ON THE ORIGIN OF A SUNQUAKE DURING THE 2014 MARCH 29 X1 FLARE^*

The FIRS data were reduced using software originally developed by Jaeggli (2011) and modified by Tom Schad (2013, private communication). The reductions followed standard procedures: correction for detector non-linearities, subtraction of dark frames, division by flat fields, co-registration of the two beams (including corrections for image rotation), polarization calibration, and de-modulation (conversion of linear combinations of Stokes parameters to individual Stokes parameters). Because the required polarization sensitivity is very high in chromospheric lines (e.g., Uitenbroek 2011), special care is needed in handling calibrations. Usual dark frames and flat fields were acquired, and a gain linearization correction was applied to the data using a curve from Jaeggli (2011). We used flats obtained with a calibration lamp that is vignetted across the detector frame, in preference to solar flats in which photospheric spectral lines are always present. This is because we analyze below the detailed profiles of the Si i line at 1082.7 nm.

Residual fringes and some detector artifacts are present in these data. Fringes are a source of systematic error. Using careful corrections for flat fields, we reduced fringing to ≲ 2 × 10⁻³I_c (peak-to-trough) where I_C is the continuum intensity, which is defined using wavelengths for each individual scan shown in Figure 2. The wavelength scales of the spectra are determined using solar flat-field scans and solar photospheric absorption lines. The spectrograph was stable at the level of 0.2 pixels in a wavelength (0.004 nm) during the observations, equivalent to a Doppler shift of 0.2 km s⁻¹.

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It is important to note that at infrared wavelengths, the enhancement of intensity during flares is moderate, quite unlike the well-known enormous UV and X-ray enhancements. All the FIRS data were obtained in the linear regime.

Figure 2 shows the mean intensity spectrum with annotated spectral features and, in the lower panels, Stokes profiles for I, Q, U, and V from left to right. The data shown in the lower panels are from the 29th scan obtained through the flare footpoint beginning at 17:46:29 UT, just as the flare was in the impulsive phase as found from RHESSI data analysis (A. Donea et al. 2014, in preparation). The line profiles of the photospheric Si i line are essentially consistent with polarization induced by the Zeeman effect, with the possible exception of those seen in the flare kernel. On the other hand, while the He i linear polarization (Q, U) profiles might initially appear to be of solar origin, the result of atomic alignment, the presence of linear polarization in the J = 0 upper level to J = 1 lower level transition at 1082.9 nm must be due to systematic errors. No atomic alignment is possible with these quantum numbers, nor is it possible for levels involving the hyperfine structure of any ³He nuclei that might be present.

In our figures, all data are taken from the dual-beam system, but we also examined single-beam data. The signals in the two beams are very similar for all wavelengths outside the helium lines, but significant differences are present in the polarized helium profiles. This is a clear sign of I − (QUV) crosstalk in the helium lines. The dual-beam corrections are clearly doing an excellent job at other wavelengths (for example, those in the Si i line core). We surmise that the helium lines are evolving in intensity at least as rapidly as the 1.2 s cadence of the modulation cycle, at the level of a few percent, thereby producing spurious signals. Faster modulation seems appropriate for flare observations of the chromosphere. We defer further analysis of the flare kernel data for He i to later work.

The noise levels of these Stokes Q, U, V data are close to 8 × 10⁻⁴I_C. The largest fringes remaining are in Stokes V at the level of 2 × 10⁻³I_C. The noise levels are more than adequate for us to attempt inversions of the Si i line data.

We used the Stokes profiles to derive several simpler quantities. From Stokes I (intensity), we compute the n = 0, 1, 2 moments M⁽ⁿ⁾ of the line profiles weighted by wavelength from the line center. We define the continuum-subtracted line intensity as

$$\begin{equation*} I^\prime _x = I_C - I_x, \end{equation*}$$

where the Doppler shift x is defined as c(λ/λ_LAB − 1) with c in km s⁻¹. Then we define

$$\begin{equation*} M^{(n)}= \int I^\prime _x x^n dx. \end{equation*}$$

The Doppler shift of the line is v = M⁽¹⁾/M⁽⁰⁾ km s⁻¹, the line width (not shown here) is $w= \sqrt{M^{(2)}/M^{(0)}}$ km s⁻¹. We also computed quick-look quantities from the IQUV Stokes parameters. These include a LOS magnetogram, which is simply the median of the ratio of Stokes V to the first derivative of Stokes I with respect to wavelength over wavelengths of significant line absorption or emission. The other magnetic parameter is the field azimuth, which is ${1}/{2} \arctan (U/Q)$ where again median values of the ratio U/Q are taken across both the Si i line and He i multiplet. The relationship of these quantities to physical parameters in the Sun arises only when the polarization is dominated by the Zeeman effect (see, e.g., Jefferies et al. 1989) and only when the polarization Q² + U² + V² ≪ 1, and when the Stokes profiles all originate from the same physical volumes underlying a given pixel. Nevertheless, such quantities as LOS magnetograms are very familiar to solar physicists who, with care, illustrate properties of the solar magnetic field.

Figures 3 and 4 show some of the quick look parameters and continuum intensity from the three scans obtained before, during, and after the impulsive phase. These we label phases "bef," "dur," and "aft." The white contours of Figure 1 are from the Si core data shown in the second row of Figure 3. Figure 4 shows parameters related to the magnetic field. Salient features of these plots include:

• 1.  
  The IR continuum shows only a weak brightening during scan dur.
• 2.  
  Both the photospheric⁶ Si i and chromospheric He i lines show considerable brightening during scan dur, in the line cores.
• 3.  
  The filament seen in the He i line core images, lying roughly along the neutral line seen in bef scan magnetograms, disappears by scan aft.
• 4.  
  The He i absorbers in the filament are seen moving upward by between 5 and 10 km s⁻¹ in scan bef. During scan aft, the filament is replaced by a diffuse region of He i line absorption.
• 5.  
  Magnetograms show only subtle changes from scans bef to dur and aft.
• 6.  
  The photospheric magnetic azimuthal angles show systematic changes as the flare progresses.
• 7.  
  The He i U/Q signals during scan dur have a coherent structure whose origin includes at least some I → (QU) crosstalk. The spectral profiles show that they certainly cannot be interpreted as a traditional Zeeman-induced linear polarization.

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During scan dur (the impulsive phase), magnetic quick-look data for the He i line cannot be trusted to represent magnetic fields because of crosstalk. We will attempt to correct for this crosstalk in a future publication, since even in the presence of atomic polarization, the 1083 nm multiplet still retains accurate information on field azimuthal angles for field strengths >10G (the strong field limit of the Hanle effect). To quantify the magnetic field changes we turn to inversions for the Si i IQUV data.

First, we will use properties of the Si i line and continuum to constrain the depth of penetration of flare energy that is sufficient to change the temperature structure in the deep chromosphere and photosphere. At a first glance, the continuum intensity appears to change very little, if at all. But both p modes and granulation modify the continuum intensity at a level of a few percent at angular resolutions similar to those of FIRS (e.g., Sánchez Cuberes et al. 2000), making relatively small changes difficult to see in slit rasters. Careful examination of the FIRS spectra obtained beginning at 17:46:16 UT, show a significant brightening of 4% ± 1% above levels in the neighboring spectra taken 13 s before and after, close to the flare kernel observed in RHESSI and line data. Evidence for this is shown in Figure 5. These data include variations in the transparency and seeing conditions of the atmosphere and hence vary significantly from row to row in the figure. As is obvious in the figure, transparency variations were strongest in the first scan, getting progressively weaker in the second and third scans. However, relative intensities along each row in each panel can be fairly compared. The uncertainty quoted above is a 1σ statistical variation of the detrended intensity measured along the rows immediately adjacent to the row containing the flare.

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The coherent streak of brightness in continuum data from 17:46:16 has all the characteristics of a genuine brightness increase associated with a white light flare. This picture is supported by HMI continuum data from the SDO spacecraft, which shows a ≈5% increase in continuum intensity during the flare. Its appearance only in one FIRS scan indicates a very rapid evolution, which is characteristic of an origin from dense photospheric material that has a radiative relaxation time of 1–2 s (Spiegel 1957).

Seismic transients from solar flares can be detected by pre-processing solar data and applying the analytical technique of helioseismic holography to Doppler measurement of the active region hosting the solar flare. A. Donea et al. (2014, in preparation) analyzed Doppler maps from the HMI instrument (Schou et al. 2012) on board the SDO. HMI measures properties of photospheric dynamics and magnetic fields every 45 s. A. Donea et al. (2014, in preparation) generated Postel projection maps of the seismic emission of NOAA 12017. We refer discussion of the principles of seismic holography to Section 4 of Lindsey & Braun (2000), with application to flare observations to Donea et al. (1999). Briefly, the seismic responses to the flare perturbations are identified through an excess of the emission power, |H₊(r, t)|². Each pixel in an image of |H₊(r, t)|² is a representation of the coherent acoustic power for waves that have propagated downward from the focus, traveled thousands of kilometers beneath the solar surface, and re-emerged into a pupil a significant distance from the focus. With this technique A. Donea et al. (2014, in preparation) uncovered a weak but significant seismic source at the footpoint shown in Figure 1. Hard X-ray emission, magnetic transients, and strong UV footpoint emission were analyzed by A. Donea et al. (2014, in preparation), confirming that the seismic source is indeed associated with the flare.

By comparing the brightness of models of continuum and Si i 1082.7 nm line to observations, we can in principle constrain the depth in the atmosphere to which significant heating from above can penetrate. Our approach is simple. We ask: what are the deepest and shallowest layers in the atmosphere heated by the flare that are compatible with the data?

To preface the model calculations below, we note that the flare Si i profile (Figure 2) resembles classical Ca ii H and K self-reversed profiles (Linsky and Avrett 1970), but with far weaker line absorption wings. The Ca ii line cores, which are much more opaque than the line of silicon, form in the chromosphere with a source function dominated by scattering. The simple observation of a self-reversed profile of Si i implies a significant column mass that is much higher than that for the calcium line. The breadth of a Doppler-broadened, self-reversed line is larger than an optically thin line formed under the same conditions by the factor ${\approx} \sqrt{\ln \tau _0}$ where τ₀ is the line center optical depth. For τ₀ = 100 this factor is more than 2.1. The self-reversal is also very narrow (FWHM ≈0.015 nm, see Figure 2), indicating turbulent speeds of FWHM/1.66 ≈2.5 km s⁻¹ where the line core forms. A profile averaged along the region with obvious Si i emission in Figure 2 is shown in Figure 6. The averaging washes out the self-reversal in the latter plot.

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We model the Si i line and neighboring IR continuum both during and outside the impulsive phase. We performed nLTE radiative transfer calculations, in several one-dimensional models of the solar atmosphere, following the tradition of Vernazza et al. (1973, 1976, 1981, henceforth VAL81). We solved nLTE statistical equilibrium equations for atoms of H, C, Si, and Fe using the program RH (Uitenbroek 2000). These atoms were chosen because UV radiation controlling the Si i spectral line at 1082.7 nm is dependent on the nLTE solutions of these abundant elements. We considered using one of several flare models (e.g., Machado et al. 1989). However, these models were constructed to identify the origin of white light emission in flares. Our goal is different—to try to see if modeling can provide a depth of penetration of flare energy into the photosphere. Therefore, we adopted a different, more straightforward strategy. We started with model C of VAL81 and explored the effects of introducing temperature plateaus of the form

$$\begin{equation*} T_e = T_0 - T^{\prime }_1 \log m, \ \ \ m_2 > m, \end{equation*}$$

where m is the column mass of the atmosphere, $T_0,T^{\prime }_1$ are non-negative constants, and m₂ is a column mass above which temperatures are changed. We have three free parameters, so our results will not be unique. However, such plateaus, with small gradients $T^{\prime }_1$, have justification at least during some phases of flaring plasmas seen in radiation hydrodynamic calculations (see the 50 s panel of Figure 3 of Allred et al. 2005, for example). The main sensitivity of the emerging spectra is to the two parameters a and m₂. Given an estimate of m₂ the height of the energy penetration follows from the m(z) relationship for the model.

We made calculations in two limits: in the calculations shown in the figures below, we allowed the atmosphere to relax to a state of hydrostatic equilibrium; in the other limit we merely solved the statistical equilibrium equations with no such adjustment. The sound crossing time of the photosphere is on the order of a few scale heights divided by 7 km s⁻¹, a minute or so, which is comparable to the duration of the flare impulsive phase. These limits probably span the behavior of intensities from an evolving atmosphere. The differences between the calculations are small in photospheric layers, but are significant for regions and spectra formed above 600 km above the photosphere. Such differences do not affect our conclusions, which depend only on the photospheric Si i line.

These calculations are not state-of-the art in terms of dynamics, rather our focus is on a careful treatment of the formation of the Si i 1082.7 nm line and of the continua formed between 125 and 180 nm for later comparisons with SDO/AIA data. We therefore took care to use modern and complete atomic data for the Si and Fe neutrals. We used atomic energy levels and transition probabilities from NIST up to and including the 4p levels in Si, and used photoionization cross sections from the OPACITY project (Seaton 1987), treated as outlined in Judge (2007). Collisions with electrons were treated using the impact approximation for permitted transitions (Seaton 1962), Seaton's semi empirical formula for direct ionization (Allen 1973), and a collision strength of 0.1 for forbidden transitions.

Figure 6 shows, in the right panels, computed and observed profiles of Si i 1082.7 nm, with all intensities normalized to quiet Sun values. These calculations are representative of two limits of the value of m₂—and hence height of penetration—used in the models.

The first class (upper panel) allows penetration of energy and enhanced temperatures down to photospheric layers—we allowed temperatures to rise down to 0 km height by adding various plateaus at such depths. Remarkably, the model shown produces an acceptable match to the observed profiles and continuum (the He i line is not modeled here). When exploring different temperature plateaus we determined that a reasonable agreement with the line and continuum observations requires the flare energy to penetrate and heat down to a height of ≳ 100 ± 100 km above the photosphere. The error bar comes from the need to produce the 4% enhancement in continuum emission (<200 km) with temperatures that can match the Si i profile, with both features spanning the region between 0 and 700 km.

The second limiting case is one where flare energy penetrates only to the mid-upper chromosphere. Downward propagating radiation enhances the cores of lines, and a typical calculation is shown in the lower panels. The line width is very narrow even though we adopted non-thermal speeds (microturbulence) of up to 8 km s⁻¹ in the middle chromosphere (close to the sound speed). The continuum, formed predominantly in the photosphere with a tiny contribution from optically thin emission in the plateau, is close to the pre-flare level. The computed continuum includes thermal photospheric emission as well as hydrogen recombination from the plateaus. These two contributions have been discussed by Machado et al. (1989); Kerr & Fletcher (2014), among others. The contribution from the latter is small in our calculations, the Balmer continuum originating from an optically thick layer near 350 km and the longer-wavelength (>364 nm) H⁻ and Paschen continua near 0 km.

The core of the Si i line during the flare is broad compared to a thermal width near 1.8 km s⁻¹ (Figure 6), and like the well-studied Ca ii H and K lines, the origin of this width appears to be most naturally explained through scattering (see above). Some decades ago there was a discussion of the Wilson–Bappu effect, which is an empirical relationship between the width of the core of the Ca ii lines and stellar luminosity, in favor of line formation in terms of scattering (Ayres 1979) and not optically thin micro- or macro-turbulence (Fosbury 1973). The presence of the narrow self-reversed core seems irrefutable evidence for the presence of scattering, and argues strongly for a deep formation of the core. Only calculations of the penetration of flare energy to the photosphere produce lines broadened by scattering and self-reversals, the latter happen to be weak in the case shown in Figure 6, but not in obvious disagreement with the observed profile.

We stress that the detailed structure of our calculations is not unique and should only be viewed as an attempt to find the depth of penetration of significant heating during the impulsive phase of the flare. Overall, our comparisons with observations of the Si i 1082.7 nm line, and considering the difficulties of tying down the continuum intensity during the flare, we conclude that heating sufficient to detectably change the photospheric temperature occurs at least to about 100 ± 100 km above the visible photosphere. Based on an exploration of values of T₀, m₂ in our model, we believe that this aspect of our calculations is robust.

We used the code MELANIE (Socas-Navarro 2003) to invert the Si i Stokes IQUV profiles to derive the vector magnetic field in the photosphere. This was done only for scans obtained before and after the impulsive phase. Codes exist for inversion of the He i multiplet (e.g., López Ariste & Casini 2002; Lagg et al. 2004; Asensio Ramos et al. 2008), but we have not attempted such inversions yet because we must deal with significant crosstalk in the He i QU profiles during the flare, and because outside the flare these profiles are mostly of low signal-to-noise ratio.

The observed Si i line—$3p4s ^3\!P^o_2 - 3p4p ^3\!P^e_2$ (lower and upper levels, respectively)—forms between ≈100 km (wings) and 600 km (core) above the photosphere in our 1D models. MELANIE solves for a solution to the Milne–Eddington equations (source function linear with optical depth) for lines with Zeeman-induced polarization, minimizing differences between observed and computed profiles. The solution includes the vector magnetic field (with its 180° ambiguity), opacity, Doppler width and shift, damping parameter, and non-magnetic filling factor. The Milne–Eddington approximation is a simplification that surely is invalid during the flare itself. But before and after the flare, its use appears reasonable, outside bright flare ribbons and below 600 km in the atmosphere. Our conclusions will be based only on the non-flaring atmosphere.

We inverted all five scans. We set the statistical uncertainty of each data point to 10⁻³I_C to evaluate values of χ², estimated using the measured fluctuations in ${QUV}$ at typical continuum wavelengths. For comparison, some of the best vector polarimetric data, the deep mode high S/N observations from the SP instrument on the Hinode spacecraft, have rms noise of 3 × 10⁻⁴I_C in the 630 nm region, for integrations of 67 s (Lites et al. 2008). Outside the flare scan, the distribution of χ² peaks near 40, showing that systematic errors are large and/or the model parameterization is poor. Given the residual fringing and other artifacts evident in the data, this does not necessarily imply that the model is poor. The reproducibility of the inversions was tested by initializing the same data set with two different random initial guesses. The resulting rms variations in the magnetic field strength B are 140 G, inclination 18°, azimuth 41°, and the LOS B 30 G.

Figures 7 and 8 show the results of inversions of the scans obtained before and after the flare, begun at 16:29:26, 16:55:58, and 18:01:55 UT. No attempt at a resolution of the 180° ambiguity in the field azimuth has been made, and the angles are defined relative to the local vertical⁷ (inclination) and in the plane of sky (azimuth, zero, and 180° being along the E–W direction). Circles show the location of the center of the acoustic source. Figure 7 shows measured changes in magnetic parameters in the two scans obtained before the flare. There are detectable differences across the bulk of the field of view in all magnetic parameters. Focusing on data in the circled region of the acoustic source, we see a significant increase in the field strength in this region, accompanied by becoming more inclined to the vertical direction (data shown in the first two rows of the figure). Note that the circled region is some 5'' from the polarity inversion line. The maps of B suggest that a channel of weak field is moved to the west by an arcsecond. Initially the field is inclined at some 130° to the vertical. But by 17:05 UT two bands of field connected in a Y-shape on its side in the image appear more inclined to the vertical. The field azimuth in the Y shape departs significantly from initially E–W to more N–S. The LOS field within the circle's center shows an increase that results from increases in B despite the decrease in inclination. It is unclear from our data if these changing fields arise from motions of field vertically (flux emergence) or horizontally (flows). There is little evidence for vertical motions from the LOS velocity measurements shown in Figure 3, but the inversion data (not shown) reveal a very small (−0.3 km s⁻¹) blue-shift pattern in the Si i data in the 10:55:58 UT scan that might conceivably be associated with the upper part only of the Y pattern seen in the magnetic data.

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The scans upon which the inversions are based are 26 minutes apart. The above changes are unremarkable when compared to the larger field of view, except that they are within the circle encompassing the acoustic source and they are significant in all magnetic parameters.

Figure 8 shows measured changes before and after the flare itself, with the scans begun 66 minutes apart. The difference panels show again an increase of B and azimuth, and a weak reduction of inclination in a band in the E-W direction cutting through the circled region, which is flanked by regions of increased inclination just to the S and N. This sheared region (differentially changing field inclinations with time) appears aligned with the bright footpoint emission seen in the core intensities of the Si i and He i lines. However, the data are noisy.

Thus, our analysis hints that magnetic fields associated with the particular acoustic source evolve to become more sheared (i.e., inclination angles diverging in time), stronger (perhaps due to flux emergence), and rotated relative to the E–W direction during the flare. These results appear to correspond to a mixture of earlier results. Wang et al. (2012a) found penumbral fields that became more vertical after flaring. In contrast, for some flares Martínez-Oliveros et al. (2008); Wang et al. (2012b) reported field lines highly inclined to the vertical after a flare-associated seismic transient. We note that the seismic source we have analyzed is unusual. It is found near a magnetic pore, emerging from a magnetically quieter area somewhere between the main two sunspots of the AR12017 (A. Donea et al. 2014, in preparation), instead of in a penumbra.

Last, if flux emergence was responsible for these measured changes in magnetic field, in one hour the plasma and magnetic field moving vertically through the compressible sub-photosphere with a surface velocity ≲  0.3 km s⁻¹ could have emerged from depths no deeper than ≈200 km. If advected by granules with 1 km s⁻¹ speeds, the flux could have emerged from no deeper than 600 km. It is interesting to consider how such changes to the immediate subsurface structure might or might not affect the generation of sunquakes.

Armed with a unique data set, we have studied the depth of penetration of flare energy down into the solar photosphere. We have shown that the detected changes in thermal structure in the atmosphere reach the photospheric level, but barely. Here we examine possible modes by which the energy might be transported through the photosphere into the deeper solar layers, thereby exciting the sunquake.

The power in the main kernel of the acoustic source measured using seismology from HMI is (A. Donea et al. 2014, in preparation):

$$\begin{equation*} P = 1.3 \pm 0.05 \times 10^{26} \,{\rm erg\, s^{-1}}. \end{equation*}$$

This power is distributed over an area including the main kernel centered at (X, Y) = (518,  264) (see Figure 1), which is the source just to the SW requiring an additional 1.0 × 10²⁶ erg s⁻¹. The main source's spatial distribution is nearly bi-Gaussian with a geometric mean FWHM of w = 4.2 HMI pixels, w ≡ 1.5 × 10⁸ cm. The peak of the power per unit area is F = P/(A = πa²) with $a=w / 2\sqrt{\ln 2}$, or

$$\begin{equation*} F=5\times 10^9 \,{\rm erg\, cm^{-2}\, s^{-1}}. \end{equation*}$$

This should be regarded as a lower limit because both the holographic technique and HMI have non-negligible angular resolutions. The area A = 2.6 × 10¹⁶ cm² is strictly an upper limit for the same reasons.

Let us consider first non-magnetic mechanisms by which energy is transported to the acoustic source. In this picture the changing magnetic field generates thermal perturbations indirectly via the end product of large coronal magnetic restructuring (conduction, particles, local downward radiative heating), channeling some flare energy into the photosphere. We can estimate energy fluxes into the acoustic source region that are compatible with our observations in several ways. First, we note that the excess thermal energy radiated from the photosphere during the few minutes of the rise phase is roughly 4%–5% (i.e., the measured continuum enhancement) of the unperturbed solar radiative flux density F_☉ = 6.33 × 10¹⁰ erg cm⁻² s⁻¹:

$$\begin{equation*} P_{{\rm RAD}} \approx 0.04 F_\odot A \lesssim7\times 10^{25} \,{\rm \ erg \,s^{-1}}. \end{equation*}$$

The radiative cooling time of photospheric plasma is 1–2 s (Spiegel 1957). Curiously, although P_RAD  ∼  P, this excess thermal energy is simply radiated into space on such timescales, and is unavailable to contribute to P. We can look at the enthalpy flux F_enth associated with bulk flows into the photosphere, for this we need a measurement of plasma motions and we turn to the Si i line core emission, which forms between 200 and 500 km in our models. We use pressures p = 2 × 10⁴ dyne cm⁻² and densities ρ = 4 × 10⁻⁸ g cm⁻², corresponding to 300 km height. These are conservatively high values for the average thermal properties of the plasma where this line is formed, favoring higher estimates of energy transport.

A careful comparison of the flare emission core and the pre-flare absorption profile of the Si i line reveals an upper limit to differential flows of roughly 0.5 wavelength pixels, 0.5 km s⁻¹. This is equivalent to 2.5σ where σ is the sensitivity of the Doppler shifts from our FIRS spectra. A Doppler photospheric signature of the flare is present in HMI data at the location of the seismic source with a shift equivalent to u ≈ 0.3–0.5 km s⁻¹ (A. Donea et al. 2014, in preparation). However, such filtergram data, scanning wavelengths in time, cannot be trusted during flaring and so we adopt the upper limit above. We then find an upper limit to the enthalpy energy flux of

$$\begin{equation*} F_{{\rm enth}} \lesssim\,\frac{5}{2}p \, u\, A \approx 6\times 10^{25} \,{\rm erg\, s^{-1}}. \end{equation*}$$

The close agreement of the upper limit for F_enth with F_rad means that the excess energy radiated by the photosphere during the flare can be supplied by a bulk flow of energy associated with a subsonic downflow of 0.5 km s⁻¹ induced (somehow) by the flare. The power in the acoustic pulse is a factor of at least two larger than our optimistic estimate of F_enth.

If, however, the pressure pulse involves high frequency phenomena (ν > c_S/H ≈ 60 mHz, where H is the pressure scale height and c_S the sound speed), the pulse would be invisible to observation except as a broadening of spectral lines to at most the sound speed (for linear waves), the lines being formed over a length ≈H in a stratified atmosphere. The WKB expression for the energy flux density (propagating both upward and downward) at the sound speed is

$$\begin{equation*} F_{{\rm wave}} = \rho c_S \langle \xi ^2 \rangle \ \ \ {\rm erg\, cm^{-2}\,s^{-1}}, \end{equation*}$$

where ξ is the velocity amplitude of the wave. We can set limits on ξ through the measured line broadening and line profiles during the flare itself. Before and after the flare the inversions yield ξ ≲ 2.8 km s⁻¹. During the flare the measured line wings are similar in shape to the pre- and post-flare profiles. The emission core of the profile has a FWHM of 0.05 nm (Figures 2 and 6). Treated as optically thin emission, this FWHM is equivalent to an e-folding Doppler broadening speed of 8 km s⁻¹. In the presence of scattering this is a strict upper limit. To estimate the energy flux available in such modes we again use ρ = 4 × 10⁻⁸ g cm⁻², and the upper limit of eight km s⁻¹. Assuming that only half the waves are emitted downward, we find

$$\begin{equation*} F_{{\rm wave}} A \lesssim2.5\times 10^{26}\ \ {\rm erg\, s^{-1}}. \end{equation*}$$

But we believe this is a gross over-estimate. First, the scattering leads to emission profiles a factor of ${\approx}\sqrt{\ln \tau _0}$ broader than mere Doppler broadening where τ₀ is the line center optical depth. We are able to reproduce the core Si i emission using microturbulent speeds of 1–2 km s⁻¹ and the full Voigt profile, reducing the estimate F_wave by a factor of at least 16. Second, within a broad emission core the line profile shows a very narrow self-reversal during the flare (lower left panel of Figure 2), indicating both scattering-induced line broadened profiles and values of ξ in the line core that are far smaller than those adopted above. Last, any high frequency waves with frequencies of a few Hz or less, are rapidly damped in the photosphere by the continuum radiative exchange processes first modeled by Spiegel (1957). The thermal perturbations associated with high frequency wave energy rapidly radiate this energy from the photosphere on a timescale of 1–2 s. Only waves with frequencies in excess of several Hz could propagate down into the interior unmodified by radiation damping. All things considered, it seems unlikely that the power of the sunquake can be provided by such high frequency waves.

We conclude that non-magnetic modes of energy transport into the interior are very unlikely to be sound waves. More likely is a coherent downward-moving plug of plasma carrying enthalpy of almost the right magnitude, but we have already set an upper limit to this process that is optimistically a factor of two smaller than needed.

Consider in turn the energetics of the Lorentz force picture. The field strength from inversions from the Si i line, before and after the flare, is of order 800 G from which the magnetic energy density is B²/8π = 2.5 × 10⁴ erg cm⁻³, about a third of the photospheric thermal energy density, 3/2p_ph (the latter is a lower limit because we neglect latent heat of ionization). The Alfvén speed c_A for a photospheric density of 2.6 × 10⁻⁷ g cm⁻³ is 4.3 km s⁻¹, so the local magnetic energy flux is at most

$$\begin{equation*} F_M A \lesssim c_A \frac{B^2}{8\pi } A = 2.4\times 10^{26} \ \ {\rm erg\, s^{-1}}, \end{equation*}$$

scaling as $B^3/\sqrt{\rho }$. This estimate is entirely a local one; it does not take into account the connections of the magnetic field throughout the entire flux system or the fact that momentum and energy is readily imparted to localities from a far larger reservoir of magnetic energy encompassing the entire volume of the active region. Thus there is naturally sufficient energy in the magnetic field of the entire active region to account for the acoustic source. But it should also be remembered that only a fraction of the total magnetic energy is available as free energy, only a fraction will penetrate into the interior, and measured changes in magnetic fields before and after flares are small relative to the ambient field. Estimating the free energy change over the entire active region is not a simple task and is fraught with problems (De Rosa et al. 2009), so is not attempted here.

Instead we assume that a magnetic force is responsible for the impulse into and below the photosphere at the acoustic source. The same argument for the enthalpy flux applies here no matter if a Lorentz-forced flow is responsible, and the same inadequate energy flux results from the stringent limit on flows found from the spectra of the Si i line. The reason is simple: magnetic fields are frozen to plasma under conditions of high magnetic Reynolds number, and motions of plasma must accompany any forcing no matter its origin. The collision time for a proton to exchange its momentum with a hydrogen atom is of order $4\times 10^{9} / \sqrt{T} n_H$ s. With T = 6000 K, n_H ≈ 10¹⁷ cm⁻³, this time is far smaller than any dynamical timescale (Gilbert et al. 2002). The partially ionized plasma behaves dynamically as a fully ionized plasma with the same charged particles, but with the additional neutral mass. Again therefore we must appeal to unresolved motions (i.e., waves). The fastest magnetosonic mode is the field-aligned modified sound wave when the plasma β > 1, so the same conclusions are drawn as for the non-magnetic case. The magnetic forces are either (a) incompatible with the observations of line profiles revealing down-flowing material with insufficient energy flux to account for the acoustic source, or (b) are in the form of modified high frequency sound waves with the same problems that the sound waves have.