A&A 375, 1075-1081 (2001)
DOI: 10.1051/0004-6361:20010877
G. A. Kovaltsov ^{1,2} - A. F. Barghouty ^{3,4} - L. Kocharov ^{1} - V. M. Ostryakov ^{5} - J. Torsti ^{1}
1 - Space Research Laboratory, Physics Department,
University of Turku, Turku, 20014, Finland
2 -
Ioffe Physical-Technical Institute, St. Petersburg 194021, Russia
3 -
California Institute of Technology, MC 220-47, Pasadena, CA 91125, USA
4 -
Physics Department, Roanoke College, Salem, VA 24153, USA
5 -
St. Petersburg State Technical University, St. Petersburg 195251, Russia
Received 6 April 2001 / Accepted 15 June 2001
Abstract
We examine the energy-dependent rates of charge-changing
processes of energetic Fe ions in the hot plasma of the solar corona.
For ionization of the Fe-ion projectile in collisions with ambient protons,
three different methods of estimating the corresponding
ionization cross sections are presented and compared.
Proton-impact ionization is found to be significant
irrespective of the particular method used.
Differences in the proton-impact ionization cross sections' estimates
are shown to have little effect in calculating highly nonequilibrium Fe
charge states during acceleration, whereas equilibrium charge states
are sensitive to such differences. A parametric study of
the Fe charge-equilibration comprises (i) impact of the ambient plasma
density, and (ii) the energy dependence impact of the acceleration rate
upon the charge-energy profiles and upon the estimated values of
the density
acceleration-time product. We emphasize potential
importance of careful measurements of charge-energy profiles
along with ion energy spectra for determining the energy dependence
of ion acceleration time and the energy dependence of the leaky-box escape time.
Key words: acceleration of particles - atomic processes - Sun: particle emission
Measurements of the energy-dependent charge states of accelerated Fe ions are expected to play a key role in parametrizing sites of acceleration and equilibration processes in solar energetic particle (SEP) events. In a number of recent papers (e.g., Mazur et al. 1999), a history of SEP observational studies is given in addition to a number of relevant and recent references. We would like to recall only the main steps in the theoretical investigation of how the observed charge states of energetic ions are established during SEP events. Luhn & Hovestadt (1987) calculated mean equilibrium charge states, , of energetic ions from carbon through silicon by averaging the cross sections of charge-changing processes over thermal electron distributions in the ion rest frame. In that study, only electron-impact ionization was taken into account and the proton-impact ionization was ignored. In application to helium and oxygen, the proton-impact ionization was incorporated into the SEP calculations by Kharchenko & Ostryakov (1987). After which study, the proton contribution to the ionization processes of solar energetic ions appears to have been ignored for more than a decade (e.g., Ruffolo 1997; Barghouty & Mewaldt 1999). Only recently this important contribution has become part of more elaborate models of SEP acceleration and charge equilibration (Kocharov et al. 2000a; Barghouty & Mewaldt 2000; Ostryakov et al. 2000), particularly in applications to solar energetic iron.
However, there are significant differences in the ionization cross sections adopted in those studies. Even though models of different research groups differ in many parameters, no systematic investigation of the parameter dependencies has been performed. For this reason, the sources of differences in the results, as well as any general regularities, are left obscured.
There is also a marked difference between results of kinetic calculations of the mean equilibrium charge states of energetic ions in a hot plasma (Luhn & Hovestadt 1987; Kocharov et al. 2000a) and a semi-empirical formula for the equilibrium charge states' dependence on energy recently proposed by Reames et al. (1999). As of yet, the source of this difference has not been fully explored.
The organization of the present paper is as follows: in Sect. 2 we present a simplified model for the investigation of the general regularities in the energy-dependent ion charge-states. The different versions of the proton-impact ionization cross sections formulations are given in Sect. 3. Results for the energy-dependent energetic iron charge states, employing the different versions of the ionization cross sections and for a variety of the model parameters are summarized in Sect. 4, followed by conclusions in Sect. 5.
Energy dependent charge states of energetic ions in a hot plasma
are established by the interplay between the charge-changing processes,
i.e., ionization and recombination, on the one hand,
and the energy-changing processes, i.e., acceleration and Coulomb losses,
on the other.
We begin by introducing the distribution function N_{i}(E) for the ions
of charge
and kinetic energy per nucleon E,
and adapt the simplest, nonequilibrium model described
by the following balance equation:
(1) |
The model incorporates a regular acceleration and Coulomb energy losses,
(2) |
(3) |
= | |||
(4) |
= | |
= |
In the present paper we are primarily interested in the mean charge of the accelerated ion as a function of energy, i.e., . For simplicity, the acceleration is considered in the escape-time approximation with a characteristic time ( ) taken to be independent of both Z and A numbers of the ion. The acceleration is assumed regular and the Coulomb energy loss < rate of acceleration for all ions, so that the accelerated ions are not allowed to revert back to their lower, pre-acceleration energies. Under these conditions, the value of will not affect the deduced energy dependence of .
The ion recombination rate comprises two terms, radiative as well as dielectronic recombinations. The relative contribution of each process and the total recombination rate depend on both the ambient plasma temperature as well as the Fe projectile energy. At low energies, the recombination rate depends only on the temperature. The energy dependence becomes critical if the Fe energy exceeds keV/n. Luhn & Hovestadt (1987), for example, show how those rates can be estimated for various ions. Here, and for applications to Fe ions, the recombination rates used are given by Kocharov et al. (2000a).
The electron ionization cross section comprises the direct ionization
part and the excitation autoionization part,
and
,
respectively. The rate coefficient is obtained by integrating the
total cross section,
,
assuming a flux of ambient particles in the rest frame of
the Fe projectile, i.e.,
(5) |
All three theoretical cross-sections' estimates are taken to be
sum over partial cross
sections corresponding to different electron subshells,
= | (6) |
u = | (7) |
(8) |
(9) |
(10) |
Figure 1: Cross sections of proton-impact ionization of Fe ions calculated using (dotted curves), (dashed curves), and (heavy solid curves). The light solid curves depict the electron-impact ionization cross section . | |
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The Bohr's (1948) cross section for the single ionization of an ion/atom by
fast, fully stripped ions (e.g., protons) can be cast into a very compact
form (Knudsen et al. 1984). In particular, for highly ionized Fe ions, i.e., Q>+8,
for which the ionization potentials are always high
,
the formula (Kocharov et al. 2000a) reads:
(11) |
(12) |
As shown in Fig. 1, the Bohr cross section exhibits the highest attainable value among the three partial cross sections' estimates. We employed the Bohr cross section to estimate proton-impact ionization of the carbon ions C^{+1}, C^{+2}, and C^{+3}, and compared the results to experimental data by Sant'Anna et al. (1998) and Goffe et al. (1979). The comparison indicates that the Bohr cross section tends to overestimate the measured maximum cross section by a factor of 1.5-2.
Among the simple, semiclassical approximations describing ionization,
the binary-encounter model (BEM) is one of the more successful schemes
(Gryzinski 1965; McGuire & Richard 1973; Peter & Meyer-ter-Vehn 1991).
Following Gryzinski (1965), the cross section for a single proton-impact ionization
from a total of n_{j} electrons in the jth subshell of the projectile
ion is given by:
(13) |
Dietrich et al. (1992) compared charge-state measurements of fast Ar and Xe ions passing through a plasma target to the results of Monte Carlo calculations incorporating the BEM cross section. It was found that for Ar and Xe the ionization cross sections calculated in the binary encounter model were too small compared to the measured total electron-loss cross sections. A comparison with experimental data for the electron loss of heavy ions showed that the BEM cross sections have the correct energy and charge-state dependences, but, on average, tend to underestimate the data by about a factor of 2.5.
Barghouty (2000) described a simple procedure to estimate proton-impact
ionization cross sections over the energy range up to tens of MeV/n.
The procedure connects, in the first Born approximation,
the partial proton-impact cross section
to the known
electron-impact cross section
using the Bates-Griffing relation (BGR).
The resultant cross section takes the form:
(14) |
(15) |
In what follows we use the designations , , and for the three versions of the proton-impact ionization cross sections: The Bohr cross section given by Eq. (11), the cross section resulting from the binary-encounter model, given by Eq. (13), and for the cross section based on the Bates-Griffing relation, given by Eq. (14), respectively. If analogy with the C, Ar, and Xe ions is valid, one may expect that experimental cross sections for the Fe ions will be about halfway the values and .
In the first set of calculations we have deduced the equilibrium
charge state distributions,
,
of Fe ions at different energies for the three different versions of
the proton-impact ionization cross sections discussed in Sect. 3.
Those distributions represent solutions of Eq. (1) when all the terms of
the equation except the square-bracketed one are neglected.
The mean equilibrium charge of Fe ions,
,
can be
calculated by averaging the ionic charge Q=i-1 over
the distribution
.
The result is shown in Fig. 2.
It is seen that the sharpness of the steps in the curve
essentially depends on the version of the proton cross-section used.
Figure 2: The equilibrium mean charge of energetic Fe ions calculated when both electron and proton-impact ionization are included (three upper curves labeled "a''). The employed versions of the proton-impact ionization cross section are (dotted curve), (dashed curve), and (heavy solid curve). For comparison we show the equilibrium charge for the cold neutral gas ("b'', Eq. (16)) and the equilibrium charge estimated using semi-empirical formulas Eq. (17) ("c'') and Eq. (18) (crosses). An additional calculation (minus signs) is explained in text. | |
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Rather than solve the complex system of charge-changing reactions,
often in astrophysical applications a simple, semi-empirical
expression (Barkas 1963; Betz et al. 1966) for the mean charge dependence on
energy is used,
(16) |
In contrast to applications to cold neutral gases, however, the simple relation between mean charge and velocity in Eq. (16) fails in applications to plasmas. Studies of fast ions in plasmas indicate significantly higher ionization of the projectile than that of the same projectile species in a cold neutral target (Nardi & Zinamon 1982; Peter & Meyer-ter-Vahn 1986). For this reason, kinetic calculations need to be performed and verified against experimental data (e.g., Dietrich et al. 1992; Peter & Meyer-ter-Vahn 1991). A similar kinetic approach is adapted in this work for energetic Fe ions in the plasma of the solar corona. With this first-principles approach, both equilibrium and nonequilibrium charge states can be calculated.
Reames et al. (1999) have recently proposed a
semi-empirical formula for the equilibrium charge states of energetic
ions in a hot plasma. The formula is essentially a slight modification
of the neutral-matter relation (Eq. (16)),
(17) |
A most likely reason for this marked discrepancy is that in expression (17) there was no correction made to account for the difference in cross sections of charge changing processes in neutral gases and those in plasmas. In particular, recombination cross sections in neutral gasses are typically very high (e.g., Table IV by Betz 1972). As an illustration of the recombination rate effect, we have calculated equilibrium charge states of Fe ions - with kinetic approach using - but with recombination rate artificially enhanced by a factor of 10 (minus signs in Fig. 2).
One can attempt to improve on Eq. (17) by accounting
for some of the salient peculiarities of charge-changing cross sections
in plasmas.
At relatively low recombination rate and high contribution of protons
to the ionization, the threshold velocity for ionization becomes close to
1/2 of the orbital electron velocity (
u=(V/v_{j})^{2}=1/4 in Sect. 3.1).
Based on this, one can argue that Fe ion retains all its electrons with
orbital velocity greater than twice the ion velocity.
Hence, Eq. (17) may be substituted with
(18) |
Figure 3: The nonequilibrium mean charge of energetic Fe ions for different values of the density acceleration-time product at S=0. The parameter is shown next to the curves in units of . The employed versions of the proton-impact ionization cross section are (dotted curves), (plus signs), and (heavy solid curves). For comparison we show the charge calculated with only electron-impact ionization included (light solid curves). The upper series in the figure illustrates also a coincidence of the mean charge inside the acceleration region (the dotted curve) and the mean charge of escaping particles (diamond signs). For the calculation of escaping particles we adopted , whereas all the curves are calculated assuming infinitely large . | |
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To calculate nonequilibrium charge state distributions,
we integrate Eq. (1) using a Monte Carlo code.
We start with the case of a constant acceleration rate, i.e., S=0 in
Eq. (3), and employ, for parameterization purposes,
the product of acceleration time, ,
and the ambient
electron/proton density, n, measured in units
(parameters appearing next to the curves in Fig. 3).
Using the Monte Carlo code we can compute
the charge-energy distribution of escaping ions,
as a time integral of the "leaky-box'' term
in Eq. (1),
as well as the charge-energy distribution
of Fe ions inside the acceleration region, as a time integral of N_{i}.
However, one can see that in the case of a regular acceleration
dominating over the Coulomb deceleration with
independent of Q, both calculations give the same mean
charge
(upper curve and points in Fig. 3
for
inside and outside the acceleration region,
respectively). For this reason,
the rest of profiles are shown for the ion distributions N_{i}
inside the acceleration region.
Nonequilibrium mean charge states are calculated for the three
different versions of the proton-impact ionization cross sections.
For comparison we also show results
with only electron-impact ionization included in the calculations.
It is seen that the differences among the versions of the proton-impact
cross sections are not all that consequential
under the assumed strong nonequilibrium conditions
(i.e.,
),
whereas inclusion of the proton-impact ionization for any version
is important.
Figure 4: The density acceleration-time product as a function of the mean charge for 30 MeV/n Fe ions at S=0. The employed versions of the proton-impact ionization cross section are (dotted curves), (plus signs), and (heavy solid curves). For comparison, we show the values calculated with only electron-impact ionization included (light solid curve). | |
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Figure 4 illustrates how the mean charge of 30 MeV/n Fe ions, , is related to the theoretical parameter , where is the characteristic acceleration time at 30 MeV/n; if S=0. It can be seen that over a wide range of the differences among the three versions of the proton-impact cross sections have little affect on estimates of the density acceleration-time product.
In the final set of simulations, we study the energy dependence effect of
the acceleration rate (S in Eq. (3)) on the charge-energy profiles
using the proton-impact cross section
.
Panel "a'' of Fig. 5 shows the charge-energy profiles for S varying
from zero to unity in the two lower members of the curves, and from
0-0.5 in the upper one. In panel "b'', values of the parameter
have been adjusted to get the same value of
the mean charge at 30 MeV/n,
,
for different dependencies of the acceleration
rate on energy,
.
The values of the parameter
are very different for different S,
because
is the characteristic acceleration time at 1 MeV/n,
not representative for 30 MeV/n. However,
the product of the density n and the time of ion acceleration
from 0.1 MeV/n to 30 MeV/n,
,
is close to
in all cases:
0.75, 0.76, 0.79 for S= 0, 0.25, 0.5, respectively.
Figure 5: The nonequilibrium mean charge of energetic Fe ions for different dependencies of the acceleration rate on energy (parameter S in Eq. (3)). Employed values of S are 0 (heavy solid curves), 0.25 (light solid curves), 0.5 (dotted curves), and 1 (dashed curves). The proton-impact cross section used is . a) Results of calculations at fixed values of the density times acceleration-time product, (shown next to the curves in units of ). b) Results of calculations at fixed value of the mean charge of 30 MeV/n Fe ions; parameter , 0.045, 0.075, 0.2 for S= 0, 0.25, 0.5, 1, respectively. | |
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We have considered a systematic of kinetic calculations for equilibrium and nonequilibrium charge states of accelerated Fe ions in the hot plasma of the solar corona. Large uncertainties exist in the proton-impact ionization cross sections' estimates (Fig. 1). Those uncertainties can affect the mean equilibrium charge (Fig. 2), but their effect on our nonequilibrium calculations is shown to be small (e.g., Fig. 3, ). This is because the high-energy structure of the ionization cross sections becomes important when the ion charge is below the equilibrium value , and the fact that the high-energy asymptotes coincide for all three versions of the cross sections' estimates used. As a result, theoretical estimates of the parameter are largely independent of the uncertainties in the cross sections' estimates (Fig. 4). We also find that estimates of are not sensitive to the energy dependence of the acceleration rate.
Based on our nonequilibrium kinetic calculations, we find the effect of proton-impact ionization on Fe charge states to be significant without regard to the particular version of the ionization cross section adopted. Inclusion of the proton-impact ionization reduces the estimated values of the by a factor of as compared with results of only electron-impact ionization included. Differences in the versions of the proton-impact ionization cross sections seem to have little effect on highly nonequilibrium Fe charge states during acceleration. In contrast, equilibrium charge states are found to be quite sensitive to the choice of the cross sections' estimates. In application to the energy dependent charge states of solar energetic ions, we estimate that uncertainties in the theoretical parameter due to differences in the proton-impact ionization cross sections' estimates are %. Further experimental studies of proton-impact ionization of Fe ions are important for more realistic modeling of the plasma ionization processes in the solar corona.
Modeling of the particle acceleration is frequently performed in the escape-time approximation (e.g., Kocharov et al. 2000b). A shape of the accelerated ion energy spectrum is ruled by both the energy dependence of acceleration time (acceleration rate) and the energy dependence of the leaky-box escape time. As one can learn from Fig. 5b, experimental measurements of the charge-energy profile possess a potentiality to deduce the energy dependence of acceleration time independently of the energy dependence of the escape time. Simultaneous use of the observed charge-energy profiles and energy spectra could enable one to decouple in empirical manner the ion acceleration time and the escape time. Such a determination of energy dependencies of the two characteristic times might provide a test/clue for theoretical acceleration models developed so far and expected in future.
Acknowledgements
We thank the Academy of Finland for the financial support. G. K. was supported by the Russian Basic Research Foundation under the grant RFFI-00-02-17031. A. F. B.'s work is supported by the US National Science Foundation grant No. 9810653, and by NASA-JOVE grant No. NAG8-1208 and Nos. NAS5-30704 and NAG5-6912 at Caltech.