THE ASTROPHYSICAL JOURNAL, 511 : 481È501, 1999 January 20
( 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.
AN EMPIRICAL MODEL OF A POLAR CORONAL HOLE AT SOLAR MINIMUM
S. R. CRANMER,1 J. L. KOHL,1 G. NOCI,2 E. ANTONUCCI,3 G. TONDELLO,4 M. C. E. HUBER,5 L. STRACHAN,1
A. V. PANASYUK,1 L. D. GARDNER,1 M. ROMOLI,2 S. FINESCHI,1 D. DOBRZYCKA,1 J. C. RAYMOND,1 P. NICOLOSI,4
O. H. W. SIEGMUND,6 D. SPADARO,7 C. BENNA,8 A. CIARAVELLA,1,9 S. GIORDANO,8 S. R. HABBAL,1 M. KAROVSKA,1
X. LI,1 R. MARTIN,10 J. G. MICHELS,1 A. MODIGLIANI,2 G. NALETTO,4 R. H. OÏNEAL,1 C. PERNECHELE,4
G. POLETTO,11 P. L. SMITH,1 AND R. M. SULEIMAN1
Received 1998 July 1 ; accepted 1998 August 25
ABSTRACT
We present a comprehensive and self-consistent empirical model for several plasma parameters in the
extended solar corona above a polar coronal hole. The model is derived from observations with the
SOHO Ultraviolet Coronagraph Spectrometer (UVCS/SOHO) during the period between 1996 November and 1997 April. We compare observations of H I Lya and O VI jj1032, 1037 emission lines with
detailed three-dimensional models of the plasma parameters and iterate for optimal consistency between
measured and synthesized observable quantities. Empirical constraints are obtained for the radial and
latitudinal distribution of density for electrons, H0, and O5`, as well as the outÑow velocity and unresolved anisotropic most probable speeds for H0 and O5`. The electron density measured by UVCS/
SOHO is consistent with previous solar minimum determinations of the white-light coronal structure ; we
also perform a statistical analysis of the distribution of polar plumes using a long time series. From the
emission lines we Ðnd that the unexpectedly large line widths of H0 atoms and O5` ions at most heights
are the result of anisotropic velocity distributions. These distributions are not consistent with purely
thermal motions or the expected motions from a combination of thermal and transverse wave velocities.
Above 2 R , the observed transverse most probable speeds for O5` are signiÐcantly larger than the
_ motions for H0, and the outÑow velocities of O5` are also signiÐcantly larger than the
corresponding
corresponding velocities of H0. Also, the latitudinal dependence of intensity constrains the geometry of
the wind velocity vectors, and superradial expansion is more consistent with observations than radial
Ñow. We discuss the constraints and implications on various theoretical models of coronal heating and
acceleration.
Subject headings : line : proÐles È solar wind È Sun : corona È Sun : UV radiation
1.
INTRODUCTION
The UVCS instrument is described in detail by Kohl et al.
(1995), and some preliminary Ðrst results have been presented by Kohl et al. (1996a, 1997a, 1997b), Noci et al. (1997),
Raymond et al. (1997), and others. The UVCS telescope
spectrometer unit contains three independent channels : (1)
the LYA (H I Lya) spectrometer channel, which primarily
observes the strong H I j1216 emission line ; (2) the OVI
spectrometer channel, which is optimized to observe the
O VI jj1032, 1037 doublet and the H I j1026 (H I Lyb) line ;
and (3) the WLC (white-light channel) visible polarimeter,
which measures the coronal polarized radiance in the 4500È
6000 AŽ wavelength band. At least 40 di†erent UV spectral
lines have been identiÐed in the solar corona by UVCS
during its Ðrst year of operation (see, e.g., Raymond et al.
1997).
The ultraviolet emission lines and polarized visible light
observed by UVCS/SOHO present a rich and varied source
of diagnostic information about the solar corona (see
reviews by Withbroe et al. 1982 ; Bely-Dubau 1994 ; Kohl et
al. 1995). The shapes of lines formed by resonance scattering
and collisional excitation are direct probes of the line-ofsight (LOS) distribution of electron, atom, and ion velocities. Integrated line intensities of resonantly scattered lines
can help determine the solar wind velocity (and other
details about the particle velocity distribution) by the
Doppler dimming method. The intensities and emission
measures of collisionally dominated EUV lines can constrain electron temperatures, densities, and elemental abundances in the extended corona. In addition, the linear
In order to explain comprehensively the heating and
acceleration of the solar wind, theories must incorporate
detailed empirical knowledge of the physical conditions in
the coronal plasma. Obtaining accurate measurements of
densities, outÑow velocities, and microscopic velocity distributions in the principal acceleration region of the solar
corona (between 1 and 5 R ) is thus of extreme importance.
_ the Ultraviolet Coronagraph
This is a primary goal of
Spectrometer (UVCS) operating aboard the Solar and
Heliospheric Observatory (SOHO) satellite. In this paper we
utilize a large ensemble of UVCS/SOHO observations to
construct an extensive empirical model of the fast solar
wind over polar coronal holes near solar minimum. Our
goal is to present in full the results initially reported by
Kohl et al. (1998).
1 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA
02138.
2 Università di Firenze, I-50125 Firenze, Italy.
3 Osservatorio Astronomico di Torino, I-10025 Pino Torinese, Italy.
4 Università di Padova, I-35131 Padova, Italy.
5 Space Science Department, ESA/ESTEC, NL-2200 AG, Noordwijk,
The Netherlands.
6 Space Sciences Laboratory, University of California, Berkeley, CA
94720.
7 Osservatorio AstroÐsico di Catania, I-95125 Catania, Italy.
8 Università di Torino, I-10125 Torino, Italy.
9 Osservatorio Astronomico di Palermo, Palermo, Italy.
10 Institut dÏAstrophysique Spatiale, F-91405 Orsay, France.
11 Osservatorio AstroÐsico di Arcetri, I-50125 Firenze, Italy.
481
482
CRANMER ET AL.
polarization of the broadband visible corona provides an
independent measurement of the electron density.
We utilize the above diagnostic techniques to produce an
empirical model of the plasma conditions in coronal holes.
The modeling procedure is initiated by deducing various
physical quantities (e.g., density, outÑow velocity, and
velocity distribution) directly from UVCS observations in a
fashion similar to that used by Kohl et al. (1997a). However,
this simple inverse determination of the plasma parameters
is not yet fully self-consistent. Synthetic observables (such as
line proÐles, intensities, and polarized white light) generated
with these initial models often do not agree perfectly with
the measured data, and small adjustments must then be
made to the plasma parameters. Repeated comparison with
the data provides guidance on how to iterate the derived
quantities until there is optimal agreement between the
empirical model and the observations.
It is important to emphasize that the empirical models
described here do not specify the processes that maintain
the coronal plasma in its assumed steady state. Thus, there
is no explicit mention of coronal heating and acceleration
mechanisms, waves and turbulent motions, and magnetic
Ðeld structure, within the models themselves. The iterated
quantities in the models depend on only observations and
well-established theory, such as the radiative transfer inherent in the line formation process. All of the resulting coronal
parameters, then, are derived straightforwardly and unambiguously from measurements and can be utilized e†ectively
to constrain theoretical models. (For certain quantities,
however, it is not possible to derive precise empirical values,
and we set reasonable limiting ranges for their values.)
The remainder of this paper is organized as follows. In ° 2
we outline the UVCS observations that are utilized to construct the empirical model. In ° 3 we begin this process by
deriving the mean three-dimensional electron density structure of coronal holes from WLC polarization data. In ° 4 we
analyze the intensities and shapes of the H I Lya lines and
build up an empirical picture of the dynamics of H0 atoms
in the corona. Similarly, in ° 5 we examine the O VI jj1032,
1037 resonance lines to deduce the state of O5` ions. In ° 6
we perform a statistical analysis of white-light and H I Lya
data designed to shed light on the distribution of dense
polar plume enhancements. Finally, in ° 7 we present a
Vol. 511
summary of the major results and properties of the UVCS
empirical model of solar coronal holes, and in ° 8 we discuss
the implications of the empirical data on theoretical models
of the corona and wind.
2.
OBSERVATIONS AND DATA REDUCTION
In this paper we model the time-averaged structure and
composition of polar coronal holes12 in the period between
1996 November and 1997 April. This time interval was very
near the activity minimum of solar cycle 22 (see, e.g., Li
1997 ; Pap et al. 1997). Although slow solar cycle variations
may have taken place over this 6 month interval, the mean
polar coronal holes have been veriÐed to remain reasonably
constant in their UV and white-light emission during this
period. In Table 1 we list the speciÐc UVCS observations
that were used in the empirical modeling process, along
with the ranges of heliocentric observing height o and position angle PA (measured counterclockwise from the projection of the north heliographic pole in the plane of the sky) in
each data set. The synoptic observations taken nearly every
day cover the entire corona in 45¡ increments of position
angle, and we primarily use the scans over the north and
south heliographic poles.
The UVCS Data Analysis Software (DAS) was used to
remove image distortion, to Ñat-Ðeld the detectors, and to
calibrate the data in wavelength and intensity (see Gardner
et al. 1996 ; Kohl et al. 1997a, 1997b ; Romoli et al. 1999).
Estimates of instrument-scattered stray light from the solar
disk have been subtracted from data in all three channels,
but the characterization and modeling of the instrument is
ongoing. The strong H I Lya line also has a signiÐcant
interplanetary contribution, especially for lines of sight over
the poles at large heights, and we model this using preliminary results from the Solar Wind Anisotropy (SWAN)
instrument aboard SOHO (Bertaux et al. 1997).
The measured UV line proÐles are broadened by various
instrumental e†ects (Panasyuk et al. 1999). The optical
12 We denote as ““ coronal holes ÏÏ the entire open magnetic Ðeld regions
above the heliographic poles at solar minimum. The issue of magnetic
connectivity between the high-speed solar wind and canonical dark (EUV
and X-ray) coronal holes at the base has been called into question recently
(Woo & Habbal 1997), but we retain the standard interpretation described
in more detail in ° 3.2.
TABLE 1
OBSERVATION LOG FOR CORONAL HOLE EMPIRICAL DATA
Start Date, Time (UT) . . . .
JD [2,450,000
1996 Nov 5, 15 :15 . . . . . . . . .
1996 Dec 28, 16 :34 . . . . . . . .
1996 Dec 29, 6 17 : 04 . . . . . .
1997 Jan 3, 18 :58 . . . . . . . . . .
1997 Jan 4, 20 :17 . . . . . . . . . .
1997 Jan 25, 15 :34 . . . . . . . . .
1997 Jan 26, 16 :18 . . . . . . . . .
1997 Jan 27, 17 : 04 . . . . . . . .
1997 Jan 28, 17 :49 . . . . . . . . .
1997 Jan 29, 18 :34 . . . . . . . . .
1997 Jan 31, 14 :28 . . . . . . . . .
1997 Feb 1, 15 :27 . . . . . . . . . .
1997 Feb 4, 16 :54 . . . . . . . . . .
1997 Apr 14, 13 :34 . . . . . . . .
1997 Apr 15, 14 :11 . . . . . . . .
Daily synoptic scans . . . . . .
393.14
446.19
447.21
452.29
453.35
474.15
475.18
476.21
477.24
478.27
480.10
481.14
484.20
553.07
554.09
...
o/R
_
1.75È2.50
1.50È3.10
1.50È3.10
1.50È3.10
1.50È3.10
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00È3.50
4.00
1.50È2.50
PA (degrees)
000È360
015
345
195
165
000
000
000
000
000
000
000
000
100È180
180
000È360
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
point-spread function of the spectrometer depends on the
slit width used, the on-board data binning, the exposed
mirror area, and the intrinsic quantization error of each
detector. This broadening is taken into account by adjusting the line widths of Gaussian Ðts to the coronal components of the data ; the data points themselves are not
corrected. However, there is also an additional instrument
characteristic that e†ectively smears the wavelengths,
adding a broad wing to most lines and reducing their
central intensities. This is modeled by a discrete zerointegral function that is numerically deconvolved from the
observed proÐles prior to any other line-shape analysis
(Kohl et al. 1997a).
3.
ELECTRON DENSITY
The electron density in the solar corona can be determined by measuring the linear polarization due to
Thomson-scattered white light from the photosphere
(Minnaert 1930 ; van de Hulst 1950). The K-coronal polarization brightness pB is proportional to the line-of-sight
(LOS) integral of the electron number density n , and it
varies as a function of observed heliocentric heighteo by the
equation
pB(o) \
P
ABC
D
`=
o 2 (1 [ u@)A(r) ] u@B(r)
3
n (x)
dx ,
p B1
e
T
_
r
1 [ u@/3
16
~=
(1)
where p \ 6.65 ] 10~25 cm2 is the Thomson scattering
T
cross section,
B1 is the mean solar-disk surface brightness,
u@ \ 0.63 is the_ linear limb-darkening coefficient in the
visible wavelengths of interest, and x \ (r2 [ o2)1@2 is the
distance measured along the LOS. The functions A(r) and
B(r) are given by Minnaert (1930) and Altschuler & Perry
(1972), and they account for scattering from the ÐniteÈsolidangle solar disk. The general problem is to invert the above
integral to solve for the three-dimensional distribution of
electron density within the corona.
Although plasma densities may also be inferred from
coronal UV emission lines, these often contain combinations of density, temperature, velocity, and ion abundances that are difficult to separate in practice. It is
important, then, for empirical models to incorporate whitelight pB measurements that depend solely on the electron
density distribution and the visible solar intensity. Because
of this, the UVCS instrument is equipped with the whitelight channel (WLC), which samples the instantaneous Ðeld
of view of the UVCS ultraviolet spectrometers.
Polarization brightness (pB) data over the extended solar
corona, over a visible spectral band of 4500È6000 AŽ for all
position angles around the SunÏs disk, have been obtained
regularly from 1996 April throughout the period of the
observations reported here. In polar coronal holes the
heights observed range between 1.5 and 4 R . The methods
_
used to reduce and calibrate the WLC measurements
are
given by Romoli et al. (1993, 1997, 1999). The polarized
radiance is determined by combining individual observations with a half-wave retarder plate oriented at 0¡, ]30¡,
and [30¡ from a linear polarizer at a Ðxed reference angle,
which eliminates unpolarized instrument contributions. In
addition, the use of pB rather than the total coronal brightness also eliminates the majority of the dust-scattered Fcorona, which is believed to be unpolarized up to distances
of D5 R (Blackwell & Petford 1966 ; Koutchmy & Lamy
_
483
1985). In °° 3.1È3.2 we analyze a set of WLC observations
and build a model of the electron density structure within a
polar coronal hole.
3.1. Radial Dependence
In Figure 1 we plot individual pB measurements, in units
of the mean white-light solar-disk surface brightness B1 , for
_
various lines of sight (with 14A spatial resolution) over the
south polar coronal hole. The data shown cover the periods
1996 December 27È30 and 1997 April 14È20 and are representative of the values and variations of pB over the poles
during similar ranges of time throughout late 1996 and
early 1997 (see Romoli et al. 1997). The error bars denote
point-to-point uncertainties that can a†ect the radial
dependence of pB, while the dashed lines show the range of
possible systematic uncertainties that a†ect only the normalization of the data. The three largest sources of point-topoint uncertainty are (1) Poisson count rate statistics, (2) an
estimated 10% uncertainty in the adopted polarized stray
light scattered from the edge of the telescope mirror, and (3)
an empirically computed 13% standard deviation in the pB
time-settling function (see below). The systematic uncertainty in the y-axis normalization of Figure 1 arises from
the photometric calibration (]25%, [50%) of the WLC
detector (Romoli et al. 1997).
Also shown in Figure 1 is a nonlinear least-squares Ðt to
the radial dependence of pB, which is given by
A B
A B
R 11.1
R 3.42
pBavg
] 6.75 _
.
(2)
\ 1570 _
o
o
10~9B1
_
Below D1.6 R , the high scatter in both the pB distribution
_
and the orientation
of the polarization plane indicates that
FIG. 1.ÈUVCS coronal polarization brightness (pB), in units of the
mean solar disk brightness B1 , as a function of observed LOS height o
_
over the south pole. Measurements
were taken during 1996 December
27È30 (squares) and 1997 April 14È20 (crosses), with error bars reÑecting
statistical and stray-light uncertainties. Also plotted is the result of a nonlinear s2 minimization Ðt (solid line), the relative uncertainty in the overall
photometric calibration (dashed lines), and an downward extension of this
uncertainty required to bring the data into agreement with earlier pB
observations (gray region ; see text).
484
CRANMER ET AL.
when the telescope mirror is very narrow, some stray light
may not be accounted for in the current instrument characterization. We thus have excluded the observed data at
o \ 1.5 R from the plot and the least-squares Ðt.
_
Assuming the distribution of electron density in the south
polar hole is spherically symmetric, the LOS pB integral can
be directly inverted to yield an analytic function of radius r,
which we compute and Ðt by
A B
A B
R 10.5
R 2.57
nsph(r)
e
] 8.69 _
\ 3890 _
r
r
105 cm~3
(3)
(for details, see van de Hulst 1950 ; Altschuler & Perry 1972 ;
Munro & Jackson 1977 ; Strachan et al. 1993). Figure 2
compares this derived density with several representative
polar coronal hole n results derived from other polarizae
tion brightness measurements. At radii larger than D2 R ,
_
the average electron density we compute is in good agreement with the determinations of Guhathakurta & Holzer
(1994) and Fisher & Guhathakurta (1995). At smaller radii,
however, our densities seem to be about a factor of 1.5È2
times larger than both the sounding rocket measurements
reported by Strachan et al. (1993) and the Spartan 201-01
measurements of Fisher & Guhathakurta (1995). Di†erences of this kind are not unexpected for measurements as
difficult and sensitive to instrument e†ects as these. These
problems are compounded in the relatively dim coronal
holes ; indeed, an equivalent comparison of electron densities in bright streamers has yielded closer agreement
between UVCS and earlier measurements at comparable
heights (see Romoli et al. 1999). There are several possible
explanations for the seemingly high UVCS coronal hole
densities at low heights :
Vol. 511
1. NonÈK-corona polarization.ÈThe polarized components of both the F-corona and the instrumental stray
light are not expected to depend on latitude. Thus, if either
of these two sources are polarized signiÐcantly more than
expected, the less-dense coronal holes should contain a
higher fraction of this polarization than the streamers. Both
the incident solar disk stray light (StenÑo et al. 1983) and the
F-corona (Koutchmy & Lamy 1985) seem not to be polarized signiÐcantly more than the 0.01%È0.03% level (at r [ 5
R ), which has a negligible e†ect on the total measured pB.
_ fractional polarization generated by the instrument,
The
arising from stray light from the solar disk, has been
modeled in detail based on laboratory and in-Ñight calibrations (Romoli et al. 1993, 1999). There is also a timedependent component of the stray light that is correlated
with the movement of the telescope mirror. This time settling has an approximately exponential behavior with a 1/e
time constant of 1.11 hr, and it primarily a†ects the derived
pB because each of the three intensity measurements (at
di†erent polarization angles) do not occur simultaneously.
Although these e†ects are accounted for in the data
reduction, there may still be some uncharacterized sources
of stray light producing instrumental polarization.
2. L ine-of-sight contamination.ÈBecause the computed
mean density at low heights lies between previous estimates
of coronal hole and streamer densities, there could have
been nonnegligible contribution from low-latitude structure
along the lines of sight assumed to contain only coronal
hole material (especially at small LOS radii). However, as
we see in ° 3.2 below, we observe no latitudinal dependence
of the electron density within polar coronal holes over large
regions of position angle, which implies that dense
streamers do not appreciably intersect these lines of sight at
solar minimum.
3. Epochal variations.ÈThe intrinsic brightness of the
inner electron-scattered K-corona may be higher during the
1996 solar minimum than during previous nonÈactive-Sun
polar measurements, which implies either a higher homogeneous density or the presence of a greater number of dense
plumes. The latter possibility may indeed be the case, since
the 1996 solar minimum seems to exhibit 2È3 times the
number of white-light polar faculae (which may be correlated with coronal plumes) than observed during the previous three solar minima (Makarov & Makarova 1996).
These issues are currently being investigated in more
detail, but in this work we assume the best-Ðt n (r) presented
e
in equation (3) accurately reÑects the large-scale
mean
density structure over the heliographic poles. In Figure 1 we
schematically extend the systematic uncertainty in pB
downward, at low heights, by the factor of D2 required to
bring these observations into agreement with previous measurements. However, it is still uncertain whether this should
be understood as an instrumental correction factor or an
indication of long-term variability that may widen the range
of possible observed pB values (see also Romoli et al. 1999).
FIG. 2.ÈRadial dependence of the mean electron density in coronal
holes, computed from various sets of pB data : UVCS WLC observations
from Fig. 1 (heavy solid line), rocket-coronagraph data from 1982 (Strachan
et al. 1993 ; Ðlled circles), interplume and plume data from the Spartan
201-01 mission, averaged between the north and south polar holes (Fisher
& Guhathakurta 1995 ; light solid lines), and Skylab and ground-based
coronameter data from 1973È1976 (Guhathakurta & Holzer 1994 ; dashed
line).
3.2. L atitudinal Density Structure
Coronal holes at the north and south solar poles are
observed to exist permanently for approximately 7 yr about
sunspot minimum, and their latitudinal extent is a strong
function of solar cycle (see, e.g., Waldmeier 1981 ; Bravo &
Stewart 1994). The present empirical model of a representative coronal hole takes advantage of the relatively simple
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
(axisymmetric) structure of the corona observed in 1996È
1997 : i.e., large polar holes with presumably open magnetic
Ðeld lines, bounded by equatorial high-density streamers
with mainly closed Ðeld lines. The open Ðeld lines are
assumed to follow the large-scale electron density structure,
expanding superradially near the solar surface and then
radially further out, eventually Ðlling a majority of the
volume of the solar wind (Hundhausen 1977 ; Low 1990 ;
Roberts & Goldstein 1998).
In this section we attempt to characterize both the shape
of the latitudinal boundary of this representative open Ðeld
region and any large-scale internal structure, leaving the
distribution of smaller scale polar plumes for ° 6. To infer
the overall hole shape, we utilize a set of UVCS pB measurements made on 1996 November 5È6. These observations
(see Table 1) consist of 360¡ scans around the limb with a
12¡ spacing in position angle, at four heliocentric heights.
We plot the pB values for these observations in Figure 3
and note that the inherent Poisson statistical uncertainty in
each measurement is about 7% in coronal holes and 2% in
streamers ; these error limits all lie within the plotted
squares. Other uncertainties, such as stray light and the
photometric calibration, should not produce variations
from observation to observation at a given radius (mirror
position). The north and south coronal holes appear as
broad and Ñat minima in pB, and the equatorial streamers
appear as sharper, often asymmetric, maxima. These observations are qualitatively consistent with the conclusions of
Guhathakurta & Holzer (1994), who found no signiÐcant
latitudinal dependence of pB and the electron density within
coronal holes.
We locate the boundaries of the coronal holes by Ðtting
the east limb and west limb streamer variations with two
Gaussians, plus a constant background pB for the coronal
0 of the holes
holes (Fig. 3, solid lines). The boundaries
FIG. 3.ÈUVCS coronal polarization brightness (pB), as a function of
height and position angle, measured in 1996 November. Open squares
represent individual data points, from top to bottom, at o \ (1.75, 2.00,
2.25, 2.50) R . Solid lines are the result of nonlinear least-squares Ðts at
each height. _
485
can then be deÐned as the latitudes at which the Ðtted
(pB [ pB ) reaches a speciÐed fraction of the streamer
0
maximum. We choose this fraction to be 15%, in accord
with the Ðndings of Guhathakurta & Holzer (1994). If we
were to choose a lower fraction (and thus deÐne the edges of
the hole by the latitudes of near-constant pB), the true size of
the hole would be systematically underestimated in the
modeling. This is because the streamer belt always deviates
slightly from ideal axisymmetry ; a portion of the brightness
near the hole-streamer boundary must arise from the projection of low-latitude streamer material along the LOS (see
also Wang et al. 1997). The axisymmetric empirical model
of the latitudinal structure removes this e†ect by adopting
Guhathakurta & HolzerÏs (1994) value of 15% as the e†ective dividing line between holes and streamers.
In Figure 4 we plot the half-angle colatitude # (r) of the
boundaries of the north and south coronal holes,0as found
from the above-described UVCS measurements (with ^6¡
error bars arising from the coarseness of the sampling in
PA). The north polar hole was consistently smaller than the
south polar hole, and the latter appears much sharper and
more distinct than the former throughout most of 1996 and
early 1997. However, because of the relatively small di†erence between the two, our empirical model shall embody an
average structure which reproduces the properties of both
observed holes reasonably well. Also plotted in Figure 4, for
comparison purposes, are measurements from a coordinated SOHO and ground-based campaign on 1996 March
7È8 that imaged the south coronal hole (DeForest et al.
1997). The shape of this curve was estimated from a composite image consisting of Fe IX j171 emission, measured by
the Extreme Ultraviolet Imaging Telescope (EIT) on SOHO
(Delaboudinière et al. 1995), and visible light polarization
brightness measured by both the HAO Mark III K-
FIG. 4.ÈHalf-angle colatitude # of the inferred boundary between the
0 equatorial streamers. Open squares
polar coronal holes and higher density
represent observations shown in Fig. 3 (with ^6¡ error bars), asterisks
correspond to the observations of Guhathakurta & Holzer (1994), and the
gray region has been deduced from the coordinated observations of
DeForest et al. (1997). The solid line shows the adopted analytic parameters in Kopp & HolzerÏs (1976) form for the superradial expansion (see
text).
486
CRANMER ET AL.
coronameter on Mauna Loa (Fisher et al. 1981) and the
Large Angle Spectroscopic Coronagraph (LASCO) on
SOHO (Brueckner et al. 1995). In Figure 4 we also plot the
boundary colatitude reported by Guhathakurta & Holzer
(1994) for the north polar hole in 1973È1974.
We model the radial variation in the total area A(r)
subtended by the coronal hole by Ðtting the boundary
colatitude # (r) in terms of this area :
0
r 2
f (r) ,
(4)
A(r) \ A(R )
_ R
_
where the superradial enhancement factor f (r) is deÐned
geometrically by
A B
1 [ cos # (r)
0
.
(5)
1 [ cos # (R )
0 _
We adopt Kopp & HolzerÏs (1976) three-parameter empirical function to model the data in Figure 4, with
f (r) \
G
H
1 [ exp [(R [ r)/p ]
_
1 ,
(6)
max
1 ] exp [(R [ r)/p ]
1
1
and we solve for # (r) using equation (5). The solid line in
0 to values # (R ) \ 28¡, f \ 6.5,
Figure 4 corresponds
0 _ of the lack
maxof reliR \ 1.5 R , and p \ 0.6 R . Because
1
_
1
_
able uncertainty limits, the adopted expansion factor given
above cannot be considered a unique Ðt to the data.
However, these parameters do seem compatible with the
coronal hole expansion observed over several solar minima
(see Fig. 4). In Table 2 we list values of f (r) for a grid of radii
between 1.5 and 4 R . We also list the expected relative
_ Ðeld B(r)/B(R ) expected for Ñux
decrease in the magnetic
_
f (r) \ 1 ] ( f
[ 1)
Vol. 511
tubes following the parameterized superradial divergence ;
for a steady state conÐguration, this quantity is given by
A(R )/A(r).
_ adopted coronal hole boundary has an asymptotic
The
(r ] O) colatitude of # \ 76¡.2, which leaves almost a
max
quarter of the volume of the heliosphere unaccounted for.
Obviously, since open magnetic Ðelds are encountered at all
latitudes in the in situ solar wind, this boundary must
encompass only a subset of the complete ““ bundle ÏÏ of open
Ñux tubes. #
is approximately 5¡È10¡ larger than the
max
angular extent of the high-speed wind observed by Ulysses
in 1995 (Goldstein et al. 1996). This implies that the latitudinally independent, low-density polar coronal holes are
probably correlated with the highest speed solar wind (see
also Wang & Sheeley 1990). It should be made clear,
however, that the empirically derived superradial expansion
function f (r) is used in this paper only to describe the latitudinal dependence of the coronal density and not as a
description of the actual magnetic Ñux tube geometry.
With this e†ective boundary of the high-speed wind
region speciÐed by equations (4)È(6), we model the density
variation with latitude by the same type of Gaussian function used to Ðt the observed pB data (see also Guhathakurta, Holzer, & MacQueen 1996), i.e.,
G
n (r, h) \ npole(r) 1 ] (S [ 1) exp
e
e
C A B DH
[
n
2
[ h /"2
2
,
(7)
where S is the relative density enhancement in the equatorial streamer and " 4 0.726[# (r) [ n/2] places the holeÏs
0 at which the integrated pB
latitudinal boundary at the point
TABLE 2
EMPIRICAL GEOMETRY AND VELOCITY DISTRIBUTION PARAMETERSa
r/R
PARAMETER
1.5
2.0
2.5
_
3.0
3.5
4.0
Geometry :
f (r) . . . . . . . . . . . . . . . . . . .
2.55
4.11
5.25
5.90
6.23
6.38
B(r)/B(R ) . . . . . . . . . . .
0.174
0.0608
0.0305
0.0188
0.0131
0.00980
_
H0 Model A1 :
u......................
...
150 (]49,[55)
186 (]37,[29)
219 (]30,[23)
248 (]27,[20)
274 (]28,[19)
w ....................
175 ^ 13b
203 ^ 11
228 ^ 16
235 ^ 20
240 ^ 53
244 ^ 90
M
w ....................
156
146
131
119
109
101
A
Kinetic (T /T ) . . . . . .
1.26
1.93
3.03
3.90
4.85
5.84
M A
H0 Model A2 :
u......................
...
145 (]71,[101)
201 (]64,[53)
270 (]55,[39)
343 (]51,[33)
413 (]59,[33)
w ....................
176 ^ 13
202 ^ 11
218 ^ 16
223 ^ 20
227 ^ 53
231 ^ 90
M
w ....................
176
202
218
223
227
231
A
Kinetic (T /T ) . . . . . .
1
1
1
1
1
1
M A
O5` Model B1 :
u......................
10.7 (]38,[10.7)
179 (]78,[76)
320 (]79,[110)
402 (]44,[68)
448 (]45,[48)
...
w ....................
85.9 ^ 15
257 ^ 60
347 ^ 80
413 ^ 110
480 ^ 110
...
M
w ....................
39.2
36.6
32.9
29.9
27.4
...
A
Kinetic (T /T ) . . . . . .
4.80
48.9
111
193
308
...
M A
O5` Model B2 :
u......................
13.3 (]68,[13.3)
294 (]138,[183)
380 (]102,[112)
436 (]83,[101)
489 (]90,[80)
...
w ....................
82.1 ^ 15
245 ^ 60
336 ^ 80
423 ^ 110
504 ^ 110
...
M
w ....................
82.1
245
99.9
58.3
72.0
...
A
Kinetic (T /T ) . . . . . .
1
1
11.3
52.6
49.1
...
M A
a All velocities expressed in km s~1.
b Uncertainties in w derived from observational error bars in 1/e line widths. For O VI, the observed spread in line widths is also incorporated into the
M
w uncertainty.
M
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
487
is D15% of the streamer maximum. Figure 3 suggests an
empirical range of S between 4 and 8, and we adopt two
alternate values for S, depending on the type of model being
constructed for the electron density distribution :
respectively. Note that, for the present work, we neglect the
relatively weak UV continuum and the Thomson electronscattered component of the H I Lya emission. The remaining line emissivities are given by
1. When modeling the mean density in coronal holes in
°° 4È5 (averaged over dense polar plumes and less-dense
interplume regions), we choose S \ 6, an average representative value.
2. When modeling the plume structure of coronal holes
explicitly (see ° 6), we choose S \ 8 as an enhancement over
the lower interplume electron density. When dense polar
plumes are added, this results in a similar range of net S
values (4È8) as are observed.
hl
j(coll) \ 0 q (T )n n /
l
4n 12 e e 1 l
Note that the exact form of equation (7) is important only
for observations near the edges of coronal holes. When used
to predict the radial variation of pB over the polar axis of
the hole, equation (7) gives the same result, to within 2%, as
a simpler model consisting of no latitudinal variations (i.e.,
S \ 1). This is mainly because of the assumed superradial
expansion, which strongly deÑects the bulk of the streamersÏ
higher density away from the poles. We thus conÐdently
adopt npole(r) \ nsph(r), as given in equation (3), as the polar
e of meane electron density in the three-dimensional
variation
empirical model.
Once the spatial distribution of density and Ñow tube
geometry are known, the conservation of mass Ñux in the
outÑowing solar wind can be used to derive empirical
values of the radial velocity. However, we defer discussion
of this until we examine the latitudinal variation of the
Doppler-dimmed H I Lya intensity, where we Ðnd that the
Ñow along individual Ñux tubes may depend sensitively on
latitude. In theoretical models of open magnetic Ðeld lines
over coronal holes (see, e.g., Wang & Sheeley 1990 ; Sittler et
al. 1997), the amount of superradial expansion may similarly be a strong function of latitude. Although not required
for reproducing only the white-light data, a more detailed
model of this Ñow tube structure may be needed to understand other diagnostics (see °° 4.2È4.3).
4.
NEUTRAL HYDROGEN PARAMETERS
The Lyman series lines of neutral hydrogen are some of
the most prominent features in coronal ultraviolet spectra.
Coronal H I Lya (j1215.67) emission is caused mainly by a
resonant scattering of chromospheric H I Lya photons, and
the higher energy transitions (e.g., Lyb and Lyc) are formed
by a combination of resonant scattering and local collisional excitation (see Gabriel 1971 ; Withbroe et al. 1982).
The shapes and strengths of these lines are determined by
the density and velocity distribution (microscopic and
macroscopic) of neutral hydrogen in the corona, which
closely reÑects the distribution of protons at low heights
(see ° 8). In this section we construct an empirical model of
these quantities from UVCS measurements of H I Lya line
proÐles.
The emergent speciÐc intensity of an emission line from
an optically thin corona is
I (nü ) \
l
P
`=
dx[ j(coll) ] j(res)] ,
l
l
(8)
~=
where nü is a unit vector pointing along the observerÏs LOS
in the x-coordinate direction, and j(coll) and j(res) are coll
lisionally excited and resonantly scattered
line lemissivities,
P Q
hl
=
j(res) \ 0 B n
dl@
l
4n 12 1
0
(9)
d)@
R(l@, nü @, l, nü )I3 {(nü @) ;
l
4n
(10)
(Mihalas 1978 ; Withbroe et al. 1982 ; Olsen, Leer, & Holzer
1994). Here, l is the rest frame line center frequency, q is
0
12
the collision rate per particle, n is the number density in
1
the lower level of the atom or ion of interest (here neutral
hydrogen, n \ n ), and B is the Einstein absorption rate
1
H
12
of the transition. The emission proÐle / is assumed to be
l
Gaussian, and the photon redistribution function R (from
incident frequency l@ and direction nü @ into observed frequency l and LOS direction nü ) is discussed in detail by, e.g.,
Allen, Habbal, & Hu (1998) and Li et al. (1998) for a general
anisotropic ““ bi-Maxwellian ÏÏ velocity distribution (see also
Olsen et al. 1994 ; Cranmer 1998). We utilize the empirical
chromospheric intensity proÐles I3 { given by Gouttebroze et
l
al. (1978) for H I Lya, in conjunction
with absolute solar
disk intensities measured by UVCS (see Raymond et al.
1997).
We have developed several independent computer codes
to calculate theoretical line proÐles by numerically integrating equations (8)È(10). We perform the integrations over
x and l@ using RombergÏs extrapolation method (Press et al.
1989), and we integrate over the solid angle of the solar disk
(d)@ \ sin h@dh@d/@) by Gauss-Legendre quadrature in h@
and equally spaced trapezoidal quadrature in /@. The small
degree of observed limb brightening of the observed intensities (Basri et al. 1979) is neglected, and the solar disk is
assumed to be uniformly bright.
We utilize the tabulated atomic data of Verner, Verner, &
Ferland (1996) for rest frame frequencies and B rates. The
12 that are
ionization, recombination, and excitation rates
used to compute (n /n ) and q are described by Raymond
H e coronal
12 conditions and assumed
et al. (1997). For typical
Maxwellian electron distributions, these rates depend primarily on the electron temperature, which can be measured
in streamers by UVCS from either the Thomson-scattered
component of H I Lya (Fineschi et al. 1997) or the relative
intensities of two lines with di†erent radiative and collisional strengths, e.g., Lya and Lyb (Maccari et al. 1997).
For the present coronal hole model, however, let us adopt
the electron temperature inferred by Ko et al. (1997) from in
situ charge state measurements in the fast solar wind made
by the SWICS instrument on Ulysses (Gloeckler et al. 1992).
We Ðt their modeled radial variation of temperature by the
simpler parameterization :
C A B
A B D
r 1.1
r ~6.6 ~1
]1.9
, (11)
T (r) \ 106 K 0.35
e
R
R
_
_
which agrees better than 5% with the values of Ko et al.
(1997) in the range 1 R \ r \ 10 R . Note, however, that
_ a slow minor
_ ion outÑow speed in
Ko et al. (1997) assumed
the corona ; Esser, Edgar, & Brickhouse (1998a) and Ko et
al. (1998) have shown that higher outÑow speeds consistent
with Doppler-dimming measurements may have an impact
488
CRANMER ET AL.
Vol. 511
on this inferred T , but the Ðnal iterated values are probably
e
not signiÐcantly di†erent from those in equation (11) above.
Resulting theoretical line proÐles from the codes
described above can be used to constrain and model various
physical quantities in coronal holes. In ° 4.1 we compare
observed proÐle shapes and widths to parameterized velocity distributions in the LOS direction ; these have direct
application to theories of heating and temperature anisotropies in the solar wind. In ° 4.2 we utilize the Dopplerdimming technique to place limits on the H I outÑow
velocities and most probable speeds of the distribution. In
° 4.3 we continue the geometrical analysis of ° 3.2 and make
further conclusions about the latitudinal structure of
coronal holes at solar minimum.
generates bulk motions along the LOS. The Doppler shifts
associated with these motions can broaden spectral lines.
2. NonÈ90¡ scattering results both in a superposition of
w and w in the line proÐles and in a systematic narrowing
M
A
of all proÐles due to incident radiation from the solar limb
(see Withbroe et al. 1982 ; Cranmer 1998). For H I Lya in
coronal holes, these e†ects are small and a†ect line shapes
only at the ^10% level.
3. The LOS may include local sites with di†erent velocity
distribution parameters. The plasma in, e.g., polar plumes
may be heated or accelerated di†erently than the ambient
coronal hole wind, and the emergent proÐles would contain
characteristics of both. The present analysis of H I Lya,
however, makes use of LOS-averaged quantities.
4.1. L ine-of-Sight V elocity Distribution
The line proÐles of H I Lya provide a direct measurement
of the total velocity distribution of neutral hydrogen atoms
along the LOS. The shape of the collisionally excited emission, as shown in equation (9) above, depends on only the
local emission proÐle / which is Gaussian for Maxwellian
l This component, however, is very
velocities along the LOS.
small for H I Lya ; it typically makes up 0.1%È0.5% of the
total coronal emission. The dominant resonantly scattered
emission is more complicated (see Allen et al. 1998 ; Li et al.
1998), but for 90¡ scattering from a point source of monochromatic incident radiation, with atoms in a general biMaxwellian velocity distribution, one can estimate
Thus, the models incorporate e†ects that are consistent
with the assumed structure and geometry of the modeled
coronal holes, but nothing additional. For simplicity and
generality, we use a bi-Maxwellian velocity distribution in
the models, but we do not include assumptions about speciÐc methods of heating or accelerating the solar wind ; see
° 8 for discussion about the implications of the models on
such theories.
Example UVCS H I Lya proÐles have been presented by
Kohl et al. (1997a, 1997b), and here we note only that the
derived coronal componentsÈover polar coronal holes,
near solar minimumÈare Ðtted adequately by single
Gaussian functions. Interplanetary hydrogen, stray disk
light, and the F-corona contribute to a narrow central line
peak that makes the total proÐle deviate from a Gaussian,
but these have been taken into account in the constrained
curve Ðtting discussed by Kohl et al. (1997a). In Figure 5 we
plot the 1/e half-widths V of coronal H I Lya lines over
1@e
the poles, from the observations
listed in Table 1. These
CA
BD C A BD
l[l 2
u 2
1
0
exp [
j(res) P
exp [
l
l w /c
w
w w
0 M
A
A M
(12)
(see also Cranmer 1998), where w and w are the most
A and perpendicular
M
probable 1/e speeds of atoms parallel
to
the magnetic Ðeld. The expression above assumes a radial
Ðeld for simplicity. These most probable speeds correspond
to independent ““ kinetic temperatures ÏÏ T and T in the
KA
KM
two directions, via
w2 4 2kT /m , w2 4 2kT /m ,
(13)
A
KA H
M
KM H
where k is BoltzmannÏs constant. In most subsequent applications, however, we retain the velocity units in these quantities because they may contain both microscopic random
motions as well as any unresolved bulk Ñuctuations along
the LOS (e.g., transverse wave velocities) which may not be
interpreted properly as temperatures.
The resonant H I Lya emissivity also depends on the
parallel macroscopic outÑow speed u ; this is the so-called
Doppler dimming e†ect, determined by the amount of
Doppler shift between the incident intensity proÐle and the
scattering line proÐle in the moving coronal plasma (Hyder
& Lites 1970 ; Beckers & Chipman 1974 ; Withbroe et al.
1982). For coronal resonance scattering, the line proÐle
shape arises mainly from the perpendicular motions, while
the variation of total intensity (Doppler dimming) arises
mainly from the parallel motions.
The above idealized conditions imply purely Gaussianshaped emission lines, but many physical e†ects can generate proÐle shapes that deviate signiÐcantly from this simple
form. In the models presented in this paper we incorporate
the most straightforward of these, for example :
1. A full speciÐcation of the vector outÑow velocity along
the superradially expanding Ðeld lines (see °° 3.2 and 6.1)
FIG. 5.ÈRadial dependence of V line-width velocity in polar coronal
holes for H I Lya. Data points from 1@e
the north (south) heliographic pole are
represented by squares (triangles), and the solid line indicates the best Ðt to
the data. Also plotted are the modeled w velocities for models A1 (dashed
M probable speed w correspondline) and A2 (dot-dashed line), and the most
e
ing to the electron temperature (dotted line).
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
widths are expressed in Doppler velocity units, where
V
*j
1@e \ 1@e
(14)
c
j
0
and *j
is the observed 1/e half-width in wavelength.
1@e
These measurements are consistent, within the uncertainties, with the V line width parameters measured at a
1@e in 1996 by Kohl et al. (1997a). There
slightly earlier period
are negligible di†erences between the line shapes in the
north and south coronal holes, so we Ðt V in Figure 5
1@e
with a single function of radius, combining the data over
both poles. For improved statistics, we have summed over
15@È25@ of data across the UVCS spatial slit ; this averages
over several plume and interplume regions and ignores the
small di†erences in line width between such structures (Noci
et al. 1997 ; Kohl et al. 1997b).
The error bars in V were computed from the results of
1@e
several sets of nonlinear least-squares Ðts of the line proÐles.
For the H I Lya proÐle measurements used here (see also
Fig. 6 for intensity measurements), Poisson count rate statistics contributed negligibly to errors in the cores of the
lines. At large heights, however, the observed proÐles are
increasingly dominated by instrumental stray light and
emission due to interplanetary hydrogenÈboth of which
are concentrated in a narrow peak near line center. We
varied the central intensity of the adopted stray light by
^30% (see Panasyuk et al. 1999) and the interplanetary
emission by ^60% (see Bertaux et al. 1997 for the full-sky
variation) and Ðtted the data at each height for all cases,
taking note of the range of resulting V widths and the
1@e parameter. The
formal ^1 p uncertainties in this Ðtting
quadrature sums (square root of the sum of the squares) of
these various uncertainties are computed and plotted as
error bars in Figure 5. Note that above 3 R the uncer_
tainties are very large ; the analysis of additional
UVCS
coronal hole data between 3 and 6 R is presently under_
way to improve and extend these measurements
(Suleiman
et al. 1998).
The values of coronal V
velocities are signiÐcantly
1@e UV instruments over the
larger than observed with early
solar poles, usually at a higher period of solar activity
(Withbroe et al. 1985, 1986 ; Strachan et al. 1993), but
similar to those measured by the Ultraviolet Coronal
Spectrometer on the Spartan 201 spacecraft (Kohl, Strachan, & Gardner 1996b). If V
were translated into a
1@e e.g., equation (13), the
hydrogen kinetic temperature using,
inferred T would greatly exceed the adopted electron temperature TK (eq. [11]) at all heights. In the present data set,
the kineticetemperature corresponding to V at o \ 1.5 R
1@e to 3.5 ] 106
_
is 2.2 ] 106 K, while at o \ 2.5 R it has risen
_
K. The kinetic temperature seems to remain roughly constant at this level out to 4 R , but the uncertainties grow
_
larger at these heights as the coronal
component becomes a
sharply decreasing fraction of the total observed proÐle.
In terms of the local coronal velocity distributions that
give rise to the proÐles, the measured V contains contri1@e (incorporated
butions from both the unresolved motions
into w and w ) and the LOS-projected macroscopic wind
A u. These
M two types of motion are separated in the
outÑow
self-consistent modeling process to be described below. It is
important to emphasize, though, that it is beyond the scope
of this paper to attempt to distinguish between the various
unresolved components making up w and w . Most
A
M
489
notably, transverse MHD wave motions, if averaged over
the time of the observation and the LOS, can increase w in
M
a manner that is extremely difficult to separate from truly
microscopic random motions (see, e.g., Esser 1990).
4.2. Doppler-dimming OutÑow and Most Probable Speeds
In Figure 6 we plot total line-integrated intensities, in
photons s~1 cm~2 sr~1, of the H I Lya observations discussed above. Di†erences between the north and south
polar coronal holes are negligible. These data points are
Ðtted by a similar radial power series as used for pB and n ,
e
and the nonlinear least-squares best Ðt is given by
I (Lya) \ 4.09 ] 1012
tot
A B
A B
R 11.3
R 6.93
_
] 3.91 ] 1011 _
.
o
o
(15)
The error bars in Figure 6 represent ^1 p uncertainties due
to radiometric calibration, photon counting statistics, and
background subtraction (see, e.g., Strachan et al. 1993 ;
Gardner et al. 1996).
At this point we have enough observational information
to build consistent empirical models of the H0 distribution
in coronal holes and to compare measured and predicted H
I Lya proÐles and intensities. We evaluate equations (8)È(10)
numerically for the adopted electron plasma parameters
(n , T ) and trial neutral hydrogen parameters (n , w , w ,
H density
A M
u).e Ate present, we utilize the smooth mean electron
in coronal holes (i.e., averaged over plumes and interplume
regions) derived in °° 3.1È3.2, and we use equation (11) for
FIG. 6.ÈT op : total line-integrated intensity over the heliographic poles
for H I Lya (solid line), O VI j1032 (squares, dashed line), and O VI j1037
(triangles, dotted line). Bottom : ratio between O VI j1032 and O VI j1037
line intensity.
490
CRANMER ET AL.
the electron temperature. The sources for the ionization
balance computation for n have been described above.
H
This leaves three free parameters
(w , w , u) to vary as
A M
functions of radius, with the goal of producing agreement
with the observed variations in V and I with height over
1@e
tot
the poles.
For all models, we specify only the radial variation of w
A
and w over the poles ; these are considered averaged over
M
any latitudinal dependence across the LOS. We iterate the
radially varying magnitude u(r), but the direction of the
outÑow velocity vector is kept Ðxed. We take the superradially diverging geometry deÐned in ° 3.2 for the entire
coronal hole and adopt it in a self-similar way to individual
Ñux tubes in the open Ðeld region (as described for polar
plumes in ° 6 ; see also Fig. 12). There will thus be a signiÐcant fraction of the outÑow directed along the LOS, which
tends to broaden the computed proÐles.
The model iteration process was initiated by assuming
w \V
(see eq. [12]) and u \ 0. Because the Doppler
M
1@e for H I Lya depends on both u and w , it is diffidimming
A
cult to determine both independently. We thus
choose
plausible extreme limits on w and vary both u and w to
A for each case. In modelMA1,
construct self-consistent models
we set a reasonable lower limit by assuming the parallel
hydrogen motions are in thermal equilibrium with the
electrons (w \ w 4 [2kT /m ]1@2). In model A2, we set
A
e assuming
e H an isotropic distribution
an upper limit
by
(w \ w ). The theoretical justiÐcations for these limits are
A
M in ° 8.
discussed
We iterate on the outÑow speed by using the approximate Doppler-dimming relation in equation (12). This
equation is consistent with the assumption that
I
tot
\ constant in u ,
(16)
exp ([u2/w2)
A
which can be transformed into an iterative solution, from
step i to step i ] 1, for the velocity,
\ Ju2 ] w2 ln (I /I ) .
(17)
i`1
i
A
tot,i obs
This relation converges when the modeled intensity I
tot,i
agrees with the observed total intensity I . Note that
obs
equation (17) does not depend critically on whether the
simple Doppler-dimming approximation (eq. [16]) holds or
not ! It merely uses its functional form to Ðnd a reasonably
efficient path to the best solution. Note also that we iterate
on the entire function u(r) simultaneously (and alter its
parameterization as needed), which automatically takes
into account LOS-integration e†ects.
The Doppler line width V is already a reasonably close
1@e most probable speed w , so
estimate for the perpendicular
we iterate on this latter variable using a much simplerMprescription : the most current value of w (r) is multiplied by
the radial variation in the ““ discrepancyM ratio ÏÏ between the
observed and computed V widths, to obtain the next iteration. This ratio initially 1@e
starts out between 0.80 and 0.95
because of broadening by components of the outÑow velocity along the LOS, and it rapidly converges to unity over
several iterations. Because of the spread in the data points
for V , we use the best-Ðt curve in Figure 5 as the Ðducial
1@e value to compare with synthesized line widths.
observed
The empirical w and u velocities for H I Lya models A1
M
and A2 each converged
to within 1% variations after Ðve
iterations, and we plot the solutions for w in Figure 5 and
M
u
Vol. 511
FIG. 7.ÈEmpirical H0 outÑow velocities over the poles, derived from H
I Lya Doppler dimming. Models A1 and A2 are denoted by heavy solid
and dashed lines, respectively. The error bars (light bracketing lines) are
derived from observational and modeling uncertainties (see text). Dotted
lines represent proton mass Ñux conservation with f \ 6.5 (upper line)
max
and f (r) \ 1 (lower line).
for u in Figure 7. Because the Doppler dimming for H I Lya
is sensitive to the electron density, which is uncertain below
D1.9 R (see ° 3.1), we present outÑow velocities only
_ radius. Above this height, note that the velocities
above this
for model A2 are signiÐcantly larger than for model A1
because a larger w requires a larger u to produce the same
amount of DopplerA dimming (eq. [12]). Also, the converged
values of w for model A1 are closer to V than for model
1@e to V
A2. This isM because u and w contribute
at the
A
10%È15% level, and their larger contamination 1@e
in model
A2 requires a slightly smaller w to produce the observed
M
V .
1@e
The solution velocities (in units of km s~1), which are
valid between 1.9 and 4 R , are given by the following
_
best-Ðt functions :
A B
C A B
A B D
A B
C A B
A B D
R 3.47
Model A1 : u(r) \ 110 ] 445 1 [ _
r
w (r) \ 174 ] 200 5.70
M
] 5.86 ] 104
R 0.499
_
r
R 14.3 ~1
_
r
R 5.46
Model A2 : u(r) \ 112 ] 1450 1 [ _
r
w (r) \ 176 ] 200 7.68
M
] 1.22 ] 106
(18)
(19)
(20)
R 0.539
_
r
R 18.9 ~1
_
r
(21)
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
(see Table 2). The uncertainties in u(r) are plotted in Figure
7 as light lines above and below the heavy lines and are also
listed in Table 2. These limits on the outÑow velocity uncertainty have been computed in a similar manner as in Strachan et al. (1993). In that work, the observational and
modeling uncertainties dI (in the total intensity) were
treated in di†erent ways ; here we assume they are statistically uncorrelated with one another and thus should
have identical e†ects on the resulting parameter uncertainty. We thus compute a total uncertainty by taking the
square root of the sum of the squares (i.e., the quadrature
sum) of the observational and modeling uncertainties. The
resulting net dI is converted into lower and upper bounds
on the outÑow velocity (u ^ du ) using equation (16) :
B
u2 [ (u ^ du )2
dI
B
.
(22)
1 < \ exp
w2
I
A
The observational uncertainties in the H I Lya intensity
have been described above (Fig. 6). The empirical modeling
uncertainties arise from three factors in the resonancescattering emissivity :
C
D
1. T he electron density (n ).È In addition to a minimum
uncertainty of ^25% in thee white-light measurements, we
also include a one-sided downward uncertainty in n arising
e
from the possible existence of uncharacterized polarized
stray light in the WLC channel (see ° 3.1 and the gray region
in Fig. 1). The total adopted uncertainty limits in n expand
e R .
from ^25% at large heights to (]25%, [45%) at 1.9
_
2. T he H0 ionization fraction (n /n ).ÈAt electron temH
e
peratures around 106 K, the neutral hydrogenÈtoÈproton
density ratio is roughly proportional to T ~1 (see, e.g.,
Gabriel 1971), so if T is known to only aboute^25%, so is
e
the ionization fraction.
3. T he incident chromospheric intensity (I3 ).ÈBecause
l
UVCS/SOHO has measured the H I Lya intensity
on the
solar disk (Kohl et al. 1997a), we are able to rely less on the
absolute intensity calibration of the instrument and use the
measured corona-to-disk ratio, which is well known with
only a ^10% uncertainty.
491
D3.0 R . This represents indirect, but reasonably Ðrm, evi_
dence that the H0 velocity distribution (along with the
protons, most probably) is anisotropic, with w [ w .
M
A
4.3. L atitudinal Structure of the Coronal Hole
In ° 3.2 we developed a general picture of the geometry
and large-scale latitudinal density dependence of polar
coronal holes near solar minimum. Here we investigate if
this derived structure is consistent with H I Lya observations as a function of position angle around the Sun. As
with the white-light data, there is a strong overall increase
in H I Lya intensity in the equatorial streamers as compared
with the poles. We focus, however, on the quantitative
shape of this variation at midlatitudes (h B 20¡È60¡) to
evaluate how both density and the velocity distribution
vary across the magnetically open regions associated with
coronal holes.
Figure 8 shows H I Lya intensity observations taken on
1997 April 14È15, at o \ 3 R and between position angles
_ intensities have been nor85¡ and 190¡. The total H I Lya
malized by their mean value at the south heliographic pole,
as well as scaled (using the relative radial variation of eq.
[15]) to account for the slight variation in heliocentric
radius across the horizontal slit. Thus, all data points refer
to 3 R , even if they were taken up to ^0.3 R away from
_
that height.
The abscissa h is deÐned as o PA [_180¡ o , or the
net angular distance from the south pole. We do not plot
the intensity in streamers because the empirical models do
not apply there.
We also plot in Figure 8 synthesized H I Lya intensities
for the empirical model A1 developed above, using two
di†erent assumptions about the large-scale geometry. The
solid line is the result of a model in which the outÑow
velocity follows the superradial Ñux tubes described in ° 3.2
(see also Fig. 12, below). The dotted line results from the
assumption that the wind velocity u(r) is oriented purely
Also plotted in Figure 7 is a gray region that shows the
expected proton outÑow velocity assuming mass Ñux conservation over the poles :
n (r)u(r)r2f (r) \ constant .
(23)
p
We use in situ measurements of the fast solar wind to evaluate the constant, with a mean mass Ñux of n u B 2 ] 108
cm~2 s~1 at r \ 215 R \ 1 AU (Goldstein et pal. 1996), and
we assume n \ 0.8n _
for a fully ionized plasma with 10%
e
helium.13 Thep lower dotted
curve in Figure 7 represents the
limit of purely radial expansion, or f \ 1 at all radii. The
upper dotted curve results from applying equation (6) for
the overall geometry of the coronal hole. It is clear that the
Ñux tube divergence is a minor factor compared to the
uncertainties in the Doppler-dimming calculation. Above
r B 2.2 R the outÑow velocity derived from model A1 is in
_
general agreement
with that assuming mass Ñux conservation and clearly not in agreement with model A2 above
13 Ulysses data also exhibit a ^30% small-scale variation in the proton
mass Ñux in the high-speed wind (Goldstein et al. 1996) ; this should be
taken into account as an uncertainty when computing the viable range of
coronal outÑow velocities from mass conservation.
FIG. 8.ÈLatitudinal dependence of H I Lya total intensity at o \ 3 R
_
vs. h \ o PA [ 180¡ o ; observations (squares), empirical model with outÑow
along nonradial Ñux tubes (solid line), and empirical model with purely
radial outÑow (dotted line).
492
CRANMER ET AL.
radially. Because Doppler dimming depends primarily on
the components of the velocity in the directions intercepting
the solar disk (i.e., mainly radial), the superradial model has
much less Doppler dimming and a higher intensity in the
middle latitudes, where the Ñux tubes are nonradially
oriented (see also Dobrzycka et al. 1998). This comparison
between the empirical model and observations at o \ 3 R
_
has also been done at 1.5, 2.0, 2.5, and 3.5 R with the same
_
general result : there is better agreement when the velocity
vectors follow the nonradial Ñux tubes. The high-speed
wind thus seems to Ñow along superradial stream lines, but
it remains reasonably constant in magnitude (at a given
radius) as a function of latitude.
Cranmer et al. (1997) investigated possible latitudinal
changes in the magnitude of u due to di†erences in the
magnetic Ðeld expansion from pole to equator. In some
magnetostatic, force-free models of the coronal magnetic
Ðeld (see, e.g., Low 1986 ; Wang & Sheeley 1990), open Ñux
tubes bordering the closed Ðeld streamer belt initially
expand much more rapidly than Ñux tubes over the poles,
and mass conservation then demands u at a given radius to
grow gradually smaller with increasing colatitude (from
pole to equator). However, synthesized H I Lya intensities
from these models do not agree with the observations ; they
predict too much intensity in the midlatitudes. Cranmer et
al. (1997) found that the most likely Ñow tube geometry in
the open Ðeld regions is one with f (r) reasonably independent of h (which results in the magnitude of u not varying
signiÐcantly with h), but the Ñow direction consistent with
the nonradial expansion of the coronal hole. These types of
conclusions, however, can be complicated by LOS e†ects
and departures from axisymmetry in the modeled corona
(see also ° 8).
5.
IONIZED OXYGEN PARAMETERS
The most surprising initial results from the Ðrst year of
operation of UVCS have been the extremely broad coronal
proÐles of such highly ionized elements as oxygen and magnesium (Kohl et al. 1997a, 1997b). In this section we develop
an empirical model of the distribution of O5` ions in
coronal holes from the bright O VI jj1031.93, 1037.62
emission-line doublet. Because of ““ pumping ÏÏ by the nearby
C II jj1036.34, 1037.02 lines, the intensity ratio between the
two O VI lines is a sensitive probe of both the outÑow
velocity and the radial most probable speed (Noci, Kohl, &
Withbroe 1987 ; Li et al. 1998).
We utilize the same line synthesis techniques described in
° 4 to compute the O VI emission from a given model. For
these lines, the collisional and radiative emissivities (eqs. [9]
and [10]) are comparable with one another, despite the fact
that the radiative part scales linearly with the density and
the collisional part scales with the square of the density. It is
fortunate that the ratio of O VI j1032 to O VI j1037 intensity is so sensitive to the ion velocity distributions because
by basing the empirical models on this ratio we need not
make assumptions about the oxygen elemental abundance
or the O5` ionization balance. Below, however, we do
investigate the coronal abundances and ionization states
once the basic self-consistent models have been constructed.
Because the absolute intensities of the oxygen lines are
smaller than H I Lya by at least an order of magnitude, the
larger associated uncertainties make it more difficult to
probe the plume/interplume structure quantitatively in this
useful diagnostic. More work in this area is underway, but
Vol. 511
in this paper we model only the mean structure, averaged
over small-scale inhomogeneities, of the polar coronal holes
(see, however, Kohl et al. 1997a, 1997b ; Noci et al. 1997 for
preliminary O VI plume results).
5.1. L ine-of-Sight Velocity Distribution
In Figure 9 we plot the radial dependence of the 1/e
widths of the O VI jj1032, 1037 lines over north and south
polar coronal holes. The spectral data have been reduced in
the same way as for H I Lya, with instrumental broadening,
the line-wing detector e†ect, and stray light from the solar
disk and F-corona all taken into account consistently (see
Kohl et al. 1997a ; Panasyuk et al. 1999). The remaining
coronal components can all be Ðtted reliably with single
Gaussian functions. The large spread in V at large heights
1@e
arises from both line width measurement uncertainties
(computed in the same way as described in ° 4.1) and a
possible long-timescale variability of this sensitive plasma
parameter in the extended corona. Because we do not
discern any systematic di†erences between the shapes of the
two components of the doublet, we Ðt the entire data set
with a single parameterized function. As in ° 4, we use this
function only as a well-behaved Ðducial mean to compare
with the synthesized proÐles.
The most striking aspect of these measurements is the
unusually large magnitude of V
at most heights, compared with H I Lya. Neither of the1@etwo most simple explanations for the origin of this line broadening (thermal motions
and transverse wave motions) can be dominant :
1. If the plasma were in thermal equilibrium, the line
widths would be dominated by a perpendicular most probable speed w that would depend on the mass of the ion to
M Thus, for equal temperatures, oxygen ions
the [1/2 power.
FIG. 9.ÈRadial dependence of V line-width velocity in polar coronal
1@e south : triangles) and O VI j1037
holes for O VI j1032 (north : squares,
(north : diamonds, south : crosses). Also plotted is the best Ðt to the data
(solid line), the most probable speed w corresponding to T (dotted line),
the maximum allowed parallel O VI emost probable speede w
(triplemax and B2
dotÈdashed line), and the modeled w for models B1 (dashed line)
M
(dot-dashed line).
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
would exhibit a velocity 4 times smaller than for hydrogen.
At the lowest heights in Figure 9, V for O VI is about 2
1@e this holds only for
times smaller than for H I Lya, but
measurements up to about 1.7 R , where the oxygen line
_
widths begin to exceed those of hydrogen.
2. If the dominant LOS velocities arose from common
transverse wave motions, all species should exhibit nearly the
same values of V (i.e., T P m ). In a collisionless plasma,
1@e of low-frequency
K
ion
the superposition
MHD waves with the
Larmor gyrations about the Ðeld lines results in transverse
particle velocities that are independent of their mass or
charge (see, e.g., Khabibrakhmanov & Mullan 1994). The
addition of collisions serves only to equalize the temperatures, thus bringing the situation closer to thermal
equilibrium.
It is clear that the V velocities for O5` ions are signiÐ1@ebe understood by the above two
cantly larger than can
scenarios (nor by any linear combination of thermal and
wave motions). In terms of kinetic temperatures, the measured line width corresponds to 9.3 ] 107 K at 2 R , and it
_
rises to 2.4 ] 108 K at 3 R . This latter value is almost
300
_
times larger than the inferred electron temperature at this
radius, and it exceeds the largest modeled H0 kinetic temperature at this radius by at least a factor of 60.
We discuss some possible theoretical explanations of
these extreme line widths in ° 8, but let us keep in mind that
V samples only the velocity distribution along the LOS,
1@e is mainly perpendicular to radii over the solar poles.
which
Doppler-dimming measurements of the relative intensity
ratio between the 1032 AŽ and 1037 AŽ components allow us
to set limits on the radial, or parallel component of the
velocity distribution, which need not be the same as in the
perpendicular direction.
5.2. Doppler-dimming OutÑow and Most Probable Speeds
In Figure 6 we plot the total line-integrated intensity of
the O VI j1032 and j1037 lines, as well as the ratio R 4
I (1032)/I (1037). As with H I Lya, we Ðnd that north/
tot
tot di†erences are negligible. We Ðt the data by
south
polar
nonlinear least-squares minimization and obtain
A B
A B
A B
A B
493
tering without Doppler dimming, R \ 4, where the above
factor of 2 is multiplied by a factor of 2 di†erence in the
incident chromospheric intensities. For the UVCS observations, the above Ðtting functions give R \ 3.1 at the
lowest measured height (1.5 R ). The ratio drops to 2 by 1.9
_
R , to 1 by 2.5 R , and to a minimum
mean value of 0.9 by
_
_
2.8 R before slowly rising again. We expect that for large
_
enough radii, R ] 2, since Doppler dimming eventually
removes the resonantly scattered component.
Noci et al. (1987) also determined that the resonant component of the O VI j1037.62 line can be pumped efficiently
by incident intensity from the C II j1037.02 line. In velocity
space, these lines are separated by 173 km s~1, which
implies that outÑow velocities near this value are resonantly
enhanced in intensity. Also, Li et al. (1998) recognized the
importance of the second C II j1036.34 line, which is
separated from the O VI j1037 line by 370 km s~1 (see
below). For completeness, we also include the weaker Fe III
j1035.77 line with a velocity separation of 534 km s~1 (see
Warren et al. 1997). Because the O VI j1037 line is in the
denominator of R, this pumping is the only known mechanism for producing O VI ratios less than 2.
The e†ective widths of the Doppler pumping resonances
in velocity space are determined by the small-scale velocity
distribution in the direction of the incoming radiation
(essentially w , but see below). For large enough values of
A
w , the 1037.62
and 1037.02 resonances are smeared
A
together for all outÑow velocities, and there is no isolated
enhancement near 150È200 km s~1. We illustrate this in
Figure 10, where we plot R versus u for modeled O VI
emissivity integrated over a polar LOS at o \ 3 R . We
_ are
model R(u) here using a range of u and w values that
A
constant in radius, along with the radial variation of w
M
from the empirical model B1 (see below).
It is evident from Figure 10 that smaller values of w
A
produce ratios R that have lower minimum values. This fact
R 14.7
I (O VI 1032) \ 1.57 ] 1012 _
tot
o
] 9.69 ] 106
R 2.88
_
,
o
(24)
R 12.8
I (O VI 1037) \ 2.31 ] 1011 _
tot
o
] 8.89 ] 107
R 4.89
_
,
o
(25)
and the error bars are computed as discussed in ° 4.2. Noci
et al. (1987) found that the ratio of these two intensities can
be used to determine the radial velocity distribution better,
and we will use this dimensionless quantity (which has the
advantage of not depending on the O5` abundance) in subsequent Doppler-dimming studies.
Theoretically, the ratio R should equal 2 for pure collisional excitation because of the factor of 2 di†erence in the
statistical weights of the transitions. For pure resonant scat-
FIG. 10.ÈModeled O VI line ratio at o \ 3 R as a function of outÑow
velocity, for a series of anisotropic models with_ constant w (labeled as
A maximum
kinetic temperature), and for a threshold w (r), which is the
max
able to agree with the observations (solid line). The perpendicular velocity
w is given by eq. (28) for model B1.
M
494
CRANMER ET AL.
can be exploited to put a Ðrm upper limit on w when the
A
ratio is less than D1.7. For example, in Figure 10 it is
impossible for the models with parallel kinetic temperatures
larger than 3 ] 107 K ever to agree with the observations of
R B 0.9 at that height, no matter the outÑow speed. Using
this constraint, we compute a threshold w \ w (r), above
A
max
which the intensity ratio could never reach the low values
observed at high heights. This limiting parallel most probable speed is plotted in Figure 9 and is given by
w
A B
(r) \ 1.49 ] 105
max
A B
R 8.27
r 3.15
_
] 1.30
km s~1 .
r
R
_
(26)
The corresponding intensity ratio is plotted in Figure 10.
The exact form of w (r) was initially estimated from the
max
grid of constant-w models in Figure 10, but when this Ðrst
A
estimate was used to synthesize proÐles and intensity ratios,
the minima of R(u) were not exactly equal to the observed
values. These di†erences arose because the strong radial
variation in w
inÑuenced the LOS integration, and three
max
subsequent iterations
were required to determine the selfconsistent form of w (r) presented above. Note from
maxR , this Ðrm upper limit on w is
Figure 9 that, for r [ 2.1
A
signiÐcantly smaller than V _ , implying strongly anisotropic
1@e
motions of O5` ions on unresolved scales at these heights.
Because of the strong anisotropy in the velocity distribution, the resonantly scattered intensity does not depend
solely on w (as in eq. [12]), but on a linear combination of
w and w A. Incident radiation from the limb of the solar
A intercepts
M
disk
a small, but signiÐcant fraction of the perpendicular motions, which broadens the Doppler-dimming
curves in Figure 10 to the point where only the C II j1036.34
line is isolated enough to cause R to dip below one (Li et al.
1998). This strongly implies that the O5` outÑow velocities
are large compared to hydrogenÈof order 400 km s~1 by
r B 2.5È3.0 R .
_
It is now possible
to model the velocity distribution of
O5` by comparing synthesized emission lines with observations. In a similar fashion as for H I Lya, two independent
observables (V
and R) are modeled by varying three
1@e (w , w , and u). Thus, as in ° 4.2, let us set
model parameters
A M on the parallel most probable
lower and upper limits
motions, and model the other quantities for these two cases.
In model B1 we assume w \ w , or thermal equilibrium
A w is
e a factor of (m /m )1@2 B
with electrons. Note that here
e
O HB2 we
4 smaller than assumed for model A1. In model
assume w is speciÐed by the smaller of w
and w at each
A
maxparallelMvelocity
radius, which
gives the maximum possible
that is less than or equal to the perpendicular velocity.
For models B1 and B2, we iterate on u and w to produce
optimal agreement with the observed values ofM R and V .
1@e
Because the O VI Doppler dimming/pumping is more complicated than for H I Lya (eq. [12]), we do not have a simple
algorithm for determining succeeding iterations in u. By
starting the modeling for models B1 and B2 at the Ðnal
iteration of model A1, however, we were able to subjectively
vary the values of u(r) and w (r) over six iterations to a
M with the observations.
converged state in agreement
Formally, this may not be a unique solution, but we are
conÐdent that it is the only one without unphysical discontinuities in radius.
The model velocities (in units of km s~1), which are valid
only between 1.5 and 3.5 R , are given by the following
_
Vol. 511
best-Ðt functions :
C A B
A B
A B D
(27)
C A B
A B D
(28)
Model B1 : u(r) \ 0.449 ] 200 0.645
] 48.2
R 6.32
_
r
] 1.47 ] 106
R 28.3 ~1
_
r
w (r) \ 71.6 ] 200 2.07
M
] 6970
R 0.324
_
r
R 1.15
_
r
R 15.6 ~1
_
r
C A B
A B D
(29)
C A B
A B
A B D
(30)
Model B2 : u(r) \ 2.99 ] 200 1.05
] 5.74 ] 104
R 19.8 ~1
_
r
w (r) \ 74.3 ] 200 0.230
M
] 4.69
R 0.747
_
r
R 0.0327
_
r
R 2.36
_
r
] 2.58 ] 105
R 22.9 ~1
_
r
(see also Table 2). The modeled perpendicular most probable speeds w are plotted in Figure 9. As in the H I Lya
models, w is Mslightly smaller than the observed V widths
because ofM the contaminating e†ects of the fast1@eoutÑow
velocities which broaden the proÐles along the LOS.
The modeled O VI outÑow velocities are plotted in Figure
11, and they can be understood in a similar way as those
derived from H I Lya. Generally, a larger e†ective w
A
requires a larger u to maintain the same amount of Doppler
dimming, but this is not strictly true when pumping by the
C II lines becomes important. We compute the velocity
uncertainties for models B1 and B2 using a similar method
as for H I Lya, but the dependence of the model parameters
on the line ratio is more complicated. Although the
modeled ratio does not depend on the abundance or ion
concentration of O5`, it is sensitive to uncertainties in the
electron density n (which determines the relative collisional
e
and radiative contribution),
the electron temperature (via
the collision rate q ), and the incident chromospheric
12
intensity I3 . The dependence
on n is also more complicated
l
e
for O VI lines because inhomogeneities
along the LOS can
a†ect the relative collisional and scattering intensities ; we
include an additional ^20% uncertainty in n to account
e are comfor these variations. The modeling uncertainties
puted by constructing new empirical models with the each
of the three error-sensitive parameters varied by a small
(1%) amount. The resulting small di†erences in R are used
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
FIG. 11.ÈSame as Fig. 7, but with empirical outÑow velocities derived
from O VI intensities and widths for models B1 (solid lines) and B2 (dashed
lines).
to compute the partial derivatives in the formal uncertainty :
C A B
A B
A BD
LR 2
LR 2
LR 2 1@2
] (dT )2
] (dI3 )2
.
dR \ (dn )2
e LT
l LI3
e Ln
e
e
l
(31)
This uncertainty is quadrature-summed with the observational uncertainty plotted in the error bars of Figure 6
and is then translated into an uncertainty in the outÑow
velocity. Note, however, that for O VI, the observational
error bars comprise the majority of the uncertainty ; the
modeling uncertainty has little e†ect. The translation from
intensity uncertainty to velocity uncertainty is more
involved than in the case of H I Lya. The individual
Doppler-dimming curves (as in Figure 10) for each height
are used to Ðnd the velocities intersected by ratios
(R ^ dR). When a single ratio can correspond to multiple
outÑow velocities, we constrain the uncertainties du to be
the closest perturbations to the model velocity u ; this is
consistent with the assumption of a monotonically increasing velocity with monotonically increasing upper and lower
uncertainty limits.
As predicted from Figure 10, the modeled outÑow velocity for O5` ions is signiÐcantly larger than the proton velocity implied by mass Ñux conservation, and also larger than
the H0 velocity from model A1 (see also Li et al. 1998). The
lowest O5` velocities pass within the uncertainty limits of
the outÑow derived from model A2. Note that the derived
outÑow velocity depends on the direction of the Ñow
vectors with respect to the direction of the incoming radiation ; if the empirical Ansatz of superradial divergence were
to be replaced by radial expansion, the derived velocities for
O5` (greater than D200 km s~1) would be lowered by
approximately 50 km s~1. The uncertainties are signiÐcant,
but it does seem probable that the outÑow velocity for
oxygen is substantially larger than that of hydrogen above
D2 R .
_
495
Spacecraft measurements have shown that the outÑow
velocity of oxygen ions at 1 AU is no more than D10%È
15% larger than that of the protons (see, e.g., Schmidt et al.
1980). However, the empirical outÑow velocities presented
here indicate that O5` ions may be Ñowing as much as
1.5È2 times faster than H0 atoms (and protons) in the
extended corona. The only reasonable way for this discrepancy to be consistent with mass Ñux conservation is for
(n 5`/n ) to vary with radius, which is not in agreement with
O
p
many theories of ““ frozen-in ÏÏ ionization for many ions in
the solar wind (see, e.g., Ko et al. 1997). SpeciÐcally, the
ionization fraction of O5` must increase from the corona
to 1 AU.
A recent in situ measurement of the O5` relative ionization in the solar wind may conÐrm this idea. Using the
SWICS instrument on Ulysses, Wimmer-Schweingruber et
al. (1998) measured (n 5`/n ) B 6 ] 10~3 in the solar wind,
O of 2O larger than predicted theoretiapproximately a factor
cally (see, e.g., Esser & Leer 1990). When this value is utilized (in concert with an oxygen elemental abundance of
n /n \ 8.5 ] 10~4 ; see Feldman 1992) to synthesize indiO H O VI jj1032, 1037 total intensities, we Ðnd that the
vidual
results do not agree with the observations. If the elemental
abundance is Ðxed, one requires approximately 6 times less
O5` in the extended corona as is observed at 1 AU to
match the observed intensities (i.e., in the corona,
n 5`/n B 1 ] 10~3). If mass Ñux is conserved, this inferred
Oof the ion concentration with radius is in agreement
O
growth
with the requirement for the O5 ` outÑow velocity enhancement (with respect to the protons) to decrease with increasing radius.
6.
PROPERTIES OF POLAR PLUMES
6.1. W hite-L ight Statistics
During times of low solar activity, coronal holes are
observed to contain bright raylike polar plumes that appear
to follow open magnetic Ðeld lines (Newkirk & Harvey
1968 ; Ahmad & Withbroe 1977 ; Wang 1994 ; and references
therein). These inhomogeneities may contain a signiÐcant
fraction of the total mass Ñux of the high-speed solar wind,
and it is important to include them in empirical studies of
the corona. The UVCS WLC, however, is not speciÐcally an
imaging coronagraph ; it samples only one 14A ] 14A spatial
element at a time. Thus, for physical and morphological
information about the distribution of plumes, we must rely
on statistical quantities.
In fact, the long time base of the SOHO mission a†ords
an unprecedented set of repeated pB measurements at high
heights, which can sample the density and plume characteristics in the extended corona, once per day over the span of
several months to years. Assuming that solar rotation and
small-scale magnetic activity e†ectively bring new plumes
into the Ðeld of view on rapid timescales (see, e.g., Lamy et
al. 1997 ; Wang 1998), these observations provide an excellent quasi-random statistical sampling of lines of sight
through the plume-Ðlled coronal holes. We thus empirically
model plumes as individual entities but utilize only statistical quantities (either from, e.g., a large number of observations or a large ensemble of randomized models) when
comparing with actual data.
Our method of simulating individual plumes follows that
of Wang & Sheeley (1995), where N identical plumes are
randomly placed in the coronal hole, and each is assumed
496
CRANMER ET AL.
to have a Gaussian enhancement in density about its Ðeld
line axis. The total electron density at r \ R is given by
_
n (R , h, /) \ n (R , h)
e _
0 _
n
N
] 1 ] p [ 1 ; exp [ [ (d /b)2] ,
i
n
0
i/1
(32)
G A
B
H
where n is a featureless interplume density (still to be
0
determined), n is the maximum density along the plume
p
axis, and d is the angular distance between an arbitrary
i
photospheric location (R , h, /) and the center of plume i
_
(see Wang & Sheeley 1995 for details). The parameter b is
the 1/e angular half-width of each Gaussian plume structure, which we set at a constant value of 0.0431 rad, corresponding to 3 ] 104 km (Ahmad & Withbroe 1977 ; Fisher
& Guhathakurta 1995). We distribute the plumes randomly
between the pole and # (R ) in colatitude h (more preci0 _
sely, uniformly between cos
# and 1 in cos h) and over all
azimuthal angles /. In radius,0 the plume ““ Ðeld lines ÏÏ are
constrained to follow colatitudes proportional to the hole
boundary at that radius : e.g., a Ðeld line at h \ 0.5# at the
0 local
solar base always follows a colatitude of 0.5 times the
# (r) (see Fig. 12). Note that this is only a simple and
0
approximate
means of locating the superradially expanding
Ñux tube boundaries within the coronal hole and should
ideally be replaced with a more realistic computation of a
magnetohydrodynamic (MHD) equilibrium conÐguration
(see ° 8).
The undetermined parameters in the above model are the
number of plumes N, the interplume density n , and the
0 individmaximum plume density n . We use a set of over 75
p
ual daily pB observations over the south heliographic pole,
taken between 1996 November 1 and 1997 February 1, to
constrain these parameters. Plumes are observed to evolve
on rapid timescales, but we assume that only the locations
of individual plumes change and that the three governing
parameters above (N, n , n ) remain Ðxed. This allows us to
0 p of measured pB values as a
use the temporal distribution
statistical sample of the constantly shifting plume population. Two well-determined quantities that we extract from
this sample are the mean SpBT and mean-square S(pB)2T
polarization brightness observed over the south pole, which
also provides the standard deviation about the mean. (We
do not attempt to extract higher moments of the distribution because of the much larger uncertainties in those
moments.)
The two measured moments SpBT and S(pB)2T are not
sufficient to specify the three undetermined parameters in
the plume model (eq. [32]) ; a further constraint must be
applied to solve simultaneously for N, n , and n . We
suggest two alternative methods which use0 di†erentp constraints :
1. If plumes are sparsely distributed, then some small
fraction of observations along a given LOS will intersect no
plumes. The minimum of the observed distribution of pB,
then, can be inverted (as in ° 3.1) to compute the assumed
interplume electron density n .
0 put constraints on the Ðlling
2. Earlier observations have
factor of dense plume material in coronal holes. Ahmad &
Withbroe (1977), for example, determined the area a in a
near-minimum coronal hole subtended by plumes (divided
by the total area) to be approximately 0.1. We can express
Vol. 511
TABLE 3
WHITE-LIGHT STATISTICAL MODELS OF POLAR PLUMES
o/R
PARAMETER
1.75
_
2.00
SpBTa . . . . . . . . . . . . . . . .
36.58
13.31
S(pB)2T1@2 . . . . . . . . . . .
36.75
13.35
S(pB)2T/SpBT2 . . . . . .
1.0088
1.0065
min (pB) . . . . . . . . . . . .
28.46
9.986
max (pB) . . . . . . . . . . .
43.59
15.35
Method 1 :
a ..................
0.515
0.866
n /nsph . . . . . . . . . . . .
0.670
0.730
0 e
n /n . . . . . . . . . . . . . .
1.388
1.261
p 0
Method 2 :
a ..................
0.100
0.100
n /nsph . . . . . . . . . . . .
0.770
0.884
0 e
n /n . . . . . . . . . . . . . .
1.887
1.761
p 0
a Polarization brightness values expressed in
(see text).
2.25
2.50
6.243
6.257
1.0043
5.379
7.261
3.161
3.169
1.0047
2.667
3.869
0.349
0.866
1.343
0.410
0.748
1.329
0.100
0.929
1.628
0.100
0.816
1.661
units of 10~10B1
_
this fractional e†ective area as a function of the number of
plumes N by
Nnb2
.
(33)
8[1 [ cos # (R )]
0 _
The above expression counts plume material as that within
an angular ““ equivalent width ÏÏ radius of d \ b(n/4)1@2
eff of b and
and assumes plumes do not overlap. For the values
# adopted above, we Ðnd that a \ 0.1 corresponds to
N0\ 16 plumes.
a4
Thus, by using either of these two methods, we reduce the
number of free parameters from three to two, and the other
two parameters are found by constructing a large grid of
models that synthesize a random ensemble of LOSintegrated pB values (e†ectively reproducing the observations and obtaining accurate moments SpBT and
S(pB)2T). For all cases considered, only one set of values for
the two varied parameters (a and n for model 1, n and n
p
0
for model 2) provides a match withp the observed moments
of the pB distribution.
Table 3 contains the statistical pB data from the observations, as well as solutions for the three model parameters
at each height between 1.75 and 2.50 R . The densities are
listed as dimensionless ratios : (n /n ) for_ the relative plume
p (n
0 /nsph) for the ratio of
to interplume enhancement, and
0 e density (eq. [3]).
interplume density to the mean observed
The level of variations in the observed pB values is more
than an order of magnitude larger than the expected Ñuctuations arising from Poisson count rate statistics, which
thus validates our use of them to determine actual coronal
inhomogeneities. (The other two sources of point-to-point
uncertainty discussed in ° 3.1 do not apply to the statistical
analysis of this set of observations because each measurement was taken in exactly the same manner each day.)
Although the results of models 1 and 2 are not completely
consistent with one another (especially for a), they present
the same general picture of inhomogeneities in the extended
corona : the dense plume component is approximately
1.3È1.8 times denser than the ambient interplume corona,
and the dense features Ðll a signiÐcant fraction of the
coronal hole volume. Other investigations of coronal inho-
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
497
TABLE 4
H I LYa STATISTICAL MODELS OF POLAR PLUMES
o/R
PARAMETER
1.50
1.75
_
2.00
2.25
2.50
SITa . . . . . . . . . . . .
650.4
153.3
48.03
18.38
8.104
SI2T . . . . . . . . . . . .
654.0
153.6
48.15
18.46
8.132
SI2T/SIT2 . . . . . .
1.0111
1.0030
1.0052
1.0088
1.0069
min (I) . . . . . . . . .
524.3
132.7
39.61
15.17
6.119
max (I) . . . . . . . . .
859.2
178.2
57.58
23.30
10.31
Method 1 :
a .............
0.278
0.476
0.539
0.342
0.890
n /nsph . . . . . . .
0.719
0.957
0.920
0.859
0.723
0 e
n /n . . . . . . . .
1.625
1.240
1.284
1.451
1.252
p 0
Method 2 :
a .............
0.100
0.100
0.100
0.100
0.100
n /nsph . . . . . . .
0.784
1.036
1.027
0.939
0.875
0 e
n /n . . . . . . . .
2.023
1.527
1.666
1.814
1.710
p 0
a H I Lya total intensities expressed in units of 108 s~1 cm~2 sr~1 (see text).
mogeneities have focused mainly on lower heights and have
often found higher density ratios and lower Ðlling factors for
plumes and other structures (see, e.g., Saito 1965 ; Ahmad &
Withbroe 1977 ; Orrall et al. 1990 ; Walker et al. 1993).
However, polar plumes are thought to arise above localized
regions with strong magnetic Ñux, and their associated Ñux
tubes may expand faster than the intervening, weak-Ðeld
corona. The Ðlling factor thus may rapidly increase with
radius (see DeForest et al. 1997).14 Strong regions of mixed
magnetic polarity should also be associated with additional
energy input to the plasma, and this has been found to drive
a larger amount of mass loss in plumes, thus increasing their
density (Wang 1994). Because this heating is primarily concentrated at the base and not extended throughout the
corona, the theoretical density ratio presented by Wang
(1994) gradually decreases from a factor of 3È5 at 1.2 R
_
down to a factor of 1.5È2 at 2.0 R , in general agreement
_
with our statistical models.
Although the present analysis of polar plume statistics is
a reasonable means of incorporating density inhomogeneities into the overall coronal hole empirical model, the
uncertainties and assumptions inherent in the process make
it unsuitable to be propagated along in the analysis of the
UV emission lines. For the H I Lya line (° 4), which depends
on the LOS integral of the density, like pB, the use of the
averaged plume/interplume density does not complicate the
analysis. For the O VI jj1032, 1037 lines (° 5), which contain
a collisionally excited component proportional to the LOS
integral of the square of the density, the use of the mean
n (r, h) described in °° 3.1È3.2 introduces a small degree
ofe uncertainty in the analysis, which has been taken into
consideration.
6.2. L ya Statistics
At present there are three general ways to probe the
plume structure of the solar corona with the emission lines
measured by UVCS : (1) direct imaging of di†erent plume
concentrations across the UV spatial slit, (2) comparison of
14 Note that in our model, a larger a is correlated with a larger number
of plumes N, but this is merely one way of increasing the total density
variance that could also be modeled by a constant number of more rapidly
expanding Ñux tubes.
intensities and widths of lines with di†erent density sensitivity, and (3) long time-base statistical sampling at a locus
of Ðxed points. We have applied the third method to synoptic data from the WLC above, and here we apply it to H I
Lya data. Analysis of polar plumes using the other two
methods is presently underway (see, e.g., Kohl et al. 1997a,
1997b ; Noci et al. 1997), and we will not pursue them
further in the basic coronal hole empirical model.
We examine the variations of H I Lya total intensity with
time for D75 daily observations between 1996 November
and 1997 February. We take the south polar (PA \ 180¡)
synoptic data at Ðve standard heliocentric distances,
o \ 1.50, 1.75, 2.00, 2.25, 2.50 R . We use the intensity in
only one spatial bin to specify_a single one-dimensional
LOS over the heliographic pole, and the mean and meansquared line intensities SIT and SI2T are computed from the
observed time distribution (see Table 4). The detector count
rates in a single spatial row of pixels in the LYA channel are
reasonably high (200È600 at all heights for the synoptic
exposures), but it is important to take account of Poisson
statistics in the standard deviation of the data, via the following correction
SI2T \ SI2T [ SIT ,
(34)
corr
obs
where intensities above are expressed as count rates, and
““ corr ÏÏ and ““ obs ÏÏ refer to corrected and observed
variances, respectively. For example, in the raw observations, the inhomogeneity variance factor SI2T/SIT2 varies
between 1.0047 and 1.0132 ; when the expected Poisson
standard deviation is removed from the data, this factor
varies between 1.0030 and 1.0111. On average, H I Lya
Poisson variations for the synoptic data are D30% of the
total at each height.
With the mean and rms H I Lya intensities known, we use
the statistical methods described in ° 6.1 to produce an
ensemble of plume models that can be searched for agreement with the data. In contrast to the pB measurements, the
H I Lya intensities contain information about the H0
density, outÑow velocity, and most probable speed. By
doing the statistical modeling under the assumption that
plumes and interplume regions have identical outÑow and
most probable speeds, the resulting density parameters will
contain Doppler-dimming information as well. Table 4 con-
498
CRANMER ET AL.
tains the relative plume enhancement (n /n ), the scaled
p 0
interplume density (n /nsph), and the fractional e†ective area
0
e
Ðlled by plumes a, derived from these H I Lya models, which
use velocity distribution parameters from model A1 (° 4.2).
As with the WLC data, we Ðnd no deÐnite radial trend in
the solution values for the three statistical parameters. The
values of (n /n ) and (n /nsph) derived from H I Lya all fall
p 0
0 e
within 10%È40% of their counterparts derived from pB.
Generally, there is no clear trend in these di†erences that
would indicate whether plumes have faster/slower outÑow,
or hotter/cooler velocity distribution widths. The main difficulty may lie in competing e†ects that produce a negligible
change, i.e., similar net Doppler-dimming factors for plumes
and interplume regions. In WangÏs (1994) model, for
example, the plumes have slower outÑow (which would
increase the Doppler-dimmed intensity) and are cooler in
the extended corona (which would decrease the Dopplerdimmed intensity). This may be consistent with our observations, although more speciÐc measurements of plume and
interplume plasma parameters are required to unambiguously determine their nature.
7.
EMPIRICAL MODEL SUMMARY
In this paper we have developed a self-consistent empirical model of the physical plasma conditions in polar
coronal holes during the period between 1996 November
and 1997 April, near minimum solar activity. This model is
organized around the primary UVCS diagnostics : electron
density (WLC, ° 3), H0 velocity distribution and outÑow
velocity (LYA, ° 4), and O5` velocity distribution and
outÑow velocity (OVI, ° 5). Here we summarize the most
pertinent results and provide a comprehensive Ðve-part
overview of the empirical model.
1. Electron density vs. radius.ÈIn ° 3.1 we computed the
radial dependence of electron density over the coronal poles
from white-light polarization brightness (pB) observations
(eq. [3]). The density measurements between 1.5 and 2 R
_
were slightly higher than some previously observed values,
and work is ongoing to determine whether this is due to
uncharacterized instrumental e†ects or an actual epochal
brightness increase.
2. Electron density vs. latitude.ÈThe geometrical distribution of the hole and streamer structure was derived
from a special WLC synoptic map made in 1996 November,
and in ° 3.2 we Ðtted these observations with a general
Gaussian function (eq. [7]). Because the empirical model
focuses on polar coronal holes, the exact form assumed for
the equatorial streamer enhancements is not extremely
important. The three-dimensional variation in the shape of
the coronal hole was modeled using Kopp & HolzerÏs
(1976) analytical superradial expansion factor (eq. [6]), and
the parameters that agree best with the data are # (R ) \
0 _that
28¡, f \ 6.5, R \ 1.5 R , and p \ 0.6 R . Note
max
1
_
1
_
this nonradial expansion need not apply to any individual
magnetic Ñux tube within the coronal hole but to only the
overall shape of the ““ bundle ÏÏ of the open-Ðeld tubes correlated with the high-speed wind.
3. Hydrogen velocity distribution.ÈIn ° 4 we modeled the
observed intensities and proÐle shapes of the bright H I Lya
line over polar coronal holes. To construct these models we
utilized electron densities from the UVCS white-light
channel and electron temperatures from in situ charge state
modeling. We self-consistently determined the radial varia-
Vol. 511
tion of outÑow velocity u and bi-Maxwellian most probable
speeds w and w over the coronal poles. Insufficient inforA
M
mation about w forced us to consider plausible upper and
A
lower limits on this speed in order to solve for u and w , but
M
mass Ñux conservation implies that the most consistent
solutions are anisotropic, with w [ w (see Table 2). We
M
A
also modeled the latitudinal variation of the H I Lya intensity and found that the best agreement with observations
was for a model in which the magnitude u(r) does not vary
with h in coronal holes, but its direction follows superradial
Ñow tubes.
4. Oxygen velocity distribution and abundance.ÈThe
observed intensities, line widths, and line ratios of the O VI
jj1032, 1037 lines were empirically modeled in ° 5. Using a
similar iteration procedure as for the neutral hydrogen
model, we determined lower and upper limits on the
outÑow velocity u and parallel most probable speed w for
A
O5` ions, and we determined the unusually large values of
the perpendicular most probable speed w needed to
produce the observed line broadening. The Mstrong anisotropy between w and w was conclusively demonstrated
M whichA put upper limits on w far below
using the line ratios,
A speeds
w at heights above D2.2 R . The O5` outÑow
M
_
were signiÐcantly larger than the hydrogen outÑow speeds
above D2.5 R .
_ structure.ÈWe statistically modeled the
5. Polar plume
distribution of small-scale raylike coronal inhomogeneities
by using large ensembles of pB and H I Lya data over the
heliographic poles. In ° 6.1 we analyzed 3 months of whitelight observations to deduce that between 1.75 and 2.50 R
_
plumes Ðll a signiÐcant fraction of the coronal hole volume
and are approximately 1.3È1.8 times denser than the surrounding interplume corona. In ° 6.2 we repeated this
analysis for the H I Lya total intensity and found similar
results. The di†erences in Doppler dimming between
plumes and interplume regions were not manifestly clear,
which indicates the need for further observations to understand better the di†erences in outÑow velocity and most
probable speed in the two regions.
For a general visual summary of the model, we plot in
Figure 12 (top) a synthesized image of LOS-integrated
polarization brightness that illustrates the latitudinal
dependence and plume characteristics derived in °° 3 and 6.
We utilize the following average plume parameters :
(n /n ) \ 1.5, (n /nsph) \ 0.8, and a \ 0.3. The strong radial
p 0
e been eliminated by dividing all data
variation
in pB0 has
pixels by the parameterization in equation (2). The darkest
shading corresponds to a relative factor of 2.4 enhancement
in pB over the lightest shading ; the greater enhancement in
the streamers is saturated for clarity. Figure 12 (bottom)
shows the electron density from a polar viewpoint, normalized by dividing by equation (3), in the transverse slice indicated in Figure 12 (top) by a dashed line (z \ 2 R ). This
_ correhelps to illustrate which individual plume sources
spond to the observed enhancements in pB. Some enhancements seem to be caused by single plumes in or near the
plane of the sky, but not all of the observed pB variations
can be assigned to individual structures. In the extended
corona (as opposed to immediately above the limb), it is not
possible to assume a priori that apparently isolated
enhancements in pB or UV line intensity are associated with
single plumeÈlike structures.
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
FIG. 12.ÈT op : Synthesized image of pB within the empirical coronal
hole, normalized by the mean radial variation in pB to clarify streamers
and plumes. The dashed line indicates the height of the density slice below.
Bottom : Two-dimensional slice of electron density at z \ 2 R , normalized
by the mean radial variation in n . The x-axis is parallel _to the obsere
vational LOS.
8.
DISCUSSION
In the empirical coronal hole model presented in this
paper, we have attempted to apply as few physical assumptions as possible to the observations and deduce only what
can be unambiguously and consistently derived from the
data. Thus, for example, we have not attempted to separate
observed velocity distributions into various unresolvable
components : e.g., thermal motions, other random microscopic motions, and transverse wave motions. We have not
attempted to speculate on the underlying heating and acceleration mechanisms that give rise to the observed distributions. In this section, however, we go a bit further in this
direction and explore the implications and constraints of
the empirical data on possible theoretical models.
499
Although the overall three-dimensional structure of
coronal holes is reasonably well understood, the precise
MHD modeling of the partially open magnetic Ðeld and
solar wind is a formidable problem (see, e.g., Pneuman &
Kopp 1971 ; Sakurai 1985 ; Wang et al. 1993). Magnetostatic
potential-Ðeld solutions are often applied to solar wind
studies, but these have proved too simplistic to reproduce
many observations. For example, the analytic force-free
model applied by Low (1986) and Charbonneau & Hundhausen (1996) would require the inner edge of the streamer
current sheet, or cusp, to be at a radius of D5È6 R to
_
reproduce the size of the base of the observed coronal hole
(# \ 28¡) ; this may be larger by several solar radii than
0
has been inferred in solar minimum streamers (Koutchmy
& Livshits 1992). Models with volumetric heating in
streamers do better at reproducing the observed streamer
shapes but with the probable elimination of truly steady
state solutions (Suess, Wang, & Wu 1996 ; Wang et al. 1998).
The Ñux tube geometry in most MHD models is incompatible with the inferred latitudinal dependence of outÑow
velocity, as has been discussed in ° 4.3. We believe these
latitudinal observations to represent important constraints
on future MHD expansion models.
The spatial distribution of small-scale inhomogeneities in
coronal holes is similarly not well established. Di†erent
diagnostics, such as radio scintillation, polarized white
light, and EUV spectroscopy seem to isolate di†erent types
of structures. We have modeled time variability in pB and
H I Lya intensity in terms of the passage and evolution of
polar plumes across the LOS, but there probably exists an
entire spectrum (in both spatial scale and density contrast)
of inhomogeneities, rather than just two separate plume and
interplume phases. Orrall et al. (1990) and Kurochka,
Matsuura, & Picazzio (1997) discuss possible observational
signatures of unresolved subtelescopic (1È10 km) density
structures in coronal plasma. If these exist, however, a statistical analysis of UVCS data very similar to that outlined
in ° 6 should still yield important information about these
inhomogeneity distributions.
The other major results of the UVCS empirical model are
the velocity distributions of H0 atoms and O5` ions. The
derived perpendicular most probable speeds (w ) of both
M thermal
species are not equal, nor does either represent
equilibrium with the electrons (see also Seely et al. 1997 ;
Esser et al. 1998b). The parallel most probable velocities
(w ) are not well constrained by direct observations, but we
areA able to place reasonable lower and upper limits on these
motions. In models A1 and B1, we equate the H0 and O5`
parallel ion kinetic temperatures with the adopted electron
temperature T . This lower limit is reliable on both obsere where in situ measurements usually show
vational grounds,
T [ T [ T (Marsch et al. 1982b ; Pilipp et al. 1987), and
ion theoretical
p
e grounds, where most postulated coronal
on
heating mechanisms (other than thermal conduction) act
preferentially on protons and heavy ions. In model A2 we
assume that w for hydrogen cannot exceed w ; this is
A by the preponderance of heating
M mechaunderstood simply
nisms that provide more perpendicular kinetic energy than
parallel. In model B2 we use a more empirically justiÐed
upper limit on w which is far below w at most heights (see
A
M
° 5.2).
The end result for the H0 velocity distribution, once mass
Ñux conservation is also considered, is that model A1 is a
more self-consistent and physically reasonable scenario.
500
CRANMER ET AL.
The H0 distribution in this model is anisotropic at all radii
between 1.5 and 4 R . Also, the neutral hydrogen motions
_ of the proton motions in the solar
should be representative
corona. Olsen et al. (1994) and Allen et al. (1998) Ðnd that
below D2.5 R there should be little decoupling between
_
H0 and H` velocities and temperatures but possibly signiÐcant di†erences in w above this radius. Yet, because these
M
di†erences are caused by the frictional dissipation of transverse waves, the kinetic or ““ e†ective ÏÏ temperatures of the
two species (owing to the sum of thermal plus wave
motions) should be very nearly equal at most radii ; this is
indeed the relevant parameter to compare with the H I Lya
line proÐle measurements.
The model velocity distribution of O5` ions has been
constrained to be highly anisotropic, and the outÑow velocity of these ions seems to be signiÐcantly larger than the H0
outÑow velocity above D2.5 R . The hydrogen velocities
_
are consistent with some conventional theoretical models
for polar wind acceleration (see, e.g., Withbroe 1988 ; Wang
1994), but the higher oxygen Ñow speeds cannot be
explained by these models. The O5` perpendicular most
probable speeds are also much larger than those of the
protons, which seems to clearly rule out thermal (common
temperature) Doppler motions and bulk transverse wave
motions along the line of sight as dominant line-broadening
mechanisms. Some additional energy deposition is required
which preferentially broadens the perpendicular velocity
distributions of the heavier ions.
The empirical models of the H0 and O5` distributions
may also provide valuable information about the transition
between collisional and collisionless plasma in the corona
(see also Esser et al. 1998b). Note from Figures 5 and 9 that
several relatively sharp variations occur around the radii
r B 1.8È2.1 R : the values of w for both hydrogen and
_ increase and diverge
M
oxygen steeply
from lower, nearconstant values, and the anisotropy between parallel and
perpendicular motions for O5` becomes deÐnite as w
max
crosses w . This small range of radii may represent the
M
location at which the thermalization and isotropization
times of various species begin to exceed the local coronal
expansion times, thus demanding a true collisionless and
kinetic treatment above this point (see also Kohl et al.
1997a). The physics of plasma transport and wave dissipation thus diverges from classical Coulomb theory in the
bulk of the coronal holes observable with UVCS/SOHO,
and a more careful description is required (see, e.g.,
Williams 1997).
The preferential ion heating and acceleration found in the
UVCS/SOHO empirical measurements must be maintained
in the corona by an extended source of momentum and
energy deposition. Traveling MHD waves have long been
suspected as a natural means of transporting this energy out
into the wind. Models containing low-frequency Alfven
wave pressure have been found to be able to accelerate the
high speed solar wind (Leer, Holzer, & Fla- 1982 ; Ofman
& Davila 1997), but it is unclear if such wave-particle
interactions are able to accelerate and heat heavier ions
preferentially.
A promising theoretical explanation for many features of
the preferential ion heating is the efficient dissipation of
high-frequency waves that are resonant with ion-cyclotron
Larmor motions about the coronal magnetic Ðeld lines (see,
Vol. 511
e.g., Abraham-Shrauner & Feldman 1977 ; Hollweg &
Turner 1978 ; Marsch, Goertz, & Richter 1982a ; McKenzie,
Banaszkiewicz, & Axford 1995 ; Fletcher & Huber 1997 ; Tu
& Marsch 1997). Wave-particle interactions of this kind in
the solar wind have been suspected since in situ measurements have shown that most heavy ions Ñow slightly faster
than protons, with di†erential speeds of the same order as
the Alfven speed (see, e.g., Marsch et al. 1981, 1982b).
Because di†erent ions have di†erent resonant frequencies,
they receive di†erent amounts of heating and acceleration
as a function of radius, which is required to understand the
di†erent features of the H0 and O5` velocity distributions.
The precise mechanisms for generating the required high
frequencies (100È5000 Hz) for the ion-cyclotron interaction
are currently unclear. Isenberg & Hollweg (1983) proposed
and developed the scenario of Alfven wave saturation followed by turbulent cascade from low to high frequencies.
Alternately, Axford & McKenzie (1992) suggested that
reconnections in small (\100 km) closed loop microÑares,
distributed Ðnely through the chromospheric network, are
able to generate high-frequency waves that can propagate
outward into the corona. These di†ering views of the origin
of the high-frequency waves (i.e., base or in situ generation)
may in principle be distinguished via a detailed comparison
between theoretical models and empirically derived plasma
parameters.
It is also possible that the TRACE (Transition Region
and Coronal Explorer ; Strong et al. 1994) instrument, in
collaboration with UVCS, may be able to distinguish better
between the several viable scenarios by obtaining precisely
co-aligned, high-resolution, and high-cadence image
sequences in the photosphere, transition region, and
corona. Current measurements of the statistical energy
spectrum of microÑare events are unclear as to whether the
smallest scale structures play a signiÐcant role in coronal
heating and acceleration (see, e.g., Porter, Fontenla, &
Simnett 1995 ; Krucker & Benz 1998). MHD waves can be
detected in coronal Ðne structures by observing either
periodic intensity Ñuctuations or oscillatory motions of Ñux
tubes ; instruments like SXT on Y ohkoh have already begun
to measure such motions in various features (see, e.g.,
McKenzie & Mullan 1997).
In any case, it does now seem possible to construct selfconsistent theories of coronal plasma heating that are constrained by coronagraphic observations of anisotropic ion
velocity distributions. Despite the fact that the observed
minor ions do not contribute signiÐcantly to the total mass
or energy Ñux of the wind, they carry unique and important
signatures of the heating and acceleration processes which
apply to the entire plasma.
The authors wish to acknowledge the contributions of A.
van Ballegooijen to the development of the UVCS instrument and the work of Brenda Bernard in the administration
of the UVCS investigation. We would also like to thank
George B. Field, Richard Frazin, John Mariska, and
Yi-Ming Wang for many valuable discussions. This work is
supported by the National Aeronautics and Space Administration under grant NAG5-3192 to the Smithsonian Astrophysical Observatory, by Agenzia Spaziale Italiana, and by
Swiss funding agencies.
No. 1, 1999
POLAR CORONAL HOLE AT SOLAR MINIMUM
501
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