arXiv:0802.0144v1 [astro-ph] 1 Feb 2008arXiv:0802.0144v1 [astro-ph] 1 Feb 2008 THE ASTROPHYSICAL...

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8THE ASTROPHYSICALJOURNAL, 679, 2008 MAY 20, IN PRESSPreprint typeset using LATEX style emulateapj v. 03/07/07

IMPROVED CONSTRAINTS ON THE PREFERENTIAL HEATING AND ACCELERATION OF OXYGEN IONS IN THEEXTENDED SOLAR CORONA

STEVEN R. CRANMER, ALEXANDER V. PANASYUK , AND JOHN L. KOHLHarvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138

Submitted 2007 November 27; accepted 2008 January 29Draft version October 28, 2018

ABSTRACTWe present a detailed analysis of oxygen ion velocity distributions in the extended solar corona, based on

observations made with the Ultraviolet Coronagraph Spectrometer (UVCS) on theSOHOspacecraft. Polarcoronal holes at solar minimum are known to exhibit broad line widths and unusual intensity ratios of the OVIλλ1032, 1037 emission line doublet. The traditional interpretation of these features has been that oxygen ionshave a strong temperature anisotropy, with the temperatureperpendicular to the magnetic field being muchlarger than the temperature parallel to the field. However, recent work by Raouafi and Solanki suggested thatit may be possible to model the observations using an isotropic velocity distribution. In this paper we analyzean expanded data set to show that the original interpretation of an anisotropic distribution is the only one thatis fully consistent with the observations. It is necessary to search the full range of ion plasma parameters todetermine the values with the highest probability of agreement with the UVCS data. The derived ion outflowspeeds and perpendicular kinetic temperatures are consistent with earlier results, and there continues to bestrong evidence for preferential ion heating and acceleration with respect to hydrogen. At heliocentric heightsabove 2.1 solar radii, every UVCS data point is more consistent with an anisotropic distribution than with anisotropic distribution. At heights above 3 solar radii, theexact probability of isotropy depends on the electrondensity chosen to simulate the line-of-sight distributionof O VI emissivity. The most realistic electron densities(which decrease steeply from 3 to 6 solar radii) produce the lowest probabilities of isotropy and most-probabletemperature anisotropy ratios that exceed 10. We also use UVCS OVI absolute intensities to compute thefrozen-in O5+ ion concentration in the extended corona; the resulting range of values is roughly consistent withrecent downward revisions in the oxygen abundance.Subject headings:line: profiles — plasmas — solar wind — Sun: corona — Sun: UV radiation — techniques:

spectroscopic

1. INTRODUCTION

The physical processes that heat the solar corona and ac-celerate the solar wind are not yet understood completely.In order to construct and test theoretical models, there mustexist accurate measurements of relevant plasma parametersin the regions that are being heated and accelerated. Inthe low-density, open-field regions that reach into interplan-etary space, the number of plasma parameters that need tobe measured increases because the plasma begins to becomecollisionless and individual particle species (e.g., protons,electrons, and heavy ions) can exhibit different properties.Such differences in particle velocity distributions are valu-able probes of “microscopic” processes of heating and accel-eration. The Ultraviolet Coronagraph Spectrometer (UVCS)operating aboard theSolar and Heliospheric Observatory(SOHO) spacecraft has measured these properties for a va-riety of open-field regions in the extended corona (Kohl et al.1995, 1997, 2006).

In this paper we focus on UVCS observations of heavyion emission lines (specifically OVI λλ1032, 1037) in po-lar coronal holes at solar minimum. One main goal is to re-solve a recent question that has arisen regarding the existenceof anisotropic ion temperatures in polar coronal holes. Sev-eral prior analyses of UVCS data have concluded that theremust be both intense preferential heating of the O5+ ions, incomparison to hydrogen, and a strong field-aligned anisotropywith a much larger temperature in the direction perpendicularto the magnetic field than in the parallel direction (see, e.g.,Kohl et al. 1997, 1998; Li et al. 1998; Cranmer et al. 1999;

Antonucci et al. 2000; Zangrilli et al. 2002; Antonucci 2006;Telloni et al. 2007). However, Raouafi & Solanki (2004,2006) and Raouafi et al. (2007) have reported that there maynot be a compelling need for O5+ anisotropy depending on theassumptions made about the other plasma properties of thecoronal hole (e.g., electron density).

The determination of O5+ preferential heating, preferentialacceleration, and temperature anisotropy has spurred a greatdeal of theoretical work (see reviews by Hollweg & Isenberg2002; Cranmer 2002a; Marsch 2005; Kohl et al. 2006). Itis thus important to resolve the question of whether theseplasma properties are definitively present on the basis of theUVCS/SOHOobservations. In this paper, we attempt to ana-lyze all possible combinations of O5+ properties (number den-sity, outflow speed, parallel temperature, and perpendiculartemperature) with the full effects of the extended line of sight(LOS) taken into account. The applicability of any particu-lar combination of ion properties is evaluated by computinga quantitative probability of agreement between the modeledset of emission lines and a given observation. Preliminary re-sults from this work were presented by Cranmer et al. (2005)and Kohl et al. (2006).

The original UVCS results of preferential ion heating andacceleration—as well as strong ion temperature anisotropy(T⊥ ≫ T‖)—were somewhat surprising, but these extreme de-partures from thermal equilibrium are qualitatively similar toconditions that have been measured for decades in high-speedstreams in the heliosphere. At their closest approaches tothe Sun (∼ 0.3 AU), theHeliosprobes measured substantial

http://arxiv.org/abs/0802.0144v1

2 CRANMER, PANASYUK, & KOHL

proton temperature anisotropies withT⊥ > T‖ (Marsch et al.1982; Feldman & Marsch 1997). In the fast wind, most ionspecies also appear to flow faster than the protons by aboutan Alfvén speed (VA), and this velocity difference decreaseswith increasing radius and decreasing proton flow velocity(e.g., Hefti et al. 1998; Reisenfeld et al. 2001). The tem-peratures of heavy ions are significantly larger than protonand electron core temperatures. In the highest-speed windstreams, ion temperatures exceed simple mass proportional-ity with protons (i.e., heavier ions have larger most-probablespeeds), with (Tion/Tp) > (mion/mp), for mion > mp (e.g., Col-lier et al. 1996). UVCS provided the first evidence that theseplasma properties are already present near the Sun.

The outline of this paper is as follows. In § 2 we present anexpanded collection of UVCS/SOHOobservational data thatis used to determine the O5+ ion properties. § 3 outlines theprocedure we have developed to produce empirical modelsof the plasma conditions in polar coronal holes and to com-pute the probability of agreement between any given set ofion properties and the observations. The resulting ranges ofion properties that are consistent with the UVCS observationsare presented in § 4 along with, in our view, a resolution ofthe controversy regarding the oxygen temperature anisotropy.Finally, § 5 gives a summary of the major results of this paperand a discussion of the implications these results may have ontheoretical models of coronal heating and solar wind acceler-ation.

2. OBSERVATIONS

The UVCS instrument contains three reflecting telescopesthat feed two ultraviolet toric-grating spectrometers andonevisible light polarimeter (Kohl et al. 1995, 1997). Light fromthe bright solar disk is blocked by external and internal occul-ters that have the same linear geometry as the spectrometerslits. The slits are oriented in the direction tangent to thesolarlimb. They can be positioned in heliocentric radiusr any-where between about 1.4 and 10 solar radii (R⊙) and rotatedaround the Sun in position angle. The slit length projectedon the sky is 40′, or approximately 2.5R⊙ in the corona, andthe slit width can be adjusted to optimize the desired spectralresolution and count rate.

The UVCS data discussed in this paper consist of a largeensemble of observations of polar coronal holes from the lastsolar minimum (1996–1997). The solar magnetic field is ob-served to exist in a nearly axisymmetric configuration at solarminimum, with open field lines emerging from the north andsouth polar regions and expanding superradially to fill a largefraction of the heliospheric volume. The plasma propertiesin polar coronal holes remain reasonably constant in the yearor two around solar minimum (see, e.g., Kohl et al. 2006),so we assemble the data over this time into a single functionof radius. In this paper, we limit ourselves to the analysis ofobservations of the OVI λλ1032, 1037 emission line doubletin these polar regions. The relevant UVCS observations aretaken from the following three sources.

1. The empirical model study of Kohl et al. (1998) andCranmer et al. (1999) covered the period between 1996November and 1997 April and took the north and southpolar coronal hole properties to be similar enough totreat them together.

2. A detailed analysis of the north polar coronal holeby Antonucci et al. (2000) coincided with the second

SOHOjoint observing program (JOP 2) on 1996 May21.

3. We searched the UVCS/SOHOarchive for any othernorth or south polar hole observations having sufficientcount-rate statistics to be able to measure the OVI linewidths at radii above 2R⊙. A total of 14 new or rean-alyzed data points were identified between 1996 Juneand 1997 July.

The remainder of this section describes the data reduction forthe third group of new data points. Table 1 provides details ofthese 14 measurements.

The criteria for identifying new UVCS data were as fol-lows. We adopted a time period from 1996 April, the be-ginning of primary science operations, to 1998 January, afterwhich the new cycle’s activity began to rise and high-latitudestreamers appeared regularly to signal the end of true solarminimum. Only measurements of the OVI lines above thepoles (i.e., position angles within±15 of the north or southpoles) and at heights above 2R⊙ were sought.1 Prior expe-rience with the count rates at large heights in coronal holesrefined the search further to use measurements only with rel-atively long exposure times (see Table 1) to gather sufficientstatistics to measure the line widths. There were two observa-tions that appeared initially to satisfy the above criteria(1997April 15, at 4.14R⊙, and 1997 July 2, at 3.10R⊙), but theywere not used because the count rate statistics were inade-quate for reliable line widths. The only point of overlap be-tween the data in Table 1 and prior analyses (e.g., Cranmeret al. 1999) concerns the end of the month-long study of thenorth polar coronal hole in 1997 January (atr ≈ 3R⊙). Thesedata were reanalyzed with a different line fitting techniqueand an improved UVCS pointing correction; the computedline widths are similar to those presented by Cranmer et al.(1999) and the intensities are given here for the first time. Forcompleteness, though, both the old and new data points arekept in the full ensemble of OVI data used below.

To achieve the lowest uncertainties in the determinations ofthe OVI λλ1032, 1037 intensities and line widths, we typ-ically integrated over 15′ to 30′ along the slit (see Table 1).This corresponds to±0.5–1 R⊙ on either side of the north-south axis. The use of such large areas implies that narrowflux-tube structures such as densepolar plumesand the less-dense interplume regions were not resolved. As long as allsteps of the analysis remain consistent with such a coarselyaveraged state (e.g., the use of a similarly averaged electrondensity), though, this need not be a problem. The derivedplasma properties thus describe the average conditions insidecoronal holes at solar minimum and do not address differencesbetween plumes and interplume regions.

Details concerning the analysis of UVCS data are given byGardner et al. (1996, 2000, 2002) and Kohl et al. (1997, 1999,2006). The UVCS Data Analysis Software (DAS) was usedto remove image distortion and to calibrate the data in wave-length and intensity. The coronal line profiles are broadenedby various instrumental effects. The optical point spread func-tion of the spectrometer depends on the slit width used (with270.3µm corresponding to 1 Å in the spectrum), the on-board

1 Below r ≈ 2R⊙, the existing data appear to be adequate, uncertaintiesare low, and there is not much of an intrinsic spread in the intensities andline widths as a function of height. Also, earlier analyses did not show thatcollisionless effects (ion temperature anisotropies, preferential ion heating, ordifferential flow) became strong until above this height.

PREFERENTIAL ION HEATING IN THE CORONA 3

TABLE 1NEWLY ANALYZED POLAR CORONAL HOLE DATA : 1996–1997

Start Date, UT Time Obs. Height Slit Lengtha Slit Width Exposure V1/e R Itot, 1032 Å line(R⊙) (arcmin) (µm) Time (hr) (km s−1) (106 phot s−1 cm−2 sr−1)

1996 Jun 21, 16:55 2.07 15.8 (N) 75 9.1 363±21.7 2.09±0.4 32.2±6.81996 Sep 29, 15:26 2.08 17.5 (N) 75 5.6 473±12.4 1.87±0.14 43.2±8.71996 Nov 07, 23:38 2.07 29.7 (N) 75 7.6 409±18.8 1.78±0.2 44.6±9.11996 Nov 10, 06:30 3.00 25.7 (N) 350 9.4 475±27.0 1.16±0.18 2.08±0.441996 Nov 10, 15:55 2.56 25.9 (N) 350 10.8 505±30.5 1.17±0.17 4.83±1.01996 Nov 16, 16:56 2.17 29.6 (N) 340 9.5 417±7.43 1.485±0.06 25.1±5.01997 Jan 05, 21:00 3.07 28.0 (N) 100 17.6 690±87.2 0.957±0.45 2.02±0.631997 Jan 10, 14:54 3.08 20.8 (N) 150 68.8 686±38.7 1.23±0.3 3.37±0.761997 Jan 24, 16:03 3.08 20.6 (N) 150 70.8 594±41.6 1.01±0.3 1.66±0.401997 Mar 09, 18:00 2.57 19.0 (S) 300 8.9 500±22.0 1.15±0.12 5.66±1.21997 Jun 04, 16:49 2.56 18.8 (N) 300 9.0 527±33.9 0.958±0.15 4.14±0.901997 Jun 08, 20:15 3.10 18.7 (N) 300 8.3 645±65.6 1.33±0.44 1.33±0.331997 Jul 01, 15:50 2.56 17.2 (N) 342 9.8 534±56.8 0.988±0.26 3.26±0.801997 Jul 04, 16:45 3.63 18.9 (N) 342 29.5 451±47.9 1.49±0.43 0.559±0.13

a Data were integrated over the specified slit length. The slitwas oriented tangent to either the north (N) or the south (S) heliographic pole,as indicated.

data binning, the exposed mirror area, and the intrinsic quan-tization error of the detector. This broadening is taken intoaccount by adjusting the line widths of Gaussian fits to thecoronal components of the data; the data points themselvesare not corrected. Tests have shown that the coronal line widthcan be recovered accurately even when the total instrumentalwidth is within about a factor of two of the width of the coro-nal component. Instrument-scattered stray light from the solardisk is modeled as an additional narrow Gaussian componentwith an intensity and profile shape constrained by the knownstray light properties of the instrument.

The analysis of the OVI emission line doublet involves fourbasic observable quantities: the total intensities of the twolines and their 1/e Gaussian half-widths∆λ1/e. The latterquantities are typically expressed in Doppler velocity units asV1/e = c∆λ1/e/λ0, whereλ0 is the rest wavelength of the lineandc is the speed of light. Rather than give the two total in-tensities, Table 1 provides the total intensityItot of the OVIλ1032 line and the ratioR of theλ1032 to theλ1037 inten-sities. The uncertainties given in Table 1 take account of bothPoisson count-rate statistics and the fact that the variousin-strumental corrections are known only to finite levels of pre-cision. Note that the ratioR does not depend on the absoluteintensity calibration of the instrument.

UVCS/SOHOhas not been able to resolve any departuresfrom Gaussian shapes for the OVI lines in large polar coronalholes, so the profiles are described by just the one parameterV1/e. For the measurements given in Table 1, we performedthe line fitting by constraining the coronal components of theλ1032 andλ1037 lines to have the same width. Thus, theV1/e values given in Table 1 are formally a weighted mean be-tween the two components. This is done mainly to lower thestatistical uncertainties but there is some observationaljusti-fication for assuming that the two components have the samewidth. In situations where the count rates are high, it is diffi-cult to see any significant or systematic difference betweentheline widths of the two components. There are various reasonswhy they may be different from one another in some regions(e.g., Cranmer 2001; Morgan & Habbal 2004), but more workneeds to be done to identify such subtle effects.

Figure 1 displays the combined ensemble of old and newUVCS O VI data for the three main observables: the linewidth V1/e, the dimensionless intensity ratioR, and the abso-

FIG. 1.— Collected UVCS polar coronal hole measurements of(a) O VIline widthsV1/e, (b) ratio of OVI λ1032 to OVI λ1037 intensities, and(c)O VI λ1032 line-integrated intensities, with symbols specifying the sourcesof the data (see labels for references). Error bars denote±1σ observationaluncertainties. Also shown (dotted lines) are the parameterized fits given byCranmer et al. (1999).SEE LAST PAGE OF PAPER FOR LARGERVERSION.

lute (line-integrated) intensity of the OVI λ1032 line. Thereare a total of 53 separate data points from the three sourcesdiscussed above, but not all of these points have all three ofthe main quantities: there are 50 values ofV1/e, 52 values ofR(with only 49 cases where bothV1/e andR exist for the samemeasurement), and 44 values ofItot. This relative paucity ofdata illustrates the difficulties of measuring the plasma param-eters at large heights in polar coronal holes.

In general, the radial dependences of the OVI quantities inFigure 1 are similar to those given by Kohl et al. (1998), Cran-mer et al. (1999), and Antonucci et al. (2000). There exists areasonably large spread in theV1/e values in Figure 1a abover ≈ 2.5R⊙. This spread exceeds the magnitude of the±1σuncertainty limits for the individual measurements, and thusseems to indicate that there is anintrinsic variability (possi-


bly temporal) of the O5+ plasma conditions in polar coronalholes above heights where the ions become collisionless. Itis possible that polar plumes and interplume regions becomecollisionless over different ranges of radius, and thus prefer-ential ion heating mechanisms may begin to broaden the OVIlines at different rates in the two regions. The observed varia-tion in line width may thus depend on the relative concentra-tions of plume and interplume plasma along the line of sightat different observation times.

3. EMPIRICAL MODEL PROCEDURE

The observable properties of the OVI line doublet dependon a nontrivial combination of various O5+ plasma parame-ters, as well as electron parameters, integrated along the opti-cally thin line of sight. In general, then, it is not possibleto de-rive accurate and self-consistent plasma parameters via a sim-ple “inversion” from the line widths and intensities. Rather,one must build up a so-calledempirical modelof the coro-nal hole—with the O5+ velocity distribution and other prop-erties as free parameters—and synthesize trial line profiles.After some procedure of varying the coronal parameters toachieve agreement between the synthesized line profiles andthe observations, the self-consistent empirical model of theion properties can be considered complete. This technique isclosely related to forward modeling approaches being used inother areas of solar physics (e.g., Judge & McIntosh 1999).

The use of the term “empirical model” has resulted in abit of confusion regarding what assumptions are embeddedin the derived plasma parameters. We emphasize that theempirical models described here do not specify the physi-cal processes that maintain the coronal plasma in its assumedsteady state. Thus, there is no explicit dependence on “the-oretical” concepts such as coronal heating and accelerationmechanisms, waves and turbulent motions, or magnetohydro-dynamics (MHD), within the empirical models. The derivedO5+ plasma parameters depend on only the observations andon well-established theory such as the radiative transfer inher-ent in the line-formation process.

In this section we summarize the forward modeling of OVIline profiles for an arbitrary set of coronal parameters (§ 3.1),then we describe how these parameters are specified and var-ied to produce various empirical model grids (§ 3.2). Finally,we present a new method of computing the probability ofagreement between a given empirical model and the obser-vations (§ 3.3), such that no regions of the possible solutionspace are neglected.

3.1. Forward Modeling

The OVI line emission in coronal holes comes from twosources of comparable magnitude: (1) collisional electronim-pact excitation followed by radiative decay, and (2) resonantscattering of photons that originate on the bright solar disk.The emergent specific intensity of an emission line from anoptically thin corona is given by

Iν =∫ +∞

−∞dx

(

jcollν + j res

ν

)

, (1)

wherex is the coordinate direction along the observer’s lineof sight (LOS) andjcoll

ν and j resν are the collisionally excited

and resonantly scattered line emissivities, respectively. Weneglect the relatively weak UV continuum and the Thomsonelectron-scattered components of the spectral lines in ques-tion. At a given point in the three-dimensional coronal hole

volume, the line emissivities are specified by

jcollν =

hν0

4πq12(Te)nen1φν (2)

j resν =

hν0

4πB12n1

∫ ∞

0dν′

∮

dΩ′

4πR(ν′, n′,ν, n) Iν′(n′) (3)

(see, e.g., Mihalas 1978). Here,ν0 is the rest-frame line-center frequency,q12 is the collision rate per particle for thetransition between atomic levels 1 and 2,n1 is the numberdensity in the lower level of the atom or ion of interest (here,the 2s 2S1/2 state of O5+), andB12 is the Einstein absorptionrate of the transition. The emission profileφν is assumed tobe Gaussian. The scattering redistribution functionR takesthe incident frequencyν′ and photon direction vectorn′ andtransforms it into the observed frequencyν along the LOSdirectionn.

The profileφν and the redistribution functionR containthe main dependences on the properties of the ion velocitydistribution. We allow for the possibility of an anisotropicO5+ velocity distribution by using a bi-Maxwellian function(e.g., Whang 1971), with the parallel and perpendicular axesoriented arbitrarily with respect to the radial direction in thecorona; see § 3.2. The emissivity profiles along the LOS aremodeled with the full effects of the bi-Maxwellian velocitydistribution and the projected components of the bulk outflowspeed along then′ andn directions. For the polar coronal holemeasurements being modeled here, we define a Cartesian co-ordinate system for which the LOS direction is denotedx andthe north-south polar axis of the Sun isz. The other coordi-natey is set to zero. General expressions for the emissivitiesare given in various levels of detail by Withbroe et al. (1982),Noci et al. (1987), Allen et al. (1998), Cranmer (1998), Li etal. (1998), Noci & Maccari (1999), Kohl et al. (2006), andAkinari (2007).

The resonantly scattered components depend sensitively onthe intensity profiles incident from the solar disk (Iν′). As inCranmer et al. (1999), we used empirically derived Gaussianprofiles with total intensities measured on the disk by UVCSat solar minimum (Raymond et al. 1997). The adopted OVIλ1032 (1031.93 Å) disk intensity is 1.94× 1013 photons s−1

cm−2 sr−1, and the total intensities of the OVI λ1037 (1037.62Å), C II λ1037.02, and CII λ1036.34 disk lines are 0.500,0.214, and 0.171 times theλ1032 intensity, respectively. Weused the profile widths as given by Noci et al. (1987); see alsothe comparative tables of Gabriel et al. (2003) and Raouafi &Solanki (2004).

The collisional components depend on how the collisionrate q12 varies with electron temperatureTe. We kept thesame tabulated values as were used by Raymond et al. (1997)and Cranmer et al. (1999). For completeness, we give a fit toq12(Te) for the OVI λ1032 transition:

log10(q12) = −0.22117t2+ 2.4565t − 14.695, (4)

wheret = log10Te, andTe andq12 are given in units of K andcm3 s−1 respectively. This expression is valid to within about± 2% over the range 5.3≤ t ≤ 6.3. The collision rate for theO VI λ1037 line is half of that of the OVI λ1032 line.

The numerical code that synthesizes line profiles by nu-merically integrating equations (1)–(3) is essentially the sameas the one used by Cranmer et al. (1999). The integrationsoverx andν′ have been simplified by replacing the adaptiveRomberg method by fixed grids, with spacings that have beenadjusted to minimize both numerical discretization errorsand


run time. The LOS integration was performed in steps of 0.1R⊙ from –15 to+15 R⊙ along thex axis. The incident fre-quency grid corresponds to a wavelength grid with a spacingof 0.03 Å inλ′. These step sizes were verified to give accurateresults by halving the step sizes and obtaining the same resultsto within a desired precision. We integrated over the solid an-gle of the solar disk (dΩ′ = sinθ′ dθ′ dφ′) by Gauss-Legendrequadrature inθ′ and equally spaced trapezoidal quadrature inφ′. The solar disk was assumed to be uniformly bright.

3.2. Parameter Selection for Line Synthesis

For the OVI doublet, there are three primary observables(Itot of λ1032,V1/e, andR) that depend on the LOS distribu-tions of four “unknown” quantities as well as a longer list ofquantities that can be considered to be known independentlyof the UVCS observations. The four unknowns are the ionfraction (essentiallyn1/ne), the O5+ bulk outflow speed alongthe magnetic field (ui‖), and the parallel and perpendicularO5+ kinetic temperatures (Ti‖ andTi⊥). The known quanti-ties include the electron densityne, the electron temperatureTe, the incident intensity from the solar disk, and the overallmagnetic geometry of the coronal hole (i.e., how to compute“parallel” and “perpendicular” at any point along the LOS).Note that both emissivities (eqs. [2]–[3]) depend linearlyonthe ion fraction, so that the total intensityItot can be used asa straightforward diagnostic of this quantity after the otherparameters have been determined. The line widths and in-tensity ratios do not depend on the ion fraction. This leavestwo observables (V1/e andR) to specify the values of three ionquantities (ui‖, Ti‖, andTi⊥). Although this system is formallyunderdetermined, we can nonetheless put some firm limits ontherangesof these quantities and compute the most probablevalues.

Below, the three O5+ velocity distribution parameters arediscussed in § 3.2.1 and the other “known” parameters arediscussed in § 3.2.2.

3.2.1. Ionized Oxygen Parameters

We treat the three unknown ion quantities as free parame-ters that are varied independently of one another. Other em-pirical modeling efforts (e.g., Cranmer et al. 1999; Antonucciet al. 2000; Raouafi & Solanki 2004, 2006) have tended touse some form of iterative refinement; i.e., they started witha specific set of initial conditions and assumptions, and theyvaried some parameters—and kept others fixed—to find themost probable values ofui‖, Ti‖, andTi⊥. The initial esti-mates tended to utilize the fact that the line widths are mostsensitive toTi⊥, whereas the line ratios depend mainly on theeffect of Doppler dimming (and Doppler pumping from theC II solar disk lines) and thus are sensitive mainly to the par-allel velocity distribution (ui‖ andTi‖). These iterative proce-dures contain the inherent possibility that some regions oftheparameter space could be neglected, and thus possibly validsolutions could be ignored. In this paper wesearch the entireparameter spaceby constructing a three-dimensional “datacube” which contains all possible combinations of the threeparameters.

The three axes of the data cube were chosen to beui‖, Ti⊥,and the anisotropy ratioTi⊥/Ti‖. The modeled ranges of thesequantities were made as wide as possible in order to avoidmissing possibly relevant regions of parameter space. Theoutflow speedui‖ was varied between 0 and 1000 km s−1 us-ing a linearly spaced grid. The perpendicular kinetic tem-

peratureTi⊥ was varied logarithmically between 5×105 and109 K. The anisotropy ratio was varied logarithmically be-tween 0.1 and 100. There were 50 values of each parameteralong the three axes of the data cube, and we synthesized 12wavelengths—spaced linearly between the line center and 2.7Å redward of line center—for both OVI lines. Thus, a datacube constructed for a specific height in the corona (z) con-sisted of 3×106 (503×24) individual LOS integrations.

For each point in a data cube, the scalar values ofui‖ andTi⊥ were assumed to be those in the plane of the sky (i.e.,x = 0). For other points along the LOS, the models usedslightly larger values that are consistent with an assumed ra-dial increase in both parameters. Mass flux conservation—using the modeledne(r) and flux tube geometry—was used tospecify the radial increase inui‖ along the LOS. Earlier em-pirical modeling results (specifically, eq. [28] of Cranmeretal. 1999) were used to specify the radial increase inTi⊥. Themodeled anisotropy ratioTi⊥/Ti‖ was assumed to remain con-stant along the LOS. It is important to note that the modeledradial increases inui‖ andTi⊥ were always taken to berelativeto the plane-of-sky values that were varied freely throughouteach data cube. Thus, there is noa priori reason for the re-sulting most-probable values of these parameters (determinedvia comparisons with observations over a range of heightsz)to exhibit similar radial increases.2

We note that the kinetic temperature quantitiesTi⊥ andTi‖may describe some combination of “thermal” microscopicmotions and any unresolved bulk motions due to waves or tur-bulence. Thus, there is a further step of interpretation requiredafter the most likely values of these kinetic temperatures havebeen derived from the empirical modeling process. Makinga definitive separation between the thermal and nonthermalcomponents of these temperatures is beyond the scope of thispaper. However, we can make some qualitative comments onthe likely ranges of magnitude of these two components basedon recent theoretical models of Alfvén waves in coronal holes;see §§ 4.2 and 4.3.

Finally, the O5+ ion fractionn1/ne was kept at a constant(and arbitrary) value in all of the models. Comparisons be-tween the observed and synthesized total intensities were usedto derive measurements of this ion fraction in the polar coro-nal holes; see § 4.5.

3.2.2. Electron and Flux Tube Parameters

The three main “known” parameters that are explored in themodels shown below (but kept constant over each data cube)are the electron densityne(r), electron temperatureTe(r), andthe macroscopic flux-tube geometry of the coronal hole. Anyother parameters that could be varied—e.g., the disk intensi-ties of the OVI and CII lines—were kept fixed at the valuesgiven above in § 3.1.

Because one main purpose of this paper is to determine whythe results of Raouafi & Solanki (2004, 2006) appear to dif-fer from earlier empirical modeling efforts, we constructedtwo main sets of electron and flux tube parameters:model R,which is designed to replicate many of the conditions assumedby Raouafi & Solanki (2004, 2006), andmodel C,which is es-sentially the same as used by Cranmer et al. (1999). Below,

2 Because the degree of radial increase inTi⊥ is relatively uncertain, weconstructed an additional set of empirical models with no radial increase inTi⊥ (i.e., where the plane-of-sky values were kept constant over the LOS).The resulting probability distributions (§ 4) were virtually identical to thosecomputed with the specified radial increase along the LOS.


we also discuss hybrid models with various combinations ofthe conditions assumed in models R and C.

Model C uses an electron temperature derived by Ko et al.(1997) from measurements of ion charge states in the fast so-lar wind made by the SWICS instrument onUlysses(Gloeck-ler et al. 1992). We utilize the fitting formula

Te(r) = 106K

[

0.35

(

rR⊙

)1.1

+ 1.9

(

rR⊙

)−6.6]−1

. (5)

For the electron density, model C uses the expression de-rived by Cranmer et al. (1999) from direct inversion ofUVCS/SOHO white-light polarization brightness (pB) dataover the poles at solar minimum; i.e.,

ne(r)105cm−3

= 3890

(

R⊙

r

)10.5

+ 8.69

(

R⊙

r

)2.57

. (6)

The above electron density is a mean value for polar coro-nal holes (intermediate between plumes and interplume re-gions) betweenr ≈ 1.5 and 4R⊙. Model C also uses thethree-parameter empirical function of Kopp & Holzer (1976)to specify the superradial expansion of a polar coronal hole.The transverse areaA(r) ∝ r2 f (r) of the entire coronal hole isspecified by

f (r) = 1+ ( fmax− 1)

1− exp[(R⊙ − r)/σ1]1+ exp[(R1 − r)/σ1]

, (7)

and Cranmer et al. (1999) determinedfmax = 6.5,R1 = 1.5R⊙,andσ1 = 0.6R⊙. Also, the area of the hole is normalized bysetting the basal colatitudeΘ0 to 28. The field lines insidethe coronal hole volume are assumed to self-similarly followcolatitudes that remain proportional to the overall boundary ofthe coronal hole at any given radius (see Cranmer et al. 1999).

Model R uses a constant electron temperature of 106 K.This value is lower than the peak of the Ko et al. (1997) model(Te ≈ 1.5× 106 K at r ≈ 1.6R⊙), and higher than the valuefrom this model at the coronal base (Te≈ 4×105 K at r = R⊙).The constant value of 106 K seems to be in closer agreementto both theoretical models that take account of strong electronheat conduction in the corona (e.g., Lie-Svendsen & Esser2005; Cranmer et al. 2007) and with SUMER/SOHOobser-vations made above the limb in coronal holes (e.g., Wilhelmet al. 1998; Doschek et al. 2001).3 For the electron density,model R uses equation (2) of Doyle et al. (1999), i.e.,

ne(r)105cm−3

= 1000

(

R⊙

r

)8

+ 0.025

(

R⊙

r

)4

+ 2.9

(

R⊙

r

)2

.

(8)To specify the superradial geometry of flux tubes in the polarcoronal hole, model R uses the analytic magnetic field modelof Banaszkiewicz et al. (1998).

Note that Raouafi & Solanki (2004, 2006) used equation (1)of Doyle et al. (1999), which is a one-parameter hydrostaticfit to various measured electron densities. Above a height ofr ≈ 6R⊙, though, the radial decrease ofne in the hydrostaticexpression becomes substantiallyshallowerthan an inverse-square radial decrease. This is not generally expected to oc-cur; i.e., in most observations and models, the radial decreasein ne goes from a rate much steeper than 1/r2 at low heights

3 The discrepancies between electron temperatures derived from spec-troscopy and from frozen-in ion charge states are not yet fully understood(see, e.g., Esser & Edgar 2000, 2001; Chen et al. 2003; Laming& Lepri2007).

FIG. 2.— Magnetic field lines in the plane of the sky for(a) the B98 (Ba-naszkiewicz et al. 1998) model, and(b) the C99 (Cranmer et al. 1999) modelthat used the Kopp & Holzer (1976) flux-tube area function. The dotted hor-izontal line denotes the position of the LOS along which various quantitiesare plotted in(c). In panel(c), the superradial angleδ is given as a functionof |x| (it is the same in the foreground and background halves of theLOS) forthe two models shown above (see labels).

to 1/r2 at large heights where the geometry is radial and thewind speed is constant. The shallow radial density decreaseina hydrostatic model is probably unphysical and could lead toan overestimated contribution from large distances along theLOS.

Figure 2 illustrates the differences between the magneticgeometries used in models R and C. Figures 2a and 2b showfield lines that are distributed evenly in polar angleθ between0 and 29 as measured on the solar surface from the northpole. The superradial angleδ characterizes the departure fromthe radial direction, and it is shown in Figure 2c as a functionof LOS distancex for a polar observing heightz = 2.5R⊙.Formally,δ is defined as the angle between the radius vectorrand the magnetic fieldB (assuming the field points outward),i.e.,

δ = cos−1

(

r ·B|r | |B|

)

. (9)

For the polar observations described here, the LOS projec-tion of any quantity that follows the magnetic field (e.g., theoutflow velocity) is given by multiplying its magnitude bysin(θ + δ). The Banaszkiewicz et al. (1998) model exhibits alarger degree of departure from radial geometry than does theCranmer et al. (1999) model. However, at the large heightsfor which the UVCS OVI anisotropy results are of interesthere, the relative differences between the two models—andalso the differences between either model and a radial geom-etry (δ = 0)—are small.


FIG. 3.— Comparison of measured electron densities in polar coronalholes. Values ofne from Guhathakurta & Holzer (1994) (thick dashed line),Fisher & Guhathakurta (1995) (dotted line & vertical bar), Guhathakurta etal. (1999) (dot-dashed line & vertical bar), and Cranmer et al. (1999) (thicksolid line) were divided by equation (8), i.e., equation (2) of Doyle etal.(1999). The polar theoretical model of Cranmer et al. (2007)is also shown(thin dashed line), as is the approximate hydrostatic fit from eq. (1) of Doyleet al. (1999) (triple-dot-dashed line). Gray-scale histogram boxes show therange ofne values from the plume statistics study of Cranmer et al. (1999),with darker shades denoting the most likely values at each height.

Figure 3 shows the range of electron densities measured byseveral instruments in polar coronal holes. The strong radialdecrease inne(r) has been removed by dividing all measure-ments by equation (8). The use of this normalization moreclearly illustrates the relative differences between the differ-ent sets of values, which Raouafi & Solanki (2004, 2006)claimed to be important in the derivation of O5+ temperatureanisotropy. The differences between plumes and interplumeregions is certainly responsible for some of the wide range ofvariation, but some of it may also be due to absolute calibra-tion uncertainties between instruments. Note, though, that thecurve representing the hydrostatic equation (1) of Doyle etal.(1999) appears to be clearly divergent from the other empiri-cal curves abover ≈ 8R⊙, with a slope that is flatter than theother measurements even several solar radii below that.

The gray-scale histogram boxes in Figure 3 illustrate thevariations due to differing concentrations of polar plumesalong a single polar LOS over three months (1996 Novem-ber 1 to 1997 February 1) using a consistent data set and largeenough count rates to make Poisson uncertainties negligible(see Table 3 of Cranmer et al. 1999). The curves from Fisher& Guhathakurta (1995) and Guhathakurta et al. (1999) showaverages of the various plume and interplume values given inthose papers, with vertical lines illustrating the relative con-trast between the densest plume-filled lines of sight and theregions with the fewest numbers of plumes. (These verticallines are shown atr ≈ 10R⊙ for clarity, but they are represen-tative of the values at the lower heights corresponding to theobserved white-light data.) Polar electron density valuesre-ported recently by Quémerais et al. (2007) are not shown, butthey are similar in radial shape to the Fisher & Guhathakurta(1995) mean curve (but with values about 10%–20% higher).Overall, the variations between data sets that appear to exceedthe plume-interplume contrast may be due to different instru-

ment calibrations.When modeling the OVI observations summarized in § 2,

it is probably incorrect to use the lowest “pure interplume”electron densities. Atz≈ 2.5–3 R⊙ in polar coronal holes,the UVCS observations were typically integrated over 15′ to30′ in the tangential direction, whereas polar plumes at theseheights have transverse sizes of only about 1′ to 2′. Thus,the most appropriate electron densities to use are those thataverageover plumes and interplume regions. The lower lim-its from Fisher & Guhathakurta (1995) and Guhathakurta etal. (1999), as well as the fitting function given by Esser etal. (1999), seem to be inappropriate to apply to the empiri-cal modeling of these UVCS OVI data. At lower heights,where the plume and interplume regions have been resolvedby UVCS (e.g., Kohl et al. 1997; Giordano et al. 2000), theuse of the full range of plume and interplume values ofnewould be warranted.

3.3. Comparison with Observations

Once a model data cube (which variesui‖, Ti⊥, andTi⊥/Ti‖along its axes) has been produced for a given observing heightz and a given set ofne, Te, and flux tube parameters, the nextstep is to compute the probability of agreement between agiven observation and each of the simulated observations inthe cube. We compute this probabilityP as the product oftwo quantities that are assumed to be independent of one an-other: (1) the probabilityPR that the observed line ratio agreeswith the simulated ratio, and (2) the probabilityPS that theobserved profile shape of the OVI λ1032 line agrees with thesimulated shape. Because the brighter OVI λ1032 line tendsto dominate the measured “weighted” line widthV1/e, we useonly the simulated OVI λ1032 line shape in the latter com-parison.

The line ratio probabilityPR is relatively straightforward tocompute. The modeled total intensities of the two compo-nents of the doublet are determined by summing up the spe-cific intensities over the 12 wavelength bins. Their ratio thusgivesRmodel. The relative distance betweenRmodel and theobserved ratioRobs, in units of the observational standard de-viation (δRobs), is the quantity that determines the probabilityof agreement. Assuming the uncertainties are normally dis-tributed, the probability is

PR = 1− erf

( |Robs−Rmodel|δRobs

√2

)

(10)

(see, e.g., Bevington & Robinson 2003). A larger argument inthe error function (“erf” above) denotes a larger discrepancybetween the modeled and observed ratios, and thus a lowerprobability of agreement.

The line shape probabilityPS is not as easy to compute asPR. An initial attempt was made to fit the simulated profileswith Gaussian functions, and then to compare the resultingV1/e widths with the observed values using a similar expres-sion as equation (10). However, there were many instanceswhere the modeled lines were far from Gaussian in shape,but the best-fitting Gaussian (which was a poor fit in an ab-solute sense) happened to agree with the observedV1/e. Thisresulted in spuriously high probabilities for wide regionsofparameter space that should have been excluded. Thus, wefound that the tabulated specific intensities (i.e., the full lineshapes) need to be compared on a wavelength-by-wavelengthbasis. This raises the issue of what to use for the “observed”line shape. As described in § 2, the UVCS/SOHOdata pointscontain a wide range of instrumental effects that were taken


into account in the line fitting process. In order to comparesimilar quantities, either these effects must be inserted into themodel profiles, or we must reconstruct “observed profile” in-formation from the extractedV1/e measurements and theδV1/euncertainties. We chose the latter option.

To determine the probability of agreement between the setof modeled specific intensities (Iλ,model) and the reconstructedobserved intensities (Iλ,obs), we computed aχ2 quantity,

χ2 =∑

λ

(

Iλ,obs− Iλ,model

δIλ,obs

)2

(11)

whereIλ,model came from the data cube, andIλ,obs was con-strained to be a Gaussian function with the observedV1/ewidth and a total intensity equal to that of the modeled profile.(The observed total intensity was not used because the com-parison being done here is only between the relative shapes.)TheδIλ,obs uncertainty was computed as a function of wave-length by comparing the idealizedIλ,obs profile with two oth-ers computed with line widthsV1/e − δV1/e andV1/e + δV1/e(with all three profiles normalized to the same modeled to-tal intensity). These three profiles exhibited a range of spe-cific intensities at each wavelength, and the standard deviationquantityδIλ,obs was defined as half of that full range. Then,theχ2 quantity above constrains the probability that the ob-served and modeled profiles are in agreement (i.e., the proba-bility that the observed and modeled specific intensity valuesare drawn from the same distribution). Assuming normallydistributed uncertainties, this probability is given by

PS ≡ Q(χ2|ν) =1

Γ(ν/2)

∫ ∞

χ2/2e−tt (ν/2)−1dt (12)

(Press et al. 1992), whereν = Nλ − 1 is the effective num-ber of degrees of freedom (forNλ = 12 wavelength points)andΓ(x) is the complete Gamma function. Whenχ2 ≪ ν theabove probability approaches unity (i.e., the modeled profileis a good match to the observed profile), and whenχ2 ≫ ν theabove probability is negligibly small.

We thus obtained the total probabilityP = PRPS as a func-tion of the three main O5+ variables of each data cube, for eachobservation at the heightzconsistent with that data cube. Thequestion of what is considered to be a large or small proba-bility is open to some interpretation. Below, we often use astandard “one sigma” probabilityP1σ = 1− erf(1/

√2) = 0.317

as a fiducial value above which the solutions are consideredto be good matches with the data.

4. EMPIRICAL MODEL RESULTS

In this section we present results for the most probable val-ues of the O5+ ion properties between 1.5 and 3.5R⊙. In§ 4.1, we show how the essential information inside the three-dimensional probability cubes can be extracted and analyzedin a manageable way. In § 4.2, the optimal values for O5+ out-flow speed and perpendicular temperature are presented formodels C and R. The resulting values ofui‖ andTi⊥ are con-sistent with earlier determinations of preferential ion heatingand acceleration with respect to protons. In § 4.3, we discussthe determination of the anisotropy ratioTi⊥/Ti‖ for models Cand R, which is less certain than the other two quantities. Wethen focus in detail on a single representative height in § 4.4in order to determine how models C and R can give rise toqualitatively different conclusions about the ion temperatureanisotropy. Finally, in § 4.5 we extract information from both

models C and R about the O5+ ion concentration in the ex-tended corona—i.e., we compute the rationO5+/nH from thecomparison of observed and modeled OVI λ1032 total inten-sities.

4.1. Deriving Ion Properties from the Data Cubes

We constructed two sets of radially dependent data cubes:one for the model C assumptions forne, Te, and flux-tube ge-ometry, and one for model R. Each set consisted of 13 datacubes with observation heightsz = 1.5, 1.6, 1.7, 1.8, 1.98,2.11, 2.3, 2.42, 2.563, 2.7, 3.0, 3.09, and 3.565R⊙. Thesevalues were chosen to align with the observed data pointsshown in Figure 1. Any discrepancies between the observedand modeled heights never exceeded±0.065R⊙, and for thewhole data set the average absolute value of the discrepancywas only 0.012R⊙. We then constructed 49 individual “prob-ability cubes” for each of the data points for which bothV1/eandR exist.

Even for just a single comparison between an observationand a data cube at one height, it is a challenge to displaythe full three-dimensional nature of the probability “cloud”P(ui‖,Ti⊥,Ti⊥/Ti‖). We limit ourselves to showing lower-dimensional projections that keep only the highest probabil-ity values taken over the axes that are not being shown. Asan example, in Figure 4 we display two-dimensional contoursof P as a function of all three unique pairings of the threeaxis-quantities of the data cube. The specific comparison isbetween the measurement shown in Table 1 from 1997 Jan-uary 5 (3.07R⊙, V1/e = 690 km s−1) and the model R datacube constructed atz= 3.09R⊙. In all three contour plots, theprobability shown at each location is a maximum taken overthe third quantity that is orthogonal to the projection plane.Thus, for regions with a low probability in these diagrams,we can be assured that there areno values of the unplottedcoordinate that can give synthetic line profiles in agreementwith the observations.

Figure 4a shows an approximate anticorrelation betweenthe ion outflow speed and the perpendicular kinetic temper-ature in the subset of generally “successful” models. Thisarises mainly because the lines can be broadened both by mi-croscopic LOS motions (roughly proportional toTi⊥) and bythe projection of the superradially flowing bulk outflow speedalong the LOS (which goes asui‖). When one of these quanti-ties goes up, the other must go down in order to match a givenobserved line profile. Figure 4b shows that the region of pa-rameter space with the larger contribution byTi⊥ (upper left)also requires a large anisotropy ratio, but the region with thelarger contribution by bulk LOS motions (lower right) maybe able to match the observations with an isotropic velocitydistribution (i.e.,Ti⊥/Ti‖ ≈ 1).

The large amount of information in contour plots like Fig-ure 4 can be collapsed down to a smaller list of parame-ters. We created three one-dimensional probability curvesasa function of each of the three main axis quantities, with themaximum values extracted from the full plane subtended bythe remaining two neglected quantities. Thus, we define thereduced probability functionsPu(ui‖), Pt(Ti⊥), andPa(Ti⊥/Ti‖)(the subscript “a” denotes the anisotropy ratio). These func-tions are generally peaked at some high value close to 1 andexhibit lower values far from the optimal solutions. Figure5shows these reduced probabilities for the same example caseshown in Figure 4. The reduced probabilities forui‖ andTi⊥are peaked relatively sharply around their most probable val-ues. Note that we plot the perpendicular kinetic temperature


FIG. 4.— Contour plots of the maximum probabilities of agreement be-tween the model R data cube and the UVCS observation from 1997January5 at r = 3.07R⊙. The three panels show probabilities as a function of eachunique pair of the three data-cube axis quantities. Contourlevels are plottedat 90% of the probability values 1, 0.3, 0.1, 0.03, and 0.01 (see gray-scalecode in panela).

in units of a most-probable speedwi⊥ = (2kBTi⊥/mi)1/2 in or-der to facilitate comparison with earlier papers. The reducedprobability for the anisotropy ratio, shown in Figure 5b, isless centrally peaked and thus the best solution for this valueis less certain. The peak value corresponds to a most-probableanisotropy ratio ofTi⊥/Ti‖ ≈ 6, but note that the probabilityof isotropy remains reasonably high at∼40%.

It is interesting to contrast the exhaustive data-cube-searchtechnique used here with the more straightforward approachestaken in earlier papers. For example, Raouafi & Solanki(2004, 2006) simulated the properties of the OVI λλ1032,1037 lines after first fixing the radial variation of the outflowspeed and ion temperature. Figure 5b shows an illustrative“cut” through the data cube atfixedvalues ofui‖ = 600 km s−1

and wi⊥ = 215 km s−1 at r = 3.09R⊙ (similar to the values

FIG. 5.— Reduced probabilities for one specific comparison between aUVCS observation atr = 3.07R⊙ and the model R data cube (see also Figs. 4,8, 10, and 11). (a) Pu versus outflow speedui‖ (solid line) andPt versus per-pendicular most-probable speedwi⊥ (dashed line). (b) Pa versus anisotropyratio Ti⊥/Ti‖ for a search of the entire data cube (solid line) and for a “slice”through the data cube with fixed values ofui‖ andwi⊥ given above (dashedline). Also shown is the threshold levelP1σ (dotted lines) defined in the text.

used by Raouafi & Solanki 2006). In this case, the most prob-able value of the anisotropy ratio is surprisingly close to 1, aswas also assumed by Raouafi & Solanki. The apparent consis-tency with the observations (i.e., a value ofPa of about 30%)may be misleading if the rest of the data cube is not searched.Thus, we can assert that any results concerning the anisotropyratio that were obtained bynotsearching the full range of pos-sibilities for the ion parameters are potentially inaccurate.

The ultimate goal of the empirical modeling process is tocharacterize the peak values and widths of the reduced prob-ability curves, in order to obtain the optimal measured values(with uncertainty limits) for the relevant O5+ plasma proper-ties. The most satisfactory outcome, of course, would be verynarrow peaks that occur far from the edges of the parameterspace, but this is not always the case. After some experimen-tation, we chose to use weighted means to obtain the peakvalues, i.e.,

〈x〉 =

∫

dxxPx(x)∫

dxPx(x)(13)

wherex denotes any of the three axis quantitiesui‖, Ti⊥, orlog(Ti⊥/Ti‖). We used the logarithm of the anisotropy ratioin equation (13) because tests showed that if the ratio itself(which spans three orders of magnitude) was used, the abovemean would be weighted strongly toward the largest valueseven when the maximum of the probability distribution is atmuch lower values.

We experimented with using the variance, or second mo-


ment, of the reduced probability distributions to characterizethe widths of “error bars” for each measurement. However,because the probability curves are generally not symmetricaround the peak values, the second moment often did notaccurately give a range of values with reasonable probabili-ties. Instead, we performed a straightforward search for therange of probabilities that are higher than the threshold valueP1σ ≈ 0.317 discussed above. The lower and upper limits ofthat range were taken to be the ends of the uncertainty boundsfor each measurement.

4.2. Preferential Ion Heating and Acceleration

Figure 6 shows the weighted mean and error-bar quantitiesfor the O5+ plasma properties, defined as in the previous sub-section, as a function of heliocentric height. The results frommodel C and model R are plotted in two different colors, withonly the error bars of model R shown for clarity. Here we fo-cus on the ion outflow speed (Fig. 6a) and the perpendicularion temperature (Fig. 6b). On average, the derived values of〈ui‖〉 and〈wi⊥〉 were consistent between the two sets of mod-els. To quantify the impact of varying the electron density,electron temperature, and flux-tube geometry, we computedratios of the model C values to the model R values for eachdata point. For the 49 data points taken together, the meanvalue of the ratio of the outflow speeds was 1.002, with a stan-dard deviation of 19%, and the mean value of the ratio of per-pendicular most-probable speeds was 0.991, with a standarddeviation of 14%. This shows that the determination of theseparameters is relatively insensitive to the choices of electrondensity, electron temperature, and flux-tube geometry.

The radial dependence of the derived〈ui‖〉 values in Fig-ure 6a is similar to that of the O5+ empirical models B1 andB2 given by Cranmer et al. (1999). Note the emergence of anatural trend of radial acceleration in〈ui‖〉, with the possibleexception of the data points atr & 3.5R⊙. This is especiallyserendipitous given that each data point was analyzed inde-pendently of all others.

The derived O5+ outflow speeds support earlier claims ofpreferential ion accelerationin coronal holes. Atr = 2.5R⊙,the range of ion outflow speeds that gives rise to high prob-abilities of agreement with the data points is approximately280–500 km s−1 At this height, these values are substantiallylarger than bulk (proton-electron) solar wind outflow speedsderived via mass flux conservation. Figure 41 of Kohl et al.(2006) showed a selection of 12 bulk outflow speed modelsderived using all possible combinations of fourne models andthree coronal-hole geometries. Atr = 2.5R⊙, these 12 modelsgave a range of bulk outflow speeds of 115–300 km s−1. De-spite the small degree of overlap between the two ranges, themean value of the O5+ range (390 km s−1) exceeds the meanvalue of the mass flux conservation range (208 km s−1) byalmost a factor of two. Also, proton outflow speeds derivedfrom H I Lyα Doppler dimming (from a selection of papersall dealing with polar coronal holes at the 1996–1997 solarminimum) were shown in Figure 41 of Kohl et al. (2006). At2.5R⊙, the range of these values is 160–260 km s−1—with amean of 210 km s−1—which is still significantly lower thanthe range of O5+ ion outflow speeds discussed above.

In Figure 6b, the trend of radial increase in〈wi⊥〉 is alsoroughly similar to that found by Cranmer et al. (1999), espe-cially below about 2.3R⊙. At larger heights, though, thereappears to be less evidence for a systematic increase than ex-isted in the model B1 and B2 curves. This could have re-

FIG. 6.— Derived outflow speeds (a), perpendicular most-probable speeds(b), and kinetic anisotropy ratios (c) for model R (red points) and model C(blue points). Symbols show the weighted means of the reduced probabilitydistributions, with styles the same as in Figure 1. Verticalbars show thefull range of parameter space with reduced probabilities greater thanP1σ (formodel R). Also shown are empirical models B1 and B2 from Cranmer et al.

(1999) (dotted lines) and Alfvén wave quantities〈δv2⊥〉

1/2x in panel (b) and

Aeff in panel (c), derived from the model of Cranmer et al. (2007) (dot-dashedlines).

sulted either from the inclusion of the new data points or fromthe more exhaustive treatment of uncertainties in the new pa-rameter determination method described above. However, ifone takes all of the derived〈wi⊥〉 values forr ≥ 2R⊙ and fitsto a straight line, the best-fitting slope is still increasing withheight at a rate of 50 km s−1 perR⊙. This is about a third ofthe∼150 km s−1 perR⊙ slope in the B1 and B2 models.

The most-probable speeds〈wi⊥〉 shown in Figure 6b, al-though slightly smaller than those given by the Cranmer etal. (1999) model B1 and B2 curves at some heights, still show


definite evidence forpreferential ion heating.The mean valueof the〈wi⊥〉 values at heightsr ≥ 2.5R⊙ in Figure 6b is 363km s−1, with a standard deviation of 73 km s−1. Between 2.5and 3R⊙, the perpendicular proton most-probable speeds de-rived from HI Lyα were about 210–240 km s−1, with a meanvalue of about 225 km s−1 (see models A1 and A2 of Cranmeret al. 1999). The fact that the O5+ mean value exceeds theproton mean value by almost two standard deviations impliesthat the O5+ kinetic temperature at this height is very likelyto be more than “mass proportional” (i.e., implying an oxy-gen kinetic temperature of 130 MK, or more than 40 timesthe proton kinetic temperature of∼3 MK.

It is important to note that the derived kinetic temperaturesare likely to be a combination of thermal and nonthermal mo-tions. The ion-to-proton kinetic temperature ratio of∼40,derived above, is likely to be alower limit to the true ratioof thermal, or microscopic temperatures. If unresolved wavemotions are deconvolved from the empirical values of〈wi⊥〉,the proton most-probable speed will be reduced by a relativelylarger amount than the O5+ speed. As an example, the the-oretical polar coronal hole model of Cranmer et al. (2007)has a LOS-projected Alfvén wave amplitude atr = 2.5R⊙ of116 km s−1 (this is also plotted in Fig. 6b). Converting thesemotions into temperature-like units and subtracting from bothvalues given above, one obtains an O5+ perpendicular temper-ature of 115 MK and a proton perpendicular temperature of2.2 MK. The ratio of ion to proton temperatures has thus in-creased from about 40 to 50. In any case, it is clear that thedominant contributor to the ion kinetic temperature is the true“thermal” temperature, with only a relatively minor impactfrom broadening due to macroscopic motions.

4.3. The Ion Anisotropy Ratio

Figure 6c illustrates the largest discrepancy between theempirical models of Cranmer et al. (1999) and the presentmodels (both C and R). Above a height of∼2.5 R⊙, modelsB1 and B2 demanded a strong anisotropy ratioTi⊥/Ti‖ > 10,but the optimal ratios derived in this paper seem to clusterbetween 2 and 10 with no discernible radial dependence. Itis important to note, though, that there is considerable over-lap of theuncertaintiesbetween the old and new ranges of〈Ti⊥/Ti‖〉. Several of the error bars shown in Figure 6c extendup into the range of ratios from models B1 and B2. Also, thedotted curves that illustrate models B1 and B2 correspond tothe “optimal” values of the anisotropy ratio from Kohl et al.(1998) and Cranmer et al. (1999); the uncertainties in thosemodels are not shown.

Because the derived values of the kinetic temperature ratio〈Ti⊥/Ti‖〉 in Figure 6c exceed unity by only a relatively smallamount, it is worthwhile to examine whether the numerator(Ti⊥) may have been enhanced by unresolved wave motionsperpendicular to the magnetic field. In other words, for a real-istic model of perpendicular wave amplitudes in polar coronalholes, we investigate whether a trulyisotropic microscopicvelocity distribution could have given an effective anisotropyratio that exceeds 1. We compute such an effective anisotropyratio as

Aeff =1

1− (〈δv2⊥〉x/w2

i⊥), (14)

where〈δv2⊥〉x is the square of the frequency-integrated Alfvén

wave amplitude divided by two to sample only the motionsalong one of the two perpendicular directions (i.e., only alongthe LOS orx axis). As above, we used the Alfvén wave prop-

FIG. 7.— Probability of ion isotropyPa(1) plotted versus heliocentric dis-tance for model R, using the same data symbols as Figures 1 and6. The filledcircle shows the probability of isotropy for the specific observation (3.07R⊙,V1/e = 690 km s−1) that is considered in detail in Figures 4, 5, 8, 10, and 11.

erties from the turbulence-driven polar coronal hole modelofCranmer et al. (2007). The model wave amplitude is plottedin Figure 6b and the quantityAeff is plotted in Figure 6c. TheconditionAeff ≈ 1 corresponds to the situation where the am-plitudes are too small to contribute to the anisotropy ratio(asdefined in the empirical models). Belowr ≈ 1.7R⊙, the curvein Figure 6c shows thatAeff does indeed exceed 1 by about theamount computed from the UVCS data. At these low heights,the derived value of〈wi⊥〉 is of the same order of magnitudeas the wave amplitude, so the latter can “contaminate” thedetermination of the true perpendicular most-probable speed.At heights larger than about 2R⊙, though, the wave ampli-tudes are small in comparison to the derived〈wi⊥〉 values, andthusAeff ≈ 1. We thus conclude that above 2R⊙, any derivedanisotropy ratio〈Ti⊥/Ti‖〉 is likely to be truly representativeof the microscopic velocity distribution and not affected bywave motions.

Despite the comparatively low values of the anisotropy ra-tio shown in Figure 6c (〈Ti⊥/Ti‖〉 ≈ 2–10), we should em-phasize that these values are often significantly differentfromunity. It is important to note thatall of the data points havetheir largest reduced probability—measured either using theweighted mean defined above or by simply locating the max-imum value—for anisotropy ratios larger than unity.

The preponderance of evidence for anisotropy is also il-lustrated in Figure 7, which shows the probability that eachmeasurement could be explained by an isotropic O5+ veloc-ity distribution. In other words, Figure 7 gives the value ofPa(1) for each probability cube. Taken together, a signifi-cant majority of the values (about 78% of the total number)fall below the fiducial one-sigma value ofP1σ, indicating thatisotropy should not be considered a “baseline” assumption.Below aboutr = 2R⊙, a few of the measurements correspondto large probabilities that an isotropic distribution can explainthe observations. Note from Figure 6c, though, that the most-probable anisotropy ratios for these measurements tend to begreater than 1, but some of the error bars extend down past


FIG. 8.— Reduced probabilityPa versus ion anisotropy ratioTi⊥/Ti‖ forone specific comparison between a UVCS observation atr = 3.07R⊙ and themodel R data cube (see also Figs. 4, 10, and 11). The probability computedwith the actual observational uncertainties (thick solid line) is compared withtrial curves computed with a range of constant factorsǫ multiplying δV1/eandδR (dashed lines). Also shown is the threshold levelP1σ (dotted line)and the probability of isotropy for the standardǫ = 1 case that is also shownin Figure 7 (filled circle).

Ti⊥/Ti‖ = 1. However, between 2.1 and 2.7R⊙ the probabilityof isotropy is very small for all of the observed data points.Above 3R⊙, some of the values ofPa(1) become large again,but we believe this may be due to the relatively high observa-tional uncertainties on the OVI intensities and line widths atthese large heights (see below).

To better understand the impact of observational uncer-tainties on the probability of isotropy, Figure 8 shows thefull Pa(Ti⊥/Ti‖) curves for one specific measurement atr =3.07R⊙ (i.e., the same measurement used in Fig. 5). Themultiple curves were constructed by multiplying the knownobservational uncertaintiesδV1/e andδR by arbitrary factorsǫ. Generally, larger uncertainties lead to lowerχ2 values whencomparing the observed and modeled line shapes, and thus tolarger probabilities of agreement between the observed andmodeled profiles. Interestingly, though, the anisotropy ratioTi⊥/Ti‖ at which the maximum probability occurs remainsroughly constant whenǫ is varied between 0.5 and 2. Thus,if future observations above 3R⊙ were to obtain the samegeneral range of values forV1/e andR but with lower uncer-tainties, it could provide stronger evidence for ion anisotropyup at these heights.

Although Figure 6c does not seem to indicate a substantialdifference between models R and C, it is useful to comparethese models in some additional detail. For all data points,the mean ratio of model C to model R anisotropy ratios was1.199, but the large standard deviation (76%) shows that themodels are often quite different from one another. Taking onlythe heights above 2.2R⊙, the mean ratio of model C to modelR anisotropy ratios increases to 1.514, indicating thaton aver-agemodel C generates larger anisotropies than model R overthe height range where anisotropies appear to be required.

Figure 9 shows the ratio of model C to model R anisotropyratios as a function of height. Belowr ≈ 2.2R⊙ the two

FIG. 9.— Ratios of model C to model R values for the weighted meananisotropy ratios〈Ti⊥/Ti‖〉 shown as a function of height and using the samedata symbols as Figures 1, 6, and 7.

models produce roughly the same result for the anisotropy ra-tio. Above that height, the solutions split into two groups:one where model C produces a substantially larger ratio (2–3 times that of model R), and one where model C producesa comparable or slightly smaller ratio than model R. Notethat the height range of 3.0–3.1R⊙—over which model Rpredicted a rise in the probability of isotropy (see Fig. 7)—strongly favors larger anisotropies for model C.

4.4. Varying the Electron Density, Electron Temperature, andGeometry

One of the main motivations for this paper was to explorewhy the results of Raouafi & Solanki (2004, 2006) were sodifferent from earlier results (e.g., Cranmer et al. 1999) re-garding the O5+ anisotropy ratio. In this subsection, we studythe differences between model R and model C in more de-tail by focusing on the shapes of the reduced probability dis-tributions for one representative data point. As in Figures4, 5, and 8, we used the probabilities generated by com-paring the UVCS/SOHOmeasurement from 1997 January 5(r = 3.07R⊙, V1/e = 690 km s−1) with data cubes constructedwith various assumptions. This data point is denoted by afilled circle in Figure 9, and it is clear that this point is rep-resentative of the majority of the data points (5 out of 7) atr ≈ 3R⊙.

Figure 10 shows a range of reduced probability curvesPa(Ti⊥/Ti‖) that were computed from data cubes constructedwith various combinations of the model R and model C pa-rameters. The three-letter names for the models denote theindividual choices forne, flux-tube geometry, andTe (in thatorder). The “pure” model R and model C cases are thus calledRRR and CCC.

Before examining the impact of the individual parameterson the reduced probability curves, we note that the modelCCC curve in Figure 10b mirrors almost exactly the resultsof Kohl et al. (1998) and Cranmer et al. (1999) atr ≈ 3R⊙:the most likely O5+ anisotropy ratio ranges between 10 and


FIG. 10.— Reduced probabilityPa versus ion anisotropy ratioTi⊥/Ti‖ forthe same data comparison as in Figures 4 and 8, but for variouscombinationsof the model R and model C parameters (see above for line styles). Theorder of the three-letter designations isne, f (r),Te. Panel(a) thus showsall models computed with the Doyle et al. (1999)ne and panel(b) showsall models computed with the Cranmer et al. (1999)ne. Also shown is thethreshold levelP1σ (horizontal dotted lines) and the probability of isotropyfor the main RRR and CCC cases (filled circles).

100, and an isotropic distribution is highly improbable. ModelCCC exhibits a most probable ion outflow speed〈ui‖〉 = 508km s−1, which is only marginally smaller than the model RRRvalue of 521 km s−1. Model CCC has an optimal solution for〈wi⊥〉, though, of 541 km s−1, which is 23% larger than thecorresponding value of 440 km s−1 for model RRR (i.e., a 51%higher value of〈Ti⊥〉 for model CCC). Model CCC tended toproduce more line broadening via “thermal” motions near theplane of the sky, and model RRR tended to produce more linebroadening via bulk outflow projected along the LOS.

The other curves shown in Figure 10 explore which of thethree varied parameters were most responsible for the differ-ences between models RRR and CCC. We see immediatelythat the choice of electron temperatureTe, which in our mod-els impacts only the collision rateq12, is relatively unimpor-tant. The 8 curves can thus be separated into 4 pairs, eachof which has the same choice forne and flux-tube geometry(i.e., RRX, RCX, CRX, and CCX, where ‘X’ denotes eitheroption forTe). The overall insensitivity to electron tempera-ture is evident from the fact that the two curves in each pairare virtually indistinguishable from one another.

Figure 10 shows that the unique features of the RRX mod-els (i.e., a higher probability of isotropy and a strong peakatTi⊥/Ti‖ < 10) are only present whenboththe electron densityand flux-tube geometry are treated using model R. The mod-

els with only one of these two parameters treated using modelR (i.e., RCX and CRX) appear more similar to the CCX mod-els. At large values of the anisotropy ratio, both the RCX andCRX models are virtually identical to the CCX models. Atlow values of the anisotropy ratio, the CRX model is roughlyintermediate between the CCX and RRX models. Generally,though, thecombinationof the model R assumptions for elec-tron density (e.g., Doyle et al. 1999) and flux-tube geometry(e.g., Banaszkiewicz et al. 1998) are needed to produce broadenough profiles via outflow speed projection along the LOS toexplain the observationswithoutthe need for extreme temper-ature anisotropies. Specifically, this enhanced LOS projectioneffect arises for two coupled reasons.

1. As seen in Figure 2c, the Banaszkiewicz et al. (1998)flux tubes are tilted to a greater degree away from theradial direction than the Cranmer et al. (1999) fluxtubes. Because of these larger values ofδ, a larger frac-tion of the outflow speedui‖ is projected into the LOSdirection (when|x|> 0) for model R.

2. Figure 3 shows that the Doyle et al. (1999) electrondensity does not drop as rapidly with increasing height(between about 3 and 10R⊙) as nearly all of the otherplottedne functions. Thus, for observation heights atabout 3R⊙, the Doyle et al. (model R) electron den-sity provides a relative enhancement for points alongthe foreground and background (|x|> 0) in comparisonto the plane of the sky (x = 0).

Note also from Figure 3 that the electron density used formodel C is about 10% to 30% larger than that used for modelR at the heights of interest (r ≈ 3–4R⊙). A higher value ofneis expected to result in emission lines that are dominated moreby the collisional component of the emissivity, which scalesas n2

e (eq. [2]), with a correspondingly weaker contributionfrom the radiative component, which scales linearly withne(eq. [3]). Because of the different density dependences, thecollisional component is not extended as far along the LOSas the radiative component. Thus, models with higher den-sities would be expected to behave more like model C (withemission dominated by the plane of the sky), and models withlower densities would be expected to behave more like modelR (with emission extended over a larger swath of the LOS).

To explore the effects of varying the electron density, we re-peated the model R data cube analysis (for the fiducial heightshown in Figs. 8 and 10) withne(r) multiplied by constant fac-tors. Figure 11 shows the resulting reduced probability curvesas a function of the O5+ temperature anisotropy. A model withhalf of the Doyle et al. (1999) electron density has a lowerpreferred value ofTi⊥/Ti‖ and a much higher probability ofisotropy than the standard model R. A model with double theDoyle et al. (1999) electron density resembles model C in thatthere is a high preferred range ofTi⊥/Ti‖ and a low probabil-ity of isotropy. Despite the large change in appearance of thePa curves as shown in Figure 11, the preferred values of theoutflow speed and perpendicular kinetic temperature do notvary by very much asne is varied up and down by a factor oftwo: 〈ui‖〉 changes by only about±8% (increasing asne de-creases), and〈wi⊥〉 changes by only about±10% (increasingasne increases). These determinations appear to be relativelyinsensitive to the choices forne and flux-tube geometry.

The ratio of collisional emissivity to the total line emissionchanges dramatically for the models shown in Figure 11. Formodel RRR, the optimal model in the data cube exhibited a


FIG. 11.— Reduced probabilityPa versus ion anisotropy ratioTi⊥/Ti‖ forthe same data comparison as in Figures 4, 5, 8, and 10, but for arange ofconstant multipliers to the Doyle et al. (1999) electron density. The basic“model R” ne (solid line) is compared to a model with half (dashed line) anddouble (dot-dashed line) this electron density function. Also shown is thethreshold levelP1σ (dotted line) and the probabilities of isotropy for the threecurves (filled and open circles).

collisional fraction of 93.7% for the OVI λ1032 line and afraction of 44.6% for the OVI λ1037 line (the latter being“Doppler pumped”). The model with half of the model R den-sity had lower collisional fractions for theλλ1032, 1037 linesof 88.2% and 28.7%, respectively. The model with double themodel R density had higher collisional fractions of 96.8% and61.7%.

It is important to note, however, that the differences in colli-sionality for the models shown in Figure 10 are not as drasticas those shown in Figure 11. Model CCC exhibited collisionalfractions for theλλ1032, 1037 lines of 90.7% and 47.0%.These values are only a few percentage points different fromthe model RRR fractions. The other intermediate models havevalues that cluster between those of models RRR and CCC.The larger value ofne in the plane of the sky for model C iscompensated—to some degree—by the slower decrease innealong the LOS for model R. Thus, despite the superficial re-semblance between the model CCC curve in Figure 10 and the“doublene” curve in Figure 11, one cannot invoke a varyingamount of collisionality to explain the differences betweenmodels R and C. The LOS extension effects discussed aboveare more subtle than simply varyingne by a constant amount.

Another way we explored the dependence of the reducedprobabilities on electron density was to produce a set ofthree other models with alternate functional forms forne(r),but the same flux-tube geometry andTe as used in modelR. These models utilized the mean electron density curvesfrom Guhathakurta & Holzer (1994), Fisher & Guhathakurta(1995), and Guhathakurta et al. (1999) (see also Fig. 3), andthe OVI data cubes were created only at the fiducial heightof 3.09R⊙.4 The reduced probabilitiesPa for these models

4 We also created a data cube for the hydrostatic equation (1) of Doyle etal. (1999), but this model exhibited an unusually strong extension along theLOS. There was a substantial contribution to the OVI emissivity even at theLOS integration limits ofx = ±15R⊙, which actually led to an extremely

FIG. 12.— Reduced probabilities of isotropy (diamonds) and weightedmean anisotropy ratios (triangles) for models having a range ofne valuesand identical flux-tube andTe properties (see text for details). All data-cubecomparisons were computed for the same fiducial data point illustrated inFigures 4, 5, 8, 10, and 11. Curves denote the best fitting quadratic relationsas a function of the optimized density quantityn1.83

3 /n6.

all fell within the general range of variation illustrated in Fig-ure 10 and are not plotted. However, the construction of thesemodels increased the number of data cubes with “model R-like” flux-tube andTe parameters to seven: i.e., these threenew ones, the three models shown in Figure 11, and modelCRR (with a model C electron density). We performed aregression analysis on the seven values of the probability ofisotropyPa(1) and the weighted mean anisotropy〈Ti⊥/Ti‖〉 tofind the optimal functional dependence on two “independent”variables that characterize the electron density:

n3 ≡ ne(3R⊙)106cm−3

, n6 ≡ ne(6R⊙)106cm−3

(15)

where the arbitrary normalizations serve only to keep thecombined quantities (discussed below) of order unity. Thequantity n3 characterizes the electron density in the planeof the sky of the observation, and the ration3/n6 character-izes the large-scale density gradient and thus the relativeen-hancement of foreground and background regions along theLOS. From the discussion above, we expect that larger val-ues of bothn3 andn3/n6 should result in lower probabilitiesof isotropy and higher most-probable values of the anisotropyratio. Indeed, the regression analysis found that the modeledvalues of these quantities exhibited the lowest combinedχ2

spread for a single independent variable that scales asn1.833 /n6

(close to the product ofn3 andn3/n6). Figure 12 shows thesevalues as well as the best-fitting quadratic functions toPa(1)and〈Ti⊥/Ti‖〉. The combined dependence on bothne and itsradial gradient appears to be a key factor in determining therelative probabilities of isotropy and strong anisotropy.

Finally, we must evaluate which sets of choices for the elec-tron density and the flux-tube geometry are most consistent

low probability of isotropy. However, we discarded this model because theshallowne at large heights is clearly unphysical.


with observed polar coronal holes. Figure 3 does seem to in-dicate that most measuredne curves (as well as one exam-ple theoretical result forne) behave more like the “model C”(Cranmer et al. 1999) case than the “model R” (Doyle et al.1999) case. Between the heights of about 3 and 10R⊙, themajority of the curves in Figure 3 exhibit asteeperradial de-crease than the Doyle et al. (1999) model. Thus, the CCX orCRX models shown in Figure 10b appear to be more consis-tent with observations than the RRX or RCX models in Fig-ure 10a. This then implies that substantial O5+ anisotropy(Ti⊥/Ti‖ & 10) is also preferred at large heights. The optimalchoice of the flux-tube geometry is less certain. Ideally, obser-vations of the nonradial shapes of polar plumes should be ableto constrain the magnetic field geometry (see, e.g., Wang et al.2007; Pasachoff et al. 2007), but it is unclear whether exist-ing plume observations would be able to distinguish the subtledifferences between, e.g., Figures 2a and 2b. In any case, thegeometry does not seem to be as major an issue as the electrondensity, since the variance between the four curves in Figure10b is not large.

4.5. Oxygen Ion Number Density

By comparing the observed and modeled total intensitiesof the OVI λ1032 line, it is possible to derive firm limits onthe combined elemental abundance and ionization fraction ofO5+. The ion concentration is useful both as a tracer of fastand slow solar wind streams (e.g., Zurbuchen et al. 2002) andas a possible diagnostic of the amount of preferential heat-ing deposited in coronal holes (Lie-Svendsen & Esser 2005).A first attempt at determining the O5+ number density fromUVCS data was made by Cranmer et al. (1999), but the “datacube search” technique developed in this paper allows a muchmore definitive set of measurements to be made.

The numerical code that computes the OVI line emissionused an arbitrary constant value for the ratiof0 = nO5+/ne

of 4.959× 10−6, which was derived from the oxygen abun-dance of Anders & Grevesse (1989) and the measured O5+

ionization fraction of Wimmer-Schweingruber et al. (1998).This is merely a fiducial value that does not affect the fi-nal determination of this ratio for a given UVCS observa-tion. When comparing the results from an empirical modeldata cube with a specific observation, the probability valuesP(ui‖,Ti⊥,Ti⊥/Ti‖) are used to construct a weighted mean ofthe modeled OVI λ1032 total intensity using equation (13), aswell as lower and upper limits using the full range of modelswith probabilities that exceedP1σ. These values are convertedinto “observed” ion concentration ratiosfobs by assuming thatthe ratio fobs/ f0 is equal to the ratio of the observed to themodeled values ofItot. By using the modeled weighted mean,lower limit, and upper limit ofItot we obtain the weightedmean, upper limit, and lower limit offobs. (Note that the lowerlimit of Itot gives the upper limit offobs and vice versa.) Fi-nally, fobs is converted into the ratio of O5+ to total hydrogennumber density (nO5+/nH) by multiplying by a factor of 1.1(assuming a helium to hydrogen number density ratio of 5%).

Figure 13 shows the resulting ion concentration ratios as afunction of height for the full range of model C data points.There is a hint of systematic radial increase at low heights.Asimilar radial increase would be predicted for ions that flowsubstantially faster than protons in the corona (by a factoroftwo) and then flow only∼10% faster than protons at 1 AU.Above 2R⊙, though, Figure 13 does not show any definitiveradial trend. Taking account of the uncertainty limits, thedataappear consistent with the O5+ ionization fraction being more

FIG. 13.— O5+ ion number density ratio (with respect to hydrogen) as afunction of heliocentric distance for model C. Symbols showthe weightedmeans of the reduced probability distributions, with styles the same as inFigure 1. Vertical bars show the full range of parameter space with reducedprobabilities greater thanP1σ. Also shown is a linear least-squares fit to thedata points (dashed line) and empirical lower and upper limits as describedin the text (gray region bounded by dotted lines).

or less “frozen in” (see, e.g., Hundhausen et al. 1968; Owockiet al. 1983; Ko et al. 1997). A linear least squares fit (usingthe logarithm ofnO5+/nH as the ordinate) is also shown, butthe relatively high uncertainties at large heights preclude anyreliable interpretation of the slope.

Figure 13 also shows a range of values that would have beenexpected from prior studies of both the oxygen abundanceand the O5+ ionization fraction. The abundance ratio (nO/nH)ranges from a relatively recent historical high of 8.5× 10−4

(Anders & Grevesse 1989) to the more recent—and some-what controversial—low of 4.6× 10−4 (Asplund et al. 2004,2005; Grevesse et al. 2007). The ionization fraction (nO5+/nO)was measured in situ by the SWICS instrument onUlysses(Wimmer-Schweingruber et al. 1998) to be about 0.0058 inthe fast solar wind. Models that include the freezing in ofheavy ions have produced values for this ratio from 0.0035(Esser & Leer 1990) to about 0.005 (Chen et al. 2003). Al-though collisional ionization equilibrium is not expectedtohold in the extended corona (for polar coronal holes), it is in-teresting to note that forTe = 106 K, both Arnaud & Rothen-flug (1985) and Mazzotta et al. (1998) give a ratio of about0.0045. This is similar to the above values, but it varies up anddown by about a factor of 50% asTe is decreased or increasedby only±30%. We thus take tentative lower and upper limitsfor nO5+/nO of 0.003 and 0.006. The horizontal lines shown inFigure 13 were computed from the products of the two lowerlimits and the two upper limits given above for

nO5+

nH=

(

nO

nH

)(

nO5+

nO

)

. (16)

The model C data points shown in Figure 13 have a meanvalue ofnO5+/nH = 1.52× 10−6 (taking a simple average) or1.39×10−6 (taking the average of the logarithms). Perform-ing the same analysis using model R yielded values that werelarger by about 5% (on average for all data points) to 20%(specifically for points at heights above∼3 R⊙). The standard


deviations of both sets of data points gave lower and upperbounds of approximately 8×10−7 and 2.4×10−6 that encom-pass the±1σ range. The prior studies of oxygen abundanceand O5+ ionization discussed above give somewhat higher val-ues, which extend from 1.4×10−6 to 5.1×10−6. Thus, boththe model C and model R data points are in reasonably goodagreement with thelower limit of the expected range, whichgives some support for the recent low oxygen abundances ofAsplund et al. (2004, 2005).

5. DISCUSSION AND CONCLUSIONS

The SOHOmission (Domingo et al. 1995) has made sig-nificant progress toward identifying and characterizing theprocesses that heat the corona and accelerate the solar wind(see also Fleck & Švestka 1997; Domingo 2002; Fleck 2004,2005). The results from the UVCS instrument regarding pref-erential heating and acceleration of heavy ions (i.e., O5+) havecontributed in a major way to these advances in understandingover the past decade, and it is important to verify and confirmthe key features of these results. Thus, this paper has ana-lyzed an expanded set of UVCS data from polar coronal holesat solar minimum with the goal of ascertaining whether iontemperature anisotropies are definitively present (as claimedby Kohl et al. 1997, 1998; Li et al. 1998; Cranmer et al. 1999;Antonucci et al. 2000; Zangrilli et al. 2002; Antonucci 2006;Telloni et al. 2007) or whether one can explain the obser-vations without such anisotropies (as claimed by Raouafi &Solanki 2004, 2006; Raouafi et al. 2007). These cases wereexemplified by two sets of empirical models: one (model R)that was designed to replicate many of the conditions assumedby Raouafi & Solanki (2004, 2006), and one (model C) thatused the same conditions as Cranmer et al. (1999).

The main conclusion of this paper is that there remainsstrong evidence in favor of both preferential O5+ heating andacceleration and significant O5+ ion temperature anisotropy(in the senseTi⊥ > Ti‖) abover ≈ 2.1R⊙ in coronal holes.More detailed conclusions, linked to the sections of the paperin which they were first discussed, are summarized as follows.

1. It is important to search the full range of possibleO5+ ion properties and not make arbitrary assumptionsabout, e.g., the ion outflow speed or the ion tempera-ture. It is clear from Figure 5b that if the comparisonwith observations is restricted to certain choices for theion parameters, the resulting conclusions about the iontemperature anisotropy can be potentially misleading.(§ 4.1)

2. The derived ion outflow speedsui‖ and perpendicu-lar kinetic temperaturesTi⊥ exhibit values similar tothose reported by Kohl et al. (1998) and Cranmer et al.(1999), independent of the choices of electron densityand flux-tube geometry. There is significant evidencefor preferential ion heating and ion acceleration withrespect to protons, although the radial rate of increaseof Ti⊥ may be slightly lower than that given by Cran-mer et al. (1999). The large values ofTi⊥ appear to bedue to true “thermal” motions and not unresolved wavemotions. (§ 4.2)

3. For heights above about 2.1R⊙, the models in this pa-per yielded higher probabilities of agreement with theUVCS observations foranisotropic velocity distribu-tionsthan for isotropic distributions. The UVCS obser-vations between the radii of 2.1 and 2.7R⊙ were found

to have probabilities of isotropy below about 10% (seeFig. 7), no matter what was assumed for the coronalelectron density or flux-tube expansion (i.e., for eithermodel R or model C). Even when using coronal proper-ties that seemed to maximize the probability of isotropy(e.g., model R), 78% of the UVCS data points exhibitedprobabilities of isotropy below our threshold one-sigmavalue of∼32%. (§ 4.3)

4. The UVCS data at heights at and above 3R⊙ can beused to put limits on the likelihood of strong O5+ tem-perature anisotropies. A key factor in discriminatingbetween empirical models that either require or do notrequire a substantial anisotropy is the degree ofexten-sion along the line of sight (LOS)of the emissivity. Thisextension is driven strongly by the rate of radial de-crease in the electron density. The relatively shallowslope ofne(r) used in model R (from eq. [8]) appears tobe an “outlier” when compared to other observationaland theoretical determinations of the electron densityprofile in coronal holes (see Fig. 3). Most otherne(r)curves exhibit a steeper radial decrease and thus a lesserdegree of LOS extension for the OVI emissivities. Ourmodel C, which utilized the empirical model parame-ters derived by Cranmer et al. (1999), had a represen-tative “steep” electron density profile and thus requireda substantial O5+ temperature anisotropy to explain theUVCS observations abover ≈ 3R⊙. (§ 4.4)

5. Models that exhibit enough LOS extension to reproducethe observed UVCS line profiles and intensitieswithouta temperature anisotropyappear to require both (1) anelectron density that decreases shallowly with increas-ing height, and (2) a highly superradial flux-tube geom-etry that projects a large fraction of the outflow speedvector into the LOS. Our model R, designed to be sim-ilar to the models used by Raouafi & Solanki (2004,2006), exhibited both of these conditions and thus hadhigher probabilities of an isotropic velocity distributionat heights abover ≈ 3R⊙. (§ 4.4)

6. At the largest heights (r & 3R⊙), the uncertainties inthe existing UVCS measurements make difficult a firmdetermination of the anisotropy ratio. The analysistechnique developed in this paper takes full accountof these observational uncertainties. Future observa-tions with smaller observational uncertainties (see Fig.8) should yield correspondingly “sharper” probabilitydistributions for the anisotropy ratio and thus better de-terminations of this quantity. (§ 4.3)

7. Total intensities of the OVI λλ1032, 1037 lines con-strain the ion concentration rationO5+/nH to be approx-imately 1.5×10−6, with at least a factor of two range ofuncertainty. If the freezing in of O5+ ions is consideredto be relatively well understood, then this value impliesa relatively low oxygen abundance in agreement withthe recent downward revision of Asplund et al. (2004,2005). (§ 4.5)

Because of existing observational uncertainties in the elec-tron density, flux tube geometry, and OVI line parameterssuch asV1/e (the line width) andR (theλ1032 toλ1037 inten-sity ratio), we cannot yet give “preferred” values for the O5+


anisotropy ratioTi⊥/Ti‖ as a detailed function of height. Be-low r ≈ 2R⊙, the observations are consistent with an isotropicvelocity distribution. Between 2.2 and 2.7R⊙, the most prob-able anisotropy ratio appears to range between 2 and 10 (seeFig. 6c). At heights aroundr ≈ 3R⊙ the uncertainties arelarge, but there does seem to be evidence that the anisotropyratio is likely to exceed 10 (see, e.g., Fig. 10b). The ratio mustincrease between 2 and 3R⊙, but we do not yet claim to knowthe exact rate of increase.

New observations are required to make further progress.For example, as seen in Figure 3, there is still some disagree-ment about the radial dependence of electron density in polarcoronal holes. Measurements of the white-light polarizationbrightness (pB) at solar minimum need to be made with loweruncertainties in the absolute radiometric calibration. Also,care must be taken to exclude time periods when high-latitudestreamers may be contaminating the LOS in order to obtaina true mean electron density for a polar coronal hole. Ex-isting measurements of the superradial geometry (as tracedby, e.g., polar plumes) tend to be limited by the fact that theshapes evident in LOS-integrated images are often assumedto be identical to the shapes of flux tubes in the plane of thesky. Better constraints on the flux-tube geometry could thusbe made by using stereoscopy (Aschwanden 2005), tomogra-phy (e.g., Frazin et al. 2007), or other time-resolved rotationaltechniques (e.g., DeForest et al. 2001) to trace the full three-dimensional shapes of the plume-filled flux tubes.

Improved ultraviolet spectroscopic measurements wouldgreatly improve our ability to determine the plasma param-eters in coronal holes. We anticipate that the UVCS instru-ment will continue to observe polar coronal holes through thepresent solar minimum (2007–2008). We do not yet knowwhether the wide spread in line widths seen a decade ago(which exceeded the observational uncertainties) was due toa changing filling factor of polar plumes along the LOS orwhether it may be connected to other kinds of time variabil-ity at the coronal base. An even wider range of variation incoronal hole properties was observed over the last solar cyclewith UVCS (e.g., Miralles et al. 2006). These observations ofhow coronal holes evolve in size and latitude have helped toconstrain the realm of possible parameter space of preferentialion heating and acceleration.

There are also observations that cannot be made withUVCS that could greatly improve our understanding of ion

energization in the solar wind acceleration region. For ex-ample, rather than having only O5+ (and, to a lesser extent,Mg9+; see Kohl et al. 1999) in coronal holes, an instrumentwith greater sensitivity and a wider spectral range could sam-ple the velocity distributions of dozens of additional ionswitha range of charges and masses. Obtaining the distribution ofderived kinetic temperatures as a function of the ion charge-to-mass ratioZ/A would put a firm constraint on the shape ofthe power spectrum of cyclotron-resonant fluctuations (e.g.,Hollweg 1999; Cranmer 2002b). A next-generation instru-ment with greater sensitivity may also be able to detect sub-tle departures from Gaussian line shapes that signal the pres-ence of specific non-Maxwellian distributions (e.g., Cranmer,1998, 2001).

The strong heating and acceleration of minor ions, as docu-mented by UVCS/SOHO, has provided significant insight intothe physics of solar wind acceleration, but the basic chain ofphysical processes is still somewhat unclear. Many theoret-ical studies have attempted to trace this chain “backwards”from the known facts of kinetic ion energization to the prop-erties of, e.g., ion cyclotron resonant waves that can providesuch energization naturally (see reviews by Hollweg & Isen-berg 2002; Marsch 2005; Kohl et al. 2006). Complementaryprogress has also been made in constraining the large-scaleproperties of the MHD fluctuations that may eventually cas-cade down to the microscopic kinetic scales (e.g., Verdini &Velli 2007; Cranmer et al. 2007). There are still many areas ofdisconnect, though, between our understanding of the macro-scopic MHD scales and the microscopic kinetic scales. Futuretheoretical work is expected to continue exploring how thecombined state of turbulent fluctuations, wave-particle inter-actions, and species-dependent heating and acceleration canbe produced and maintained.

The authors would like to thank Adriaan van Ballegooijen,Mari Paz Miralles, Leonard Strachan, and John Raymond forvaluable discussions. This work has been supported by theNational Aeronautics and Space Administration (NASA) un-der grants NNG04GE84G, NNG05GG38G, NNG06GI88G,NNX06AG95G, and NNX07AL72G to the Smithsonian As-trophysical Observatory, by Agenzia Spaziale Italiana, and bythe Swiss contribution to the ESA PRODEX program.

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FIG. 1.— Collected UVCS polar coronal hole measurements of(a) O VI line widthsV1/e, (b) ratio of OVI λ1032 to OVI λ1037 intensities, and(c) O VIλ1032 line-integrated intensities, with symbols specifying the sources of the data (see labels for references). Error bars denote±1σ observational uncertainties.Also shown (dotted lines) are the parameterized fits given by Cranmer et al. (1999).

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