Observations of Thomson Scattering from a Loop-prominence System

The K component of the white-light continuum observed from coronal structures consists of Thomson-scattered photospheric continuum, with the resulting linear polarization (e.g., Billings 1966). This component of coronal brightness scales directly with N_e (the column electron density) and hence the electron density n_e , while collisional atomic emissions (collisional bremsstrahlung) scale as n_e n_i . The total free electron number ${{ \mathcal N }}_{e}$ then determines the mass of the scattering material, if the abundances are known and if the location of the source justifies the "plane-of-the-sky" approximation. At high densities the atomic emissions processes may dominate and even produce an optically thick continuum (Hiei et al. 1992; Jejčič et al. 2018). This may obscure a part of the polarization source; accordingly the mass estimate represents a lower limit. Such an effect is likely to be a small one because of the efficiency of the thermal collapse (Field 1965). The dust-scattered F corona is not important in the lower corona. Thus where Thomson scattering dominates, its brightness directly determines the mass of the scattering material, if the photospheric radiation field is known (Saint-Hilaire et al. 2021):

$$\begin{eqnarray}&&M=\mu {m}_{p}{{ \mathcal N }}_{e}=\displaystyle \frac{{F}_{s}}{{F}_{\lambda }}\cdot \displaystyle \frac{\mu {m}_{p}(\pi {\text{}}{R}_{\odot }^{2})}{{\sigma }_{T}W}\ ,\end{eqnarray} \tag{ 1 }$$

where μ is the mean molecular weight, m_p the mass of the proton, ${{ \mathcal N }}_{e}$ the number of scattering electrons, F_s /F_λ the ratio of scattered irradiance to the total solar spectral irradiance at the HMI wavelength, W a geometrical factor as estimated by Saint-Hilaire et al. (2014), and σ_T = 6.65 × 10⁻²⁵ cm⁻² the Thomson cross section. This estimate assumes fully ionized plasma and does not depend upon the source density or filling factor η, which we discuss in the following sections. Note that we use ${{ \mathcal N }}_{e}$ for the total number of scattering electrons, N_e for the column density (cm⁻²), and n_e for the number density (cm⁻³) here and below.

With further assumptions we can estimate the density of the scattering source and its filling factor. The basic idea of this technique is to derive the Thomson-scattering contribution of the measured flux using the ratio between the observed linear polarization and the total intensity data. Following Rybicki & Hummer (1994) and Jejčič & Heinzel (2009) the density n_e of an off-limb source emitting by Thomson scattering is:

$$\begin{eqnarray}&&{n}_{e}=\frac{j(\nu )}{{I}_{0}(\nu )}\frac{1}{W}\frac{1}{{\sigma }_{T}}=\frac{{\varepsilon }_{I}}{{I}_{0}}\frac{1}{{\sigma }_{T}}\frac{1}{{G}_{I}}\,\end{eqnarray} \tag{ 2 }$$

(Equation (14) in Saint-Hilaire et al. 2021), where j(ν) is the volume emissivity at a given prominence location, I₀(ν) is the emitted intensity from the solar disk center, W = G_I is a geometrical dilution factor dependent on the height over the limb, which represents our knowledge of the source structure. Saint-Hilaire et al. (2014) described how to use the theoretical values from Minnaert (1930) in order to rewrite Equation (2) in terms of the effective Thomson-scattering cross section at a given height H over the solar limb σ_T (H) = σ_T · G_I (H) ⁶ , the line-of-sight thickness D, and the ratio between the scattered intensity from the source at the given height ΔI(H), understood as the intensity excess value above the baseline, and the intensity from the solar disk center such that:

$$\begin{eqnarray}&&{n}_{e}(H)=\displaystyle \frac{{\rm{\Delta }}I(H)}{{I}_{0}}\displaystyle \frac{1}{{\sigma }_{T}(H)}\displaystyle \frac{1}{D}\ .\end{eqnarray} \tag{ 3 }$$

This allows us to estimate the electron density of a source, with ΔI being the intensity produced by only the scattered-light component. The observed ΔI, however, also may have a component due to optically thick sources seen in emission. As these latter sources are unpolarized and the expected degree of polarization due to Thomson scattering is known as a function of height (e.g., Saint-Hilaire et al. 2014), the two components can be separated using the actual measured degree of polarization P_m and the expected one P_T . Taking this factor into account in Equation (3), it is possible to calculate the density of the material responsible for the Thomson-scattered signal:

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{{\rm{\Delta }}I}{{I}_{0}}\displaystyle \frac{1}{{\sigma }_{T}(H)}\displaystyle \frac{{P}_{m}}{{P}_{T}}\displaystyle \frac{1}{D}\ .\end{eqnarray} \tag{ 4 }$$

The column density N (electrons cm⁻²) is implicit in Equation (4); it is the integral of the electron density along the line of sight through the emitting volume. Here we estimate the column depth by taking the width of the 70% image contour of the RHESSI source. This is shown as the box dimension in Figure 2. Thus, the column density, with the polarization-factor correction, becomes:

$$\begin{eqnarray}&&{N}_{e}=\displaystyle \frac{{\rm{\Delta }}I}{{I}_{0}}\displaystyle \frac{1}{{\sigma }_{T}(H)}\displaystyle \frac{{P}_{m}}{{P}_{T}}\ .\end{eqnarray} \tag{ 5 }$$

The upper panels of Figure 4 show the column densities for regions A and B, respectively, as calculated using Equation (5). If the depolarization factor is not applied, i.e., if the flux originated entirely from Thomson scattering, the column densities would follow the dashed curves. The correction factor in each box increases with time, as expected for sources cooling and recombining. For region A, which is the closest to the photosphere and therefore is reached first by the white-light structure seen in HMI/Stokes I (see Figure 2), the uncorrected column density is as large as 10²¹ cm⁻². However, when the column density is corrected by the polarization, gray curve in the top left panel, it drops by almost an order of magnitude as this represents the actual Thomson-scattered component only. For region B, there is no significant variation between the uncorrected and corrected column densities initially, meaning that the source is mainly dominated by Thomson scattering until the observed structure reaches it at a later stage. Finally, the bottom panels shows the expected electron densities derived from the corrected column densities using Equation (4) and assuming the line-of-sight thickness D between 2.9 and 40.7 Mm for regions A and B. Although the electron densities for the two regions are quite similar during the observation, region A shows a slightly higher density, which is expected as it is closest to the photosphere and consequently influenced by the material just above the bright loops. We performed an error propagation analysis to determine the uncertainty in our calculation of the column density and hence the electron density, finding this relative error to be about 11%, and to be dominated by the observational error of Q'. An additional and ill-defined uncertainty is due to the assumption of a value for the column depth.

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This geometrical uncertainty does not affect the mass M estimated via Equation (1), using the excess flux F_S observed from each source region. From the first entry in Table 1, we find that F_s /F_⊙ = 6.7 × 10⁻⁹ for box A at 16:19 UT. This corresponds to an electron number 4 × 10³⁸ and a (total) mass 8 × 10¹⁴ g. This mass estimate is interesting as it is a simple to measure the gravitational potential energy of the scattering source, here roughly 5 × 10²⁸ erg. For further discussion see Section 4.3.

The polarized continuum observed by the HMI relates directly to the free–free and free–bound continua observed in the microwave and X-ray bands. The same electrons contribute both to the emission measure ($\propto {n}_{e}^{2}$) and to the polarized flux (∝ n_e ). The X-ray continuum also has significant free–bound and bound–bound contributions, whereas the microwaves do not. Each can serve to estimate the value of n_e n_i η V, and all three wavelength ranges should agree for a hot isothermal source according to Equation (6) (e.g., Hudson & Ohki 1972):

$$\begin{eqnarray}&&{f}_{\nu }\propto ({n}_{e}{n}_{i}V){\rm{\Lambda }}({n}_{e},T)\cdot \displaystyle \frac{{e}^{-h\nu /{kT}}}{\sqrt{T}}\ ,\end{eqnarray} \tag{ 6 }$$

where f_ν is the spectral flux in erg (cm² s Hz)⁻¹, including the slowly varying Coulomb logarithm Λ(n_e , T). Note that the microwave flux has only a weak dependence upon the source temperature, but it may have low-frequency cutoffs due to opacity or the Razin–Tsytovich effect (e.g., Krucker et al. 2013); furthermore the relatively weak thermal-microwave continuum also may have to compete with other emission mechanisms such as gyrosynchrotron or plasma radiations.

Figure 5 shows single-frequency data from the San Vito site of the Radio Solar Telescope Network (RSTN). Based on Equation (6), we expect to see a long-wavelength extension of the X-ray and white-light continua through the IR, millimeter-wave, and longer radio wavelengths. This extension should have only the weak frequency dependence of the Gaunt factor, and so ${f}_{\nu }\approx \mathrm{const}.$—a flat spectrum. What we see in the figure is different from this expectation, mainly because of gyrosynchrotron emission. The major time-series peak prior to 16:20 UT is likely to be from this mechanism, and the emission after about 16:25 UT may also come from a competing source. We interpret the minimum at around 16:20-25 UT in terms of the thermal signature of the hot plasma detected by RHESSI, but only as an upper limit. The spectral character of the minimum-flux epoch is consistent with optically thin free–free emission (Equation (6), which could reflect the long-wavelength extension of the RHESSI hot plasma and the polarized white-light continuum. The microwave flux does indeed have the right magnitude for the free–free interpretation, but this is just a consistency check.

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The two boxes A and B introduced in Figure 2 have been chosen to coincide with the main X-ray sources seen by RHESSI. We defer the details of the RHESSI measurements to Section 3.4 but report the result for the thermal emission measure of the source in box A here. From the thermal component of the spectral fits we can derive a volumetric emission measure n_e n_i η V cm⁻³ for the hot component in terms of the volumetric filling factor η. We compare this with the same quantity derived from the whole-Sun fluxes observed by GOES/XRS. Finally we can also interpret the RSTN ⁷ whole-Sun microwave observations for this parameter, noting that the lack of temperature sensitivity for the microwave thermal component makes it possible to check for low-temperature flare sources not visible in soft X-rays (e.g., Penn et al. 2016). Table 2 lists all three of these estimates.

This agreement between microwave and X-ray emission measures confirms the identification of the sources and precludes the presence of substantial flare emission at low temperatures, according to Equation (6), this would destroy the agreement between microwave and X-ray emission-measure estimates displayed in the "loci plot" of Figure 6.

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The agreement of the HMI estimate of column density N_e with the radio and X-ray values also allows us, uniquely for this flare, to estimate the line-of-sight thickness of the source η D in terms of the observed volumetric filling factor emission measure and the total electron number ${ \mathcal N }$ as derived in Section 3. The emission measure implies an electron content ${ \mathcal N }^{\prime} ={n}_{e}^{{\prime} }V^{\prime} =\sqrt{{EM}\cdot \eta {DA}}$, which we can combine with the true electron number to give the estimate for the source thickness

$$\begin{eqnarray}&&\eta D=\displaystyle \frac{{{ \mathcal N }}^{2}}{A\cdot {EM}},\end{eqnarray} \tag{ 7 }$$

based on simple estimates of the source area A and depending on the volumetric filling factor η.

The final information needed for characterizing the polarization sources comes from the RHESSI imaging, from which we can obtain estimates of the projected area of the joint X-ray/Thomson-scattering/thermal-microwave source identified by the agreement in Table 2.

At the time of the observations of the September 10 flare, the RHESSI spectrometer had been operating in space for more than 16 yr and only four of the nine detectors were functioning. Even for these four detectors (detectors one, three, six, and eight), radiation damage had substantially affected the spectral resolution and effective area. Because radiation damage affects each detector differently, we performed spectral fitting for each detector separately and then compared the derived parameters to judge their consistency.

The RHESSI observations ended at about 16:17:40 UT when the RHESSI spacecraft entered the South Atlantic Anomaly. We selected the interval from 16:16:30 to 16:17:20 UT for our comparison with the HMI observations taken at 16:19 UT. The orbit-night background from 15:49 to 15:52 UT has been subtracted before the standard RHESSI spectral fitting of a thermal component plus a nonthermal component was done. The thermal fits dominate the total count spectrum for energies below 25 keV, so we take the 12–18 keV images shown in Figure 1 to represent the thermal sources. The derived temperatures from the four different detectors range from 18.2 to 19.3 MK in good agreement. The corresponding emission measures vary from 2.6 to 3.5 × 10⁵⁰ cm⁻³, with the average value of 3.3 ± 0.5 × 10⁵⁰ cm⁻³ as quoted above in Table 2. The temperature and emission measure derived from GOES-15 are 19.1 MK and 3.0 ± 0.1 × 10⁵⁰ cm⁻³, respectively, in surprisingly good agreement with RHESSI, considering the effects of radiation damage to the effective area of the RHESSI detectors.

The RHESSI parameters derived above refer to the total counting rates. We can also use imaging spectroscopy to estimate temperature and emission measure for the RHESSI source contained within box A (e.g., Krucker & Lin 2002). The fluxes from within box A are derived from CLEAN images at 1 keV resolution, which were reconstructed in the same way as the images shown in Figure 1. Using the energy range 13–18 keV gave parameters [21.0 ± 1.1 MK, 1.2 ± 0.4 × 10⁵⁰ cm⁻³]. Here the errors have been derived by adding Gaussian random noise at 10% rms in a Monte Carlo procedure; this gave parameters [19.2–22.6 MK, and 0.6–2.0 × 10⁵⁰ cm⁻³]. The derived temperature of the main source is thus slightly hotter than the flare-averaged value. Considering the uncertainties the difference is of minor significance. The emission measure of the bright source is about a factor of 3 smaller, indicating that the fainter sources outside box A cannot be neglected and indeed affect the flare-averaged fit parameters.

In the final step of the RHESSI analysis, we estimate the source area of the thermal hard X-ray source. We use CLEAN images to estimate the source size of the main compact RHESSI source (source A). From the RHESSI clean image shown in Figure 1 (top), we derive a FWHM source size of the compact part of the source of 158. The image is reconstructed with a CLEAN beam of 122. Hence, we get a deconvolved source size of 100.

RHESSI exited the South Atlantic Anomaly around 16:39 UT; the second interval (box B) was fully observed by RHESSI with both attenuators still inserted. The same analysis is applied to the later interval as has been done for the earlier interval using the orbit-night background from 16:55 to 16:58 UT. The nonthermal fit is marginal at this time, with very steep spectral slopes (≈8). The temperature ranges from 16.0 to 17.7 MK with an average value of 17.0 ± 0.6 MK; the emission measures vary between 0.7 and 1.3 × 10⁵⁰ cm⁻³, with an average value of 1.0 ± 0.2 × 10⁵⁰ cm⁻³. GOES at this time gives [T, EM] = [15.5 MK, 1.3 × 10⁵⁰ cm⁻³], again in good agreement with RHESSI. Imaging-spectroscopy fits for the main compact source give higher temperatures (22.8 ± 1.1 MK). As the temperature of the main source is higher than the flare-averaged temperature, a significantly lower emission measure of 0.10 ± 0.04 × 10⁵⁰ cm⁻³ of the compact source already produces a good fit with the observed hard X-ray flux in the box B. As extreme values for the emission measure we found 0.04 × 10⁵⁰ cm⁻³ and 0.3 × 10⁵⁰ cm⁻³.

The CLEAN image shown in Figure 1 has an area with the 50% contour level of 185 ×185 giving a deconvolved source diameter of 140. Compared to the earlier interval around 16:19 UT, the source area apparently had roughly doubled in area.

As described above, the measurement of scattered flux directly determines the total number of free electrons, with the plausible assumptions of plane-of-the-sky location and nominal limb-darkened solar radiance (Saint-Hilaire et al. 2021). We find 4 × 10³⁸ for the total electron number. This in turn determines the coronal mass at the time of observation, plus its gravitational potential energy. The values we have derived for the box A snapshot are 8 × 10¹⁴ g and 5 × 10²⁸ erg, respectively.

The right panel of Figure 6 summarizes the constraints imposed by the observed value of η D = 4 × 10⁹ cm. To this we have added constraints from the Razin–Tsytovich effect and from free–free opacity (Belkora 1996). These limits are based on the assumption that the RSTN three-point spectrum at the box A epoch (Figure 5) does not show a spectral break. The free–free limit definitely restricts the density estimate for the unity filling factor, shown in the figure as the thick solid line (the depth-dependence of mean density for η = 1). For some part of the range of possible source thicknesses, we can exclude this limit. We would expect a filling factor below unity in any case from the fine structure in the developing arcade, but there is no real constraint from the microwave or X-ray imaging. From the diagram we conclude that the probable value for the mean source density does not exceed 2 × 10¹⁰ cm⁻³ for line-of-sight depths comparable to the dimension of the integration box.

This remarkable solar flare (SOL 2017-09-10) has allowed us to consider in detail the information about hot coronal emission sources in a major flare. These sources consist of hot plasma, and we have shown for the first time that observations of linear polarization in the optical continuum match the extrapolated soft X-ray thermal emission spectrum right down to the microwave range. This convincingly establishes that flares do not require the injection into the corona of cool material (below about 1 MK), which would readily be detectable in the Paschen continuum and in the microwave free–free emission in the Rayleigh–Jeans spectrum (e.g., Penn et al. 2016). The absence of few-mega Kelvin temperatures is also consistent with the AIA-derived emission-measure distributions for this event, as reported by Warren et al. (2018). The polarization brightness is directly proportional to the mass of the hot coronal cloud; this information cannot be obtained otherwise except in model-dependent ways. The coronal mass, 8 × 10¹⁴ g, estimated some 30 minutes after the onset of the flare's impulsive phase, suggest that the magnetically trapped mass can rival that of the most massive coronal mass ejection (CME), often quoted as about 10¹⁶ g. The potential energy estimate, however, is less than a percent of total energy estimates for such an event, either a flare or CME (e.g., Emslie et al. 2012).

The standard picture of the development of a major eruptive flare envisions a "magnetic explosion" (e.g., Moore et al. 2001) creating a CME, followed by an extended period of magnetic reconnection that forms the loop-prominence system finally seen in chromospheric emission lines. The plasma that fills these loops, according to this well-accepted picture, originated via chromospheric "evaporation" and appears as the hot cloud our observations show and as depicted in many cartoon descriptions. This hot mass has been driven into the corona and heated, although the details of how this happens remain unclear. Our data are consistent with the idea that the heating can be identified with particle acceleration leading to plasma β of order unity (Krucker et al. 2010). The standard picture, dating back at least to Bruzek (1964), involves hot mass in larger and larger loops cooling sequentially, implying that the instantaneous mass we infer at the epoch of box A is a lower limit to the total mass for the entire flare process. After cooling, the ionized plasma recombines and may become optically thick, radiating strongly via collisional processes (Jejčič et al. 2018) at densities much higher than we infer for the hot plasma.