Astron. Astrophys. 331, 121-146 (1998)
Appendix A: the timescales for the generation of the electric field and for the attainment of ionization equilibrium
We define
![[EQUATION]](img478.gif)
so that Eq. (10) is
![[EQUATION]](img479.gif)
Using (20b) and Eqs. (6) through (9) we find
![[EQUATION]](img480.gif)
with
![[EQUATION]](img481.gif)
Clearly, in the absence of charged grains the e - folding growth
timescale of the electric field would be given
by
![[EQUATION]](img482.gif)
We also define
![[EQUATION]](img483.gif)
and
![[EQUATION]](img484.gif)
Then from (A4) to (A7)
![[EQUATION]](img485.gif)
and each of , , and
is an upper bound to
.
The following estimates are based on various conditions which are
discussed in Appendix C.
To estimate , , and
we must estimate ,
, , and
. We find from (1) and (2b) and evaluations of
's that to a good approximation for cases that
we have studied
![[EQUATION]](img491.gif)
and
![[EQUATION]](img492.gif)
and that the timescale for and
to reach local steady state equilibrium values
are less than about
![[EQUATION]](img493.gif)
![[EQUATION]](img494.gif)
respectively. and
are the average specific mass densities inside the small and big
grains.
From Eqs. (4), (5), (15), and assuming as
done to derive (15), we find
![[EQUATION]](img496.gif)
with
![[EQUATION]](img497.gif)
Taking only the first and second term on the right hand side of Eq.
(A13) into account and neglecting all the other terms, we get in the
case of local equilibrium
![[EQUATION]](img498.gif)
In fact, as will be discussed in more detail in Appendix C, (A15)
is a good approximation for our 1 A.U. standard parameters and
standard parameters for the dust enriched subdisk. For our 5 A.U.
standard parameters the term may be a
significant fraction of the term decreasing
signicantly as compared to approximation
(A15), but even in this case (A15) can still be considered to be an
order of magnitude estimate.
Approximating and estimating
from Eq. (8) with we
find
![[EQUATION]](img503.gif)
From (3) and the evaluations of the 's and
's we see that for cases in which parameters
differ little from the standard ones once equilibrium is reached
![[EQUATION]](img506.gif)
and small grains carrying charges with a magnitude larger than one
elementary charge can be neglected. Approximation (A17) is a good
approximation for our 1 A.U. standard parameters and for our
standard parameters for the dust enriched subdisk. For our 5 A.U.
standard parameters (A17) is less accurate but still an order of
magnitude estimate. The local equilibrium distribution of small grain
charges is reached on a timescale of
![[EQUATION]](img507.gif)
With the use of the local equilibrium values of
, ,
, , and
and expressions for
and evaluated under the assumption that the
motions of the dust grains relative to the neutrals are very
subthermal we obtain the following approximations which are reliable
for parameters near the standard ones.
![[EQUATION]](img512.gif)
where A is about 0.3 when and
and becomes larger for E and
becoming smaller.
![[EQUATION]](img515.gif)
and
![[EQUATION]](img516.gif)
Conditions on which approximations (A19) to (A21) are based are
discussed in more detail in Appendix C. These conditions are fulfilled
only if local charge equilibrium prevails. The timescale
, being an upper bound to
, can be approximated by (A19) if the gas phase
ions and electrons have reached local equilibrium for their number
densities even if the grains are not in local charge equilibrium. This
condition should be fulfilled if the amount of time which has passed
during the evolution of a dynamic structure is large compared to
as defined and approximated by (A11), or in
our stationary models, for lengthscales z large compared to
with . (For electrons
we have used the approximation of local equilibrium anyway, i.e. Eq.
(2b) in place of (2a)). Analogously, the timescale
, being an upper bound to
, can be approximated by (A20) if the big
grains have reached local charge equilibrium even if the small grains
are not in local charge equilibrium. This condition should be
fulfilled for lengthscales z large compared to
. Finally, the timescale
can be approximated by (A21) only if the small
grains have reached local charge equilibrium, which should occur only
at distances large compared to . In fact,
because local charge equilibrium for the small grains is reached on
lengthscales greater than the thickness of the protosolar nebula for
many sets of parameters and because local equilibrium electric fields
are usually weaker than those at which discharge is induced, the
's in many storage volumes capable of powering
discharges that could have formed chondrules cannot be estimated by
(A8) if is approximated by (A21). However,
approximation (A21) for the timescale is
useful for the interpretation of physical processes causing the
variation of the local equilibrium electric field with various
parameters.
Appendix B: upper bounds and the equilibrium values of the electric field
For the electric field E as a function of z we have
at the position z where E
reaches a peak value, and also at sufficiently large distances
z where the medium reaches local equilibrium. Taking
, (A2) and (A3) give for these cases the
electric field strength
![[EQUATION]](img523.gif)
For the adopted standard values of relevant parameters grains carry
far more of the charge than gas phase ions or electrons do, except for
a small region near the boundary in z
- space. Thus, (A1) and (A2) imply that for
![[EQUATION]](img524.gif)
implies together with (B1) and (B2) that at
positions where
![[EQUATION]](img526.gif)
In fact, (B3) holds approximately whenever
and we will use (B3) as an approximation for
the electric field variation for distances .
Using approximation (A19) we get ,
, , and
for Figs. 1, 3, 5, and 6, respectively.
According to (A8) we have , which yields
with (B3)
![[EQUATION]](img535.gif)
where is defined as
![[EQUATION]](img536.gif)
We define
![[EQUATION]](img537.gif)
For our standard parameters we have .
According to (A15) we have in case of local
charge equilibrium and for our standard parameters. Inserting (B6) and
(B5) into (B4) yields
![[EQUATION]](img540.gif)
When local equilibrium obtains, and for values of the parameters
near the standard ones that we have adopted
may be estimated by as given by approximation
(A21) and may be estimated by (A15). Hence,
(B3) yields a local equilibrium electric field strength which is
roughly
![[EQUATION]](img541.gif)
(B8) holds only for a restricted range of values for the relevant
parameters. For parameter values significantly different from our
standard ones, may not be estimated by
but possibly better by
or , and conditions on which (A19) to (A21) are
based may not be fulfilled. Conditions on which approximations (A19)
to (A21) are based and also conditions which allow the estimation of
by are discussed in
more detail in Appendix C.
Appendix C: conditions for the approximations for time scales and equilibrium values
We discuss in more detail the conditions on which approximations in
Appendices A and B are based.
The motions of dust grains relative to neutrals are assumed to be
very subthermal, i.e.
![[EQUATION]](img542.gif)
with d = b, sk, which is well fulfilled for
our standard parameters, so that approximation
![[EQUATION]](img543.gif)
for d = b, s can be used (see Draine, 1986).
For the big grains the magnitude of the electric force is assumed
to be small compared to the magnitude of the gravitational force, i.e.
![[EQUATION]](img544.gif)
or
![[EQUATION]](img545.gif)
so that with (8)
![[EQUATION]](img546.gif)
For our standard parameters and for a local equilibrium for which
, condition (C3) is fairly well fulfilled. If
local equilibrium does not obtain condition (C3) is still barely
fulfilled for our standard parameters. We also assume
![[EQUATION]](img548.gif)
With (C2), (8), (9), and assumption (C5)
gives
![[EQUATION]](img550.gif)
or
![[EQUATION]](img551.gif)
(C7) is fulfilled in the case of a local equilibrium for which
charged dust grains with dominate the
electrical conductivity of the medium and . For
and condition (C7) is
fairly well fulfilled for our standard 1 A.U. and 5 A.U.
parameters but barely fulfilled for our standard parameters for the
dust enriched subdisk.
We evaluate the 's with
and from results given
by Havnes et al. (1987). For the electrons we assume
![[EQUATION]](img556.gif)
which is equivalent with
![[EQUATION]](img557.gif)
if (C1) together with is fulfilled.
Condition (C8b) is fulfilled for an electric field strength
as is the case for our standard parameters for
an electric field strength up to since
grows sufficiently fast with the electric field
strength up to . A condition for the ions
analogous to (C8b) is fulfilled for for which
. This is the case for our standard parameters
for local equilibrium. However, for an electric field strength
relevant for inducing lightning, i.e. for , the
condition for ions analogous to (C8b) is strongly violated since
for an electric field strength up to
and we do not require this condition.
For an estimate of with j = i, e, we
assume
![[EQUATION]](img567.gif)
or
![[EQUATION]](img568.gif)
at least for . For the electrons conditions
(C9) and (C10) are usually fulfilled. For the ions and for
they are usually well fulfilled only for the
1 A.U. standard parameters but may be barely fulfilled for our
standard 5 A.U. and dust enriched subdisk parameters due to the
low ion temperature there. For the big grains we proceed from the
assumption
![[EQUATION]](img570.gif)
or
![[EQUATION]](img571.gif)
We find from our numerical results that condition (C12) is well
fulfilled for our standard parameters, at least for distances
. We assume that
![[EQUATION]](img572.gif)
![[EQUATION]](img573.gif)
which is consistent with assumption (C9) and
and with our standard parameters. For local
equilibrium, (1) and (C14) yield (A9), and (2b) and (C13) yield
(A10).
We now consider the conditions under which we can neglect the terms
and in (A13) compared
to .
From (A10) and evaluating and
with (C8a) through (C12) we get
![[EQUATION]](img579.gif)
with
![[EQUATION]](img580.gif)
and
![[EQUATION]](img581.gif)
According to our calculations of the electron temperature,
increases with E from
at up to
at , implying with
(C17) a variation of with E. For local
equilibrium we have for our standard
1 A.U. parameters, for our standard
5 A.U. parameters, and for our standard
parameters for the dust enriched subdisk so that we find from (C17)
; ; and
for our standard 1 A.U., 5 A.U., and
dust enriched subdisk parameters, respectively. Taking
as derived from approximation (A15), we find
that the factor in (C15) is 0.15; 0.47; and
0.44, respectively.
Requiring
![[EQUATION]](img594.gif)
we get with (C1), (C4), (C5), definition (A14) and approximation
(C15) that
![[EQUATION]](img595.gif)
Requiring
![[EQUATION]](img596.gif)
we get with (C1) through (C7) and with (C15) that
![[EQUATION]](img597.gif)
In addition, in (A13) we can neglect
compared to if
![[EQUATION]](img599.gif)
With (A9) we get that
![[EQUATION]](img600.gif)
With (C11) and (C23) we estimate that
![[EQUATION]](img601.gif)
If
![[EQUATION]](img602.gif)
which with (C15) is equivalent with
![[EQUATION]](img603.gif)
then condition (C22) is fulfilled. From (C17) and
we see that for parameters with values near
the standard ones (C26) is fulfilled if can be
approximated by (A15) which is justified below.
In order to arrive at approximation (A15) we neglect the term
in (A13), requiring that
![[EQUATION]](img605.gif)
Using definition (A14), approximation (C15) and assumptions (C1)
through (C7) we get from (C27) that
![[EQUATION]](img606.gif)
From our numerical results we find that conditions (C19), (C21) and
(C26) are fairly well fulfilled for our standard parameters, at least
for distances . The left hand sides are usually
lower than the corresponding right hand sides by an order of magnitude
or more, except for (C19) in the case of the standard 1 A.U.
parameters (corresponding to the upper curve in Fig. 1) where E
is lower than the right hand side of (C19) only by a factor 3 near the
peak of the electric field and except for (C21) in the case of the
standard parameters for the dust enriched subdisk (corresponding to
Fig. 6) where the left hand side is lower than the right hand side
only by a factor less than 2 for a large distance range. Condition
(C28) is well fulfilled for our standard 1 A.U. and dust enriched
subdisk parameters. For our standard 5 A.U. parameters it is also
well fulfilled for distances up to and
somewhat beyond where the medium has not yet reached charge
equilibrium but is only barely fulfilled for local equilibrium.
In order to find approximation (A17) we consider an estimate for
the frequencies . From (A13), (C18), (C20), and
(C22) we get for local equilibrium that
![[EQUATION]](img607.gif)
(C29) gives a somewhat more accurate estimate for
than (A15) where the term
decreases as compared
to approximation (A15) due to electron impact on the big grains. In
addition, we require that
![[EQUATION]](img609.gif)
which with (C29) we see is fulfilled if (C18) is fulfilled.
Requirement (C30) implies that we can neglect electrostatic
polarization of grain material in (16b), (17b), and (18) so that
(16a), (16b), and (18) yield with (C30)
![[EQUATION]](img610.gif)
From (C31), (16a), (C29), (C15), (A14), and (C1) through (C7),
![[EQUATION]](img611.gif)
That is, for parameters near the standard ones and for local
equilibrium the average charge carried away by a small grain in a
collision with a big grain is negative and its magnitude is small
compared to one elementary charge.
We require now
![[EQUATION]](img612.gif)
which yields together with (C1) through (C4) that
![[EQUATION]](img613.gif)
Condition (C34), which is well fulfilled for our standard
1 A.U. and dust enriched subdisk parameters and barely fulfilled
for our standard 5 A.U. parameters, implies with (C32) that
![[EQUATION]](img614.gif)
Although (C34) is only barely fulfilled for our standard
5 A.U. parameters condition (C35) holds also in this case.
From (C31), (C35) and definitions of given
in Sect. 3 we get , ,
and with (17a) and (17c) ,
. Then, we get from (19b) immediately that for
or
![[EQUATION]](img622.gif)
and from (19a) with (16a), (C29), (C30), (A14), and (C5)
![[EQUATION]](img623.gif)
![[EQUATION]](img624.gif)
From (C37) and (C38) we get that
![[EQUATION]](img625.gif)
Eq. (3) gives for local equilibrium and for
or together with (C36), (C38) and (C39)
![[EQUATION]](img628.gif)
where is the Kronecker
and where . With (A9),
(A10), and (C13) we get that
![[EQUATION]](img632.gif)
implying with (C33)
![[EQUATION]](img633.gif)
Finally, we assume that
![[EQUATION]](img634.gif)
and
![[EQUATION]](img635.gif)
Inserting (A9), (A10), (C13), (C42) to (C44) into (C40) we get that
![[EQUATION]](img636.gif)
(C43), (C44), and (C45) yield (A17).
Approximation (A19) is based on (A9) and (A10) and neglect of
compared to which is
correct within an order of magnitude for
according to (C16). (A19) should be valid for a wide range of
parameter values within an order of magnitude, at least for electric
field strengths E varying from 0.3 to
and where A varies from about 0.8 to
0.3.
Besides the conditions detailed above on which (A19) to (A21) are
based, Eq. (B8) is based also on the assumption
which holds only if the following conditions
are both fulfilled
![[EQUATION]](img643.gif)
![[EQUATION]](img644.gif)
or, with the use of (A19) to (A21),
![[EQUATION]](img645.gif)
and
![[EQUATION]](img646.gif)
For example, for local equilibrium and for our standard parameter
values except for which is varied, condition
(C48) is only valid if ,
, and for the parameter
values given in column 1, column 2 and column 3 of Table 1,
respectively. For larger values of ,
in (B3) should be approximated by
as given by (A20) yielding a local equilibrium
electric field strength given by (B7) if " "
is replaced by " ".
![[TABLE]](img652.gif)
Table 2. List of symbols
![[TABLE]](img653.gif)
Table 2. (continued)
![[TABLE]](img654.gif)
Table 2. (continued)
© European Southern Observatory (ESO) 1998
Online publication: February 4, 1998
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