DETERMINATION OF CORONAL MASS EJECTION PHYSICAL PARAMETERS FROM A COMBINATION OF POLARIZED VISIBLE LIGHT AND UV Lyα OBSERVATIONS

In the event of 2000 November 8 (hereafter Event 1), which is classified as a narrow (angular width of $\sim 59^\circ$), slow CME, the CME leading edge entered in the LASCO-C2 FOV at 19:27 UT with an estimated plane-of-the-sky (POS) speed of ∼160 km s⁻¹. The CME was associated with a prominence eruption and propagated in the southeast quadrant at a latitude of $\sim 30^\circ {\rm{S}}$ showing the classical, quite symmetrical three-part structure (see Figure 1, top row). The CME launch time, derived by extrapolating the LASCO-C2 measurements to the projected height of 1 ${R}_{\odot }$, was approximately at 17:19 UT, in agreement with MLSO/Mark-iv image sequences (Figure 2). LASCO-C2 acquired a polarimetric sequence for the measurement of the polarized visible light between 20:59 and 21:03 UT. SOHO/EIT 304 and 195 Å images show that the CME probably originated from active region (AR) NOAA 09227 (located at 10°S 55°E): prominence material laying above the source AR appears to be ejected between 13:19 UT and 19:19 UT, as showed by the only two EIT 304 Å images available before and after the estimated CME launch time. The eruption is barely visible in corresponding EIT 195 Å images, where a weak evacuation of plasma followed by a rearrangement of the magnetic configuration above the source AR can be noticed. Although an M2.9 class flare was detected by the GOES satellite at 16.30 UT—i.e., very close to the CME start time—this flare occurred at the NE limb, as shown by TRACE images.

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The event of 2000 December 25 (hereafter Event 2) consists of a small-scale eruption of plasma along a coronal streamer triggered by an adjacent CME (see Bemporad et al. 2010, for similar CME-induced eruptions). The coronal configuration above the NE limb before the event was characterized by the presence of a faint, quite stable streamer located at a latitude of $\sim 60^\circ {\rm{N}}$ as shown by the LASCO-C2 white-light images. The first sign of ejection appeared in the LASCO-C2 FOV at 18:54 UT as a brightening at the visible base of the streamer; the emerging plasma "blob" later on traveled along the streamer with an estimated velocity of ∼250 km s⁻¹ showing a complex configuration with evidence of a small-scale filamentary structure (see Figure 1, middle row). The ejection start time, as extrapolated from the LASCO height-time diagrams, was at 17:34 UT. The polarimetric sequence for the measurement of the pB was acquired between 20:52 and 21:00 UT. At 21:04 UT, when the leading edge of the ejected material was above ∼4.5 ${R}_{\odot }$, the hemispherical front of a faint CME entered the LASCO-C2 FOV propagating slightly northward to the streamer, at a latitude of $\sim 52^\circ {\rm{N}}$, with approximately the same speed (as reported in the LASCO CME catalog). As the CME progressed, no clear signature of a bright core inside was detected in white light. EIT 195 and 304 Å images show little evidence of an eruption that occurred above the NE limb starting around 17:00 UT, quite consistent with the successive appearance of ejected plasma as seen by LASCO. The eruption is not associated with flares or EUV brightenings on the disk, hence the source region was most likely located behind the visible disk.

In the event of 2003 May 2–3 (hereafter Event 3), the first appearance of the CME leading edge in the LASCO-C2 FOV was at 23:50 UT on May 2. The initial velocity, estimated from a second-order polynomial fit, was ∼100 km s⁻¹. The CME propagated inside a large and bright mid-latitude coronal streamer at the latitude of $\sim 45^\circ {\rm{N}}$ and exhibited a three-part structure with a faint hemispherical front and clear evidence of an inner core (see Figure 1, bottom row). At 02:58 UT LASCO-C2 started to acquire the sequence of three polarimetric images for the measurement of the pB, until 03:03 UT. At that time, the CME front velocity was around ∼220 km s⁻¹. EIT 304/195 Å images did not show remarkable phenomena occurring below the coronal region where the CME propagated, and the ejection was not associated with a disk event; therefore, it most likely originated on the back side of the Sun, probably from ARs 10354/10355, which became visible at the NE limb in EIT 195 Å images on May 4.

During the three events considered here, the UVCS FOV (a 42 arcmin long slit) performed a radial scan of the coronal region crossed by the CME. Details about the observations are listed in Table 1. Aside from the date and time, we report the radial scan altitude range, the polar angle (PA, measured counter-clockwise from north pole) where the slit was centered, the projected altitude of the slit (ρ) at the time of the pB measurement made by LASCO, the exposure time for each spectrum, the projected slit width, and the spatial bin size. In all cases, the detected spectral range includes the H i Lyα λ1215.6 Å line (acquired using the redundant path of the O vi channel with spectral binning of 2 pixels $\simeq 0.1830$ Å), the O vi $\lambda \lambda$ 1031.9/1037.6 Å doublet lines (acquired with spectral binning of 1 pixel $\simeq \,0.0991$ Å for Events 1 and 2 pixels $\simeq \,0.1983$ Å for Events 2 and 3), and other minor lines (not considered in this work). Standard calibration and data reduction was made with the latest version available of the UVCS Data Analysis Software (DAS 51).

Table 1.  Summary of UVCS Observations

Event	Date	Timea	Altitude Range	PA	ρb	${t}_{\exp }$ c	Projected Slit Width	Spatial Bin Size
			(${R}_{\odot }$)	(degree)	(${R}_{\odot }$)	(s)	(arcsec)	(arcsec)
1	2000 Nov 08	18:02 UT	1.5–4.0	113	2.5	300	28	42
2	2000 Dec 25	20:58 UT	1.6–3.0d	45	3.0	120	42	21
3	2003 May 02	02:48 UT	1.6–3.5d	45	3.5	120	42	21

Notes.

^aInitial time of UVCS observations. ^bHeliocentric distance of the UVCS slit at the time of LASCO-C2 pB measurement. ^cExposure time. ^dFor this event, the radial scan was performed from higher to lower heights.

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In what follows, our analysis focuses on the H i Lyα intensities and profiles. For each event, we averaged the two exposures acquired around the time of the LASCO pB measurement, in order to increase the signal-to-noise ratio; for Events 2 and 3, we also averaged the data over two spatial bins ($=42$ arcsec). We then fitted a single Gaussian to the observed Lyα spectrum at each position along the slit, getting line intensities and widths. Usually spectra in the optically thin corona must be corrected for the ambient-corona contribution before spectral integration, in order to isolate the CME emission with respect to the emission of the surrounding corona aligned with the CME along the same line of sight. However, in all three cases, observations by UVCS started when the CME was already crossing the instrument FOV, making it impossible to get a reliable pre-event Lyα spectrum to be subtracted from the CME exposures to remove the quiet-corona contribution. To account for that, after integration of the Lyα spectra, we modeled the quiet-corona intensity at each position along the slit using a power-law extrapolation (with a ${r}^{-\alpha }$ dependence on the heliocentric distance), constraining the fit to the intensity measured in the slit regions not crossed by the CME. The intensity from the fit extrapolated at the spatial region crossed by the CME was thus subtracted from the measured one.

The line widths ${\rm{\Delta }}{\lambda }_{1/e}$ obtained from the Gaussian fit to the Lyα line profiles were corrected for the instrumental profile broadening and converted into effective temperatures with the usual relationship

$$\begin{eqnarray}&&{\rm{\Delta }}{\lambda }_{1/e}=\displaystyle \frac{{\lambda }_{0}}{c}\sqrt{\displaystyle \frac{2{k}_{{\rm{B}}}{T}_{\mathrm{eff}}}{{m}_{{\rm{H}}}}},\end{eqnarray} \tag{ 1 }$$

where ${\lambda }_{0}$ is the spectral line central wavelength at rest, c is the light speed, k_B is the Boltzmann constant, m_H is the hydrogen mass, and ${T}_{\mathrm{eff}}$ is the so-called effective temperature. Note that the observed profiles are likely broadened by non-thermal motions as well as by the line-of-sight component of the plasma bulk velocity; moreover, since we were not able to subtract a pre-event spectrum, they include a component relevant to the quiet corona as well. For these reasons, the derived effective temperatures are only an upper-limit estimate of the real hydrogen kinetic temperature.

The additional analysis of the O vi $\lambda \lambda 1032/1037$ Å doublet lines detected by UVCS could, in principle, provide further information on the CME plasma outflow velocity. In fact, the O vi intensity ratio, $R={I}_{1032}/{I}_{1037}$, is sensitive to the radial component of the plasma outward speed through the Doppler-dimming effect (see, e.g., Noci et al. 1987); the outflow velocity can be thus inferred from the comparison of the measured ratios with those predicted by models. In general, values of $R\gtrsim 2$ imply outflow velocities around or lower than 100 km s⁻¹, while at higher speeds, above ∼150 km s⁻¹, pumping of the O vi $\lambda 1037$ Å by the C ii $\lambda \lambda 1036/1037$ Å lines causes R to be lower than two (see, e.g., Raymond & Ciaravella 2004; Ciaravella et al. 2005). Moreover, the dependence of R on the outflow velocity is a function of the electron density and temperature (see Dobrzycka et al. 2003).

Nevertheless, two reasons led us to neglect results based on the analysis of the O vi line ratios. First, the uncertainties on the O vi line intensities are larger than those affecting the Lyα line intensity, due to lower count statistics; moreover, since we were not able to correctly evaluate the coronal background intensity, we have larger uncertainties in the measurement of R (of the order of ∼30% for Event 1 and ∼50% for Events 2 and 3). The consequence is that in all the three cases, the intensity ratio does not differ significantly from 2 in most of the UVCS FOV (according to Raymond & Ciaravella 2004, we considered as statistically significant only departures of the ratio from the value of 2 larger than $2\sigma$). Second, as we will discuss in the next sections, the analysis of the white-light emission measured by LASCO-C2 provided electron densities that are of the order or larger than 10⁶ cm⁻³ in all cases. As clearly demonstrated by Dobrzycka et al. (2003) (for the 1999 April 15 CME at 1.98 ${R}_{\odot }$), the O vi intensity ratio at typical coronal temperatures and for electron densities higher than 10⁶ cm⁻³ is $R\simeq 2$ in a wide range of velocities, so that it is impossible to identify a unique value for the CME outflow speed owing to those conditions. Therefore, we could not place reliable constraints on this parameter using these results and we preferred to estimate the outflow plasma velocity using the analysis of the CME dynamics in LASCO-C2 VL images (see below).

The coronal visible light (K-corona) is produced by the Thomson scattering of the photospheric incident radiation off coronal free electrons; VL images can provide the electron column density ${N}_{e}={\int }_{\mathrm{los}}{n}_{e}\,{dz}$ using the method described, for instance, by Vourlidas et al. (2000): the number of scattering electrons along the LOS (i.e., the column density N_e, in units of cm⁻²) is given by the ratio of the observed total brightness B over the brightness ${B}_{\exp }(z)$ expected for a single electron located at distance z from the plane of the sky. Usually this computation is done pixel-by-pixel in the CME exposure under the assumption that the scattering electrons are located on the plane of the sky (z = 0). In this way, a lower limit estimate of the column density N_e is obtained. However, in their quantitative comparison between the real 3D structure of a flux-rope CME obtained from MHD simulations and that derived via the polarization-ratio technique, Pagano et al. (2015) recently demonstrated that if ${B}_{\exp }$ is calculated by assuming $z=\langle {z}_{\mathrm{cme}}\rangle$ with $\langle {z}_{\mathrm{cme}}\rangle$ center of mass of the LOS plasma distribution, the resulting value of N_e is much closer to the real one, with uncertainties that are generally lower.

An estimate of the location of the emitting plasma along the LOS, $\langle {z}_{\mathrm{cme}}\rangle$, can be obtained with the so-called polarization-ratio technique, first proposed by Moran & Davila (2004) to infer from a single-viewpoint pB image of a CME the three-dimensional structure of the ejected plasma. The degree of polarization introduced by the Thomson scattering is a function of the scattering angle between the incident photon direction and the direction toward the observer (see Billings 1966), and, in turn, of the distance z of the scattering electron from the POS. As shown by Moran & Davila (2004), the dependence of the ratio $p={pB}/B=p(z)$ for a single electron on the distance from the plane of the sky can be exploited to determine, pixel by pixel in the CME image, the mean position $\langle {z}_{\mathrm{cme}}\rangle$ of the CME plasma. The more the region where the electron density is locally increased by the CME is spatially limited along the LOS, the more accurate this determination is (see Bemporad & Pagano 2015). On the other hand, owing to the forward/backward symmetry of the Thomson-scattering process, it is impossible to establish with this method if the emitting plasma is actually located in front of or behind the plane of the sky, because $\langle {z}_{\mathrm{cme}}\rangle$ can be derived aside from its sign. Hence, additional considerations are needed to discriminate between these two possible cases.

A key point in applying the polarization-ratio technique is the coronal background removal, which is necessary to exclude from the observed emission contributions coming from both the surrounding ambient corona and the dust-scattered (F-corona) component that affects the unpolarized part of the total brightness. These contributions, that sum together with the genuine CME emission because of the optical thinness of the solar corona, would result in an erroneous determination of the CME 3D structure.

The background removal is usually performed by subtracting a convenient pre-CME image from the frame that is analyzed; for the polarized visible light, this operation can be achieved in several ways: for instance by subtracting the pre-event Stokes parameters from the Stokes images of the CME, or by subtractig pre-event B and pB images directly from the corresponding B and pB images of the CME (see, e.g., Dere et al. 2005), or even by subtracting pre-event frames separately from all the polarized images before they are combined to obtain the pB (see Moran & Davila 2004). On the other hand, Mierla et al. (2009) suggested a different approach that consists in subtracting from the exposure containing the CME a minimum-intensity image instead of a single pre-event frame. The minimum-intensity image is created by taking, pixel by pixel, the minimum brightness value over a sequence of exposures acquired before and during the event.

In our analysis, we followed the methods described in Moran & Davila (2004) and Mierla et al. (2009) within certain limitations. As stated above, LASCO-C2 usually acquires one or two sequences of three polarimetric images per day. With this quite poor temporal cadence, the minimum-intensity background image cannot be created with more than one pre-CME exposure, because the configuration of the ambient corona two days before each event can be different from that existing during the CME. For this reason, after standard calibration of the VL images (which includes correction for vignetting and offset bias, and the radiometric calibration; see Llebaria & Lamy 2008, for further details about the LASCO-C2 calibration), we computed a minimum-intensity image (for each orientation of the C2 polarizers: 0°, $+60^\circ$, and $-60^\circ$) using the exposures acquired soon before, during, and just after each event. In this way, the removal of steady structures not affected by the transit of the CME, such as streamers, turns out to be better than considering only the pre-event and the event images. In the specific case of Event 3, we also took into account the different calibration of the CME images that were acquired with the "blue" filter of the LASCO-C2 instrument, unlike all the other images considered in this work, which were taken with the "orange" one.

Another issue to be accounted for is the effect of the CME motion between the polarimetric exposures, which can be a source of additional uncertainties in the determination of the polarized brightness because of the partial misalignment between the same CME structures in different images. This is a minor concern in our case, however, because the events analyzed here were slow or very slow CMEs: we estimated that in the worst case (i.e., Event 2, where the maximum projected velocity was $\approx 250$ km s⁻¹) the CME front moved less than 2 pixels between consecutive exposures. In order to compensate for this displacement, and following the same procedure adopted by Moran & Davila (2004), we smoothed each polarized exposure using a 3 pixel by 3 pixel median filter. In this way, information at spatial scales equal or lower than 3 pixels ($=71.4$ arcsec) is lost, but this also significantly reduces the errors associated with the CME motion.

The polarimetric images were then used to compute the Stokes parameters (I, Q, and U) that were corrected for the small instrumental elliptical polarization produced by the two folding mirrors in the beam path, which was removed with the use of the Müller matrix formalism following the method described in Moran et al. (2006). Finally, the B and pB images of the CME were obtained (see Figure 3, left column) with the standard relationships $B\equiv I$ and ${pB}\equiv \sqrt{{Q}^{2}+{U}^{2}}$ (see Landi Degl'Innocenti & Landolfi 2004). In this way, the radiometric calibration of the polarized exposures is not strictly necessary when applying the polarization-ratio technique, because the ratio pB/B is used to determine the position of the emitting plasma along the LOS; conversely, it is essential for the derivation of the electron density (see below). As explained by Frazin et al. (2012), the uncertainty in the standard radiometric calibration of C2 can be considered reasonably within ±15% of the measured pB and B values, so we used this value for the estimate of the errors associated with all the other derived quantities.

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For each pixel in the background-subtracted CME pB image, the mean distance from the POS of the plasma responsible for the emission in that pixel was estimated by comparing the observed ratio ${p}_{\mathrm{obs}}$ with the theoretical ratio ${p}_{\exp }$ expected for a single electron, according to the Thomson-scattering theory. We used the SolarSoft routine eltheory.pro to compute the values of ${p}_{\exp }$ for a range of distances z from the POS and found the value $\langle {z}_{\mathrm{cme}}\rangle$ for which ${p}_{\mathrm{obs}}={p}_{\exp }$. Note that, as shown by Bemporad & Pagano (2015), the resulting $\langle {z}_{\mathrm{cme}}\rangle$ actually represents the location along the LOS of the center of mass of a "folded" density distribution, given by reflecting and summing in front of the POS the fraction of the real density distribution located behind the POS. This leads to over/underestimates of the real distance from the POS of the emitting plasma and the derived $\langle {z}_{\mathrm{cme}}\rangle$ can be considered more accurate only when the emitting plasma is at an angle of $\sim 20^\circ$ from the plane of the sky.

The topographical z-maps derived for the three events are shown in Figure 3 (center column) together with the background-subtracted pB maps (filtered using the NRGF filter to highlight the latitudinal structures; left column). Note that the calculation of $\langle {z}_{\mathrm{cme}}\rangle$ was restricted only to pixels with values of pB and B above 10% of the mean, in order to prevent artifacts in the 3D reconstruction.

The map relevant to Event 1 shows that the CME front and core plasma is concentrated at an average distance of ∼1.6 ${R}_{\odot }$ from the POS (i.e., at an average angle of $\sim 30^\circ$), while in the regions surrounding the front and in the cavity (the low-brightness region inside the CME) the emission appears to be associated with plasma located at greater distances from the POS. This result can be interpreted in light of what is found by Bemporad & Pagano (2015): the density distribution of the plasma in the void of the CME and in the surrounding quiet regions is most probably extended along the LOS more than that in the front and in the core, which are more spatially confined; therefore, in the former regions, the tail of the density distribution behind the POS is reflected back in front of the POS when the polarization-ratio technique is applied (due to the sign ambiguity of the method, as explained above) and this leads in turn to an overestimate of $\langle {z}_{\mathrm{cme}}\rangle$.

The z-map of Event 2 shows that around 21:00 UT the leading edge of the ejected plasma, located at a heliocentric distance of ∼4.5 ${R}_{\odot }$, had correspondingly reached the maximum average distance from the POS of ∼3.5 ${R}_{\odot }$, consistent with a blob of plasma travelling in a direction forming an angle of $\sim 40^\circ$ from the POS. At lower altitudes, other small-scale structures located approximately along the same direction with respect to the plane of the sky can be recognized in the map. Note that, as in the previous case, the brightest features identifiable in the pB map are located systematically closer to the POS in the z-map.

Concerning Event 3, the CME plasma appears to be located at an average distance of about 1.5 ${R}_{\odot }$ from the POS in the southern leg of the front, while at higher distances, $\gt 3$ ${R}_{\odot }$, in the northern one. Note that the southern part of the CME front was mostly superimposed to the pre-event coronal streamer, as evidenced by LASCO total-brightness images, so the difference in the position along the LOS between the two legs could be due to the contribution of the spurious white-light emission coming from the streamer.

Finally, the resulting z-maps of LOS plasma distribution were used to derive the corrected column density maps (see Pagano et al. 2015), shown in Figure 3 (right column). The column density can be then converted to volume density (units of cm⁻³) if the thickness L of the CME plasma along the LOS is known; in fact, it is

$$\begin{eqnarray}&&{N}_{e}\equiv {\int }_{\mathrm{los}}{n}_{e}\,{dz}\approx \langle {n}_{e}\rangle \cdot L,\end{eqnarray} \tag{ 2 }$$

where $\langle {n}_{e}\rangle$ is the average electron density of the CME plasma (see also Quémerais & Lamy 2002, for a discussion on the derivation of electron density maps from the inversion of VL images.). Unfortunately, the parameter L cannot be directly estimated from VL images, nor derived with the polarization-ratio technique, but it must be reasonably assumed. For each of our events, we derive the average electron density assuming for L the two values of 0.25 and 0.5 ${R}_{\odot }$. We report in Figure 4 the density profiles interpolated along the UVCS slit, together with the corresponding Lyα intensity profiles derived with UVCS.

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The electron densities obtained in the three cases are quite consistent with those usually measured in the core of CMEs with other techniques ($\sim {10}^{6}\mbox{--}{10}^{7}$ cm⁻³; see, e.g., Ciaravella et al. 2000, 2001; Dobrzycka et al. 2003; Bemporad et al. 2007). Event 2, which turns out to be the faintest in white light, accordingly exhibits the lowest densities (up to $\sim 8\times {10}^{6}$ cm⁻³). Event 3 shows a noticeable latitudinal correlation between the distribution of the electron density and that of the Lyα intensity. Conversely, for Events 1 and 2, the locations where the density has local maxima appear to be less correlated with the Lyα intensity peaks, in particular, those located around 29°N for Event 2 and around 20°S for Event 1.

We used the electron densities derived from the analysis of LASCO VL images to evaluate the Lyα line intensities expected along the UVCS FOV in the three cases; comparison with the intensities actually measured by UVCS have been used to constrain possible ranges of electron temperatures of the CME plasma under certain assumptions, as we will discuss in the following.

The mechanism of formation of many strong coronal lines in the UV (such as the H i Lyα or the O vi doublet lines) is a combination of radiative and collisional excitations, followed by spontaneous emission. The total intensity of such lines is then the sum of a radiative and a collisional component, $I={I}_{\mathrm{rad}}+{I}_{\mathrm{col}}$. The radiative component is produced by resonant scattering of the chromospheric radiation by coronal ions and can be affected by the outflows of the scattering atoms through the Doppler-dimming effect. We refer the reader to Kohl & Withbroe (1982) and Noci et al. (1987) for a complete treatment of the resonant scattering and Doppler-dimming processes in coronal plasmas.

An approximate expression for the radiative component, holding for coronal structures that are spatially limited along the LOS (e.g., coronal streamers or CMEs; see also Kohl et al. 2006), is

$$\begin{eqnarray}&&{I}_{\mathrm{rad}}\simeq \displaystyle \frac{h}{16\pi }{B}_{12}\cdot b\cdot {\lambda }_{0}\cdot {\rm{\Omega }}\cdot {F}_{D}({v}_{\mathrm{out}}){\int }_{\mathrm{los}}{n}_{i}\,{dz},\end{eqnarray} \tag{ 3 }$$

where h is the Planck constant, B₁₂ is the Einstein coefficient for absorption, b is the branching ratio for radiative de-excitation, ${\lambda }_{0}$ is the wavelength of the line transition, Ω is the solid angle subtended by the solar disk at the scattering location, n_i is the ion number density, and

$$\begin{eqnarray}&&{F}_{D}({v}_{\mathrm{out}})={\int }_{0}^{\infty }{I}_{\odot }(\lambda -\delta \lambda ){\rm{\Phi }}(\lambda -{\lambda }_{0})d\lambda \end{eqnarray} \tag{ 4 }$$

is the so-called Doppler-dimming factor, which is a function of the intensity line profile of the chromospheric radiation, ${I}_{\odot }(\lambda -\delta \lambda )$—shifted by the quantity $\delta \lambda =({v}_{\mathrm{out}}/c)\cdot {\lambda }_{0}$, i.e., the Doppler shift introduced by the radial component of the plasma outflow velocity ${v}_{\mathrm{out}}$—and the normalized coronal absorption profile along the direction of the incident radiation, ${\rm{\Phi }}(\lambda -{\lambda }_{0})$.

The collisional component is due to the de-excitation of a coronal ion previously excited by collision with a free electron. Approximately, it is given by

$$\begin{eqnarray}&&{I}_{\mathrm{col}}\simeq \displaystyle \frac{1}{4\pi }b\cdot {q}_{\mathrm{col}}({T}_{e}){\int }_{\mathrm{los}}{n}_{e}\cdot {n}_{i}\,{dz},\end{eqnarray} \tag{ 5 }$$

where the term

$$\begin{eqnarray}&&{q}_{\mathrm{col}}({T}_{e})=2.73\times {10}^{-15}\cdot \bar{g}\cdot {f}_{12}\cdot \displaystyle \frac{1}{{E}_{12}}\cdot \displaystyle \frac{1}{\sqrt{{T}_{e}}}{e}^{-\tfrac{{E}_{12}}{{k}_{{\rm{B}}}{T}_{e}}}\end{eqnarray} \tag{ 6 }$$

is the collisional excitation rate, which depends on the average Gaunt factor, $\bar{g}$, the oscillator strength of the transition, f₁₂, the transition energy, ${E}_{12}={hc}/{\lambda }_{0}$, and the electron temperature, T_e.

In both Equations (3) and (5), the numerical density of the emitting ions can be approximated using the relationship (see, e.g., Withbroe et al. 1982)

$$\begin{eqnarray}&&{n}_{i}\approx 0.83\cdot A\cdot R({T}_{e})\cdot {n}_{e},\end{eqnarray} \tag{ 7 }$$

where the factor 0.83 is the ratio of proton to electron density in the case of fully ionized plasma with 90% of H and 10% of He (as in typical coronal plasma conditions), A is the element abundance relative to hydrogen (A = 1 for ${{\rm{H}}}^{0}$ ions), and $R({T}_{e})$ is the element ionization fraction, which is a function of the electron temperature. Therefore, if L is the thickness along the LOS of the emitting plasma, the above equations simplify to

$$\begin{eqnarray}&&{I}_{\mathrm{rad}}\propto R({T}_{e})\cdot {F}_{D}({v}_{\mathrm{out}})\cdot \langle {n}_{e}\rangle \cdot L\ \mathrm{and}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{I}_{\mathrm{col}}\propto R({T}_{e})\cdot {q}_{\mathrm{col}}({T}_{e})\cdot {\langle {n}_{e}\rangle }^{2}\cdot L.\end{eqnarray} \tag{ 9 }$$

Equations (8) and (9) can be applied to the case of the H i Lyα emission and used to constrain the electron temperature by the excitation rates needed to account for the observed line intensity.

As an example, we report in Figure 5 the prediction of the radiative and collisional components of the H i Lyα intensity as functions of the electron temperature T_e, for two representative values of the the plasma outflow velocity (100 and 400 km s⁻¹), three values of the electron density (10⁶, 10⁷, and 10⁸ cm⁻³), and two values of the thickness along the LOS (L = 0.25 ${R}_{\odot }$ and L = 0.5 ${R}_{\odot }$). These estimates were made using standard values for all the atomic coefficients and the other quantities entering Equations (8) and (9), and by adopting the ionization equilibrium provided by the CHIANTI atomic database (version 7) for hydrogen. The idea of our analysis is to derive the expected total H i Lyα intensity ${I}_{\exp }({T}_{e})$, as a function of T_e, given by ${I}_{\exp }({T}_{e})={I}_{\mathrm{rad}}({T}_{e})+{I}_{\mathrm{col}}({T}_{e})$ and then to determine T_e from a comparison between ${I}_{\exp }({T}_{e})$ and the real observed intensity.

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Nevertheless, the relative importance of radiative and collisional contributions is also related to the plasma outflow speed. For the Lyα transition, the radiative component is usually dominant in coronal plasmas; however, at low velocities, due to the Doppler-dimming effect, it becomes comparable to the collisional one at coronal temperatures (${10}^{5}\mbox{--}{10}^{6}$ K) when the electron density is higher than 10⁸ cm⁻³. At higher velocities, the radiative component is significantly dimmed, and it can be even lower than the collisional one when the electron density exceeds 10⁷ cm⁻³. It is worth noting that the variation of the LOS thickness L of the emitting plasma has a much lower effect on the Lyα intensity than the variation of the outflow velocity and/or the electron density.

Therefore, the determination of CME plasma temperatures requires the constraint of the outflow plasma velocity as well, in order to estimate the Doppler-dimming factor F_D. A first estimate is simply provided by the POS component of the CME speed, ${v}_{\mathrm{pos}}$, measured with LASCO. We used the two consecutive total-brightness exposures acquired before and after the time of the pB measurement, to determine the radial component of the CME front projected on the POS, by locating homologous points along the front at the two different times and computing ${v}_{\mathrm{pos}}={\rm{\Delta }}h/{\rm{\Delta }}t$. This allowed us to derive the profile of the radial velocity in different points located along the UCVS slit FOV (i.e., at different latitudes). Nevertheless, the ${v}_{\mathrm{pos}}$ is an underestimate of ${v}_{\mathrm{out}}$ (which is the real radial component of the plasma outflow velocity) because of the projection effect. It is possible to correct for this effect and obtain a better estimate of real ${v}_{\mathrm{out}}$ by using information on the angle φ between the POS and the reconstructed position of a CME plasma element derived with the polarization-ratio technique, owing to the relationship

$$\begin{eqnarray}&&\cos \varphi =\displaystyle \frac{\rho }{\sqrt{{\rho }^{2}+{\langle {z}_{\mathrm{cme}}\rangle }^{2}}}.\end{eqnarray} \tag{ 10 }$$

Then, for any position along the UVCS slit, it is ${v}_{\mathrm{out}}={v}_{\mathrm{pos}}/\cos \varphi ;$ this also shows how the polarization-ratio technique can be used to derive from single view-points (as those that will be provided with Metis) the real unprojected speed and CME propagation direction, two very important parameters to help constrain the CME propagation time and for Space Weather forecasting purposes.

Once the unprojected velocity has been determined, it can be used to estimate the corresponding Doppler-dimming factor ${F}_{D}({v}_{\mathrm{out}});$ the assumptions we made for the calculation of this parameter (entering Equation (8)) deserve some consideration. The choice of the incident radiation profile from the lower atmosphere, ${I}_{\odot }(\lambda )$, is crucial; we adopted here the Lyα disk profile reported by Lemaire et al. (2002) and measured with SOHO/SUMER on 2000 November 12, i.e., very close to the dates of Events 1 and 2; note that this profile was reconstructed with a maximum underestimation of 1%, as stated by the authors. We also used the same line profile for Event 3 because, although the integrated Lyα flux may vary by 80% between the solar maximum and minimum (e.g., Tobiska et al. 1997), the spectral line profile does not appear to change significantly over the solar cycle (see Lemaire et al. 2002). The line profile was scaled so that the total integrated line intensity matched the Lyα irradiance measured by the UARS/SOLTICE (Woods et al. 2000) instrument at the times of Events 1 and 2 ($5.34\times {10}^{11}$ and $5.94\times {10}^{11}$ photons cm⁻² s⁻¹, respectively) and by the TIMED/SEE instrument at the time of Event 3 ($4.70\times {10}^{11}$ photons cm⁻² s⁻¹). Note that these values are more than a factor of approximately two higher than the average quiet-Sun Lyα flux reported by Vernazza & Reeves (1978), which is often used as a reference; this difference is consistent with the fact that the value reported in Vernazza & Reeves (1978) was measured during the solar minimum, while our events are close to the maximum.

The absorption profile ${\rm{\Phi }}(\lambda -{\lambda }_{0})$ was assumed to be Gaussian with a $1/e$ line width equal to the average Lyα line width measured by UVCS in the three cases. Since the measured line width is actually an upper limit, because of the possible sources of line broadening that have been neglected, the calculated Doppler-dimming factor is underestimated, and this leads in turn to an underestimate of the radiative component of the Lyα line.

Once the Doppler-dimming factors are computed, given the average electron densities derived from LASCO in the specific cases of our events, we evaluated for each position along the UVCS FOV, the radiative and collisional component of the Lyα line for a range of electron temperatures, and obtained the expected total intensity as a function of the temperature, ${I}_{\exp }({T}_{e})$. We then found, through inspection, the temperature at which the expected intensity matches the observed one. In this way, we determined, at the same time, not only the electron temperatures T_e, but also the relative contributions of the radiative and collisional components needed to reproduce the observed total intensity.