Sun-as-a-star Analysis of H$\alpha$ Spectra of a Solar Flare Observed by SMART/SDDI: Time Evolution of Red Asymmetry and Line Broadening

We analyzed an M4.2-class solar flare which occurred in active region 12673 (AR 12673) at 01:05 UT on 2017 September 5. AR 12673 is known as the most flare-productive AR in solar cycle 24, producing many M- and X-class flares (Yan et al., 2018; Shen et al., 2018; Chertok et al., 2018; Soni et al., 2020; Yamasaki et al., 2021). Our target solar flare occurred when the AR 12673 was not far from the disk center of the Sun as in Figure 1. The detailed H$\alpha$ map is shown in Figure 2 (see, Section 2.2) and its magnetic field map is shown in Figure 3 (see, Section 2.3). We selected this event for the following reasons: (i) It was observed with the SMART telescope at Hida Observatory under good weather condition. (ii) The event occurred near the disk center. (iii) The flare emission is dominant, and the associated eruption phenomena and coronal rain are weak. Because the event satisfied the above criteria, we consider the event as a favorable solar flare to understand the nature of the flare ribbon in the Sun-as-a-star view.

Here we introduce the Sun-as-a-star analysis method of H$\alpha$ flare spectrum taken by SMART/SDDI in more detail than Namekata et al. (2022a). SDDI has been conducting a monitoring observation of the Sun since 2016. It takes the full-disk Sun images at 73 wavelength points at every 0.25 Å from $\lambda_{\rm cen}$$-$9.0 Å to $\lambda_{\rm cen}$+9.0 Å, where $\lambda_{\rm cen}$ is the H$\alpha$ line center wavelength. For this events, each set of images is observed with a time cadence of 15 seconds and a pixel size of 1.2 arcsec. Figure 1 is the full disk image of SDDI at 6553.8 Å. Figure 2 shows the partial images of the M4.2-class solar flare on 2017 September 5 for $\lambda_{\rm cen}$– 2 Å, $\lambda_{\rm cen}$, and $\lambda_{\rm cen}$ + 2 Å. The rightmost panels show the red asymmetry “RA”, which is defined as

$\displaystyle RA=\frac{I({\lambda_{\rm cen}+1\,{\rm\AA}},t,x,y)-I({\lambda_{\rm cen}-1\,{\rm\AA}},t,x,y)}{I({\lambda_{\rm cen}+1\,{\rm\AA}},t,x,y)+I({\lambda_{\rm cen}-1\,{\rm\AA}},t,x,y)},$

(1)

where $I(\lambda,t,x,y)$ is the intensity at the position ($x$,$y$), a wavelength $\lambda$, and time $t$ (cf., Asai et al., 2012).

In principle, the “real” Sun-as-a-star spectrum will be obtained by simply performing a full-disk integration at each wavelength. However, in the case of the full-disk integration, we could not obtain a clear spectral variation of the M4.2-class solar flare. This could be not only due to intrinsic brightness variation of the solar surface, but also due to noise in spectral profiles induced by atmospheric fluctuations and weather variations since each wavelength are not exactly observed at the same time (15 sec lag between $-$9 Å and $+$9 Å). Therefore, in our study, we spatially integrate the intensity inside a partial region of the SDDI images that are large enough to cover the visible phenomena (the magenta region is Figure 1, hereafter referred to as Flare Region “FR”). This is not exactly a Sun-as-a-star analysis, but it can be virtually treated as a Sun-as-a-star by finally normalizing it with the solar luminosity (also referred to as “virtual” Sun-as-a-star in Namekata et al., 2022a).

In the following, we will introduce the method of this “virtual” Sun-as-a-star analysis: First, we performed a pixel-level position correction for every image using non-flaring spots in AR 12673 as a marker in order to correct position changes due to solar rotation and seeing fluctuation. Before going to the spatial integration, each image was normalized by the value of median of the reference region (the green region in Figure 1) for each wavelength in order to cancel the variation due to the weather condition and solar altitude,

$\displaystyle I^{\prime}(\lambda,t,x,y)=\frac{I(\lambda,t,x,y)}{median_{x,y\in\rm ref.}({I}(\lambda,t,x,y))},$

(2)

where “ref.” means the reference region. Here, $I^{\prime}(\lambda,t,x,y)$ is 2D-image normalized by the median value of “each wavelength” of the reference region. Then, the 2D-image spectrum normalized by the median value of “continuum level” of the reference region $I_{\rm n}$($\lambda$, $t$, $x$, $y$) can be obtained as

	$\displaystyle I_{\rm n}(\lambda,t,x,y)$	$\displaystyle=$	$\displaystyle I^{\prime}(\lambda,t,x,y){\rm F_{\rm n}}(\lambda)$		(3)
		$\displaystyle=$	$\displaystyle\frac{I(\lambda,t,x,y)}{median_{x,y\in\rm ref.}({I}(\lambda,t,x,y))}{\rm F_{\rm n}}(\lambda),$		(4)

where F_n($\lambda$) is a template solar quiescent H$\alpha$ spectrum which is convolved with SDDI instrumental profile and normalized by the continuum level (Seki et al., 2017). Here, we define a spectrum normalized by the median value of continuum level of the reference region as $L(\lambda,t,{\rm region})$. We integrated the intensity inside the Flare Region “FR” (the magenta region in Figure 1) to obtain the spatially-averaged H$\alpha$ spectrum $L(\lambda,t,{\rm FR})$ as

$\displaystyle L(\lambda,t,{\rm FR})=\sum_{x,y\in\rm FR}{I_{\rm n}}(\lambda,t,x,y).$

(5)

In the same manner, the “real” Sun-as-a-star spectra (normalized by the median of the reference region continuum) can be obtained as

$\displaystyle L(\lambda,t,{\rm full\,disk})=\sum_{x,y\in{\rm full\,disk}}{I_{\rm n}}(\lambda,t,x,y),$

(6)

Again, it is difficult to measure a solar flare component with Equation 6 as mentioned above, so this full disk spectrum is used just to scale the Equation 5 to Sun-as-a-star. The correction factor to convert Equation 5 to the full disk continuum unit can be obtained as $L(\lambda_{\rm cont},t_{0},{\rm full\,disk})$, where $\lambda_{\rm cont}$ = $\lambda_{\rm cen}$ $-$ 9 Å is a wavelength of the continuum level, $\lambda_{\rm cen}$ is the H$\alpha$ center wavelength, and $t_{0}$ is a time in pre-flare phase. Here, a pre-flare-subtracted flaring spectrum at the time $t$ (normalized by the median of the reference region) can be expressed as $L(\lambda,t,{\rm FR})-L(\lambda,t_{0},{\rm FR})$ ($\approx$ $L(\lambda,t,{\rm full\,disk})-L(\lambda,t_{0},{\rm full\,disk})$). Therefore, a pre-flare-subtracted flaring spectrum normalized by the full disk continuum level, $\Delta S_{(}\lambda,t)$, can be expressed as

	$\displaystyle\Delta S(\lambda,t)$	$\displaystyle=$	$\displaystyle\frac{L(\lambda,t,{\rm FR})-L(\lambda,t_{0},{\rm FR})}{L(\lambda_{\rm cont},t_{0},{\rm full\,disk})}$		(7)
	$\displaystyle\Biggl{(}$	$\displaystyle\approx$	$\displaystyle\frac{L(\lambda,t,{\rm full\,disk})-L(\lambda,t_{0},{\rm full\,disk})}{L(\lambda_{\rm cont},t_{0},{\rm full\,disk})}\Biggr{)}.$		(8)

If we add a full disk spectrum (normalized by full disk continuum) to Equation 7, we can obtain the Sun-as-a-star spectrum of the H$\alpha$. However, the variations of flare is too small compared to the original solar full disk spectrum (an order of $\sim$10^-4). Therefore, we show Equation 7 as a flaring spectrum in this study, and hereafter, we sometimes call $\Delta S(\lambda,t)$ a Sun-as-a-star (flare) spectrum by omitting “pre-flare-subtracted” for simplicity. Note that the analysis method described above is the same as those in Namekata et al. (2022a)²²2Note that Equation 1 in Namekata et al. (2022a) has a typo, and $t_{0}$ is a correct expression in its denominator. We correct it here., but the expression of equations has been a bit improved in this paper. We calculated the equivalent width EW = $\int\Delta S(\lambda,t)d\lambda$ in the unit of Å. We also calculated the H$\alpha$ flare radiated energy by simply multiplying the EW by solar contiuum luminosity around H$\alpha$ (cf., Namekata et al., 2022a).

In this section, we briefly explain the analysis of the data obtained by the Helioseismic and Magnetic Imager (HMI) and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory ($SDO$). The purpose of analyzing these data is to overview the magnetic field configuration in which this event occurred and to investigate how this solar flare behaved in white light and UV in the Sun-as-a-star view.

Figure 3 shows the magnetic field configuration around the time of the flare (5 min after the onset) and the intensity of AIA 1600 Å. Four flare ribbons are observed in AIA 1600 Å (A, B, C, and D in the figure). According to the NLFF calculation of Yamasaki et al. (2021), the flare ribbons of A and B, B and C, and C and D are connected by magnetic field lines, and most of the flare radiation is emitted from flare ribbons B and C in AIA 1600 Å. Figure 2 shows that also in H$\alpha$, flare ribbons B and C are dominant, while flare ribbon A is visible.

Next, we show the Sun-as-a-star analysis of the associated white-light flare with the use of the SDO/HMI continuum (Namekata et al., 2017b). The colormap in Figure 3(d) shows the enhanced white-light flare emission (white region) with the AIA 1600 Å contour (red). We first made a time-integrated AIA 1600 Å image and made a mask image where the intensity is over 10 % of the peak intensity. By integrating the white-light enhancement inside the mask, we have obtained the total counts of the white-light flare $\rm DN_{\rm flare}$. After that, the white light enhancement as Sun-as-a-star ($\rm DN_{\rm flare}$/$\rm DN_{\rm\odot}$) was obtained by dividing it by the full disk count of the SDO/HMI continuum $\rm DN_{\rm\odot}$ (see Figure 6). The flare flux and energy were calculated in the same way as Namekata et al. (2017b) by assuming the 10,000 K blackbody radiation (Kretzschmar et al., 2010; Kretzschmar, 2011).

We also made a Sun-as-a-star light curve of AIA 1600 Å. The radiation source of AIA 1600 Å band is the continuum and emission lines such as C IV, which originates from the transition region and upper photosphere (Simões et al., 2019). Although the formation heigh of AIA 1600 Å is complicated with the contribution from multiple emission lines during flares (mostly from transition region via C IV lines; Simões et al., 2019), the rising phase of AIA 1600 Å and GOES X-ray light curves shows a relation of Neupert effect (Nishizuka et al., 2009; Qiu et al., 2013; Dai et al., 2018). Hence, the AIA 1600 Å light curve is plotted as a reference for impulsive energy release in the lower atmosphere (i.e. a proxy of non-thermal heating) in this paper (see Section 3.1).

We have employed two fitting methods to measure the line shift and line width of H$\alpha$ flaring emission using the pre-flare-subtracted spectra, $\Delta S(\lambda,t)$, defined in Section 2.2: (1) a two component fitting method (refer to as “2 Component” in each figure) and (2) a single wing component fitting (refer to as “Wing Component” in each figure). The method (1) is often used in stellar flare observations (Vida et al., 2019; Maehara et al., 2021; Wu et al., 2022), and the method (2) is similar to that used in solar flare observations (cf., Ichimoto & Kurokawa, 1984). The method (1) is the following procedure (see Figure 4 as example): First, the data on the short wavelength side of the H$\alpha$ center is fitted with the Voigt function to obtain the H$\alpha$ center component (orange line in Figure 4). At this time, the line width of the Voigt function is determined. Then, the remaining redside excess component is fitted with a Gaussian function to obtain its redshift velocity and intensity (green line in Figure 4). The line width is determined as widths of one tenths of peak intensity for the central Voigt function. The method (2) is the following procedure: Only the line wing component is fitted with a single-velocity Voigt function, and its line width (one tenths of peak intensity level) and shifted velocity are obtained. The wing criterion used for the fit is $<$0.2 of continuum level for the Sun-as-a-star data and $<$0.25 of continuum level for the spatially resolved data. The line width is simply determined as widths of one tenths of peak intensity of spectrum.

Figure 4 shows the test for the fitting methods for a modeled pre-flare-subtracted spectrum (a black solid line), which is calculated with the Becker’s cloud model (Beckers, 1964) with optically thickness $\tau$ of 5, velocity of 70 km s^-1, and source function $S/I_{0}$ of 1.2 (original spectrum is indicated with blue line). We applied the above fitting methods to the pre-flare-subtracted spectra with the aim of an application to stellar flares. In the case of the Sun where the background H$\alpha$ spectrum is usually absorption, the pre-flare-subtracted spectrum is usually shaped as a sum of the shape of flare spectrum (blue line) and that of the inverse of background spectrum (black dashed line). This results in an apparent enhancement of the central component in the pre-flare-subtracted spectrum (a black solid line), but would not significantly affect the estimation of shifted and/or broadened component. For this modeled spectrum in Figure 4, the estimated velocity was 83.2 km s^-1 and 62.8 km s^-1 with method (1) and (2), respectively. After the tests for several cases, we noticed that both methods have their own advantages and disadvantages as follows: The method (1) is a good fitting method when the flaring atmosphere has both stationary and redshift components (e.g., seen in numerical simulations; Kowalski et al., 2015). However, it should be noted that the subtraction of the stationary component can improperly remove the blue wing of the red-shift component. Therefore, the method can overestimate the velocity due to the overfitting of the central component when flaring atmosphere is a single-velocity atmosphere and/or when the velocity is low. The method (2) gives correct solution if both wings of Ha emission are produced by the redshift component, but underestimates the velocity in the case of multiple atmospheric components. We have tested how the methods (1) and (2) can estimate the input velocity with a simple optically-thick H$\alpha$ flaring spectrum for several cases, but it is difficult to say which method is better to obtain the real velocity as in Figure 4. Hence, we have applied both methods in this paper (see, Section 3.2, 3.3, and 3.4).

Figure 5 shows the time evolution of the Sun-as-a-star H$\alpha$ spectrum $\Delta S(\lambda,t)$ for the flare of interest obtained by the SDDI. We found that the spectra show the enhanced red wing enhancement even in the Sun-as-a-star view, in most of the periods. This is the same as the previous studies performing spatially-resolved analyses (e.g., Ichimoto & Kurokawa, 1984). In addition, the Sun-as-a-star spectrum shows the significant line broadenings during the flare. Qualitatively, both the redshift and broadening are prominent especially in early 0.5$\sim$4.5 minute (i.e., the impulsive phase), but as the flare decays, they become gradually weaker (see Section 3.2 for quantitative results).

Left panel of Figure 6 shows Sun-as-a-star light curves of the solar flare in H$\alpha$, white light, GOES X-ray, and AIA 1600 Å. The flaring amplitude in Sun-as-a-star H$\alpha$ EW is $\sim 4\times 10^{-4}$ Å. This is approximately the same order of the impulsive C-class flare event, i.e. $\sim 3\times 10^{-4}$ Å, in our previous study (see, Supplementary Figure 9 in Namekata et al., 2022a). The duration of the H$\alpha$ flare was approximately 10 minutes (FWHM duration is 4.7 minutes), and that of white-light flare is approximately 7 minutes (FWHM duration is 3.0 minutes). The AIA 1600 Å flare peak came first, followed by the H$\alpha$ and white light about a minute later, and then the GOES X-rays about a minute later. Rising phase of the AIA 1600 Å and GOES X-ray light curves have a relation suggested by the Neupert effect (see Figure 14 in Appendix A), so the AIA 1600 Å light curve can be regarded as a proxy of time profile of energy release of the flare.

The right panel of Figure 6 shows the light curve of flare luminosity in H$\alpha$, white light, and GOES X-ray (1–8 Å). The flare luminosity of H$\alpha$ (8.4$\times 10^{25}$ erg s^-1) and GOES X-ray (6.1$\times 10^{25}$ erg s^-1) have almost the same order ($\sim$10²⁶ erg s^-1). The flare luminosity of white light (7.2$\times 10^{27}$ erg s^-1) is an about two orders of magnitude larger than those of H$\alpha$ and GOES X-ray (1–8 Å). The integrated flare radiated energy are estimated as $1.6\times$10³⁰ erg in white light and $2.8\times$ 10²⁸ erg in H$\alpha$ (i.e. 1.8 % of the white-light emission). Since few studies have measured the energy of white light and H$\alpha$ in solar flares, such values are worth to be compared with those of stellar flares (e.g., Osten & Wolk, 2015; Guarcello et al., 2019). For example, the ratio of H$\alpha$ to white-light energy $\sim$1.8 % are comparable to those of stellar superflares on solar-type stars (0.85$\sim$1.5 %; Namekata et al., 2022a, b).

Figure 7 shows the example of a result of the spectral fitting for a Sun-as-a-star flare spectrum with the two fitting methods introduced in Section 2.4 and Figure 4. In the case of Figure 7, the 2 component fitting methods estimates the redshift velocity as 82.7 km s^-1, and the single wing component fitting methods estimates as 47 km s^-1. Figure 8 shows the time evolution of each parameter for Sun-as-a-star H$\alpha$ spectrum in comparison with H$\alpha$, white-light, and AIA 1600 Å light curve. The line width is ranging from 2.5 Å to 7.5 Å, and the redshift velocity is from 0 km s^-1 to 95 km s^-1 during the solar flare (see, Section 2.4). The two fitting methods result in different absolute values, but the qualitative properties are roughly similar. In particular, the Sun-as-a-star spectra shows the following properties:

• •

  The redshift velocity peaks out and decays more rapidly than the H$\alpha$ EW, and the time evolution is similar to near-ultraviolet luminosity (1600 Å), rather than H$\alpha$ and white light (Figure 8(b,d)).

• •

  The H$\alpha$ line width also peaks out and decays more rapidly than the H$\alpha$ EW, and the time evolution is nearly consistent to white-light luminosity, rather than H$\alpha$ and near ultraviolet (1600 Å) (Figure 8(a,c)).

• •

  The time evolution of EW of redshifted component is similar to that of EW of the central component (Figure 8(e)), and the ratio of redshifted component to central component is 14$\sim$54 % during the flare.

Figure 9 shows an example of the pre-flare subtraction of the spatially-resolved H$\alpha$ spectrum at a given flare kernel and a given time. As explained in Section 2.4, the pre-flare subtraction can apparently enhance the central component of the pre-flare-subtracted H$\alpha$ spectrum of flares (a black solid line) because the background (a black dashed line) is absorption, but would not significantly affect the evaluation of shifted and/or broadened component (see Section 2.4). We also analyzed the pre-flare-subtracted spectrum for spectral fitting in this spatially-resolved case. Figure 10 shows the spatial distribution of the line width (a,c) and redshift velocity (b,d) of H$\alpha$ lines in the flare ribbon obtained by applying the two methods for each pixel. We fitted the H$\alpha$ spectrum for each pixel and each time with the different two methods, and derived the temporal evolution of the spatial distribution. Figure 11 shows the line width (a,c) and redshift velocity (b,d) of H$\alpha$ lines as a function of H$\alpha$ line center intensity. The estimated line width is ranging from $\sim$2.5 Å to $\sim$14 Å, and the redshift velocity is from 0 km s^-1 to $\sim$230 km s^-1. The cross and triangle symbols indicate the time evolution of the pixels indicated in Figure 10 with the same symbols ((x,y) = (31,33) and (30,42), respectively). Also, the color of each dot corresponds to the phase of the flare in each pixel (see the color bar). Histograms for earlier and later phases are shown on the right and on the top with red and blue lines.

We found that the line width has a correlation with the intensity of H$\alpha$ (correlation coefficient of 0.68 and 0.45 for the two methods), but the redshift velocities have almost no correlation with the intensity of H$\alpha$ (correlation coefficient of 0.17 and $-$0.18 for the two methods). Asai et al. (2012) also showed that red asymmetry (RA; Equation 1) is independent of intensity, although they did not derive the velocity itself. On the other hand, if we look at each pixel individually, we can see that there is an evolutionary track through the flare phases. For example, if we look at the triangle and cross symbols, the evolutionary track is that line width and redshift velocity increase first, then intensity increases later, and finally they gradually decay. This is apparent in Figure 11(b,c,d), but not apparent in Figure 11(a). Especially for redshift velocity, Asai et al. (2012) and Ichimoto & Kurokawa (1984) reported the same property, i.e., redshift velocity peaks out and decays more rapidly than intensity.

Here we compare line width and redshift calculated from Sun-as-a-star (Section 3.2) spectra and spatially-resolved spectra (Section 3.3). Qualitatively, the variations of both spectra show very similar evolutionary track: line width and redshift velocity peak out and decay more rapidly than EW. On the other hand, the spatially resolved case has significantly wide range of values, while the Sun-as-a-star case has relatively smaller values than the upper limit of the spatially-resolved case. This would be because the Sun-as-a-star spectrum is seen as a superposition as discussed in Section 4.1 in more detail.

Then, which regions of the flare represent the Sun-as-a-star spectrum? The left panel of Figure 12 shows the comparison between Sun-as-a-star spectrum (black line) and integrated spectra for a different H$\alpha$ intensity levels at a given time. This shows that the Sun-as-a-star spectrum is quite far from the spectrum from the region of high intensity level ($>$45 % of the maximum intensity, N_pixel=35 pixels), which shows strong redshift and line broadening. On the other hand, the Sun-as-a-star spectrum seems to correspond fairly well with the spectrum from the region where the intensity level is weak, especially 15-30 % intensity level (N_pixel=65 pixels). The right panel of Figure 12 is the same as the left panel, but it is normalized by the peak of Sun-as-a-star spectrum. This shows that where the intensity level is weak, the area of the region (number of pixels) is large, so the contribution to the Sun-as-a-star spectrum is high. Therefore, as a result of the superposition, the Sun-as-a-star spectrum becomes close to the shape of the spectrum from the region with the weak intensity level (in this case, 15-30 %).

Figure 13 shows the time evolution of line width and redshift velocity for each intensity levels. The comparison shows that the Sun-as-a-star spectrum well corresponds to the spectrum from 15-30 % (or, 5-15 %) intensity level. It is also found that the line width of the Sun-as-a-star spectrum is about a factor of 2 (2-5 Å) smaller than the strong-intensity region of the flare ribbon (e.g. $>$ 60 %). Moreover, in the case of the two component fitting, the redshift velocity of the Sun-as-a-star spectrum is about 20-40 km s^-1 slower than that of the core region of the flare ribbon (e.g. $>$ 60 %). Note that the redshift velocities obtained by single wing component fitting correspond well to the Sun-as-a-star spectra in all regions except the brightest core at the flare onset.

Section 3.4 shows that the region with weak intensity level (here we call “halo”) is more significantly contributing to Sun-as-a-star spectrum than the region with strong intensity level (here we call “core”). It is true that the core of the flare ribbon is bright, but its contribution to the Sun-as-a-star radiation is not necessarily large. This indicates that the red asymmetry and line width that can be observed in stars do not represent the bright “core” component but “halo” component, and underestimates the “core” value by a factor of $\sim$2. Our first message to stellar flare observation is that we need to keep in mind the above point when we interpret stellar flare spectra and compare with numerical model (e.g., Kowalski et al., 2017; Namekata et al., 2020; Wu et al., 2022). Note that only for the redshift velocity calculated with the single-wing-component fitting gives good correspondence between Sun-as-a-star values and spatially-resolved values (see Figure 13(d)). This may indicate that the single-wing-component fitting can better estimates the evolutions of flare ribbons in stellar flare observations, although it still may miss the highest velocity component at the onset of flares.

In Section 3.2, we show the evolutionary track of the redshift velocity of the Sun-as-a-star H$\alpha$ spectra, i.e., redshift velocity peaks out and decays more rapidly than H$\alpha$ intensity or EW. Section 3.3 and 3.4 shows the same feature seen even in the spatially-resolved spectrum, which was also previously reported not only in H$\alpha$ (e.g., Ichimoto & Kurokawa, 1984) but also in other chromospheric and transition-region lines (e.g., Zhou et al., 2020). These mean that Sun-as-a-star H$\alpha$ spectra of solar flares well reflect the feature of the spatially-resolved H$\alpha$ spectrum, even though it is formed as a result of superposition of different positions as discussed in Section 4.1. According to previous studies, the fact that the increase in redshift velocity precedes the increase in intensity is thought to represent the properties of chromospheric condensation process, where thickness of the chromospheric condensation increase with the rapid decrease of its velocity (Livshits et al., 1981; Somov et al., 1982; Ichimoto & Kurokawa, 1984; Kowalski et al., 2022). Our study suggests that even the Sun-as-a-star spectra retain the characteristics of chromospheric condensation. This predicts that we can investigate occurrences of chromospheric condensation even in stellar flares from the time-resolved H$\alpha$ spectroscopic observations.

The redshift velocity shows similar changes to Sun-as-a-star light curve of the near-ultraviolet rays at 1600 Å, rather than Sun-as-a-star light curve of white-light flare. The AIA 1600 Å has a sensitivity to the transition region radiation (C IV lines) and its time evolution shows very impulsive peak during the impulsive phase, which is represented by the time derivative of the GOES X-ray (Nishizuka et al., 2009; Qiu et al., 2013; Dai et al., 2018). Our study suggests that the time evolution of redshift velocity can be a good proxy for heating in the upper chromosphere and transition region (or possibly non-thermal heating) in the Sun-as-a-star view. Although hard X-rays, an indicator of non-thermal heating, is not observable in stellar flares, the near-ultraviolet wavelength can be observed in stellar flares by the Ultraviolet Optical Telescope onboard the Swift spacecraft (Osten et al., 2010) and the Optical/UV Monitor Telescope onboard XMM-Newton (cf., Mitra-Kraev et al., 2005), so we propose that these features could be tested in stellar flares in the future.

Recently, there are reports on red asymmetry on stellar flares (Houdebine et al., 1993; Vida et al., 2019; Koller et al., 2020; Wu et al., 2022), but its origin has not been well understood partly because of the lack of time-resolved spectroscopic observations or multi-wavelength observations are limited. Our study can provide a basis for interpreting the observed red asymmetry on stellar flares. If the time variation of redshift of stellar flares is the same as that of this result, we can conclude that it comes from flare ribbon radiation, while if it is not, we can suggest the possibility of other sources of redshift radiation, such as flare loop or prominence activation. For example, Wu et al. (2022) reported that redshifted velocity of H$\alpha$ line asymmetry during a superflare on an M-type star showed more rapid decay than H$\alpha$ EW. This observation indicates that the observed red asymmetry of the superflare may originate from chromospheric condensation region. Here, we need to be a bit careful when applying our result on the Sun to other stars whose background H$\alpha$ line shows emission (active M-type stars) since we performed an analysis of pre-flare subtraction of spectra in this study. If the background is emission, pre-flare-subtraction is thought to subtract the H$\alpha$ line center component of the flare more than necessary. In this case, although the discussion of the qualitative nature of line asymmetry/broadening (velocity and line width) is applicable to stars (see Section 2.4), the ratio of line center intensity to red asymmetry intensity is expected to be different from those obtained in this solar study.

Sun-as-a-star H$\alpha$ spectra shows that the line width decays more rapidly than EW (Section 3.2), and the spatially-resolved spectra show the same feature (Section 3.3 and 3.4). This also supports the suggestion that Sun-as-a-star H$\alpha$ spectra of solar flares well inform the feature of the spatially-resolved H$\alpha$ spectrum even after the superposition. Similar time evolution has been reported on M-dwarf flares (e.g., Namekata et al., 2020; Wu et al., 2022), and this study suggests that the chromospheres on the M-dwarf flares could be heating in the same way as the M4.2-class solar flare analyzed here.

It is also found that the line width peaks out and decays in the similar way to the Sun-as-a-star light curve of white light flare (or near-ultraviolet flare), rather than H$\alpha$ EW. In the case of solar flares, line broadening is thought to be caused mainly by the statistical Stark effect (Kowalski et al., 2017, 2022), and the line width is sensitive to electron density $n_{e}$ in chromospheric condensation ($\propto n_{e}^{2/3}$). Also, if we assume that the white-light emission is radiated from the same chromospheric condensation via hydrogen free-bound or free-free process, its radiation intensity reflects the chromospheric density ($\propto n_{e}^{2}$). Therefore, the nearly similar time evolution of white-light flare and line width may be understood as both are related to the density of chromospheric condensation. Note, however, that the origin of white-light emission is still controversial even in solar physics, further studies are necessary for the physical interpretation.