THE EVOLUTION OF PLASMA PARAMETERS ON A CORONAL SOURCE SURFACE AT 2.3 R_☉ DURING SOLAR MINIMUM

The ambient solar wind is known to have essentially two states: a fast and a slow wind component (Hundhausen 1972; McComas et al. 2000). The fast solar wind escapes from open magnetic field regions found in coronal holes (Krieger et al. 1973), but the slow solar wind probably has multiple sources. The primary source of the slow solar wind seems to be near coronal streamers, i.e., from the streamer tips or from streamer/coronal hole boundaries (e.g., Feldman et al. 1981; Woo & Martin 1997; Habbal et al. 1997; Wang et al. 1998; Strachan et al. 2000; Antonucci et al. 2005). Other candidates for source regions of the slow wind are active regions and small equatorial coronal holes (e.g., see Kojima et al. 1999; Liewer et al. 2004; Miralles et al. 2004; Woo & Habbal 2005; Fisk & Zurbuchen 2006). Part of the uncertainty in identifying the origin of the solar wind (particularly the slow wind) is that most measurements of the wind speed are made at heights far above the source regions. Mapping the streams back to the Sun is not always straightforward due to differences in the speed of the wind and the interactions between fast and slow wind as the wind flows away from the Sun. Early attempts to measure the solar wind speeds near the Sun relied on in situ velocity measurements that were extrapolated back to the Sun using potential field magnetic models (e.g., see Arge & Pizzo 2000, and references therein). These extrapolations also depend on assumptions made about the expansion factors for the diverging coronal magnetic field, which serves as streamlines for the outflowing plasma.

Solar wind velocity maps have been made for more than two decades using radio interplanetary scintillation (IPS) techniques. For example, Kojima & Kakinuma (1987), Kojima et al. (1998), Breen et al. (2000), and Tokumaru et al. (2010) have made maps of outflow speeds on a source surface by interpreting the scintillation pattern from distant radio sources such as quasars. The scintillations carry information about the speeds of small scale structures moving away from the Sun in the solar wind. There are several different tomographic methods that can be used to extract the three-dimensional structure from the scintillation data (e.g., see the reviews by Kojima et al. 2007; Jackson et al. 2011). However, near-Sun maps from these data must be created carefully, since the minimum distance of the source–receiver lines of sight for these measurements are typically at ∼20 R_☉ from the Sun (depending on frequency). In order to make the velocity maps at ∼2.5 R_☉, some type of extrapolation must be used, which can have its own uncertainties as already stated. Despite these limitations, an important contribution of the IPS measurements is that the velocity maps produced from this technique cover all heliographic latitudes. These results are the most similar to the work presented in this paper and thus invites a comparison with our results, which will be described below.

Another method for determining outflow velocities near the Sun is tracking brightness inhomogeneities (or blobs) that appear in white light coronagraph images. This technique has been used successfully with the Large Angle Spectroscopic Coronagraph (LASCO; Brueckner et al. 1995) and more recently with the SECCHI³ suite of coronagraphs and heliospheric imagers (Howard et al. 2008) on the Solar Terrestrial Relations Observatory mission. If the blobs are considered to be tracers of the solar wind (Sheeley et al. 1997), then their outflow speeds can be used to measure the bulk outflow velocity (projected onto the plane of the sky). Movies of the difference images of successive frames show structures moving out from the LASCO-C2 inner field of view at ∼2 R_☉ to the LASCO-C3 outer field of view at ∼30 R_☉. Sheeley et al. (1997) used scatter plots of many blobs to show that outflow speeds start out near zero at ∼3 R_☉, rise rapidly with height, and then gradually increase to 300 km s⁻¹ at and beyond ∼20 R_☉. Because the blobs appear to emerge from streamer tips and the termination speeds are similar to that expected for the slow solar wind, the data suggest that streamers are likely sources of the slow speed wind. If the blob material does indeed form part of the solar wind, it is likely that it contains mostly material from the streamer legs, i.e., the bright regions in O vi emission on either side of the central core. This is suggested by the work of Raymond et al. (1997), which shows that the minor ion abundances in the streamer legs are similar to those measured in the slow solar wind at 1 AU. However, there is still some debate about whether the blob material comes from streamer evaporation at the tips or from foot point reconnections at the streamer base (see Wang et al. 2000; Jones & Davila 2009). While feature tracking works well for determining outflows from streamers, it has been less successful in tracking features in the much dimmer coronal holes where the fast solar wind originates.

With this new work we produce a two-dimensional map of coronal outflow velocities covering the entire Sun at a fixed height by using a Doppler dimming analysis of coronal emission lines (e.g., see Withbroe et al. 1982; Noci et al. 1987; Li et al. 1998; Cranmer et al. 1999). The present work builds on the techniques developed for providing outflow velocity measurements as a function of height and latitude that were described in our previous study (see Strachan et al. 2002). The advantages of using the Doppler dimming technique is that the outflow velocities can be determined much closer to the Sun. In fact, the most sensitive height range for observing coronal UV emission lines with the Ultraviolet Coronagraph Spectrometer (UVCS; Kohl et al. 1995) is in the heliocentric height range r = 1.5–3.5 R_☉. This is the region where UVCS observations have shown that much of the coronal heating and solar wind acceleration takes place (Kohl et al. 1998; Cranmer et al. 1999). The outflow velocities are defined on a reference sphere at 2.3 R_☉, which we will refer to as a source surface. The choice for the reference height was made so that the outflow velocity measurements would be as close as practical to the traditional magnetic field source surface at 2.5 R_☉ (e.g., see Schatten et al. 1969). The maps, which are organized by Carrington rotations (CRs), were not constructed at exactly 2.5 R_☉ since there are significant gaps in the UVCS synoptic observations at this height.

The solar minimum period is used as a starting point for the velocity maps since the corona is in its simplest state at that time in the solar cycle. It is also an ideal time for comparing conditions in the corona with predictions from magnetohydrodynamic (MHD) models. For the purposes of this study, we define the time between 1996 May and 1997 January as the period that is characteristic of low solar activity (the "solar minimum period"). Harvey & White (1999) considered the same time period in their analysis to define the minimum of cycle 23. The first date is the time when the smoothed monthly sunspot number was at a minimum and the second date is when the sunspot number in the old cycle was equal to number in the new cycle. By using seven different solar parameters they determined that the actual minimum of solar activity for cycle 23 occurred in 1996 September. The maps produced in the present paper cover the period from 1996 May to 1998 June, which includes this period of low solar activity as well as part of the rising phase that started in 1997.

This paper describes the evolution of the outflow velocity and density structures in the corona with a discussion of a few derived parameters such as the particle flux and two types of solar wind expansion factors. The paper is organized as follows. In Section 2, we describe the line of sight coronal model that is used to produce the coronal outflow velocities. Maps of outflow velocity and electron density and their variations over the solar minimum period are presented in Section 3. We also compare our outflow velocity maps with those produced by IPS observations at the end of this section. Section 4 contains a discussion of the latitudinal and temporal variation of the derived solar wind particle flux and flux tube expansion factor. A summary of the results and a discussion of future work are provided in Section 5.

The coronal outflow velocities for O^{5 +} ions are determined from a self-consistent model of the observed O vi 103.2 and 103.7 nm intensities. The method uses a spectral line synthesis coronal code called CORPRO to compute modeled O vi 103.2 and 103.7 nm spectral emissivities, given empirical constraints for the incident O vi radiation from the solar disk, the coronal electron densities, and the kinetic temperatures (which includes both thermal and non-thermal motions) for the electrons and ions. The code is similar to previous codes that have been used in the past (e.g., Withbroe et al. 1982; Li et al. 1998; Cranmer et al. 1999; Noci & Maccari 1999; Strachan et al. 2000; Akinari 2007); however, upgrades were required to produce the outflow velocity maps. These include a complete rewrite of the original code so that it runs more efficiently and the construction of new algorithms for sorting the data and handling the two-dimensional aspects of the inputs and outputs.

In principle, coronal outflow velocities are computed by finding the velocities that produce modeled intensities and line widths that are consistent with the observed resonantly scattered profiles. However, for the O vi doublet, there is a collisional component to each line, which complicates the analysis. Instead we use the O vi 103.2 nm and 103.7 nm intensity ratios to isolate the resonantly scattered component that is sensitive to the outflow velocities. As described below, the model intensity ratios are derived from the line-of-sight integrated emissivities for each line. Another advantage of using the line ratios is that the absolute abundance for oxygen is not needed since this quantity cancels out when the ratio is performed.

Following Strachan et al. (2000), the coronal emissivities for the O vi lines, where the subscript i = 1 or 2 for the 103.2 nm or 103.7 nm line, can be expressed as the sum of a collisional component and a resonantly scattered component: E_i = E_i^col + E_i^res. The expressions for the collisional and resonantly scattered components, in units of photons s⁻¹ cm⁻³ sr⁻¹, are given below. For the collisional component, E_i^col, we have

$$\begin{equation} {E_i}^{{\rm col}} = {\int _{- \infty }^{\infty } \frac{1}{4 \pi } N_1({\rm O^{5+}}) N_e q_{{\rm col}} \phi _{\rm c}(\lambda - \lambda _i) d \lambda }, \end{equation} \tag{ 1 }$$

where N₁(O^{5 +}) is the ground state number density for O^{5 +}, N_e is the electron density, q_col(T_e) is the collisional excitation rate, as a function of electron temperature, T_e, and ϕ_c(λ − λ_i) is the coronal line profile, with the line center wavelength specified by λ_i. For the resonantly scattered component, E_i^res, we have

$$\begin{eqnarray} {E_i}^{{\rm res}} & = & {\int _{- \infty }^{\infty } \int _{- \infty }^{\infty } \int _{\Omega }} C N_1({\rm O^{5+}}) \phi _r(\lambda - \lambda _i) \nonumber \\ &&\times I_D(\lambda ^{\prime } - \delta \lambda ^{\prime }, \theta _i) p(\theta _s) d \Omega d \lambda ^{\prime } d \lambda. \end{eqnarray} \tag{ 2 }$$

The parameters in the above equation are: the constant C = B₁₂hλ_i⁻¹, where B₁₂ is the Einstein absorption coefficient, h is Planck's constant, and λ_i is the line center wavelength. N₁(O^{5 +}) is the ground state number density for O^{5 +}. The O^{5 +} number density can be defined in terms of the electron density, N_e, by using the following: N₁(O^{5 +}) = 0.8A_OR_i(T_e)N_e, where A_O is the total oxygen abundance relative to hydrogen and R_i(T_e) is the ionization balance term for O^{5 +}. The ionization balance term is a function of the electron temperature (a true thermal temperature), T_e. Using the above substitutions in the equation for the combined emissivity, it can be shown that the abundance and ionization balance terms cancel in taking the emissivity ratio E₁/E₂. This is an advantage when determining the outflow velocity from the emissivity ratio, since the uncertainties in these quantities are not propagated to the uncertainties in the final velocity determinations.

The third term, ϕ_r(λ − λ_i), is the resonantly scattered coronal line profile, which is a function of the small-scale ion velocity distribution in the direction of the incoming radiation, f(w). Also specified are the absolute intensities of the incident disk radiation, I_D, as a function of wavelength, λ', and the angle of incidence, θ_i, from the disk. The Doppler shift δλ' is a function of the coronal outflow velocity V_o (through the usual relation δλ' = λ'V_o/c). The phase function parameter p(θ_s) is the angular dependence of the scattering process, where θ_s is the angle between the incident and scattered radiation. The total resonance scattered emissivity involves an integration over the solid angle, Ω, which is the angle subtended by the solar disk. The wavelength integrations are performed over the incident profile from the disk (primed) and the scattered (unprimed) profile in the corona.

In performing the computations for the coronal emission line profiles, it is more useful to work with velocity units instead of wavelength units. In doing so, the observed line profile is treated as a velocity distribution that has a line width equal to the 1/e half-width of the summed velocity distributions of all of the scatterers along the line of sight. Some of the line broadening in the direction along the line of sight is due to the projection of the bulk outflow along the line of sight. This effect is included in the determination of the true line widths. In a large polar coronal hole for example, the line profiles far from the plane of the sky will be red shifted if they are behind the plane of the sky and blue shifted if they are in the foreground. This additional effect on the profiles is taken into account with the CORPRO model.

We assume that the coronal velocity distributions are not isotropic but instead are bi-Maxwellian with 1/e velocity half-widths, $w_\Vert$ and w_⊥ (e.g., Cranmer et al. 1999; Akinari 2007). The velocity distributions, w, have contributions from both thermal and non-thermal motions, such as turbulence or wave sloshing. In general, the velocity distribution width in the direction parallel ($w_\Vert$) to the local magnetic field is not the same as the velocity distribution width in the perpendicular direction (w_⊥; see Kohl et al. 1998). We see from above that it is the $w_\Vert$ component that provides the sensitivity for determining the coronal outflow velocity. Although $w_\Vert$ is not measured directly, it has been found to be highly constrained by several authors (e.g., Cranmer et al. 1999; Li et al. 1998; Frazin et al. 2003). In particular for coronal holes, Cranmer et al. (2008) found that by using an exhaustive search in parameter space that $T_\bot / T_\Vert = 6$ was the most probable anisotropy ratio for the oxygen kinetic temperatures, defined as T_{⊥, ||} = mw²_{⊥, ||}/(2k). We use this value to determine the outflow velocities in coronal holes but show below (in Section 3.5) the effect on the outflow velocities if other values are used. We use the same anisotropic temperature ratio for all bins in coronal hole regions, defined where the electron density is below 2 × 10⁵ cm⁻³. For coronal streamers, where the higher densities imply that collisions should reduce the anisotropy, we use $T_\bot / T_\Vert = 1$ (e.g., see Strachan et al. 2002). We also assume that the temperature for all species (protons, electrons, and O^{5 +}) are the same for streamers at our source surface height. Frazin et al. (2003) provide evidence for temperature anisotropy at heights above 2.3 R_☉ in streamers, but only in mid-latitude streamers near solar minimum.

Before the outflow velocities can be produced, one needs to know the electron temperatures and the electron densities in the observed coronal regions in order to establish the baseline emissivities in Equations (1) and (2). We originally used an adjustable electron temperature for each region but found that this made a very small difference in the results, so instead we fixed this parameter to T_e = 1 × 10⁶ K for all regions. The electron densities for this work are derived from an inversion of the LASCO-C2 polarized brightness measurements of the white light corona (van de Hulst 1950). The specific implementation used the axisymmetric model of Quémerais & Lamy (2002) which is appropriate for this phase of the solar cycle (near solar minimum). A more sophisticated implementation (e.g., Saez et al. 2007) can be used for obtaining the coronal electron densities during more active phases of the solar cycle. This is not warranted for the current work since the high spatial resolution of the LASCO-C2 images has been reduced to match the spatial bin size used for UVCS Carrington maps. In addition, the original LASCO density maps for 2.5 R_☉ were adjusted to estimate the densities at the same height (2.3 R_☉) that was used for the outflow velocity calculations. These densities were obtained by computing density ratios N_e(2.3 R_☉)/N_e(2.5 R_☉) from previously studied streamers and coronal holes. The average value for this ratio was found to be 1.4 for streamers and 1.5 for coronal holes. Susino et al. (2008) suggest that LASCO-derived densities in streamers may be may be too large compared with local densities derived using the O vi intensities. While this may be true for the high latitude streamer that they examined, this correction is not applicable when looking through a horizontal streamer belt that exists during the solar minimum phase.

The reported outflow velocities are computed on a grid of 30 × 28 (latitude × longitude) bins using only west limb data. The outflow velocity in each bin is determined by creating a series of O vi profiles for a trial set of 26 outflow velocities that lie between 0 and 500 km s⁻¹. The resonantly scattered and collisional emissivities are computed along the line of sight at each spatial bin, where we assume that the bulk of the emission comes from within 1 R_☉ from the plane of the sky. After the emissivities are summed to form the separate 103.2 nm and 103.6 nm intensities, we compute the intensity ratios that are compared to the observed O vi ratios. One advantage of using the line ratio is that the oxygen abundance drops out of the calculations. The most likely outflow velocity value for the spatial bin is the one that produces an O vi ratio that matches the observations. All specified parameters are held fixed along the line of sight in order to reduce the number of iterations. As is usual with forward modeling, the final outflow velocities may not be unique.

In the following sections, we present electron density maps made from white light polarized brightness data and, for the first time, outflow velocity maps obtained with the new Doppler dimming diagnostic code (see Section 2). LASCO and UVCS daily synoptic images of the corona were used to make Carrington maps at selected heights. Constructing the electron density maps is straightforward since the data reduction starts with fully two-dimensional images from the LASCO-C2 coronagraph (e.g., Biesecker et al. 1999). For each day, white light intensities or polarized brightness values are recorded, at a fixed height, for a complete 360° (in position angle) around the Sun. A full Carrington map for a single rotation period can be made when data are placed in a rectangular grid with latitude and longitude bins. To do this, the observation times are converted to Carrington longitudes (assuming solid body rotation of the corona). The approximately one-day interval between each observation corresponds to about 135 in longitude, but we actually use an interval that depends on the variable speed of the Solar and Heliospheric Observatory (SOHO) spacecraft in its orbit around the Sun.

Constructing the UVCS maps of spectroscopic parameters used to infer the outflow velocities is not as straightforward as preparing the LASCO maps. The main reason is that UVCS uses a slit spectrometer with a narrow (∼1' × 40') field of view of the corona. An additional step requires the construction of two-dimensional plane-of-the-sky "images" of the corona for the UVCS observations. To build up a complete coronal image, O vi and H i Lyα profile measurements are obtained with the UVCS field of view positioned at several different heights between 1.5 and 3.5 R_☉ while the instrument remains at a fixed roll angle (i.e., position angle). A complete raster of the full corona (called a "synoptic image") is produced by making similar height scans at a total of eight roll angles about the Sun, with 45° intervals between each roll. Fits are made to the O vi profiles in each UVCS spatial bin so that intensity and line width information can be obtained for each point in the sky. The UVCS data (intensities and line widths) are then interpolated along a pole-to-pole arc at a fixed distance from Sun-center in order to make a uniform data set with equal latitude intervals. It should be noted that the coronal line widths are corrected for the instrument profile function and the spectrometer silt width. At this stage, the data from successive daily image maps are time tagged and converted to Carrington longitude just as for the density Carrington maps. More details about the preparation of the UVCS Carrington maps are described in Strachan (1997) and Panasyuk (1999).

Once the Carrington maps of the UV profile data (intensities and 1/e line widths only) and the electron densities are computed, there is enough information to estimate the bulk outflow velocities for each latitude/longitude bin of the Carrington maps. The outflow velocities are computed using the Doppler dimming model described in Section 2. The only other information needed is the known atomic parameters for the collisional and radiative excitation rates (Withbroe 1970; Noci et al. 1987), the ionization equilibrium values for the atomic levels (Mazzotta et al. 1998), and the intensities and line widths for the disk spectral lines (Noci et al. 1987; Li et al. 1998).

3.1. Carrington Maps for CRs 1912 and 1932

Figure 1 shows two representative maps⁴ for the electron densities and O^{5 +} outflow speeds at 2.3 R_☉. Panels (a) and (b) are for CR 1912 (near solar minimum at the start of cycle 23) while panels (c) and (d) are for CR 1931 (17 months later) when solar activity has increased and the coronal magnetic current sheet has become more warped. The color bar above the density plots indicates that the coronal hole regions have number densities less than 5 × 10⁵ cm⁻³ at the selected height. Coronal streamers, which form the bright orange/red band in both panels (a) and (c), can have densities as high as 2 × 10⁶ cm⁻³. All of the densities are shown for 2.3 R_☉ and were computed by scaling the density values from the LASCO density maps at 2.5 R_☉ as previously described above.

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In these figures, the LASCO Carrington maps have been resampled to match the resolution of the UVCS Carrington maps, which use a binning of 30 × 28 (latitude × longitude). Note that the apparent break in the streamer belt near 270° longitude in Figure 1(a) is caused by the deflection of the heliospheric current sheet by an active region on the disk. The apparent tilt of the diagonal streamer arms at mid latitudes, e.g., in Figure 1(a) between 180° and 270°, has been shown by Wang et al. (1997) to be related to the tilt angle of the solar rotation axis. The direction of the tilt depends on whether the solar rotation axis is in front of or behind the plane of the Sun and on whether the map is constructed from East-limb or West-limb data. While the tilts for CR 1931 are faint in the density plots (Figure 1(c)), they are very pronounced in the velocity maps (Figure 1(d)).

The outflow velocities for CRs 1912 and 1931 are shown in Figures 1(b) and (d), respectively. The velocity maps have been interpolated to fill in for some days where there were no observations from either UVCS or LASCO. The missing days (columns) are identifiable on the maps by the columns with black bins at both the top and bottom rows (at 90° and −90°). Interpolations for missing data are linearly applied in both the longitudinal (temporal) and latitudinal directions. Note that the dark, low-velocity regions in the outflow velocity maps correspond to the light colored, high-density regions in the density maps. This shows that, already by 2.3 R_☉, the highest speed outflows come from the broad coronal holes surrounding both poles. An interesting result is that the latitudinal width of the slow speed wind is considerably wider at CR 1931 than it is at CR 1912 (the heart of the solar minimum); however, the thickness of the streamer belt (in the density) for both time periods is comparable. We will return to this point in Section 3.2 below.

3.2. Contiguous Carrington Maps for Cycle 23 Minimum

In order to provide a global perspective of the period near the minimum at the start of cycle 23, we place all of the available Carrington maps for electron densities and outflow velocities side-by-side. The maps for 2.3 R_☉ are shown in Figures 2 for the electron densities (left side) and for the coronal outflow velocities (right side). The entire period is divided into three parts: CRs 1909–1918 (top panels), CRs 1919–1928 (middle panels), and CRs 1929–1937 (bottom panels). To preserve the standard longitude orientation for each rotation period (longitude increases from left to right on the horizontal axis), the maps were constructed so that observation time increases to the left. The discontinuities in the maps are due to extended periods (>2 days) without valid data. The longest time gaps (e.g., CRs 1915–1916) are due to periods when the SOHO spacecraft temporarily lost its Sun-pointing attitude control and therefore, no observations were made. Other shorter gaps (∼2 days in duration) are caused by several factors. Some of these are for periods when the UVCS synoptic observations were not made. Other gaps, especially those that extend in the latitude direction, are the result of instances where the data were missing or corrupted.

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Although the vertical height of each map in Figure 2 is the same as that for Figure 1, the horizontal axis has been compressed considerably. The maps reveal the evolution of the density and velocity structures in the transition from solar minimum to the rising phase of the solar cycle. The figures clearly show a gradual drift toward the poles of the equatorial high density, slow speed region as is expected during the progression away from solar minimum. This latitudinal spread of the low velocity streamer belt toward the poles is not uniform and there is an indication that the minimum width of the slow speed belt did not occur during sunspot minimum in CR 1909 in 1996 May (SIDC-Team 2010) but possibly as late as CR 1922 (shown in the middle panels). The latitudinal width of the low speed belt appears nearly constant from approximately CR 1921 to CR 1926, indicating a broad minimum in the belt thickness. This broad minimum is most easily seen in the middle density map shown in Figure 2.

Before describing how the corona evolves in time, we present the mean properties (as a function of latitude) of several solar wind parameters in the plots on the left side of Figure 3. These data are useful for providing a quantitative characterization for the mean conditions during the solar minimum period at the start of cycle 23. Figure 3 shows the following: (a) the solar wind outflow velocity, (b) electron density, and (c) the particle flux (all averaged over CR periods 1909–1925) for each latitude bin. The statistical 1σ variations for the data points in each plot are shown as vertical lines and are generally small, i.e., about the size of the symbols for each plot. The error bars give an indication of the variation of the data but not the overall uncertainties in the parameters, which are discussed in Section 3.5.

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At the heart of solar minimum (CRs 1909–1925), we see a nearly symmetrical corona in both the outflow velocity and electron densities as a function of latitude. The plots are not inverses of each other; there are some subtle differences between them. First of all, it should be noticed that the outflow velocity minimum is flat compared with the sharper peak in the density data. This is somewhat artificial since the averages were calculated by using only the nonzero velocity values in the streamer belt (we are mainly interested in the open field regions with solar wind outflow). Another noticeable difference is that there is a steeper outflow velocity slope in the transition between the polar coronal holes and the low latitude streamer belt when compared with the density transition. The implication is that, at least at the selected source surface height, both polar coronal holes have a gradient in their velocity profile. The smoother density transition could also be attributed to the line of sight integration.

It is interesting to compare the velocity versus latitude plot in Figure 3(a) with the Ulysses proton velocity versus latitude plot in McComas et al. (2000). The Ulysses plot has a much larger range of latitudes with fast wind since the coronal holes are still over expanding above 2.3 R_☉. The velocity gradient for the fast solar wind (V > 700 km s⁻¹) is 0.95 km s⁻¹ deg⁻¹ above 36° latitude. Our data have a mean velocity gradient (using data from both poles) of 0.68 km s⁻¹ deg⁻¹ for latitudes above 54°, which is outside of the slow wind belt. The gradient in the south pole alone is 0.85 km s⁻¹ deg⁻¹ (negative). This is larger than the velocity gradient for the north pole using our data and is closer to the Ulysses value.

Also notice in Figure 3(c) that the mean particle flux for the escaping solar wind is approximately 1.8 × 10¹² cm⁻² s⁻¹ for all latitudes, which includes the streamer belt and the polar coronal holes. The bump in the particle flux between −90° and −30° is probably real. There appears to be a smaller one in the northern hemisphere as well. We will comment more about these features in Section 4, when we discuss how the plasma parameters for one rotation (CR 1909) compares to a theoretical model.

3.3. Evolution of the Source Surface Parameters at Different Latitudes

In this section we describe the spatial and temporal variation of the coronal data with the emphasis mainly on the outflow velocities since these are the newer results. The corona had a simple structure during the cycle 23 minimum with a nearly continuous streamer belt and two large polar coronal holes that were present throughout the period of this study. In order to better observe the long-term trends, we use averages for each CR at three selected latitudes: 90°, 60°, and 30°. The north and south hemisphere data are similar except for higher densities at the south pole (see Figure 3 above). In Figure 4 we present plots for the mean values of the outflow velocity (V), plasma density (N), and proton kinetic temperature (T_p) in the three stated latitude bands. The data for 0° are not plotted since most of the values obtained for the outflow velocity were too small to measure. The large number of unmeasurable small outflow velocities makes the concept of an average value not very useful at the equator.

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The rotation-averaged outflow velocities at high latitudes (60° and 90°) are fairly constant up to CR 1925, after which they start to decline with time. At 30° latitude, the decline in the outflow velocity starts earlier, at about CR 1920. The decrease in the solar wind outflow velocity may be related to the appearance of dense streamers at low latitudes (see the density plot at 30° in the middle panel). Contrary to the density rise for the 30° plot, the high latitude plots of density versus time are nearly flat. The nearly constant density with the corresponding increasing proton temperatures (bottom panel) for the high latitude plots may partially explain the decrease in outflow velocity over the poles. If one assumes a time steady average energy input into the coronal holes, it appears that over time, more energy is converted into heating the coronal plasma rather that increasing the outflow velocity. On the other hand, the bottom panel in Figure 4 shows that the proton kinetic temperatures in the high density streamer regions decrease with time, especially after CR 1925. The lower kinetic temperatures could be explained by the fact that the increase in density of the low latitude regions leads to an increase in the radiative cooling of the streamer plasmas. Another effect that could be important is that the denser streamer plasmas have a smaller non-thermal temperature component (compared with coronal holes) and this reduces the overall proton kinetic temperature in these regions. This reduction of the non-thermal component in streamers is qualitatively consistent with a one-fluid model that shows that the non-thermal perpendicular velocities (produced by Alfvén waves in this case) are lower in the streamer legs compared with the values in a coronal hole (see Figure 11 in Cranmer et al. 2007).

3.4. Solar Wind Source Regions

In order to better understand changes of the intrinsic properties of the solar wind source regions, we need a method to identify coronal hole regions that are uncontaminated by streamer material that may be along the same line of sight. This can be accomplished by using an outflow velocity criterion or a density criterion to distinguish coronal hole regions from streamers. We show below that both types of criteria give similar results for defining the pure coronal hole regions. For the first method, we examine the distribution of outflow velocities from all spatial bins on the source surface maps for the entire period from CRs 1909–1937. Figure 5 shows a histogram plot of the outflow velocities. The distribution is clearly double peaked with a low velocity peak centered at ∼30 km s⁻¹ and a high velocity peak centered at ∼150 km s⁻¹. For convenience, we assume that the two velocity groups represent the fast and slow wind components at the 2.3 R_☉ source surface. It should be noted that the number of counts in the lowest outflow velocity bin do not include pixels with V = 0 km s⁻¹.

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The overlap between the two distributions occurs at roughly ∼90 km s⁻¹ and so we will use this as the criterion for separating the fast and slow speed source regions. Since we know that the fastest solar wind originates in polar holes, we can use the velocity break point as the separation between coronal hole and non-coronal hole source regions. We can compute the mean latitude bin for the transition from fast to slow wind by starting at the pole deep inside of the coronal hole and then move to progressively lower latitude bins until we reach a bin where the outflow velocity falls below 90 km s⁻¹. The mean latitude of this boundary for each CR is shown as the filled circles connected by the solid lines in Figure 6. The uncertainty of the latitude location of the boundary is computed by combining the 1σ uncertainty due to the latitude binning of the data (±3°) with the uncertainty in the velocity at the cutoff location.

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For comparison, the open square symbols show the coronal hole boundary (CHB) using the method of Guhathakurta et al. (1996), where the boundary is found by plotting the electron density profile from pole to pole (along a constant longitude). The latitudinal profile for the density is approximately Gaussian with the peak density near the equator in the streamer belt and the base of the profile in the polar coronal holes (see, for example, the electron density profile in Figure 3(b)). The coronal hole densities form a baseline above zero which is approximately 15% of the peak density in the streamer belt (Guhathakurta et al. 1996). It is remarkable that the coronal hole boundary determined with this density threshold is nearly identical to the boundary determined by using the outflow velocities. The fact that the results from the two techniques are in close agreement provides some confidence in the identification of the coronal hole boundaries for our Carrington maps.

Once the coronal hole boundary is defined at 2.3 R_☉, in this case for the northern hemisphere, we can examine how the size of the coronal hole changes over time. Figure 6 shows that the CHB was fairly constant at ∼30° latitude until CR 1925; after this the boundary increases rapidly in latitude to about 60° by CR 1937. Therefore, the best period for characterizing the steady-state plasma properties of the coronal hole for the case of a relatively stable geometric configuration is during the period CRs 1909–1925. After CR 1925, the latitude of the CHB increases rapidly and so the intrinsic changes of the coronal hole plasma properties would be more difficult to interpret since they could be masked by the fact that the volume of the coronal hole is also changing.

Also plotted in Figure 6 is the mean latitude of the coronal hole boundary on the disk (at r ≈ 1 R_☉). The boundary is indicated by open circles using data obtained from He i 1083.0 nm observations (Harvey & Recely 2002). The uncertainties for these data are not shown since they are smaller than the symbol size. The authors estimate that the boundaries are determined to within 1° for most of the data and to within 3°–5° for times when the boundary is less clear. As an average uncertainty, we use 2° for these data. There is a clear indication that the polar coronal hole has a super-radial expansion since the CHB on the disk is nearly constant at ∼60° latitude, while the same CHB at 2.3 R_☉ (solid circles) is at much lower latitudes. However, this is not true for the entire period. At the time of CR 1937, the coronal hole expansion from the disk to the source surface becomes essentially radial since the CHB at both heights are at approximately the same latitude.

The surface area of the coronal hole on the disk and at 2.3 R_☉, both as a function of time, are shown explicitly in Figure 7. The plot in the left panel shows the rapid decline in the coronal hole surface area (in square solar radii) after the time period near CR 1925. A factor of 10 multiplier is used to plot the area of the coronal hole on the disk in order to show the details of its variation with time. The error bars indicate the estimated 1σ uncertainty in the calculation for the areas. These are determined by combining, in quadrature, the uncertainties in the observation height with the uncertainties for the CHB latitude. Again, the error bars for the disk data are omitted as they are smaller than the plotted symbols.

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Munro & Jackson (1977) define the areal expansion factor f_A at the source surface height, r_ss, as

$$\begin{equation} f_{A}(r_{{\rm ss}}) = \frac{A_{{\rm CH}}(r_{{\rm ss}})}{A_{{\rm CH}}(r_{\rm o})} \frac{{r_{\rm o}}^2}{{r_{{\rm ss}}}^2}, \end{equation} \tag{ 3 }$$

where A_CH(r_ss) is the coronal hole area on a sphere of radius r_ss and A_CH(r_o) is the coronal hole area at the base height, r_o ≈ 1 R_☉. (Note that we take r_o to be above the canopy structures at the coronal base where there is an initial rapid expansion of the flux tubes rising from the photosphere.) The right panel of Figure 7 shows the calculated values of f_A for the north coronal hole using the data for A_CH(r_ss) and A_CH(r_o) shown in the left panel.

An interesting feature of the areal expansion factor plot, is that f_A decreases with an approximately linear relationship with time, starting midway through the data period. It reaches a value of ∼1 at the end of the period under study. The data suggest that the areal expansion factors of polar coronal holes are not constant and there is at least one period in the solar cycle where f_A = 1, i.e., the expansion with height goes as r². However, unlike the suggestion of Woo & Habbal (1999), our data show that the areal expansion of the polar coronal hole is super-radial during the deepest part of solar minimum, when the coronal holes are the largest.

A summary of the changes with time of the plasma parameters and mean size of the northern polar coronal hole is presented in Table 1. Mean and standard deviations for the outflow velocity, electron density, and proton kinetic temperature are shown for selected latitudes and CRs. (The standard deviations are deviations from the mean values calculated by averaging over all longitude bins for the fixed latitude and specified CR.) As previously mentioned, there is little variation in the mean values of the plasma parameters between CR 1909 and CR 1925 (see Figure 4). Consequently, we have omitted any intermediate CRs between these two endpoints. We suggest that the changes that occur after CR 1925 can be characterized as the start of the rising phase of the solar cycle. The data for the plasma parameters are tabulated for 30°, 60°, and 90° latitudes to show their latitudinal dependences. Mean parameters for the southern latitudes are similar except at the south pole which appears to be contaminated by foreground/background high density structures (possibly plumes or streamers). The geometric parameters that describe the mean latitude of the coronal hole boundary, the coronal hole size, and its areal expansion factor (from the disk to the source surface) are shown in the last three rows of the table. These parameters are shown for 2.3 R_☉ in the last three lines of Table 1. Although the ± uncertainties for each quantity are sometimes unequal, only the average of the positive and negative uncertainties are shown in parentheses next to the mean quantities.

Table 1. Plasma and Geometric Properties at 2.3 R_☉ versus Time and Latitude

Parameter	Latitude	Mean (and Standard Deviation) for Five Periods
		CR 1909	CR 1925	CR 1929	CR 1934	CR 1936
V(km s−1)	+90°	200.(10.)	176.(7.)	151.(7.)	152.(5.)	140.(6.)
	+60°	191.(9.)	160.(7.)	132.(11.)	110.(10.)	73.(8.)
	+30°	92.(10.)	61.(8.)	38.(4.)	40.(4.)	37.(4.)
N(105 cm−3)	+90°	1.09 (0.01)	1.03 (0.01)	1.24 (0.02)	1.23 (0.01)	1.16 (0.01)
	+60°	0.96 (0.01)	0.95 (0.02)	1.33 (0.08)	1.44 (0.09)	2.02 (0.14)
	+30°	2.38 (0.31)	2.25 (0.14)	4.54 (0.29)	4.65 (0.28)	5.61 (0.54)
Tp(106 K)	+90°	2.61 (0.06)	2.61 (0.05)	2.91 (0.07)	3.00 (0.07)	3.14 (0.05)
	+60°	2.57 (0.03)	2.52 (0.00)	2.73 (0.07)	2.72 (0.07)	2.63 (0.11)
	+30°	2.30 (0.06)	2.21 (0.06)	2.04 (0.06)	2.00 (0.07)	2.09 (0.08)
θB(deg)	⋅⋅⋅	24. (7.4)	33. (4.8)	48. (4.6)	43. (5.3)	57. (4.7)
Anch(R☉2)	⋅⋅⋅	19.4 (3.9)	15.0 (2,4)	8.1 (1.8)	10.3 (2.2)	5.1 (1.5)
fA	⋅⋅⋅	4.5 (1.1)	3.3 (0.7)	2.7 (0.7)	2.7 (0.7)	2.2 (0.73)

Notes. The quantities shown are averaged over an entire rotation with their estimated standard deviations shown within the parentheses. The parameter T_p is the proton kinetic temperature that includes both thermal and non-thermal (e.g., wave) heating of the protons. See the main text of details.

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3.5. Uncertainties of the Plasma Parameters

3.6. Comparisons to IPS Velocity Maps

It is interesting to compare our velocity maps with those obtained with IPS observations. The three most relevant IPS studies that have time periods that overlap with the present work are by Kojima et al. (1999), Tokumaru et al. (2010), and Breen et al. (1999). The first two references use data from the Solar-Terrestrial Environment Laboratory, Nagoya (STELab) and the last one uses data from the European Incoherent Scatter array (EISCAT). Kojima et al. (1999) have IPS velocity data from the period starting from CR 1895 and ending in CR 1917. They project their outflow velocities onto a source surface at 2.5 R_☉, which is ideal for making comparisons to the velocity structures in our outflow velocity maps. In their Figure 1, they show velocity maps for the whole period, each with contour lines at 300, 350, 400, and 500 km s⁻¹. The maps are restricted to latitudes between ±30° because their work emphasized the low speed regions around the streamer belt. We show in Figure 8 the IPS contours only for V < 500 km s⁻¹ (shown in white) superimposed on our outflow velocity maps for CRs 1909, 1912, and 1916. The maps for each rotation in their paper are produced with data from three rotations centered on the primary rotation period. Figure 8(a) shows that the IPS low velocity contours follow our low velocity regions (black and dark blue) very well. In addition, the changes in the structures from one rotation to the next is similar for both data sets. This comparison provides another confirmation that the slow wind in the heliosphere maps back to the streamer belt during solar minimum as expected.

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The major differences between our results and the IPS results in general is the magnitude of the outflow velocities. The IPS velocities are based on reconstructions that use line of sight data from many elongation angles from the Sun. The minimum inner boundary of these reconstructions is around 0.2 AU for these full coverage maps (Kojima et al. 1999). Since a constant velocity approximation is used to extrapolate the IPS velocity reconstructions back to the source surface, the IPS velocities at 2.5 R_☉ will be identical to the velocities at their minimum reconstruction distance. This appears to be the case for the maps produced by Tokumaru et al. (2010) as well. Their map for the 1996 solar minimum (Figure 1(c) in their paper) shows "low speed" regions mapping back to the streamer belt and "high speed" regions (mapping back to coronal holes. Again, while the structure is correct, these outflow velocity values are more typical of 1 AU solar wind speeds than the outflow velocities in the inner corona.

The EISCAT radio telescope array was used to produce IPS velocity maps during CRs 1912–1913 for the First Whole Sun Month (see Breen et al. 1999, 2000). Because EISCAT operates at the relatively high frequency of 931.5 MHz, it can observe as close as ∼15 R_☉ from the Sun. The disadvantage with this array is that there are fewer radio sources that can be observed at this frequency and so the EISCAT maps have velocity measurements at relatively few locations. The EISCAT outflow velocities are generally lower than those obtained with the STELab telescope, but their velocities are still much larger than our results. For example, regions of 40–100 km s⁻¹ on our maps are 200–300 km s⁻¹ on the EISCAT maps; and likewise regions of 130–170 km s⁻¹ on our maps are from 600 to >800 km s⁻¹. Clearly the solar wind is still accelerating beyond our source surface height.

The Carrington maps produced for this work are useful for providing constraints on parameters that are used in theoretical models of the corona and the solar wind. For example, many solar wind models propose some form of mass, momentum, and energy input at the coronal base. In some cases, these inputs are used as free parameters in the model. However, with the data presented here, we can provide empirically determined constraints on many of these quantities. We mention only two here: (1) the solar wind mass flux and (2) the derived expansion factors in the corona. These two topics are briefly discussed below.

The plots for the particle flux as a function of latitude, which are shown in Figure 3 (panels (c) and (f)), contrast with what is measured in the distant heliosphere (see, for example, the data from Ulysses first full orbit McComas et al. 2000). The 1 AU scaled latitudinal profile of the particle flux is essentially constant (to within the measurement errors) from 30° latitude all the way to the poles. On the other hand, our data show an increase from mid-latitudes toward both poles. This difference is partly explained by the fact that the coronal hole expands super-radially so that low latitudes at Ulysses map back to higher latitudes near the poles at 2.3 R_☉. Also the gradients along the field lines in both the density and outflow velocity are different for the high latitude fast solar wind and the low latitude sources of the slow solar wind. The different radial gradients could explain how the different particle fluxes that are determined in the high and low latitude regions close to the Sun become nearly the same when measured at larger distances.

Realistic models of the solar wind will have to explain the differences between the behavior of the fast and slow solar wind in the regions near the Sun. To test one such model by Cranmer et al. (2007), we show its results for V, N, and NV in the plots on the right-hand side of Figure 3. The model results are plotted with dashed lines in the figure. Note that this is a model for the open magnetic field regions only and so there is no prediction for the plasma parameters in closed-field regions around the equator. The model is a self-consistent, single-fluid, wave-driven MHD model of an axisymmetric corona. It requires relatively few inputs, which include a definition of the Alfvén wave spectrum determined by measurements of the magnetic footpoint motions in the photosphere. It includes a coupled chromosphere model that is heated by acoustic waves and cooled primarily by radiation. The geometry of the specified magnetic field controls the form of the wave damping that leads to heating and acceleration of the solar wind.

The qualitative agreement between the data and the model is remarkable considering that no adjustments were made to the model. There are obvious differences near the streamer boundaries but this is most likely due to line of sight effects that are not included in this axisymmetric model. There also appears to be some asymmetry in the data with the south pole showing a higher particle flux than the north. This is most likely a real feature and not an artifact, since it remains present even after averaging over a dozen rotations as shown in Figure 3(c). We will report on a more complete comparison between this model and our data in a future paper.

The next topic concerns the so-called expansion factors of the solar wind, which describe the geometric spreading of structures in the solar corona. This expansion can be measured in two different ways. The first method, which has already been presented, uses the areal expansion factor of the entire coronal hole. A second method relates to the local expansion factors of magnetic flux tubes embedded in the solar wind. Ideally the flux tube expansion factors can be determined directly from measurements of the coronal magnetic field, however, such measurements do not routinely exist. In the past, estimates for the flux tube expansion factors were determined by using estimates of the magnetic field determined by potential field models (e.g., see Wang & Sheeley 1990) or by MHD models (e.g., see Riley et al. 2010). However, for the present work, we will use conservation of mass (or particle) flux arguments to compute the flux tube expansion factors from the coronal base to the source surface. The particle fluxes are derived from our determinations of density and velocity at the source surface. In the future, we plan to incorporate a magnetic field model to use with the current maps of plasma parameters.

First we investigate how the outflow velocity in the coronal hole at the north pole varies with the areal expansion factor. Recall that the areal expansion factor, f_A (defined in Equation (3)) is a global expansion factor for the entire coronal hole. Because f_A is only a function of radial height, r, there is no dependence on latitude or longitude as is normally used for the magnetic flux tube expansion factors. We considered using an average outflow velocity for the entire coronal hole but this could possibly dilute any relationship between f_A and V. Instead we use the empirical outflow velocities V(90°), determined at the north pole for each CR, and plot these as a function of the coronal hole expansion factor. Figure 9 shows that outflow velocity at the center of the coronal hole tends to increase with larger expansion factors for the coronal hole.

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The actual relationship between V(90°) and f_A may not have a constant slope for the entire range of expansion factors plotted. For example, the slope for the data with f_A ⩽ 3 may be considerably less steep than the portion of the curve with f_A > 3. However, considering the size of the error bars we will use the same slope for the entire data set in the figure. Figure 9 shows that larger coronal holes have the largest expansion factors and the fastest solar wind. While this result is not new for solar wind measurements at 1 AU, it is a new result for quantitative velocity measurements made close to the Sun. An empirical fit to the V(90°)–f_A relationship is shown in the figure. Note that the relationship will have different values for the fitting parameters at a different source surface height used for calculating V(90°) and f_A. This fit does not necessarily contradict the idea that flux tube expansion factors have an inverse relationship with solar wind speed since, to state again, we are using a global expansion factor for the entire coronal hole.

Our results are similar the results of Nolte et al. (1976), which showed that the maximum solar wind speed (at 1 AU) had a positive correlation with increasing area of the coronal hole source regions on the disk. However, there are differences between the two measurements. Nolte et al. (1976) looked at equatorial coronal holes and they used solar wind speed measurements at 1 AU, which could be affected by the transit of the wind through the interplanetary medium. The data in Figure 9 are for a polar coronal hole with velocity measurements made much closer to the Sun at 2.3 R_☉. If the coronal hole area on the disk is used, our data give a linear correlation coefficient for a liner fit of V and A_CH of only 0.42. The correlation coefficient using V and the coronal hole area on the source surface has a value of 0.72, which is significantly better. This is another way of saying that while the coronal hole area on the disk is important, the areal expansion factor is a much more important factor in determining the solar wind speed. The linear correlation coefficient for the fit in Figure 9 is 0.83.

We now turn to a discussion of the local (versus global) expansion of the polar coronal hole. In order to estimate the expansion factors for the local magnetic flux tubes (i.e., the flux tube expansion factors, f_exp) we will use mass flux conservation (or actually we use particle flux conservation by dividing the mass flux by the particle mass, m_p). The expression for the particle flux conservation for material flowing from the base of the corona (r_o ≈ 1 R_☉) to the source surface height (r_ss = 2.3 R_☉) is

$$\begin{equation} N_{{\rm ss}} V_{{\rm ss}} A_{{\rm ss}} = N_{\rm o} V_{\rm o} A_{\rm o}, \end{equation} \tag{ 4 }$$

where N, V, and A are the electron density, outflow velocity, and flux tube area respectively at the source surface (designated with subscript "ss") and at the coronal base (with subscript "o"). The areas, A_o and A_ss, are now no longer the entire coronal hole area at the respective heights but are the elemental flux tube areas for the plasma flow. We now define the traditional expansion factor, f_exp, which gives the expansion of a flux tube in going from the coronal base to the source surface height, from the following expression:

$$\begin{equation} \frac{A_{{\rm ss}}}{A_{\rm o}} = f_{{\rm exp}} \frac{{r_{{\rm ss}}}^2}{{r_{\rm o}}^2}. \end{equation} \tag{ 5 }$$

Combining Equations (4) and (5) and rearranging terms gives the final expression for the expansion factor at the source surface height:

$$\begin{equation} f_{{\rm exp}} = \frac{N_{\rm o} V_{\rm o} }{ N_{{\rm ss}} V_{{\rm ss}}{r_{{\rm ss}}}^2 }. \end{equation} \tag{ 6 }$$

As expected, we now see that the outflow velocity at the source surface, V_ss, is inversely related to the flux tube expansion factor, f_exp.

We can compute values for the particle flux (N_ssV_ss) for any element on our selected source surface height by using the independent determinations of density and outflow velocity for each CR in our data set. The particle flux at the coronal base (N_oV_o) can be estimated by assuming that f_exp = 1 on the axis of a polar coronal hole and using Equation (6). By taking the mean of the particle flux at the north pole we obtain a value for the particle flux at the coronal base N_oV_o = 1.1 × 10¹³ cm⁻² s⁻¹. We have confidence in this value for the base flux since it is identical to the value found by using the high latitude proton particle flux from Ulysses during its first polar pass at solar minimum (McComas et al. 2000). We also assume that the base particle flux, N_oV_o, is constant for all latitudes on the Sun where there is open flux (magnetic field lines that make it out to the extended corona). This last point is a working assumption only but is the most simple argument for the coronal base. Later in this section, we will discuss the consequences of relaxing this condition for a constant base particle flux.

In order to understand how the flux tube expansion factors, f_exp, vary in the corona, we could compute the particle fluxes for every spatial bin in each of the CR maps. Variations of the particle flux and expansion factor with latitude can then be found. However we find that it is difficult to determine trends in this way because of the large scatter in the data that tends to mask any correlations. There is also the problem that the streamer belt is not constant and its spatial locations vary with time for each map. One solution to these problems is to use density instead of latitude as the independent parameter. This choice has the advantage that the density is a more suitable physical parameter for describing different coronal structures. Since the electron density changes monotonically from the pole to the streamer belt/current sheet, it can be used as a proxy for different types of coronal structures regardless of their actual map locations. In Figure 10(a), we show particle fluxes at the source surface as a function of electron density for CRs 1909–1922. This period was chosen because the size of the north coronal hole was relatively stable during this time. The data points are mean values for the discrete density bins for each Carrington map. The density bins are determined by finding the minimum and maximum densities for each map and dividing this range into 10 intervals. The lowest density bin has a fixed range from 0 to 2 × 10⁵ cm⁻³ while the remaining nine bins are equally divided from 2 × 10⁵ cm⁻³ to the upper density limit. The density intervals for each Carrington are slightly different because the upper densities are not identical for each map. Once the density intervals are calculated for each Carrington map, the mean velocities and particle fluxes are then calculated using the data for the same spatial bins in each density interval.

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Figure 10(a) is a scatter plot for the variation of the particle flux with density at the source surface height. Even though there are some outlier points, there is clearly a downward trend in the data, i.e., the particle flux decreases toward the high density streamer belt. Despite the relatively large scatter and a standard deviation of 40% about the mean particle fluxes, there is still a statistically significant difference between the particle fluxes in the coronal holes and in the high density streamer regions. The particle fluxes in the coronal hole at 2.3 R_☉ are at least two times higher than the particle fluxes from the high density streamers at the same height. This is in contrast to the 1 AU particle fluxes measured using other techniques that suggest a higher particle (mass) flux over the poles (e.g., see Quémerais & Lamy 2006; Quémerais et al. 2007). The differences are most likely due to the assumed solar wind acceleration and the expansion factors for the magnetic flux tubes above our source surface, which we do not address here. The topic of the solar wind expansion from the source surface to 1 AU will be addressed in the future.

The spatial bins located in the coronal hole regions have a density which is less than 2 × 10⁵ cm⁻³ and this cut-off is indicated in the figure by a vertical mark labeled "CH." Note again that we have excluded the data from closed field regions in the center of the streamer belt, where the outflow velocities are essentially zero. The remaining high density regions include the edges of streamers where some solar wind outflow is still detected as well as the quiet regions between the polar coronal hole boundary and the streamer belt. Panel (b) of Figure 10 shows the flux tube expansion factors plotted as a function of density for the same time period. The flux tube expansion factors, f_exp, are calculated by using Equation (6). Note how f_exp is relatively constant at low densities but then it increases for density values above 6 × 10⁵ cm⁻³. We believe that this is a real effect since the uncertainty in f_exp is only ∼ ± 1 or less.

The particle fluxes, NV, and expansion factors, f_exp, on the source surface have also been computed for later CRs, i.e., CR 1923 through CR 1936. These data are plotted in Figure 11. There are some similarities as well as differences when these plots are compared to the plots of the earlier CRs in Figure 10. Close examination of data in the two figures shows that both the particle flux and expansion factors are nearly the same for the low density coronal hole regions in both time periods. This indicates that the density and outflow velocity in the coronal holes compensate for each other for the entire data period from CR 1909 to 1936. The same is true for the surrounding quiet regions (i.e., densities between 2 and 4 × 10⁵ cm⁻³), where there is very little difference between the earlier and later CRs. However, for the higher density (e.g., N > 5 × 10⁵ cm⁻³) regions surrounding the streamer belt there is a much greater sensitivity of the expansion factors to changes in the electron density during the earlier time period. Figure 11(a) shows a trend with higher expansion factors at higher electron densities. This differs significantly from the trend in Figure 11(b), which shows a tight dependence of f_exp with coronal density. In other words, during CRs 1923–1936, f_exp is essentially independent of N, with a value for f_exp slightly greater than 1.

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The differences between the expansion factors in Figures 10 and 11 suggest that the large expansion factors at the edges of streamers, as suggested by Wang & Sheeley (1990) and others, may exist only at solar minimum. This is the period with the largest coronal holes and a high density streamer belt confined to a very narrow latitude range. At later times in the solar cycle there is no evidence of such large expansion factors using our data. It should be noted that the Wang & Sheeley (1990) expansion factors are calculated differently since they use extrapolated values of the coronal magnetic field at the source surface. Also, care must be used in comparing numerical values by different authors since some researchers calculate f_exp by evaluating the expansion from some lower height to 1 AU, as opposed to our approach of calculating the expansion from the base of the corona to the source surface.

The expansion factors in our study have a maximum value of about 3–4 times an r² expansion from the coronal base to 2.3 R_☉. Some coronal models have even higher expansion factors near the coronal hole/streamer edges e.g., Cranmer et al. (2007). These higher expansion factors can be accommodated if, for example, we relax the choice for a constant particle flux at the corona base. If N_oV_o is not constant but instead increases from the coronal hole axis to the streamer belt, then the expansion factors near the streamer belt can be larger than the factor of four shown by our data. For example, an increase of the base particle flux near the streamer boundary by a factor of two over the coronal hole value would make the expansion factor f_exp ∼ 8 near the streamer boundary.

Another interesting result of this study is the fact that f_exp is nearly constant for most of the corona with values between 1 and 2. This suggests that there is very little super-radial expansion inside of coronal holes and the surrounding quiet regions (not including the boundaries of the streamer belt). A possible explanation for this is that there may be two different types of slow solar wind, as suggested by Abbo et al. (2010). Deep in the solar minimum period, the slow wind is produced by the super-radial expansion of the magnetic flux tubes anchored in the corona. Later on in the solar cycle when f_exp is nearly the same everywhere, the slow wind is probably a result of the fact that the solar wind source regions are denser and this therefore requires more energy to accelerate the wind to the same speed as in the less dense coronal holes.

A different way to show the relationships between the outflow velocity, density, and particle flux is shown in Figure 12. Once again we show that the outflow velocity structure of the corona at the deepest part of solar minimum (when the coronal holes are the largest) is indeed different from the more radial structure that is present when the corona is in its rising phase. Panel (b) shows that the data from later in the solar minimum period (CRs 1923–1936) can be fit using a simple equation of the form N_ssV_ss = 1.9 × 10¹² cm⁻² s⁻¹. The fit works because the f_exp is approximately constant for all densities during this period. The data from deep in solar minimum (panel (a)) do not follow the same fit (solid curve). A better fit is the dashed curve, which is the equation N_ssV_ss = 1.5 × 10¹² cm⁻² s⁻¹. However, neither fit is particularly good especially at densities above ∼6 × 10⁵ cm⁻³. The outflow velocities corresponding to the spatial bins with these densities are much lower than either fit, which is another way of showing that the expansion factors f_exp become larger with increasing density during the deepest part of solar minimum. Since we know that the highest densities correspond to regions in the streamer belt, the best explanation is that there is a super-radial expansion of the solar wind flow near the boundary of the streamer belt.

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This is the first paper on our efforts to produce complete outflow velocity maps in the corona using over a decade of UVCS and LASCO observations of the corona. We use the data from these observations along with the well-established Doppler dimming analysis of UV spectral lines to make the outflow velocity maps. We present maps for both outflow velocity and density from the period of low solar activity at the start of cycle 23. This period (CRs 1909–1937) was selected because it shows a clear distinction between the fast and slow solar wind source regions.

The outflow velocity and density maps are used to reveal new information about the expansion of the solar wind from the coronal base to our selected source surface height of 2.3 R_☉. We find the following four main results using the data from this work.

• 1.  
  The boundary of the large polar coronal hole that was studied has a well-defined signature using an outflow velocity threshold that agrees well with estimates of the boundary based on a density threshold.
• 2.  
  Our results are consistent with the expected super-radial expansion with height for the area of polar coronal holes during most of the solar minimum period, however, there are periods when the areal expansion factor f_A is ≈1 in the rising phase of the solar cycle.
• 3.  
  The flux tube expansion factors f_exp (this is different from f_A) at the source surface height have values between 1 and 2 in the interior of coronal holes. However, the regions of increased densities near the streamer belt have a maximum value of f_exp ≈ 4 at 2.3 R_☉. The larger expansion factors may explain the decrease in the solar wind speed at the streamer belt, however it is found that these large factors are not present when the overall size of the polar coronal holes start to shrink during the rising phase of the solar cycle. This suggests that the expansion factors may not be the only controlling factor that governs the production of the slow speed wind. A constant particle flux coupled with an increased particle density can produce the same effect of lowering the wind speed.
• 4.  
  Finally, the comparison of our outflow velocity maps with those derived from IPS measurements show that while the IPS maps have the same general structure of the velocity features as our maps, the absolute magnitudes of the IPS outflow velocities are always much larger than the velocities determined by Doppler dimming. This is true in both coronal holes and streamer regions. We conclude from this that there is additional acceleration of both the fast and slow solar wind beyond 2.3 R_☉.

What is new about this work is that the expansion factors are calculated using only the outflow velocity and density measurements made in the extended corona. Our approach is different from studies of coronal expansion factors that use magnetic field measurements at the base or at 1 AU, with models to extrapolate the magnetic field values for the regions in between. In the future it would be useful to compare the results for the expansion factors presented here with results from these models.

The long-term objective for this work is to produce constraints for coronal and solar wind models that are as free as possible from assumptions and thus are tied closely to the observational data. One way to do this is to use observations made in the solar wind source regions so that there is no need to use extrapolations from more distant regions in the heliosphere. Another important aspect of this work is to make the case that the results of theoretical models to be tested should not be included in the determination of the physical parameters that are to be used as the constraints, or vice versa. With the production of the outflow velocity and density maps for this work we have accomplished the first step. The next step is to include these maps in the ongoing detailed comparisons between coronal observations and the latest coronal/solar wind models. We anticipate both our own independent comparisons of our data with models, as well as tests performed by the modeling groups themselves. As a first example, we made a preliminary comparison of our data with a model (in Figure 3) to show that extended heating and momentum deposition from Alfvén waves may be important. A more rigorous quantitative comparison will be reported in a future paper. Also, for future studies we would like to include the entire period of the UVCS and LASCO synoptic data set, which includes the solar maximum and the declining phases of cycle 23.

This work was supported in part by NASA grants NNG06GE74G, NNX07AB98G, and NNX08AQ96G to the Smithsonian Astrophysical Observatory. The LASCO-C2 project at the Laboratoire d'Astrophysique de Marseille (formerly Laboratoire d'Astronomie Spatiale) is funded by the Centre National d'Etudes Spatiales (CNES). UVCS and LASCO are part of SOHO, which is a project of international cooperation between ESA and NASA. The authors acknowledge the use of sunspot data from the World Data Center for the Sunspot Index, Royal Observatory of Belgium; and coronal source surface magnetic field data from the Wilcox Solar Observatory Web site at http://wso.stanford.edu/synsourcel.html. We also thank S. R. Cranmer for supplying the results of his theoretical model for comparing to our data and an anonymous referee for providing useful comments on the manuscript.

Facilities: SOHO(UVCS,LASCO) - Solar Heliospheric Observatory satellite