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Results & Discussion

Figure: Variation of the electron density above the polar coronal hole. The circles represent data from SUMER, diamonds data from LASCO and triangles data from UVCS. The solid line represents Eq. (1) with T_eff = 1.2 10⁶ K and the dashed line represents the polynomial Eq. (2).

Figure: Variation of the half width in km s^-1 at 1/e of the peak intensity in the coronal hole. The circles represent data from SUMER and triangles data from UVCS.

In Fig. 2, we plot the electron density as derived from Si VIII, coupled with N_e from UVCS (Kohl et al. 1998) and LASCO (Lamy et al. 1997). The LASCO data was acquired in February 1996, while the UVCS images were time-averaged from November 1996 to April 1997.

Guhathakurta & Fisher (1995, 1998) estimated the electron density distribution in the coronal hole from observations made with the space based SPARTAN 201-01,03 instrument. At that phase of the descending solar magnetic activity cycle the coronal conditions were different from our observing period (during our observing period the solar activity was in the minimum phase), so we don't compare our results with SPARTAN, but the essential density profile has a striking similarity. The above authors obtained the density profile from polarized brightness (pB) measurements between 1.16 - 6 $R_{\odot}$. They give an analytic prescription of the density profile as,

$${N(r) \over N(R_{\odot})} = {\rm exp}\left[ -{\mu m_{p}g_{\o... ...k_{B}(T_{e}+T_{p})/2} \left(1-{R_{\odot}\over r}\right)\right]$$ (1)

where N is the number density (= $\ N_e \ + \ N_p$), $\mu$ is mean atomic weight (= 0.62), m_p the proton mass, k_B the Boltzmann constant, $g_{\odot}$ the solar gravity and the effective temperature is given by T_eff=(T_e+T_p)/2, where T_e and T_p are the electron and proton temperature, respectively. The line fit as a function of $(1-{R_{\odot}\over r})$ for all densities out to 8 $R_{\odot}$ is shown as the solid line in Fig. 2 for T_eff = 1.2 10⁶ K. Thus for a locally isothermal fully ionized two-fluid coronal plasma solution we can either estimate T_e or T_p. Using T_e of Si VIII as 8 10⁵ K (i.e. its formation temperature in ionization equilibrium), we find an estimate of T_p as 1.6 10⁶ K. Furthermore this formula predicts an electron density of $4.5 \ 10^3 \$cm^-3 at 8 $R_{\odot}$ in excellent agreement with observations.

Our results for the ion temperature are in agreement with observations obtained by UVCS (Kohl et al. 1997; Cranmer et al. 1999) who find that at larger distances from the limb the ions in a coronal hole are extremely `hot' and the electrons are much `cooler'. Thus our observations re-confirms that in the coronal hole plasma, particularly at larger distances from the limb the assumption of collisional ionization equilibrium can not be used any more. This is also consistent with a study of several ions by Tu et al. (1998) who found ion temperatures perhaps as high as 2-3 times the formation temperature in ionization equilibrium even at heights of only a few tens of arcsecs. Recently, Esser et al. (1999) have combined data from SPARTAN, White light coronagraph (WLC) from 1.5 - 5.5 $R_{\odot}$ in April '93, with Mauna Loa observations between 1.16 - 1.5 $R_{\odot}$ (Fisher & Guhathakurta, 1995). We are unable to fit the data in Fig. 2 with their polynomial as our data-set is more complete, extending from 1.02 - 8 $R_{\odot}$. In-fact, the expression given by Esser et al. predicts a rather high electron density at the limb and a density three orders of magnitude smaller at 1AU. Instead, the data in Fig. 2 suggests a fall-off in density proportional to r^-8 from 1 to 2 $R_{\odot}$, then r^-4 from 2 to 4 $R_{\odot}$ and finally as r^-2. A polynomial fit of the form

$$N_{e} = {1\times10^{8}\over r^{8}} + {2.5\times10^{3}\over r^{4}} + {2.9\times10^{5} \over r^{2}}$$ (2)

provides an excellent fit to the data as shown by the dotted line in Fig. 2. The first coefficient in the R.H.S of Eq. (2) is determined by the electron density close to the limb (i.e. from our Si VIII SUMER data), the last coefficient determined by the electron density at 1 AU, while only the middle coefficient was varied in order to provide the best fit.

In Fig. 3, we plot the half width at 1/e of the peak intensity (V_1/e) in km s^-1 as measured from Si VIII SUMER data plus O VI UVCS data (Kohl et al. 1998). The effective (ion kinetic) temperature (T_k) can be obtained from (Kohl et al. 1998),

$${V_{1/e}} = {\frac{c \Delta \lambda_{1/e}}{\lambda_{0}}} = \sqrt{\frac{2k_{B}T_{k}}{M}}$$ (3)

where M is the ion mass and $\Delta \lambda_{1/e}$ is the observed 1/e half width. We have applied a mass correction factor of 1.32 to the Si VIII data to make it consistent with O VI. Note that the kinetic temperature include contributions both from microscopic thermal motions and unresolved transverse wave motions. As shown in Fig. 3, this data suggests a small increase from 1 to 1.2 $R_{\odot}$, then a plateau up to 1.5 $R_{\odot}$, followed by a sharp increase up to 2 $R_{\odot}$, then a more gradual increase further out.

Now we turn our attention to the question of the non-thermal velocity. The analysis of DBP and BTDW showed that the non-thermal line-of-sight velocity as derived from the Si VIII line widths increases above the limb while the electron density decreases. On a closer inspection, the observations revealed that the non-thermal velocities were inversely proportional to the quadratic root of the electron density, in excellent agreement with that predicted for undamped radially propagating Alfvén waves. In the WKB approximation the rms wave velocity amplitude and density are related by (Hollweg 1990),

$$<\delta v^2>^{1/2} \ \propto \ \rho^{-1/4}$$ (4)

The non-thermal velocity can be deduced from the standard equation for an optically thin line broadened by thermal broadening caused by the ion temperature T_i and broadening caused by non-thermal motions as given by,

$${FWHM}= \left[ 4ln 2 \left(\frac{\lambda}{c}\right)^2 \left({\frac{2 {k_{B}} T_i}{M}} + \xi^2\right) \right]^{1/2}$$ (5)

where M is the ion mass, $\xi$ is the non-thermal speed, related to the wave amplitude by $\xi^2 = {\frac{1}{2}}<\delta v^2>$, where the factor of 2 accounts for the polarization and direction of propagation of a wave relative to the line of sight.

Measuring the FWHM of the Si VIII 1445 line and using Eq. (5) with T_i $\sim 1.6~10^6$ K (as suggested from the line fitting to N_e), the diamond symbols in Fig. 4 represents the observed non-thermal velocities at different heights. Now If we assume the same ion temperature as Si VIII for the 721.23Å line (as suggested by the intensity fall-off compared to Si VIII), the Si VIII non-thermal widths implies an ion mass of $\sim$53 for this new feature. This latter assumption is probably valid as different emission lines are generated in the same column mass as seen by the spectrograph. Although, this can not be taken as absolute proof it does suggest that the line may be due to Fe, probably ionization stages VII-VIII.

In Fig. 4, the (+) represents the velocities calculated on the basis of the Eq. (4), using the measured N_e from Si VIII. We have used a magnetic field strength of B= 6G and a proportionality constant so as to match the energy flux density (see BTDW for details). In both cases, we find excellent agreement in the inner corona, but for the outer corona $\gg$ 200 arc sec (see Fig. 4), the (+) symbols starts to deviate from those calculated from the observed FWHM(diamond symbols in Fig. 4). It is also possible to estimate the errors in the calculated non-thermal velocity (+); for the last 4 points, errors of 3, 3, 5 and 15 km s^-1 have been estimated. Despite these large errors we have no overlap with the error-bars of the observed non-thermal velocities, suggesting an effective break-down of Eq. (4) above 1.2 $R_{\odot}$.

Lou & Rosner (1994) have already pointed out that waves can be reflected against gradients in the Alfvén speed and the WKB approximation fails. The reflection is important as it increases the momentum transfered from the Alfvén wave to the medium. Recently Torkelsson et al. (1998) have shown that the Alfvén waves steepen and produce current sheets in the non-linear regime. These waves are strongly damped by non-linear steepening. They have also reported density oscillations in their simulations, which are consistent with observations by Ofman et al. (1997) who reported quasi periodic variations in the polarized brightness in the polar coronal holes between 1.9-2.45 $R_{\odot}$, as observed from the white light channel (WLC) of UVCS. It would be interesting to see whether these density oscillations are also present in the inner corona which can be due to the presence of non-linear high amplitude compressional waves as reported by Ofman et al. (1997). For that one needs a high cadence dataset. We propose that somewhere around 1.2 - 1.3 $R_{\odot}$ this non-linearity becomes important (as indicated by Figs. 3 & 4). The Alfvén waves with an amplitude of 30-50 km s^-1 (as observed) at the base of the coronal hole can generate non-linear solitary type of waves, which can contribute significantly to solar wind acceleration in open magnetic field structures.

Figure: Variation of the non-thermal velocity with height in the north polar coronal hole. The diamonds represent those measured with Si VIII, the plus symbols correspond to theoretical estimates based on Eq. (4) and the measured N_e (see text) and squared boxes represent those derived from the 721.23Å line assuming it's due to an Fe ion.

Note from Fig. 3 that relatively sharp variations occur around 1.5 $R_{\odot}$. This may be the location where the thermalization and the isotropization times of various species begins to exceed the local coronal expansion time. Esser et al. (1999) from a study of Mg X and O VI lines (observed with SOHO/UVCS) found a transition from collisional to collisionless plasma between 1.75 to 2.1 $R_{\odot}$ in a polar coronal hole. Cranmer et al. (1999) have presented an empirical model of H I and O VI distributions, which also indicates the presence of this transition. In their model they found a sharp variation between 1.8 $R_{\odot}$ - 2.1 $R_{\odot}$ (see their Figs. 5&9). It is also interesting to note that the electron density variation at 2 $R_{\odot}$ has changed to r^-4 from its earlier r^-8 fall-off. For line width measurements, UVCS data are not available below 1.5 $R_{\odot}$, so we feel that the combined SUMER and UVCS datasets allows us to locate this transition point with better precision suggesting that the physics of the plasma transport and wave dissipation diverges from classical Coulomb theory at heights beyond 1.5 $R_{\odot}$. At larger distances, e.g. above 2 $R_{\odot}$, the large V_1/e can also be due in part to the ion-cyclotron resonant acceleration by high frequency MHD waves (McKenzie et al. 1995). We hope that our results will provide more precise input parameters at the base of the coronal hole for future solar wind models.