
Fig. 4 gives results for the electric field E for models
appropriate for a position at 1 A.U. from the Sun, when the mean
specific mass density of the big grains is varied but their mass is
kept constant at . Very low mean specific mass
densities correspond to very porous grains. Again, all other input
parameters have their standard values given in the first column of
Table 1, and all boundary conditions are the standard ones. Fig.
4a gives the electric field as a function of distance for specific
mass densities of and of
as indicated at the corresponding curves. The
upper curve, which is not marked, is the same as the upper curve in
Fig. 1, corresponding to standard 1 A.U. parameters
(Table 1). The thin horizontal lines at the right hand side of
the figure indicate the respective electric field for the local
equilibrium.
![[FIGURE]](img267.gif) |
Fig. 4. The Electric Field for Further Models Appropriate for a Position at 1 A.U. from the Sun. The density of the material of which the big grains is composed, , is varied and the radius of the big grains is assumed to be . All other parameters have the values given in the first column of Table 1. a The electric field as a function of distance. The curves marked , , are for models for which the specific mass density of big grains has the indicated value. The upper curve which is not marked is for the standard parameter , (i.e. the same curve as the upper curve in Fig. 1) and is given for comparison. The thin horizontal lines at the right hand side of the figure indicate the respective electric field for the local equilibrium. b Local equilibrium values of the electric field where , is the independent variable. As in Fig. 2 the dashed line marks the electric field for the standard choice of the independent parameter which is
|
Fig. 4b gives results for the local equilibrium value of E
for a larger range of values for the parameter
. For Fig. 4b, (B8) holds nearly for the total
parameter range considered where E is about constant due to
similar reasons as for Fig. 2c, except for
where reduction of by charged big grains
becomes significant.
From Fig. 4a it can be seen that the electric field strength is
generally reduced when decreases. This
reduction of the electric field strength is predominantly a transient
effect which becomes less significant for large distances where the
medium approaches local equilibrium in accordance with Fig. 4b. As can
be seen from Eq. (A16) in Appendix A, the time scale
, which is the time scale for big grains to
reach their equilibrium charge (as given by (A15)) increases with
decreasing when simultaneously
(= constant grain mass). In addition, the
absolute value of the average charge (with
) carried away by the small grains in
collisions with big grains increases when is
reduced relative to its equilibrium value (see Eq. (18)), resulting in
a larger number of charged small grains and a higher electrical
conductivity. These are the two prime effects responsible for the
decrease of the electric field during the transient phase.
As for the upper curve in Fig. 1 the total current is negligible at
least for distances and affects the electric
field by a factor of only order unity for smaller distances.
Fig. 5 displays the electric field as a function of distance for
the conditions that obtained in the protosolar nebula at what is
presently Jupiter's orbit (5 A.U.). The curve is for the standard
5 A.U. parameters listed in Table 1 and the standard
boundary conditions. The scale height of the nebula was probably less
than . The thin horizontal line at the right
hand side of the figure again indicates the electric field for the
local equilibrium.
![[FIGURE]](img276.gif) |
Fig. 5. The Electric Field as a Function of Distance for a Model Appropriate for a Position at 5 A.U. from the Sun. All parameters have the values given in the second column of Table 1. The boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium
|
Here, the effect of the total current on the electric field is
again small at least for distances although it
is larger than for the conditions of Figs. 1 and 4a. It is of order
unity for smaller distances. The ratio of the total current J
to the sum S shows a local maximum of about 0.2 at the distance
and it is about 0.1 at
where E reaches its maximum so that the current J should
increase the maximum electric field by a factor 1.1.
The curve in Fig. 6 gives the electric field as a function of
distance for the standard parameters for the dust enriched subdisk in
Table 1 and standard boundary conditions. The radial extent of
the subdisk may have been as much as almost ,
but its thickness was probably about or less
(Dubrulle et al. 1995). The thin horizontal line at the right hand
side of the figure, as usual, indicates the electric field for the
local equilibrium. Here, the effect of the total current J on
the electric field is negligible everywhere.
![[FIGURE]](img282.gif) |
Fig. 6. The Electric Field as a Function of Distance for a Model Appropriate for a Dusty Subdisk. All parameters have the values given in the third column of Table 1. The boundary conditions are the standard ones. The thin horizontal line at the right hand side of the figure indicates the electric field for the local equilibrium
|
The physical processes operating to produce the electric field
variations as shown in Fig. 3, by the two lower curves in Fig. 4a, and
in Figs. 5 and 6 are qualitatively similar to those discussed for the
upper curve in Fig. 1 although quantitative details are different. As
is the upper curve in Fig. 1, the curves in Figs. 3, 4a and 6 are
plotted up to distances well beyond (i.e. well
beyond for Figs. 1, 3, 4a and
for Fig. 6) where the electric field almost
reaches local equilibrium. However, the curve shown in Fig. 5 is not
plotted up to a distance beyond , which here is
, because numerical solution proved to become
difficult beyond . Therefore, the curve is
still significantly above the local equilibrium value of the electric
field at the largest distances shown in the figure.
Fig. 7 shows the local equilibrium electric field for models
appropriate for a dusty subdisk. The ratio is
the independent variable in panel a, and the ratio
is the independent variable in panel b.
is assumed. All other parameters have the
standard values in column 3 of Table 1.
![[FIGURE]](img293.gif) |
Fig. 7. Local Equilibrium Values of the Electric Field for Models Appropriate for a Dusty Subdisk. In panel a is the independent variable; in panel b is the independent variable. For results in both panels . All other parameters have the standard values given in the third column of Table 1. Just as in Figs. 2 and 4b the dashed lines indicate the result obtained when all input parameters have their standard values
|
For Fig. 7a, (B8) is fulfilled when roughly
, where the increase of E with
increasing results largely from the increase
of the current of charged big grains as
(for ) and, in
addition, when from the increase of
. The increase of is
caused by the increase of the rate of discharging for small grains due
to collisions with big grains with increasing
decreasing the number density of charged small grains in spite of the
fact that also increases with
. However, the increase of
with increasing as
predicted by (A21) becomes neglibible at as,
at the same time, increases nearly as
. In fact, even
slightly decreases with increasing at
probably because there the decrease of
with the increase of the electric field
results in a decrease of the discharging rate for small grains as
compared to what is expected if electric forces are neglected for
determining .
(B8) breaks down for and
where the electric field as shown in Fig. 7a
is larger in the first range and smaller in the second range than that
estimated by (B8). For very small values of
the assumptions for estimating the number densities of charged small
grains by (A17) break down, mainly because here small grains are
discharged also significantly by ion and electron impact in addition
to being discharged by collisions with big grains. For large numbers
of charged big grains dominate
, i.e. the electric field approaches
as defined by (B5).
For Fig. 7b, (B8) is fulfilled when , where
the variation of E observed in the figure results from the
variation of with as
given by (A21) and the assumed variation of
with whereas is
roughly constant.
(B8) breaks down at because there number
densities of gas phase ions and electrons become so large due to their
small recombination rates that the positive charge on big grains is
drastically reduced by electron impact and, at
gas phase ions and electrons dominate the electrical conductivity of
the medium.
© European Southern Observatory (ESO) 1998
Online publication: February 4, 1998
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