Charge States of C and O from Coronal Holes:
Non-Maxwellian Distribution vs. Unequal Ion Speeds
1
S. P. Owocki
2
and Y.-K. Ko
1Bartol Research Institute, University of Delaware, Newark, DE 19716
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138
Abstract. The Solar Wind Ion Composition Spectrometer (SWICS) on board Ulysses has compiled an
extensive collection of ion charge state measurements in high-speed-wind streams. These provide important
diagnostic constraints for the acceleration region of the large south polar coronal hole in which these charge
states were \frozen-in". Initial analyses of these data have inferred that the coronal electron distribution
may deviate modestly from a Maxwellian (1) or that the coronal outow speeds of heavy ions may vary
with the ion mass (2) Here we apply a simple freezing-in approximation to examine the robustness and
uniqueness of these inferences. In particular, we emphasize that careful attention to the ionization states
of both Oxygen and Carbon provides the best potential diagnostic for a non-Maxwellian distribution of
coronal electrons, since the similarity in their overall rate coecients suggests a similar freezing-in location,
while dierences in their (comparitively high) ionization potentials provide a dierential sensitivity to a
high-energy electron tail. We also note the possibility that the freezing-in of the ionization state of these
elements may begin in the underlying transition region of their source coronal hole.
INTRODUCTION
The Solar Wind Ion Composition Spectrometer
(SWICS) on Ulysses (3) measured a relatively steady
ionization in high-speed solar wind emanating from
the south polar coronal hole. Tables I and II of Ko et
al. (2) summarize the data. Here we discuss the theoretical interpretation of these data in terms of the
\freezing-in" approximaton of the ionization state in
the coronal expansion. The aim is to gain a better
physical understanding of previous data-tting results obtained by Ko et al. (1, 2, 4), which found evidence for a mildly non-Maxwellian electron distribution, and/or unequal ion ow speeds, in the coronalhole source region. Our analysis indicates that the
relative freezing-in of the ion stages of Carbon and
Oxygen are particularly key for these inferences of a
non-Maxwellian distribution and/or non-uniform ion
outow.
FREEZING-IN ANALYSIS
For steady, radial outow of ions at a common
species speed us , the number density ni of ions of
ionization stage i evolves according to:
i
us dn
dr = ne (ni;1Ci;1 ;ni (Ci +Ri)+ni+1 Ri+1); (1)
where ne is the number density of electrons and Ci
and Ri are the ionization and recombination coefcients (cm3 =s) for rates out of the ith ionization
stage. In the inner corona, the left side can be neglected, yielding the ionization equilbrium:
ni Ri+1
(2)
ni+1 = Ci
For a Maxwellian electron distribution, Ri+1 =
Ri+1(Te ) and Ci = Ci (Te ), where Te is the electron
temperature. Then for any given ion ratio ni=ni+1 ,
solution of (2) yields an unique ratio temperature,
Ti=i+1 . In the outer corona and solar wind, the
right side of eq. (1) becomes negligible, yielding the
frozen-in condition (5, 6):
ni
(3)
ni+1 ! constant
The transition occurs at a freezing-in radius rf , dened roughly where the expansion time, exp ne =us=(dne=dr), equals an ion exchange time,
i$i+1 1=ne(Ri+1 + Ci ). If the ions ow at the
same speed as the electron-proton plasma, then in
Charge States of C and O from Coronal Holes: Non-Maxwellian Distribution vs. Unequal Ion SpeedsFebruary 17, 19991
situ measurements of the 1 AU mass ux (nu)E con-
strain the freezing-in density, nf ne (rf ) as
n3f
(nu)E (rE =rf )2 fexp ;
(dne=dr)f
Ri+1 + Ci
(4)
where fexp accounts for a possible faster-than-radial
ow expansion between the coronal base and 1 AU.
The rate coecients can be evaluated using the temperature inferred from the observed ion ratio, following eq. (1).
1.6
1.4
Tf
(MK)
κ=∞
-Si8
-Si9
Si7
Si10- Mg8 O6
Mg9
Mg7 - - Mg6
C4
C5
-
a.
1.2
1
-Fe11
Fe10
- - Fe9
-
0.8
0.6
ratio temperatures, Tf , as inferred from the SWICS
data, vs. rf . Ionization and radiative recombination
rates are from the recent compilation by (8). For dielectronic recombination, we use rates from (9) for
Fe ions, from (11) for non-Fe H and He isoelectronic
sequences, and from (10) for other ions.
These rates have some minor dierences from
those used in the previous detailed data-ts by Ko et
al. (2). The principal trends and inferences remain,
however. For example, there seems clear evidence
that the maximum temperature occurs at a radius
Rmax 1:5R, somewhat above the coronal base
R . However there is also considerable scatter in inferred temperatures at a given radius, some of which
clearly exceeds the estimated cumulative errors in
rates and measurements. In particular, there is a
systematic dierence between the freezing temperatures for Si vs. Mg ion stages, but since the reasons
for this are not yet understood, we must defer its
further discussion to future work.
0.4
1
1.6
Tf
(MK)
1.2
b.
1.4
1.2
1
0.8
1.4
1.6
1.8
2
2.2
2.4
rf
κ=5
Si8
- - Si9
- - -Si7
Si10 Mg7-- Mg8
- Mg6
Mg9 -
1.2
O6
Tf 1
C5
(MK)
- - Fe9
Fe10
Fe11
--
κ=∞
C4
O6
κ=5
0.8
C4
C5
O6
--
0.6
C4
C5
0.6
0.4
1
1.2
1.4
1.6
1.8
2
2.2
2.4
rf
FIGURE 1. a. Ratio temperatures vs. freezing-in ra-
dius for SWICS measurements in high-speed wind from
the south coronal hole, assuming a Maxwellian distribution ( ! 1). Each ion is labeled according to the lower
stage, e.g. O6 for O+6 =O+7 . The bars denote estimated
error ranges, as given by (2). b. Same, except assuming
a xed Kappa distribution with = 5.
Given an independent measurement of the coronal electron density prole, ne (r), these freezing-in
densities can be inverted to estimate freezing-in radii
rf . Here we apply the electron density prole derived
from the white-light coronagraph measurement of the
south polar coronal hole during the SPARTAN 20101 Mission together with the ground-based observation by HAO Mk-III K-coronameter (7). Taking also,
for simplicity, a constant expansion factor fexp = 7
in the corona, and assuming all ions ow at the proton speed, us = up, Figure 1a plots the frozen-in
0.4
1
1.1
1.2
rf
FIGURE 2. Blow up of ratio temperature vs. freezingin radius for Oxygen and Carbon charge stages, again
comparing results for ! 1 and = 5.
Figure 2 focuses on the especially signicant disagreement for O vs. C, which have relatively small
errors. In particular, the O+6 =O+7 and C +5 =C +6
have nearly identical freezing-in radii, but ratio temperatures which dier by much more than the individual errors. In previous formal data tting analyses by Ko et al. (1, 2, 4), this one disagreement is
a signicant contributor to the increased chi-square
that requires consideration of models with either
a non-Maxwellian distribution (1) or unequal ion
speeds (2).
Charge States of C and O from Coronal Holes: Non-Maxwellian Distribution vs. Unequal Ion SpeedsFebruary 17, 19992
FREEZING-IN TEMPERATURES
FOR NON-MAXWELLIAN
DISTRIBUTIONS
3 a.
m
(5)
The major eect is to bring the C and O ionization
temperatures into much better agreement, with both
now being well below 106 K.
Figure 3a compares the ratio temperatures derived for the Maxwellian case ( = 1) with three
cases = 10; 5; 3 with progressively stronger highenergy tails. The plotting vs. the inverse of the
ionization potential shows that the stages with highest ionization potentials have the greatest sensitivity
to a non-Maxwellian tail. Figure 3a shows this sensitivity in terms of the temperature overestimation
factor from assuming a Maxwellian distribution. Figure 3b shows that the freezing-in radius also tends to
vary with the inverse ionization potential. (Higher implies a smaller fraction of the electrons can ionize, giving a lower intrinsic rate coecient, and so
a lower freezing-in radius.) Together, Figures 3ab
show that ionization stages most sensitive to a nonMaxwellian distribution are also those that freeze-in
at lower radii. This implies an inherent ambiguity
between inferring a non-Maxwellian distribution and
variations in the coronal temperature gradients.
EFFECT OF UNEQUAL ION FLOW
SPEEDS
An alternative for bringing the C and O ionization stages into better agreement is to assume the
two species ow at dierent speeds. Ko et al. (2)
assumed ions ow speeds of the form
dC5
2
cO6
dC4
cC5
cC4
bC5
bC4 dSi10
dMg9
cSi10
dSi7
cMg9 dSi9
cSi7
dMg8
cMg8
bSi10
dFe11
dSi8
cSi8
bMg9 cSi9
dMg6
cFe11
cMg6
cMg7
bSi9
bMg8
dMg7
cFe10
bSi7
bSi8
dFe9
cFe9
bFe11dFe10
bMg7
bMg6
bFe10
bFe9
3
4
5
bO6
1
2
1/χ (keV)
3=2
;( + 1) 2(;3=2)kT
f (E) = ;(
; 1=2) h1 + E i+1 :
(;3=2)kT
d: κ=3
c: κ=5
b: κ=10
T(κ=∞)
T(κ)
Figure 1b and the lower portion of gure 2 show
the corresponding freezing-in temperatures for a nonMaxwellian electron distribution, specically using a
ve-Maxwellian t (1) to a Kappa-distribution (12)
with = 5,
dO6
; R ps km=s
us(r) = 1 + 699 r 9R
(6)
For their best-t temperature model, Figures 4 a and
b compare the radial variation of ratio temperatures
of C and O for models with (a) pC = pO = 1:93 and
(b) pC = 1:6 and pO = 1:93.
b.
2.4
rf
Fe9
Fe11 Fe10
2.2
2
Mg6
Si7
Mg7
Si8
1.8
Si9
Mg8
1.6
Si10
1.4
Mg9
1.2
1
C4
O6
1
C5
2
3
4
5
1/χ (keV)
FIGURE 3. a. Ratio temperature overestimation factor from assuming a Maxwellian distribution, plotted vs.
inverse ionization potential for = 10; 5; 3, labeled respectively with b,c,d. b. Freezing-in radius vs. inverse
ionization potential. The plot here is for the Maxwellian
case (; > 1), but is very similar for nite .
Model a with equal ow speeds gives too high a
freezing-in temperature for both Carbon stages. In
contrast, the higher Carbon ow speed in Model b
causes it to freeze-in at a lower radius, thus lowering
the freezing-in temperature to a level in good agreement with the observations.
SUMMARY AND CONCLUSIONS
A simple freezing-in analysis suggests that the
ionization stages of C and O should have similar freezing-in radii, and thus similar freezing-in
temperatures.
Inferred dierences in the freezing-in temperature of C and O can be explained either by a
non-Maxwellian distribution, or by a higher outow speed for C. Independent measurements of
the ion ow speeds are needed to resolve this
ambiguity (13).
Charge States of C and O from Coronal Holes: Non-Maxwellian Distribution vs. Unequal Ion SpeedsFebruary 17, 19993
ACKNOWLEDGMENTS
a.
1.5
Te
We acknowledge support of NASA grant NAG56470.
u C~u O~z 1.93
C4
T
(MK)
O6
C5
1.0
0.5
1
2
b.
1.5
3
u ~z 1.6
Te
C
u ~z 1.93
O
T
(MK)
O6
C4
1.0
C5
0.5
1
2
3
r/R*
FIGURE 4. a. Radial variation of electron tempera-
ture (heavy solid curve) compared with the local ion ratio temperatures for C and O, computed by integrating
eq. (1) using the Ko et al. (2) model with equal ion ow
speeds. b. Same for model in which the velocity power
index for Carbon is lower than for Oxygen, resulting in
higher Carbon ow speeds that cause its ionization stages
to freeze-in at a lower height and temperature.
Ions with a higher ionization potential have a
greater sensitivity to a non-Maxwellian electron
distribution, with O+6 showing the greatest sensitivity.
Ions with a higher ionization potential also
freeze-in lower in the coronal expansion, with
O+6 freezing-in lowest in a simple model. This
implies an ambiguity between changes in the
temperature gradient and inferences of a nonMaxwellian tail.
If the electron distribution has a modestly enhanced non-Maxwellian tail (i.e. if 5), then
the actual freezing-in temperatures of C and O
are quite low, i.e. < 106 K. This suggests that
the ionization balances of C and O may actually
be xed below the coronal base, i.e. in the upper
Transition Region. (See Esser et al. (13).)
REFERENCES
1. Ko, Y.-K., Fisk, L., Geiss, J., and Gloeckler, G. 1996,
\Limitations on Suprathermal Tails of Electrons in the
Lower Solar Corona", Geophysical Research Letters,
23, 2785-2788.
2. Ko, Y.-K., G., Fisk, L., Geiss, J., Gloeckler, G., and
Guhathakurta, M. 1997, \An Empirical Study of the
Electron Temperature and Heavy Ion Velocities in the
South Polar Coronal Hole", Solar Physics, 171, 345361.
3. Gloeckler, G. et al. 1992, \The Solar Wind Ion Composition Spectrometer", Astron. Astrophys. Suppl.
Ser.,92, 267-289.
4. Ko, Y.-K., Geiss, J., and Gloeckler, G. 1998, \On the
Dierential Ion Velocity in the Inner Solar Corona and
the Observed Solar Wind Ionic Charge States", Journal of Geophysical Research, 103, 14,539-14,545.
5. Hundhausen, A. J., Gilbert, H. E., and Bame, S. J.
1968, \The State of Ionization of Oxygen in the Solar
Wind", Astrophys. J., 152, L3-L5.
6. Owocki, S. P., Holzer, T. E., and Hundhausen, A.
J. 1984, \The Solar Wind Ionization State as a Coronal Temperature Diagnostic", Astrophys. J., 275, 354356.
7. Fisher, R., and Guhathakurta, M. 1995, \Physical
Properties of Polar Coronal Rays and Holes as Observed with SPARTAN 201-01 Coronagraph", Astrophys. J., 447, L139-L142.
8. Verner, D. 1998, Fortran codes posted on
www.pa.uky.edu/ verner/atom.html.
9. Arnaud, M., ad Raymond, J. C. 1992, \Iron Ionization and Recombination Rates and Ionization Equilibrium", Astrophys. J., 398, 394-406.
10. Romanik, C. J., 1996, \Dielectronic Recombination
in Astrophysical Plasmas", Ph.D. Dissertation, University of California, Berkeley.
11. Aldrovandi, S. M. V., and Pequignot, D. 1973, \Radiative and Dielectronic Recombination Coecients
for Complex Ions", Astron. Astrophys.,25, 137-140.
12. Olbert, S. 1969, \Summary of Experimental Results
from M.I.T. Detector on IMP-1", in Physics of Magnetospheres, R. Carovillano et al., eds., (Reidel: Hingham, MA), p. 641-659.
13. Esser, R., Edgar, R., and Brickhouse, N. 1998, \High
Minor Ion Outow Speeds in the Inner Corona and
Observed Ion Charge States in Interplanetary Space",
Astrophys. J., 498, 448-457.
Charge States of C and O from Coronal Holes: Non-Maxwellian Distribution vs. Unequal Ion SpeedsFebruary 17, 19994