Sensitivity of solar off-limb line profiles to electron density
  stratification and the velocity distribution anisotropy
Astronomy & Astrophysics manuscript no. 2568˙AA November 30, 2018
(DOI: will be inserted by hand later)
Sensitivity of solar off-limb line profiles to electron density
stratification and the velocity distribution anisotropy
N.-E. Raouafi1,2 and S. K. Solanki1
1 Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Straße 2, 37191 Katlenburg-Lindau, Germany
2 National Solar Observatory, 950 North Cherry Avenue, Tucson, AZ 85726, USA
e-mail: nraouafi@nso.edu ; solanki@mps.mpg.de
Received ; accepted
Abstract. The effect of the electron density stratification on the intensity profiles of the H I Ly-α line and the O VI and Mg X
doublets formed in solar coronal holes is investigated. We employ an analytical 2-D model of the large scale coronal magnetic
field that provides a good representation of the corona at the minimum of solar activity. We use the mass-flux conservation
equation to determine the outflow speed of the solar wind at any location in the solar corona and take into account the integration
along the line of sight (LOS). The main assumption we make is that no anisotropy in the kinetic temperature of the coronal
species is considered. We find that at distances greater than 1 R⊙ from the solar surface the widths of the emitted lines of O VI
and Mg X are sensitive to the details of the adopted electron density stratification. However, Ly-α, which is a pure radiative
line, is hardly affected. The calculated total intensities of Ly-α and the O VI doublet depend to a lesser degree on the density
stratification and are comparable to the observed ones for most of the considered density models. The widths of the observed
profiles of Ly-α and Mg X are well reproduced by most of the considered electron density stratifications, while for the O VI
doublet only few stratifications give satisfying results. The densities deduced from SOHO data result in O VI profiles whose
widths and intensity ratio are relatively close to the values observed by UVCS although only isotropic velocity distributions
are employed. These density profiles also reproduce the other considered observables with good accuracy. Thus the need for
a strong anisotropy of the velocity distribution (i.e. a temperature anisotropy) is not so clear cut as previous investigations of
UVCS data suggested. However, these results do not rule completely out the existence of some degree of anisotropy in the
corona. The results of the present computations also suggest that the data can also be reproduced if protons, heavy ions and
electrons have a common temperature, if the hydrogen and heavy-ion spectral lines are also non-thermally broadened by a
roughly equal amount.
Key words. Line: profiles – Scattering – Sun: corona – Magnetic fields – Sun: solar wind – Sun: UV radiation
1. Introduction
Extreme-ultraviolet (EUV) spectral lines provide valuable in-
formation on the plasma conditions in the solar wind accel-
eration zone. Information on the densities, the temperatures,
the bulk velocities and the velocity distributions of different
species, can be obtained from the total intensity, Doppler width
and shift of multiple line profiles with the appropriate instru-
ment. Basically three techniques are used to deduce both bulk
velocity and the velocity distribution.
The first is the usual Doppler effect that provides infor-
mation on the LOS velocity of the emitting source. The sec-
ond is the Doppler dimming which yields information on
(microscopic and macroscopic) velocity components not di-
rected along the LOS, on the basis of the technique proposed
by Rompolt (1967 & 1969) and Hyder & Lites (1970). See,
e.g., Beckers & Chipman (1974) for a description. Note that
Doppler dimming acts only on the radiative component of a
Send offprint requests to: N.-E. Raouafi
line. Rompolt (1967, 1969 & 1980), Heinzel & Rompolt (1987)
and Gontikakis et al. (1997a,b) used the Doppler dimming of
Ly-α to study moving prominences. It has later been considered
to diagnose the solar wind (Kohl & Withbroe 1982; Withbroe
et al. 1982; Strachan et al. 1989 & 1993; etc.).
Optical pumping is a third process. It happens when, due to
the Doppler effect, the absorption profile of the moving atom
overlaps with a neighboring incident spectral line. This is the
case for the coronal O VI 1037.61 Å line that can be excited by
the chromospheric C II doublet at 1036.3367 and 1037.0182 Å
when the solar wind speed reaches values between ∼ 100 and
∼ 500 km s−1 (Noci et al. 1987; Kohl et al. 1998; Li et al.
1998). Finally, Cranmer et al. (1999a) used optical pumping
by Fe III 1035.77 Å to diagnose wind speeds around ∼ 530
km s−1.
UVCS (the UltraViolet Coronagraph Spectrometer; Kohl et
al. 1995 & 1997) on the SOHO (the SOlar and Heliospheric
Observatory; Domingo et al. 1995) mission yielded valuable
data on the solar corona, including the polar coronal holes up
http://arxiv.org/abs/0704.1127v1
2 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles
to more than 3.5R⊙ (Kohl et al. 1997 & 1998; Noci et al. 1997;
Habbal et al. 1997; Li et al. 1998; etc.). One of the most excit-
ing results obtained from data recorded by UVCS concerns the
very broad lines emitted by heavy ions (namely O VI and Mg X)
in polar coronal holes during the minimum of the solar activ-
ity cycle. From the analysis of profiles of coronal lines (mainly
the Ly-α, O VI and Mg X doublets) Kohl et al. (1997 & 1998),
Noci et al. (1997) and others concluded that the evidence for
highly anisotropic velocity distributions of the cited species is
strong. For O VI and Mg X the ratio of kinetic temperatures in
the directions perpendicular and parallel to the coronal mag-
netic field, respectively, is found to range from 10 to more than
100 (Kohl et al. 1997 & 1998; Cranmer et al. 1999b; etc.). In
addition, the heavy ions are deduced to be significantly hotter
and faster than the protons (Li et al. 1997).
In the present paper we extend the work of Raouafi &
Solanki (2004; hereafter abbreviated as RS04) in different
ways. Firstly, we consider a more complete set of published
density profiles. Secondly, we study the effect of the density
stratification on spectral lines having very different formation
mechanisms: Ly-α is purely radiative; Mg X lines are exclu-
sively collisional and the O VI doublet results from both mech-
anisms. In addition, we also calculate the LOS integrated total
intensities of the Ly-α and O VI lines.
2. Line formation in the corona
In an optically thin medium, the general form of the contribu-
tion to the emissivity of a given point on the LOS, assuming a
Maxwellian velocity distribution and a Gaussian shaped inci-
dent profile, is given by
I(ν) =
NXn+(θ)Ne(θ)αlu
NXn+(θ)Blu IR
(4 π)
dΩ [f ×D × P ×M] (Ω, θ)
where NXn+(θ) and Ne(θ) are densities of scattering ions
Xn+ and electrons (cm−3), respectively. Blu and αlu are the
Einstein coefficient for absorption between the two atomic lev-
els l and u1 and the coefficient of electronic collisions, respec-
tively. The width of the microscopic velocity distribution αs
is assumed to be isotropic (no anisotropy in the kinetic tem-
perature of scattering atoms/ions is considered throughout this
paper)). αs includes the contribution due to thermal motions.
δν = ν − ν0, where ν0 is the rest frequency of the reemitted
line. The geometry of the scattering process is given by Fig. 1 in
RS04. The other terms entering Eq. (1) are defined in the same
paper. The different values of the coefficient Wul in the expres-
sion of M(Ω, θ) and the corresponding spectral lines studied
here are given in Table 1. Eq. (1) is general and applies to any
spectral line irrespective of whether it is formed by collisions
and/or by radiation. An exception is when the incident profiles
deviates from a Gaussian (see Sect. 5.1 on how such a case can
be dealt with).
1 In the present paper, we consider the assumption of a two level
atom.
Table 1. Values of Wul for the spectral lines considered here.
Jl Ju Wul Examples of spectral lines
Mg X 624.941 Å, O VI 1037.61 Å
H I 1215.6736 Å
Mg X 609.793 Å, O VI 1031.92 Å
H I 1215.6682 Å
Fig. 1. Electron density plotted as a function of the distance to
Sun center for different empirical models (solid curve: DKL
(Doyle et al. 1999a,b; Kohl et al. 1998; and Lamy et al. 1997)
dotted: Guhathakurta et al. 1999; Dashes: Guhathakurta &
Holzer 1994; Dot-dashes: Cranmer et al. 1999c; Triple-dot-
dashes: Esser et al. 1999).
According to Eq. (1), the profiles created by collisions and
by resonant scattering at a given point along the LOS all have
Gaussian shapes. However, there are significant differences be-
tween the two components. The main differences are:
1) The intensity of the collisional component is proportional to
the square of the electron density, where as the radiative com-
ponent is only proportional to the density. Thus, the collisional
component suffers the radial density drop more than the radia-
tive one.
2) The radiative component is affected by the outflow speed of
the scattering ions through the Doppler dimming effect, while
the collisional component is not. Thus, the contributions to the
collisional component from sections along the LOS supporting
large outflow speeds are more important than the contributions
to the radiative one, since the latter gets progressively out of
resonance due to the increase in outflow speed of the solar wind
with the distance from Sun center.
3) For a given point on the LOS, the Doppler shift of the col-
lisional component reflects the real LOS speed uZ of the scat-
tering atoms/ions. The shift of the radiative component, how-
ever, corresponds to only a fraction of this speed. This point is
very important for the broadening of the line profile. Note that
in frame of the complete redistribution approximation, there is
no shift difference between collisionally and radiatively excited
profiles (see Sahal-Bréchot & Raouafi 2005).
Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 3
3. Atmospheric parameters
The realistic synthesis of spectral line profiles requires the
modeling of all the parameters entering the formation process
of a given spectral line. According to Eq. (1), this includes
the magnetic field, the solar wind outflow velocity, the den-
sity stratifications of electrons and of atoms/ions, etc., at every
point along the LOS. For the large scale magnetic field of the
corona and the solar wind velocity we adopt the same descrip-
tions as RS04, while for the density stratifications we consider
a wider range of models.
The coronal magnetic field is described by the model of
Banaszkiewicz et al. (1998; see Fig. 1 of RS04). Cranmer et al.
(1999c) used the same model to constrain the strength and su-
perradial expansion of the magnetic field in the corona. The so-
lar wind outflow speed is obtained through the mass-flux con-
servation equation
V (r, θ) =
Ne(R⊙)
Ne(r, θ)
B(r, θ)
B(R⊙, θ⊙)
V (R⊙, θ⊙), (2)
where Ne(R⊙), V (R⊙, θ⊙) and B(R⊙, θ⊙) are the electron
density, the outflow speed of the ions and the coronal magnetic
field at the base of the solar corona (solar surface), respectively.
Ne(r, θ), V (r, θ) and B(r, θ) are the same quantities at coor-
dinates (r, θ), with r > R⊙. The angles θ and θ⊙ are related
to each other by the requirement that Eq. (2) is valid along
a field line. This also implies that the velocity vector is par-
allel to the magnetic field vector (needed for the calculation
of the dimming rate of the radiative component and also for
the determination of the Doppler shift of the reemitted spec-
tral line). In order to be consistent with Ulysses observations
that the fast solar wind varies by less than a few percent with
latitude (Neugebauer et al. 1998; McComas et al. 2000; Zhang
et al. 2002), we adopt V (R⊙, θ⊙) ∝ B(R⊙, θ⊙) as bound-
ary condition on the outflow speed on the solar surface (see
Table 2). We consider different values of V⊙ for the different
species. We also consider different V⊙ for the different density
stratification models in order to get as close as possible to the
observed values of widths, total intensities and intensity ratios.
RS04 have shown that the density stratification is a key
quantity for determining the off-limb profiles of collision-
dominated lines, in particular the line width. Here we con-
sider a more complete set of empirical density stratifications
than in that paper. In addition to the electron density profiles
from (Doyle et al. 1999a,b; hereafter abbreviated as DKL)
determined empirically by combining the results obtained by
three instruments on SOHO (SUMER: Doyle et al. 1999a,b;
UVCS: Kohl et al. 1999; LASCO: Lamy et al. 1997) and from
Guhathakurta & Holzer (1994; SKYLAB) which were consid-
ered by RS04, we also employ the models of Guhathakurta
et al. (1999; SPARTAN), Cranmer et al. (1999c) and Esser et
al. (1999; SPARTAN and Mauna Loa; Fisher & Guhathakurta
1995). The absolute values of the density in all these models
are not very different. They are generally within the error bars
of the coronal densities which are typically 20 - 30 % of the
measured values (the main exception is the Esser et al. model).
Except for the DKL model, the data for the other models has
been fitted by power series functions based on the fact that the
Fig. 2. Outflow speed of the O VI ions along the polar axis and
field line arising from 70 degree latitude, respectively, for the
different density models considered. The Boundary condition
on the outflow speed of the coronal ions is chosen to be pro-
portional to the solar surface magnetic field strength (see text).
In order to get as close as possible to the measured values,
we adopt different values of the proportionality coefficient of
V (R⊙, θ⊙) with respect to B(R⊙, θ⊙) for the different den-
sity models. The outflow speed of other species can be obtained
by multiplying the present ones by a constant.
density drops very fast close to the Sun and follows an ∼ r−2
function sufficiently far away. The description given by Doyle
et al. for the DKL electron density stratification (Eq. (13) of
RS04) assumes that the coronal gas is isothermal and in hydro-
static equilibrium (see Guhathakurta & Fisher 1995 & 1998).
The absolute value of the electron density directly affects
the intensity of the emitted line at any point along the LOS.
However, the importance of the density stratification resides
therein that it also influences the LOS-integrated profile indi-
rectly through the solar wind speed, which is determined via
mass-flux conservation and thus depends on both the density
stratification and the magnetic field. Even somewhat different
density stratifications give quite different solar wind speeds and
therefore very different line profiles. A more thorough discus-
sion is given in the following Sects. The effect of the density
stratification on the outflow speed and LOS velocity has been
illustrated by RS04; see their Fig. 5.
4 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles
Table 2. Proportionality coefficient, κ, of V (R⊙, θ⊙) toB(R⊙, θ⊙)
V (R⊙, θ⊙) = κ×B(R⊙, θ⊙)
used for the calculations
of the different spectral lines considered in the present paper and corresponding to the different density models.
Guhathakurta & Holzer Guhathakurta et al. Cranmer et al. Esser et al.
(1994) (1999) (1999c) (1999)
H I 0.63 0.85 0.35 0.30 0.13
O VI 0.85 0.88 0.50 0.35 0.13
Mg X 0.85 0.88 0.50 0.35 0.13
The solar wind speed profiles obtained for the combination
of the chosen magnetic structure and the density stratifications
are plotted in Fig. 2 for two extreme cases. In Fig. 2(a) the wind
speed along the polar axis is displayed, while Fig. 2(b) shows
the speed along the field line arising from 70 degree latitude on
the solar surface. The LOS at 3.5R⊙ intersects this field line at
6 R⊙, which is the distance to which we integrate the profiles.
The small decrease in the solar wind speed at high altitudes
(solid line in panel a) indicates some mismatch between the
DKL model and the Banaszkiewicz et al. magnetic structure
right above the pole. This decrease does not affect our con-
clusions since it occurs at altitudes above 4 R⊙, whereas the
furthest LOS we consider passes at 3.5 R⊙ above the pole.
The scaling factors of the solar wind speed at the solar sur-
face are listed in Table 2. Clearly although the scaling factors of
the heavy elements are not the same as for hydrogen, they differ
by less than a factor of 1.5, with the hydrogen atoms moving
more slowly.
In order to compute the total intensities of the lines emit-
ted in the polar coronal holes, we need the absolute density of
each ion NXn+ , a parameter not considered by RS04. It can be
written as follows
NXn+ =
Ne, (3)
where (NXn+/NX) is obtained from the ionization balance of
the X species, (NX/NH) is the abundance of X relative to hy-
drogen and (NH/Ne) is the abundance of Hydrogen relative
to electrons. Here we adopt a plasma with 10% of helium (the
cosmic value) so that NH/Ne = 0.83. However, the coronal
helium abundance could be considerably larger and could vary
significantly through the region considered here (see Joselyn &
Holzer 1978; Hansteen, Holzer & Leer 1993, Hansteen, Leer &
Holzer 1994a,b & 1997). The considered fraction of neutral hy-
drogen is 2.667 10−7 (John Raymond, private communication).
NO5+/NO = 4.78 10
−3 (Nahar 1999) and the abundance of
oxygen relative to hydrogen is taken to be 8.7 (Asplund et al.
2004).
4. Effect of the integration along the LOS on the
collisional and radiative components
The results that will be presented in the present Sect. are ob-
tained by using the DKL density stratification. The use of any
of the other density models does not change the conclusions.
UVCS samples three types of lines: the Mg X lines at
609.793 Å and 624.941 Å are only excited by electron colli-
sions, while the Ly-α line is exclusively created by resonant
scattering of the solar disk radiation. The contribution of the
collisional component is negligible (less than 1% according
to Raymond et al. 1997). The O VI lines at 1031.92 Å and
1037.61 Å are excited by the combination of collisions and
scattering of the transition region radiation.
Here we consider separately the collisional and radiative
components of the considered coronal spectral lines in order
to illustrate better the effects of the solar wind outflow veloc-
ity, density stratification and the integration along the LOS on
each of them. The behavior of a particular line depends on the
relative strengths of these two components.
Figs. 3 and 4 display the profiles emitted from different
sections along the LOS for two different projected heliocen-
tric distances 1.5 and 3.5 R⊙ (i.e. different rays) for spectral
lines created solely by electron collisions and purely from reso-
nant scattering of the solar disk radiation, respectively. For the
sake of clarity, we used constant αs to illustrate the effect of
the LOS integration on the profiles in Figs. 3, 4 & 5. When
comparing with observed profiles in Sect. 5, however, we only
consider computations with height-dependent αs.
At low altitudes (< 2 R⊙) the main contribution is given
by the central part of the LOS (i.e. near the polar axis). This is
due to the fast drop of the densities as a function of the helio-
centric distance (Fig.1). However, when further from the solar
disk, significant contributions are obtained from larger sections
(larger |Z|) along the LOS, due to the relatively slow decrease
of the densities far from the solar limb ( Fig. 1). Doppler dim-
ming, which is largest at greater heights, concentrates the con-
tribution of the radiative component to the observed line profile
toward small |Z| (compare the bottom panels of Figs. 3 & 4).
Another striking difference is seen in the Doppler shift behav-
ior of the two components. The Doppler shift of the collisional
profiles increases when moving away from the polar axis along
the LOS (Fig. 3). The Doppler sift of the radiative component
also increases at small |Z|, but ever more slowly at larger |Z|.
Depending on the profile of the LOS velocity, the line shift can
reach a limiting value characteristic of that LOS, or can even
start to decrease again at greater |Z| (Fig. 4).
The behavior seen in Figs. 3 & 4 can be understood as fol-
lows. At a given Z , the Doppler shift of the collisional compo-
nent is equal to the LOS speed uZ of the scattering atoms/ions
(see first term of the RHS of Eq. (1)). However, for the radiative
component the Doppler shift is given by (uZ − αsi nZ u · n).
For small |Z| (close to the pole where the field lines are only
slightly inclined with respect to the polar axis), nZ is very
small and the Doppler shift is approximately equal to uZ , so
that collisional and radiative line profiles arising at small |Z|
Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 5
Fig. 3. Contributions to the final line profile emanating from
different sections of the LOS (different values of Z) for a spec-
tral line excited only by electron collisions (namely a Mg X
doublet line) for LOSs coming to within 0.5 R⊙ and 2.5 R⊙
of the solar surface (top and bottom frames, respectively). The
LOS integrated profile is the sum of all the individual profiles
emitted by the different sections along the LOS. The marked
Z values are in units of R⊙. Blue Doppler shifts correspond to
positive LOS speeds (toward the observer) that are obtained for
positive values of Z .
are similar. However, further away from the polar axis, nZ in-
creases (in absolute value) so that, depending on nZ and αsi
(= α2s/(α
i + α
s)), the expression (uZ − αsi nZ u · n) either
saturates or even decreases again (αi is the Doppler width of
the incident line profile). This effect is more marked at higher
altitudes where the contribution from LOS sections with large
|Z| are important (see bottom panel of Fig. 4).
In Fig. 5 profiles for a purely collisionally (a) and purely
radiatively (b) excited line are displayed. The plotted lines are
obtained by integrating out to different distances along the LOS
for a projected heliocentric distance of 3.5R⊙. Profiles of col-
lisional lines turn out to be sensitive to the atmospheric parame-
Fig. 4. Equivalent results as in Fig.3 but for a purely radiatively
excited line.
ters, in particular LOS velocity, out to |Z| = 6R⊙. The profile
shape of the radiatively excited line is hardly altered by LOS
integration, although the total intensity is significantly affected
(not visible in Fig. 5 due to the normalization).
5. Application to coronal lines
Among the spectral lines observed by UVCS in polar coronal
holes three sets are of particular interest. The first is the most
intense line emitted in the solar corona, Ly-α. This line is pro-
duced by the resonant scattering of the radiation emitted by
the solar disk. The few electron collisions are negligible for
this line (Raymond et al. 1997). The second group of lines is
the O VI doublet. These lines are excited by both, the radia-
tion coming from the underlying chromosphere-corona transi-
tion zone and by isotropic electron collisions. The third class
is composed of the Mg X doublet, whose members are emitted
following excitations produced almost exclusively by electron
collisions (Wilhelm et al. 2004).
6 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles
Fig. 5. Line profiles obtained by integrating over different distances between −Z0 and Z0 along the LOS for a spectral line
excited only by electron collisions (a) and only by radiation (b). These profiles are obtained for a point of closest approach of
3.5 R⊙ from Sun center and for αs = 200 km s−1 (αi = 100 km s−1 for the radiative component). The effect of the LOS
integration on the radiatively excited line is rather small compared to the collisional line.
Fig. 6. Disk emission (+ signs) of Ly-α during the minimum of
solar activity (Lemaire et al. 1998). The solid curve is a four-
Gaussian fit to the data. The individual Gaussians contributing
to the fit are given by the dotted, dashed, dot-dashed and triple-
dot-dashed lines.
The dependence of αs on r and θ is computed consistently.
We assume that αs depends only on r, so that its value changes
along the LOS. In the next subsections, we describe calcula-
tions of the profiles of these three sets of lines and their com-
parison with the observed ones.
5.1. The Ly-α line of hydrogen
To calculate the coronal emission H Ly-α, it is necessary
to know the shape of the line profile coming from the so-
lar disk (Fig. 6). A four Gaussian fit plus a constant back-
ground is applied to the complex shape of the observed pro-
file. Numerically, we assume that coronal hydrogen atoms are
illuminated by the photons of four incident Gaussian profiles
Table 3. Parameters of the different Gaussians used to fit the
Ly-α line. The Gaussians’ centers are given in (km s−1) with
respect to the line center as obtained from the data (+ signs in
Fig. 6). The amplitudes are in (1012 photons/cm2/s/(km s−1)
and the widths are in (km s−1).
Gaussian Amplitude Position Width
Dotted 0.22 -94.01 16.81
Dashed 2.24 -54.79 34.39
Dot-dashed 2.70 7.41 125.28
Triple-dot-dashed 2.13 65.13 34.91
emitted by the solar disk, with of course different parame-
ters, as listed in Table 3. We do not consider any center-to-
limb variation in the intensity emitted by the solar disk in Ly-α
(f(Ω, θ) = 1), which is in accordance with the filter images of
Bonnet et al. (1980).
A single Gaussian is sufficient to fit the calculated off-
limb Ly-α profiles with good accuracy. Fig. 7(a) displays the
e-folding (Doppler) widths of the LOS integrated Ly-α profiles
as a function of the projected heliocentric distance on the plane
of the sky (the widths of the best-fit Gaussian are plotted). The
values of αs used to reproduce these profiles are also given in
the same Figure (dot-dashed line) together with the best fit to
the observational data (solid line) by Cranmer et al. (1999a)
(see also Kohl et al. 1998).
The computed Ly-α profiles are relatively in-sensitive to
the details of the electron density stratification. For most of the
density stratifications given in Fig. 1, the observed line width
of Ly-α in polar coronal holes at different heights above the so-
lar limb are comparable to the calculated ones in the presence
of an isotropic velocity distribution and a gradual rise in αs
with projected distance to the limb. The displayed results are
all located in the 1-σ error bars of measured widths except for
the widths obtained at low altitudes by using densities given by
Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 7
Fig. 7. Widths (a) and total intensities (b) of the LOS-integrated
Ly-α line profile as a function of the projected heliocentric
distance obtained for the different density stratification mod-
els presented in Fig. 1. The correspondence symbols to density
models are given in the top panel. The solid lines is the best fits
to the observations recorded by UVCS and the dot-dashed line
(top panel) is the turbulence velocity distribution of the hydro-
gen atoms. The vertical lines are the statistic error bars of the
measurements.
Guhathakurta et al. (1999) or between 2.3 and 2.7 R⊙ based
on the Esser et al. densities. The reasons for these departures
are visible in Fig. 2; note the high/low outflow speeds obtained
through these density models at these altitudes. Note that the
outflow speed of the hydrogen atoms at the solar surface is a
free parameter. By lowering the outflow speed at 1 R⊙ (solar
surface) for Guhathakurta et al. (1999) we can obtain a bet-
ter correspondence with the measured widths a low altitudes.
However, this occurs only at the cost of lower widths also at
higher altitudes, so that the overall correspondence with the
data is not improved.
Fig. 7(a) shows how the contribution to the line width from
the small-scale velocity distribution, given by αs, which domi-
nates at small r is replaced by contributions from the outflow-
ing solar wind at larger r, as the line width significantly exceeds
the αs value.
Fig. 7(a) also demonstrates that a more rapid drop in den-
sity with height leads to broader line profiles (compare profiles
resulting from the DKL stratification with that of Guhathakurta
et al. 1999). This means that of the two oppositely directed ef-
fects introduced by a steeper density gradient, a more rapidly
increasing wind speed and a smaller contribution to the mea-
sured line profile from large |Z| values, the former dominates.
Fig. 7(b) displays the total intensity of Ly-α as a function
of the projected heliocentric distance for the different density
stratifications considered in the present paper. The best fit to the
observed intensities (solid curve) and the error bars of the mea-
surements (vertical bars) are also plotted (see Cranmer et al.
1999a). All the density models considered give total intensities
that are close to the observed ones (within a factor of 2− 3).
5.2. O VI doublet
We compute the intensity profiles of the lines of the O VI
doublet at 1031.92 Å
2s 2S1/2 − 2p 2P3/2
and 1037.61 Å
2s 2S1/2 − 2p 2P1/2
. In the solar corona, the O VI ions in-
teract with electrons (with isotropic velocity distribution) and
with the radiation coming from the underlying transition re-
gion. The effects of the ions’ motion are taken into account via
the Doppler dimming, the Doppler shift and the optical pump-
ing of the O VI 1037.61 Å line by the chromospheric C II dou-
blet (1036.3367 Å & 1037.0182 Å). This last effect is impor-
tant above ∼ 2R⊙ where the solar wind outflow speed reaches
values that enable optical pumping to occur (the exact radius at
which this occurs depends on the adopted solar wind profile).
The incident solar disk spectrum considered for the calcula-
tions of the coronal O VI lines is shown in Fig. 2 of RS04. All
the line profiles are assumed to be Gaussians. In the present
computations, the radiances of the individual members of the
O VI doublet on the disk spectrum are equated to 305 and 152.5
erg cm−2 s−1 sr−1, respectively. Both O VI incident lines are
assumed to have the same widths, corresponding to 35 km s−1.
We consider also two identical profiles for the C II doublet with
widths of 25 km s−1 and radiances of 52 erg cm−2 s−1 sr−1
each. We use the limb-brightening measured by Raouafi et al.
(2002) for the O VI line and no limb-brightening for the C II
lines (see Warren et al. 1998). For comparison, Table 1 of RS04
presents different values of these parameters measured from
different observations.
The top panel of Fig. 8 displays the widths of the O VI
1031.92 Å line calculated for rays (LOSs) corresponding to dif-
ferent projected heliocentric distances and for different density
stratification models. These widths are obtained by applying a
Gaussian fit to each calculated profile. All profiles are well rep-
resented by a Gaussian. The profiles at the furthest considered
ray (3.5 R⊙) show the strongest departure from a Gaussian
shape. They are plotted for all considered density models in
Fig. 10. The solid line represents the best fit to the UVCS data
(Cranmer et al. 1999a). Note that no anisotropy in the kinetic
temperature of the emitting ions is considered.
8 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles
Fig. 8. The symbols represent the variation of the width (top
panel) and intensity ratio (bottom panel) of the LOS integrated
synthetic line profiles of O VI as a function of the heliocentric
distance. The correspondence between symbols and models is
indicated in the grey frame. The dot-dashed curve displays the
used values of αs. The observed dependence is given by the
solid curves (best fits to the observations; see Cranmer et al.
1999c), with the vertical lines being error bars.
At small heights (< 2 R⊙) the widths of the calculated
profile are comparable for most of the density models except
for the one by Guhathakurta et al. (1999). For this model the
boundary condition on the solar wind speed at the solar surface
is chosen to fit the widths at high latitudes. This gives higher
solar wind speeds already at 1.5 R⊙ and explains the broader
profiles resulting from this model at low altitudes. At larger
heights (> 2.0 R⊙) the line widths are very sensitive to the
details of the electron density stratification. This can be easily
seen by the difference in the width of the different profiles ob-
tained through slightly different density stratification models.
The model by DKL gives line widths comparable to the ones
obtained from the data, except between 2 and 2.7 R⊙, where
the obtained widths are slightly smaller than the observed ones.
The bottom panel of Fig. 8 displays the ratio of total in-
tensities of the O VI doublet lines (I1032/I1037) as a function a
the projected heliocentric distance. This ratio exhibits a marked
dependence on radial distance being well over 2 close to the
Fig. 9. Total intensities of the O VI lines as a function of the
projected heliocentric distance. The correspondence to the dif-
ferent density stratifications is given by the symbols in the top
panel. The solid curves are the best fits to the observed inten-
sities by UVCS and the vertical bars represent estimates of the
accuracy of the observations.
Sun, then dropping rapidly (except for the density models by
Esser et al. 1999, which produces larger ratios compared to
the ones given by the other density models and which do not
fit the observations in particular at distances between 2.3 and
3 R⊙). All the other models lead to a minimum in the ratio at
r ≈ 2.5−3R⊙, which then increases again at larger r (a num-
ber of the considered models exhibit a slightly different behav-
ior, showing a slight decrease in the ratio out to r = 3.5 R⊙).
Generally, most of the calculated intensity ratios (in particular
those obtained from the DKL density stratification) are within
the error bars of the ones observed by UVCS.
Fig. 9 displays the computed total intensities of the O VI
lines as a function of the projected heliocentric distance. The
density model of DKL, Cranmer et al., Guhathakurta & Holzer,
and for r ≥ 2 R⊙, Esser et al. give comparable intensities of
the O VI doublet that agree relatively reasonably with respect to
the observations. All models produce a slightly less steep drop
in intensity with altitude than suggested by the observations,
however. The density models by Guhathakurta et al. (1999) and
Esser et al. (1999) give low intensities at low altitudes. This is
due the fast drop of the electron density at low altitudes for
these two models.
Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles 9
Fig. 10. LOS integrated profiles of the O VI 1031.92 Å line
obtained at a projected heliocentric distance of 3.5 R⊙ for
the different electron density stratifications considered in the
present paper. The obtained line profiles are all approxima-
tively Gaussian shaped, although they differ strongly in width.
5.3. Mg X doublet
The coronal Mg X ion emits a doublet at 609.793 Å and
624.941 Å that corresponds to the atomic transitions 2s 2S1/2−
2p 2P3/2 and 2s
2S1/2 − 2p 2P1/2, respectively. They are
formed in the solar corona almost exclusively by electron col-
lisions (the disk emission is almost non existent; see Curdt et
al. 2001). Due to the weakness of the coronal emission in these
lines, UVCS was only able to record data in the polar coronal
holes up to a height of 2 R⊙ from Sun center. The members of
the doublet have almost identical widths, so that a single value
suffices to describe both of them. The widths of the observed
profiles are displayed in Fig. 11 together with their error bars.
Note the small, but systematic difference between the values
determined by Esser et al. (1999) and Kohl et al. (1999). The
widths of the calculated profiles and the values of microscopic
velocities αs used to reproduce these profiles are plotted in the
same Figure. The obtained synthetic profile widths are compa-
rable with the observed widths within the accuracy of the data,
in particular if we consider the scatter of the data point. Note
that we consider only the widths obtained from single Gaussian
fits to the data, i.e. assuming no anisotropy of the velocity dis-
tribution. The line profiles obtained at 3.5 R⊙ for the differ-
ent density models present a slight flattening at the central fre-
quency which makes their one-Gaussian-fitting less accurate.
The corresponding line widths in Fig. 11 are therefore more
uncertain.
At heights above 2R⊙ the variation of the width of the syn-
thetic profiles of Mg X as a function of the projected heliocen-
tric distance shows how sensitive the collisional line profile is
to the details of the electron density stratification. This explains
why the profiles emitted by heavy ions in the corona, namely
O VI and Mg X, which are partially or totally collisional, are
larger compared to the ones of Ly-α (pure radiative lines).
Fig. 11. Widths of the LOS integrated line profiles of the Mg X
doublet plotted as a function of the projected heliocentric dis-
tance for the different density stratification models considered
in the present paper. The dot-dashed line gives αs as a func-
tion of the heliocentric distance. Also plotted are the measured
widths of the Mg X lines together with the error bars. For the
sake of clarity they are plotted slightly to the right (full circles;
data of Kohl et al. 1999) and to the left (full diamonds; data of
Esser et al. 1999) from their respective x-coordinates.
A comparison with Fig. 8 shows that the line widths ob-
tained for the Mg X lines are only slightly larger than for the
O VI lines. This suggests that already for O VI the excitation
through electron collisions dominates over resonant scattering.
6. Discussion
We have considered the effect of the density stratification on the
strongest lines observed by UVCS in the polar holes of the solar
corona in the presence of a consistent model of the large scale
magnetic field and solar wind structure. We consider lines with
different formation processes (pure scattering lines, purely col-
lisionally excited lines, and lines for which both processes are
important; represented by Ly-α, the Mg X and O VI doublets,
respectively). We calculate the line profiles, total intensities and
(for the O VI lines) intensity ratio of the spectral lines emitted at
different altitudes in the polar coronal holes. Integration along
the LOS is taken into account. The velocity distributions of the
reemitting atoms/ions are considered to be simple Maxwellians
with a drift macroscopic velocity vector that is identical to the
solar wind outflow velocity, which is calculated according to
mass-flux conservation. We note that the main assumption un-
derlying this investigation is that no anisotropy in the kinetic
temperature of the scattering atoms/ions is considered.
10 Raouafi & Solanki: Density Stratification Effect on Solar Off-limb Line Profiles
It is found that whereas the width of the Ly-α profile reacts
little to the choice of density stratification, the widths of the
O VI and Mg X lines are strongly dependent. The Ly-α total
intensities obtained from the different density stratifications are
comparable and reasonably fit the observed intensities by most
models at most heights.
The line widths of the O VI and Mg X doublets obtained
at different heights in the polar coronal holes are found to
be very sensitive to details of the electron density stratifica-
tion. At low altitudes in the polar coronal holes, the calcu-
lated profiles of the lines of a given doublet all have compa-
rable widths, which is due to the fact that the main contribu-
tion comes from the area around the polar axis, due to the fast
drop of the electron density with height. At greater heights, the
density drops more slowly and contributions to the LOS inte-
grated profile from sections of the LOS with large |Z| are in-
creasingly important. At these relatively large heights, where
the solar wind speed reaches values that leave the O VI line out
of resonance, the reemitted lines behave like pure collisional
lines. This is confirmed by the fact that the calculated O VI and
Mg X lines exhibit similar, large line widths at these altitudes.
Their widths are very sensitive to the LOS solar wind speed,
which is strongly dependent on the density stratification, so that
the differences between the profiles obtained through different
density stratifications start to show up clearly at greater heights.
The intensity ratios of the O VI doublet drops from values
above 2 at heights of 1.5 R⊙ to reach a minimal value that
is very close to unity at a height that depends on the density
model. For most of the considered density stratification mod-
els, the obtained intensity ratios are within the error bars out-
lining the scatter of the observations. The LOS integrated total
intensities of the O VI lines also show only a weak dependence
on the density stratification. Generally, the obtained values are
reasonable compared to the observations for both lines of the
O VI doublet. Note that for the total intensity the absolute val-
ues of the electron densities are as important as their stratifica-
tion, which may explain the small dependence of the obtained
intensities on the density stratification.
For the DKL density stratification the calculated parame-
ters are close to those obtained from observations carried out by
UVCS, in spite of the fact that we only consider an isotropic ki-
netic temperature of the scattering atoms and ions. Our analysis
does not aim to decide between density models, since the exact
results depend on, e.g, the details of the magnetic structure, or
on how well the contribution of the polar plumes or foreground
coronal material has been separated from the part of the coronal
hole giving rise to the fast wind. However, it shows that a rea-
sonable combination of the magnetic, solar wind and density
structures exist that reproduce a wide variety of the observed
parameters. Our analysis confirms and strengthens the conclu-
sion of RS04 that the need for anisotropic velocity distributions
(i.e. anisotropy of kinetic temperature of the heavy ions) in the
solar corona may not be so pressing as previously concluded,
although we stress that the current results do not rule out such
anisotropies.
It is beyond the scope of the present paper to determine the
implications of this result for the mechanisms of heating and
acceleration of different species in the polar coronal holes. In
particular, it remains to be shown to what extend the ion cy-
clotron waves proposed to heat the outer corona in numerous
studies (Cranmer et al. 1999c; Isenberg et al. 2000 & 2001;
Isenberg 2001 & 2004; Markovskii & Hollweg 2004; etc.) are
consistent with an isotropic non-thermal broadening velocity
(which is known to be present in all layers of the solar atmo-
sphere).
We note that the difference between the kinetic temper-
atures of heavy ions and protons, found earlier by Li et al.
(1997), is also present in our analysis. If, however, we consider
the more realistic case that the αs values obtain a contribution
both from thermal and non-thermal broadening, then the pro-
tons and heavy ions give more consistent results. If we further
assume both protons and heavy ions to be affected by a temper-
ature of 0.9 106 K, which is typical of the electron temperature
in the polar coronal holes (David et al. 1998), we find that the
non-thermal broadening for the O VI lines varies between ∼ 85
km s−1 at 1.5 R⊙ to ∼ 210 km s−1 at 3.5 R⊙. For Ly-α these
values are ∼ 165 km s−1 and ∼ 185 km s−1, respectively;
i.e. they lie between the values deduced from the O VI lines.
This relative consistency between the non-thermal broadening
of protons and heavy ions is all the more surprising since we
did not in any way optimize the computations with such an
aim. We note that an extended model including the influence of
polar plumes (Raouafi & Solanki, in preparation) suggests that
the difference in the non-thermal broadening at low altitudes is
at least partly due to the neglect of polar plumes in the present
computations. Once more, we stress that the current work in
no way rules out alternative interpretations; it simply opens up
the possibility of considering models in which protons, heavy
ions and electrons all have the same temperature. In agreement
with earlier results we find a somewhat lower outflow speed for
protons than heavy ions, however.
Acknowledgements. The authors would like to thank Dr. T. Holzer
for the very constructive and encouraging report on the manuscript.
We are grateful to Steven Cranmer, Bernhard Fleck, Bernd Inhester,
Eckart Marsch, Klaus Wilhelm, Giannina Poletto and Jean-Claude
Vial for helpful discussions and critical comments that greatly im-
proved the paper.
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	Introduction
	Line formation in the corona
	Atmospheric parameters
	Effect of the integration along the LOS on the collisional and radiative components
	Application to coronal lines
	The Ly- line of hydrogen
	Ovi doublet
	Mgx doublet
	Discussion
