The morphology of average solar flare time profiles from observations of the Sun's lower atmosphere
Larisa K. Kashapova, Anne-Marie Broomhall, Alena I. Larionova, Elena G. Kupriyanova, Ilya D. Motyk
aa r X i v : . [ a s t r o - ph . S R ] F e b MNRAS , 1–8 (2020) Preprint 5 February 2021 Compiled using MNRAS L A TEX style file v3.0
The morphology of average solar flare time profiles fromobservations of the Sun’s lower atmosphere
Larisa K. Kashapova, Anne-Marie Broomhall ★ , Alena I. Larionova, Elena G. Kupriyanova , and Ilya D. Motyk Institute of Solar-Terrestrial Physics, SB Russian Academy of Sciences, Lermontov str. 126a, 664033, Irkutsk, Russia Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, UK Department of Radio Astronomical Research, Central Astronomical Observatory at Pulkovo of the RAS,Pulkovskoe Shosse 65/1, Saint Petersburg 196158, Russian Federation Faculty of Physics, Irkutsk State University, 20 Gagarin Blvd., Irkutsk, 664003, Russia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study the decay phase of solar flares in several spectral bands using a method based on thatsuccessfully applied to white light flares observed on an M4 dwarf. We selected and processed102 events detected in the Sun-as-a-star flux obtained with SDO/AIA images in the 1600 Åand 304 Å channels and 54 events detected in the 1700 Å channel. The main criterion for theselection of time profiles was a slow, continuous flux decay without significant new bursts.The obtained averaged time profiles were fitted with analytical templates, using different timeintervals, that consisted of a combination of two independent exponents or a broken powerlaw. The average flare profile observed in the 1700 Å channel decayed more slowly than theaverage flare profile observed on the M4 dwarf. As the 1700 Å emission is associated with asimilar temperature to that usually ascribed to M dwarf flares, this implies that the M dwarfflare emission comes from a more dense layer than solar flare emission in the 1700 Å band.The cooling processes in solar flares were best described by the two exponents model, fittedover the intervals t1=[0, 0.5] 𝑡 / and t2=[3, 10] 𝑡 / where 𝑡 / is time taken for the profile todecay to half the maximum value. The broken power law model provided a good fit to the firstdecay phase, as it was able to account for the impact of chromospheric plasma evaporation,but it did not successfully fit the second decay phase. Key words:
Sun: flares – Sun: atmosphere – Sun: photosphere – Sun: Chromosphere
Flares are explosive events that occur in solar and stellar atmo-spheres. Flares are observed across a wide range of wavelengths,such as radio, visible, X-rays and gamma rays and the emissionresponsible for these observations originates from many differentregions within solar and stellar atmospheres, from photospheres tocoronae. It is generally believed that both solar and stellar flaresare produced by the same mechanism, namely magnetic reconnec-tion (Shibata & Magara 2011). However, discrepancies are readilyobserved, most notably the energy of the flares, which, for stellarflares, can often be several orders of magnitudes larger than eventhe largest solar flares.While stellar flares have been studied for decades, it is onlyrecently, through observations made by NASA’s
Kepler mission(Borucki et al. 2010), that a statistically large sample of flares for asingle star has been obtained. Davenport et al. (2014) detected over ★ E-mail: [email protected]
Kepler data.Using a subset, containing 885 flares, which were classified as “clas-sical” because they contained just a single peak, Davenport et al.created an empirical flare template, that was well-represented bya polynomial in the fast rising phase, and two exponential decays,one each for the impulsive and gradual decay phases respectively.The first decay phase corresponds to radiative cooling losses andthe second one is thought to be related to thermal conduction losses(Aschwanden 2004). Considering the large number of flares used,the scatter around this flare template is remarkably low.
Kepler obtained broadband white light observations (between4200 and 9000 Å) of stellar intensity and has vastly increased thenumber of solar-like stars on which flares have been observed,prompting a number of statistical studies (e.g. Maehara et al. 2015;Notsu et al. 2019), many of which consider flaring rates, with a par-ticular interest in determining how likely the Sun is to produce anenergetic superflare. While flares are a common feature of
Kepler light curves, the same cannot be said for white light flare obser-vations of the Sun. Solar white light flares are rare because they © Kashapova et al. tend to be relatively short in duration (a few minutes) and have alow contrast, making “Sun-as-a-star” observations of white lightflares particularly challenging. Nevertheless, Kretzschmar (2011)performed a statistical study of Sun-as-a-star flares observed in anumber of data sets, spanning a range of wavelengths, includingTotal Solar Irradiance (TSI). Kretzschmar demonstrated that whitelight emission is ubiquitous in solar flares, regardless of the en-ergy of that flare, and that, while there is some dependence onflare strength, the blackbody temperature of solar flares is between8 , ,
000 K, which is similar to estimates for M dwarf flaretemperatures (Hawley & Fisher 1992). In a similar analysis to thatof Davenport et al., Kretzschmar produced average flare profiles,using 2,100 flares, and found that the TSI profile only containedthe impulsive phase, with no evidence of the gradual decay phase.Kretzschmar (2011) speculated that this could be because the grad-ual decay was below the noise level.However, using resolved observations of the Sun, it has beenshown that white light flares are common even for relatively weakflares (e.g. Matthews et al. 2003; Hudson et al. 2006; Jess et al.2008). Kawate et al. (2016) use highly resolved white light obser-vations of a single flare to produce a large number of light curveswithin the flaring region. The authors find that 58% of the lightcurves can be well represented by a single-component decay phase,while the other 42% are better represented by two components, akinto the morphology found by Davenport et al. (2014). Kawate et al.also found that, where two phases are favoured, the cooling timescorrespond to those expected for the chromosphere and corona re-spectively (Xu et al. 2006).Namekata et al. (2017) used the
Helioseismic and MagneticImager (HMI) onboard the
Solar Dynamics Observatory (SDO)to produce white light flare light curves, finding that, for a givenenergy, solar white light flares are an order of magnitude longerthan stellar superflares observed with
Kepler . Although, it is worthnoting that the energies of the solar and stellar flares do not over-lap, and so this result is based on extrapolations. Furthermore, thesolar light curves were obtained by selecting spatial regions corre-sponding to the flare, rather than using Sun-as-a-star data. Althoughpotential explanations for the discrepancy are proposed in terms ofdifferences in cooling or magnetic field strength, the authors remainunderstandably cautious, stating that a better understanding of themechanisms responsible for white light emission would help clarifythe situation.The exact emission mechanisms responsible for white lightflares are not yet totally clear (see discussions in Kleint et al. 2016;Kawate et al. 2016, and references therein). It is generally assumedthat continuum white light emission originates from the region en-compassed by the mid-photosphere and lower chromosphere. Whitelight flares can show good temporal correspondence with hard X-ray emission (e.g. Fang & Ding 1995), implying that energetic elec-trons play an important role. While these electrons are capable ofreaching the chromosphere, the mechanism by which the energy istransported to the photosphere is still debated. To determine howanalogous solar flare morphology is to the white light stellar flaresobserved by Davenport et al. (2014) this article considers 304 Å,1600 Å and 1700 Å data from the
Atmospheric Image Assembly (AIA; Lemen et al. 2012) instrument onboard the SDO. Althoughthese wavelengths are not traditionally considered as white light,and are outside the waveband observed by
Kepler , the lines areused here because they originate from various locations within thephotosphere and chromosphere (as described in Section 2).Although TSI and Sun-as-a-star SDO/HMI data may be moreakin to Kepler data, solar flares are infrequently observed in these data, even in the case of large eruptive events. As a result, analysesoften rely on assumed theoretical models. For example, Emslie et al.(2012) had to complement the direct measurements of bolometricirradiance by estimations based on modelling. Moreover, the tem-poral resolution of SDO/AIA data is less than 1 minute, in contrastto TSI. This condition is important for the analysis of time profilesof solar flares which are more dynamic than stellar ones.A better characterisation of the underlying shape of a flarepermits studies of more transient flare light curve features, such asquasi-periodic pulsations (QPPs; see Van Doorsselaere et al. 2016;Kupriyanova et al. 2020, for recent reviews), which are notoriouslydifficult to detect and characterise robustly. Detection mechanismsthat rely on detrending or model fitting (e.g. Dominique et al. 2018;Pascoe et al. 2017; Broomhall et al. 2019) would benefit from in-formation concerning the underlying flare shape.This article aims to investigate the solar-stellar flare connec-tion, and emission mechanisms associated with white light flaresby comparing the flare profile associated with various layers ofthe Sun’s lower atmosphere with the flare profile associated withwhite light observations of flares on an M dwarf obtained byDavenport et al. (2014). The target of the study was to providean instrument for the analysis of cooling during the decay phaseand revealing cases related to additional sources of energy release.As previously mentioned, we use SDO/AIA data from the 1600 Å,1700 Å and 304 Å channels. In Section 2, we describe the data inmore detail, including how they were combined to determine me-dian flare profiles for each channel. These median flare profiles arethen fitted with both a combination of two exponential functions anda broken power law, as described in Section 3. The fits are discussedin detailed Section 4, and the main conclusions are summarised inSection 5.
We obtained the total flux of the Sun-as-a-star using the imagesobtained by AIA (Lemen et al. 2012) on-board SDO. The channelsused here demonstrate the chromospheric and photospheric emis-sion (O’Dwyer et al. 2010): 1600 Å (transition region and upperphotosphere), 1700 Å (temperature minimum, photosphere) and304 Å (chromosphere, transition region). The image cadence is 24 sfor the 1600 Å and 1700 Å channels and 12 s for the 304 Å channel.The initial flare selection was performed using the GOES flarecatalogue . Each event was visually inspected and was required tohave a “classical" flare time profile, consisting of a fast rise followedby a slow decay without any flattening or additional peaks in softX-rays (SXR). When determining which flares to include in oursample the time interval onset was taken to be about 10 minutesbefore the flare maximum in GOES X-ray flux and the durationwas defined as the length of time taken for the flux to decrease tothe pre-flare level. More rigorous time scales were defined in thesubsequent analysis, as described below. The initial list consisted of359 flares from B5 to X9.3 GOES class.After processing the AIA images with the standard package ofSolarSoftware (Bentely & Freeland 1998; Freeland & Handy 1998,SSW), we obtained the total flux of the whole image for each timemoment and produced a preliminary time profile of the flare. Thus,we processed the AIA images as if they were Sun-as-a-star or with-out an extracting the flare area and the resultant flux was equivalent https://hesperia.gsfc.nasa.gov/goes/goes_event_listings/ MNRAS000
We obtained the total flux of the Sun-as-a-star using the imagesobtained by AIA (Lemen et al. 2012) on-board SDO. The channelsused here demonstrate the chromospheric and photospheric emis-sion (O’Dwyer et al. 2010): 1600 Å (transition region and upperphotosphere), 1700 Å (temperature minimum, photosphere) and304 Å (chromosphere, transition region). The image cadence is 24 sfor the 1600 Å and 1700 Å channels and 12 s for the 304 Å channel.The initial flare selection was performed using the GOES flarecatalogue . Each event was visually inspected and was required tohave a “classical" flare time profile, consisting of a fast rise followedby a slow decay without any flattening or additional peaks in softX-rays (SXR). When determining which flares to include in oursample the time interval onset was taken to be about 10 minutesbefore the flare maximum in GOES X-ray flux and the durationwas defined as the length of time taken for the flux to decrease tothe pre-flare level. More rigorous time scales were defined in thesubsequent analysis, as described below. The initial list consisted of359 flares from B5 to X9.3 GOES class.After processing the AIA images with the standard package ofSolarSoftware (Bentely & Freeland 1998; Freeland & Handy 1998,SSW), we obtained the total flux of the whole image for each timemoment and produced a preliminary time profile of the flare. Thus,we processed the AIA images as if they were Sun-as-a-star or with-out an extracting the flare area and the resultant flux was equivalent https://hesperia.gsfc.nasa.gov/goes/goes_event_listings/ MNRAS000 , 1–8 (2020) verage solar flare time profile Figure 1.
Time profile of solar flare emission at 1600 Å. Red data points arethe individual data from the 102 flares in the sample, thick blue line showsthe median values and the thin green lines show the interquartile ranges. to data of instruments observing without spatial resolution. To en-able a combined analysis of the flares of different strength andduration, the profiles were normalised in both flux and time. First,each time profile was normalised using the flux maximum observedin each flare, meaning the maximum in normalised flux was unityfor all flares in the sample. Second, we defined the time moment ofthe flux maximum as zero, so the time of the rise phase had negativevalues and the decay phase time had positive values. Finally, to getthe same time scale for events of different duration, we presentedthe time series in normalised units defined by the time taken forthe intensity to decrease to half the maximum for each time profile( 𝑡 / ), following the methodology of Davenport et al. (2014). Thenthe duration of each time profile was limited to a range of –5 to10 time bins ( 𝑡 / ). The time profiles in each channel were againchecked to ensure there were no additional peaks during the decayphase that were missed when selecting the initial sample. The cri-teria for inclusion in the final sample was that any fluctuations inthe time profile should be less than 30% of the flux value. We usedthe time profile derivative for control of this criterion. Applicationof this criterion also minimises uncertainties in the determinationof the 𝑡 / value. As we processed solar flares from X to B GOESclass and saturation in emission could not be excluded, this currentcriteria allowed us to escape significant contribution from saturationwhen forming time profiles.The final sample contained 105 events in total from the 1600 Åand 304 Å channels (104 from 1600 Å and 102 from 304 Å). Allprofiles are presented on the same plot, with data shown as redcrosses in Figures 1 and 3 for the 1600 Å and 304 Å channels,correspondingly. The X class flare ratio is 6%, the M class flareratio is 58%, the C class flare ratio is 35% and B class flare ratio is1%. Only 53 time profiles of emission in the 1700 Å channel showedboth significant response and satisfied the all criteria outlined above(see Figure 2).To construct the average time profile we re-sampled time pro-files to a time resolution of 0.001 𝑡 / using linear interpolation, as Figure 2.
Time profile of solar flare emission at 1700 Å. Line colours anddata symbols are as described in Figure 1, except only 54 flare profiles wereincluded.
Figure 3.
Time profile of solar flare emission at 304 Å. Line colours anddata symbols are as described in Figure 1. was done by Davenport et al. (2014). The average time profiles weredefined as the median flux value computed at each time bin and areshown in Figures 1–3 as blue lines. One can see that the dispersionof initial data is significantly higher in the 1700 Å band plot, despitecontaining a smaller number of flare profiles.The standard errors, as shown in Figures 1–3, are given bythe median of values above and below the average value for high( 𝑒𝑟 ℎ ) and low values ( 𝑒𝑟 𝑙 ), correspondingly. This is also referredto as the interquartile range. As these errors are mainly asymmetric MNRAS , 1–8 (2020)
Kashapova et al.
Figure 4.
Comparison of the obtained time profiles of solar flare emissionwith the time profile of the M4 dwarf. during the decay phase we used half of the sum of these two values( 𝜎 = ( 𝑒𝑟 ℎ + 𝑒𝑟 𝑙 )/
2) as the error for the function fitting. The time-averaged time profiles obtained for the solar flares demonstrate thesimilar behaviour to the M4 dwarf flare time profile obtained byDavenport et al. (2014) (see in Figure 4). During the decay phase,the evolution is fully agreed up to 𝑡 / equal about 1 . After thismoment, the solar intensity decays slower in all three wavelengthbands relative to the M4 dwarf flare emission. Gryciuk et al. (2017) determined an analytic template of the decayphase of a solar flare composed of a single exponent with various co-efficients taking into account peculiarities of the flare profile shape.The template was derived and applied to soft X-ray observationsthat describe coronal plasmas. However, the decay phase of a solarflare is a composition of the cooling process and various heatingprocesses, especially in lower layers of the solar atmosphere. As ourtime profiles clearly demonstrated a steep initial decay followed bya slower decline in flux, we decided to apply the two-phase model.This model was successfully used by Davenport et al. (2014) for theflares observed on an M4 dwarf. Despite the difference in the atmo-spheric structure (temperature and density stratification) betweenthe Sun and M dwarfs, the flare evolution is similar in both cases(Allred et al. 2006).The average time profiles, obtained using the natural logarithmof flux, are presented in Figures 5–7. The flux data were fitted usingtwo straight lines, 𝑓 ( 𝑡 ) = 𝑎 exp 𝑏 𝑡 / and 𝑓 ( 𝑡 ) = 𝑎 exp 𝑏 𝑡 / ,corresponding to the two different decay phases. Initially, the sametime regions as Davenport et al. (2014) were used for the fitting.These ranges are 0 < 𝑡 / < . 𝑓 ( 𝑡 ) and 3 < 𝑡 / < 𝑓 ( 𝑡 ) . The results are shown in red in Figures 5–7. In addition,alternative time regions were chosen to improve the fitting of eachphase: 0 < 𝑡 / < . 𝑓 ( 𝑡 ) (shown in blue) and 3 < 𝑡 / < Figure 5.
Average (median) time profile of solar flare emission at 1600 Å(black with error bars) fitted by two exponential functions for differentperiods. Red, dashed lines and red text in the legend correspond to thet1=[0, 0.5] 𝑡 / and t2=[3, 6] 𝑡 / . Blue dashed lines and blue text in thelegend correspond to t1=[0, 1.5] 𝑡 / , and light green solid lines and lightgreen text in the legend correspond to t2=[3, 10] 𝑡 / . Table 1.
Results of fitting by two exponential functions 𝑓 = 𝑎 exp − 𝑏 𝑡 / and 𝑓 = 𝑎 exp − 𝑏 𝑡 / .Fit interval 1600 Å 1700 Å 304 Å 𝑡 / <0.5 𝑎 ± ± ± 𝑡 / <1.5 𝑎 ± ± ± 𝑡 / <0.5 𝑏 ± ± ± 𝑡 / <1.5 𝑏 ± ± ± 𝑡 / <6 𝑎 ± ± ± 𝑡 / <10 𝑎 ± ± ± 𝑡 / <6 𝑏 ± ± ± 𝑡 / <10 𝑏 ± ± ± for 𝑓 ( 𝑡 ) (shown in green). The results of the fitting are shown inTable 1.We also fitted the time profiles with a broken power-law modelto reveal the time of the phase change and with the aim of finding abetter description of the average time profiles. The function was: 𝑓 ( 𝑡 ) = 𝐴 ( exp index × 𝑡 / for 𝑡 < 𝑡 break , exp index × 𝑡 / × exp ( index − index )× 𝑡 / for 𝑡 > 𝑡 break , (1)where 𝑡 break , index and index are all parameters determined by thefitting process.Table 2 shows the results of fitting equation 1 to the median flareprofiles, with 𝜒 values. A comparison of fitting a broken power-lawmodel with the previous fitting, of two independent exponents, ispresented in Figure 8. One can see that the broken power-law modeldescribing the decay phase of the average time profiles providesa good fit to the data until 𝑡 = 𝑡 / . We note that the brokenpower-law function is a better fit to the flare profile at around 2 𝑡 / . MNRAS000
Results of fitting by two exponential functions 𝑓 = 𝑎 exp − 𝑏 𝑡 / and 𝑓 = 𝑎 exp − 𝑏 𝑡 / .Fit interval 1600 Å 1700 Å 304 Å 𝑡 / <0.5 𝑎 ± ± ± 𝑡 / <1.5 𝑎 ± ± ± 𝑡 / <0.5 𝑏 ± ± ± 𝑡 / <1.5 𝑏 ± ± ± 𝑡 / <6 𝑎 ± ± ± 𝑡 / <10 𝑎 ± ± ± 𝑡 / <6 𝑏 ± ± ± 𝑡 / <10 𝑏 ± ± ± for 𝑓 ( 𝑡 ) (shown in green). The results of the fitting are shown inTable 1.We also fitted the time profiles with a broken power-law modelto reveal the time of the phase change and with the aim of finding abetter description of the average time profiles. The function was: 𝑓 ( 𝑡 ) = 𝐴 ( exp index × 𝑡 / for 𝑡 < 𝑡 break , exp index × 𝑡 / × exp ( index − index )× 𝑡 / for 𝑡 > 𝑡 break , (1)where 𝑡 break , index and index are all parameters determined by thefitting process.Table 2 shows the results of fitting equation 1 to the median flareprofiles, with 𝜒 values. A comparison of fitting a broken power-lawmodel with the previous fitting, of two independent exponents, ispresented in Figure 8. One can see that the broken power-law modeldescribing the decay phase of the average time profiles providesa good fit to the data until 𝑡 = 𝑡 / . We note that the brokenpower-law function is a better fit to the flare profile at around 2 𝑡 / . MNRAS000 , 1–8 (2020) verage solar flare time profile Figure 6.
Average (median) time profile of solar flare emission at 1700 Åfitted by two exponential functions for different periods. The line and textcolours correspond to the same period as on Figure 5.
Figure 7.
Average (median) time profile of solar flare emission at 304 Åtwo exponential functions for different periods. The line and text colourscorrespond to the same period as on Figure 5.
Table 2.
Results of fitting with a broken power law function1600 Å 1700 Å 304 Åindex ± ± ± ± ± ± break ± ± ± 𝜒 Figure 8.
Average (median) time profiles (black) fitted by the broken power-law model (green line) and two exponential functions as shown in Figures5–7 (dashed blue lines) for t1=[0, 0.5] 𝑡 / and t2=[3, 6] 𝑡 / . The panelsfrom the top to the bottom present 1600 Å, 1700 Å and 304 Å bands,correspondingly. The red solid line marks the time interval of the secondphase fitting. In this study we obtained averaged time profiles and fitted templatesdescribing the decay phase of a solar flare for different spectralbands. As previously mentioned, there are two main mechanismsdetermining the cooling processes of a flare decay: radiative cool-ing and thermal conduction (Aschwanden 2004). According to themodel of plasma cooling, thermal conduction losses initially dom-inates during the first part of the decay phase. Later we observethe domination of radiation losses during the second phase (see,Cargill et al. 1995; Aschwanden & Tsiklauri 2009). Cargill et al.(1995) also noted that if radiative losses dominate during the initialphase of decay, they still dominate during the entire decaying phase.However, here we can discriminate between the steep (called impul-sive by Davenport et al. 2014) and gradual phases during the decayphase of the analysed time profiles. For this reason, we focusedour study on the fitting of the averaged time profiles with modelsconsisting of two components only, as done in previous studies (seee.g. Davenport et al. 2014).Domination of thermal conduction or radiative losses relates
MNRAS , 1–8 (2020)
Kashapova et al. to the cooling times due to these processes, and depends on tem-perature, density and loop length in the case of thermal conduction.Thus, the flux behaviour during the decay phase depends on twofactors — the ratio between cooling by radiation and cooling by ther-mal conduction and the ratio between the temperature and densityof the generating emission region.The temperature of spectral lines that dominate in the formationof emission of the chosen spectral bands depends on the formationheight of the emission. According to O’Dwyer et al. (2010), emis-sion in the 304 Å band is mainly formed by the doublet H II line(303.78Å) that has log ( 𝑇 ) = . ( 𝑇 ) > ( 𝑇 ) = . ( 𝑇 ) = .
85 contributes to flare emission.However, its fraction of total emission was estimated as 0.05.The quiet Sun emission in the 1600 Å band is generated pre-dominantly by the C IV line and continuum emission, both withlog ( 𝑇 ) =
5. This temperature value implies that the level wherethe emission of this band is formed is higher than the formationheight of the 304 Å band. The continuum dominates the emissionof the 1700 Å band for quiet Sun regions. Thus, the emission isformed at log ( 𝑇 ) = .
7, and it typically originates from the pho-tosphere. The results obtained by Simões et al. (2019) refined thecontribution of the spectral lines emitted within the 1600 Å and1700 Å bands. The authors confirmed the share of the C IV doubletemission to the 1600 Å band, but they also revealed the dominationof the C I 1656 Å multiplet contribution over the continuum forthe 1700 Å band. This means that emission in the 1700 Å band ismostly formed at log ( 𝑇 ) = ( 𝑇 ) ≈ 𝑏 of decay for solar fluxis lower than that obtained by Davenport et al. for the M4 dwarfeven in the case of the 1700 Å band (0.800 vs 𝑡 / decreasedthe coefficient 𝑏 to 0.668 for the 1700 Å band. We note that thedifference between the coefficients obtained for different time inter-vals exceeds the error bars. Therefore, cooling during the first steepphase was found to be even slower for solar flares than for flareson the M4 dwarf. As the 1700 Å channel and M dwarf flares areassociated with similar temperatures this implies that the plasmaresponsible for M4 dwarf flare emission is denser than that asso-ciated with the 1700 Å solar flares, and so potentially originatesfrom a deeper layer. The fact that the coefficient decreases whenthe fitting range is extended could be related to the impact of chro-mospheric evaporation, which would cause an increase in flux ataround 1–3 𝑡 / .Fitting a broken power law to the 1700 Å band gave the decaycoefficient index equal to 0.714 and the time of the phase changeas about 1.3 𝑡 / . This implies that a new phase with a different rate Figure 9.
Comparison of the obtained time profile of solar flare emissionat 304Å with result of modelling by Allred et al. (2006). of cooling onsets after 1.3 𝑡 / . The coefficient index for this phase,according to the fitting by a broken power law, is about 0.2. Thisvalue is close to the value obtained for M4 dwarf flare (0.29) butit does not fit time profile values above 5 𝑡 / . This could be due tothe impact of the evaporation of the heated chromospheric plasmato the flare time profile (see, for example, Fletcher et al. 2011) . Aspredicted by the modelling of Allred et al. (2006) for the 304 Åband, the contribution of the evaporation for solar flare flux is moresignificant than for M dwarfs. Figure 9 contains a comparison of the304 Å band and the appropriate model from Allred et al. (2006),where we can see the model flux drop below the observed flux ataround 𝑡 = 𝑡 / . Then the impact of chromospheric evaporationis seen in the form of an increase in the modelled flux. Althoughwe note the model flux again drops below the observed flux for 𝑡 > 𝑡 / .In the case of fitting for t1=[0, 0.5] 𝑡 / , the relation betweenthe decay coefficients of the first phase for the 1600 Å, 304 Åand 1700 Å bands confirms that the temperature of the emissionformation is as discussed above. The value of 𝑡 break in the brokenpower-law fitting characterises the transfer from the radiative cool-ing to the contribution of chromospheric evaporation. We obtainedthe lowest value of 𝑡 break for the 1700 Å band (1.282) and the highestvalue for the 1600 Å band. This is the same ordering as suggestedby the temperature-height model of the solar flare atmosphere. Analternative way to specify a temperature-height model is throughan analysis of the delay between maximums of flare emission indifferent spectral ranges. The delay between signals obtained inthe UV and EUV bands are actively used for analysis of height-temperature dependence of solar atmosphere over the sunspot (see,Reznikova et al. 2012). As one can see in Table A1, the delay be-tween the maximum of flares in 304Å band and 1600Å band canbe both positive and negative. The most common time delay was17 seconds but a group of events demonstrated a negative delay ofabout 7 seconds. The uncertainties related to an image cadence of 24seconds are 14 seconds (using the same approach as Reznikova et al.(2012)). Thus we can conclude that negative delays are within error MNRAS , 1–8 (2020) verage solar flare time profile bars, and most of the positive delays are around this value. Suchsmall delays, close to the level of uncertainties, agree with classicalF1 and F2 flare models by Machado et al. (1980) suggesting a heightdifference between the layers with temperatures corresponding to304Å and 1600Å bands of about 3 km.When fitting the two exponents to the median flare profiles,we obtained two sets of parameters for each of the first and secondphases by fitting over different time intervals (see Table 1). Thedifference between the parameters obtained for the same phase, butfor the different time intervals, exceeds the parameter uncertainty.Thus, it is necessary to determine which time intervals should befitted to best represent the cooling processes. We believe that thedecay during the second phase should be fitted using parametersobtained for t2 = [3, 10] 𝑡 / because the flux takes longer to decayin the solar atmosphere compared to the M dwarf flare profile. To re-veal the decaying related to radiative and thermal conduction lossesduring the first phase, we should avoid the chromospheric evapora-tion contribution to emission. As mentioned above, the breakpoint, 𝑡 break in the broken power-law fitting parameters characterises thetransfer from the radiative cooling to the contribution of the chro-mospheric evaporation. For both the 1700 Å and 304 Å bands, theobtained values of 𝑡 break were below 1.5. While, for these bands, theparameters obtained by fitting the different time intervals showeda significant difference, the parameters obtained by fitting the dif-ferent time intervals for the 1600 Å band, where 𝑡 break was above1.5, did not demonstrate a significant difference. All these facts in-dicate the impact of chromospheric evaporation to time profiles isnot negligible (depending on observational wavelength), meaningthat fitting over the interval t1 = [0, 1.5] 𝑡 / is not appropriate andthe range t1 = [0, 0.5] 𝑡 / is favourable.Based on a comparative analysis of the fitting results for solarand M dwarf flare templates, we can conclude that, for the Sun,the optimal template describing cooling processes consists of twoexponents, fitted for t1 = [0, 0.5] 𝑡 / and t2 = [3, 10] 𝑡 / respectively.We note that the second cooling phase of solar flares turnedout to be more complicated than for M4 dwarf flares. The coolingof the 304 Å band during the second phase is marginally fasterthan for the 1600 Å band, which is hotter and originates from lessdense plasma. It is possible that, during cooling the relative con-tribution of spectral lines with different temperatures to the emis-sion of the 1600 Å band changes with time, which results in afaster decrease of emission flux. We also would like to note thatthe 304 Å band time profile demonstrated unusual behaviour. Asemission of this band mostly forms by the emission of a singlespectral line (O’Dwyer et al. 2010), its time profile should be moreakin to density evolution (Aschwanden & Tsiklauri 2009). This factis confirmed by modelling (see Allred et al. 2006). However, theobserved time profile shows an exponential decay, which is morecharacteristic of temperature evolution.As mentioned above, there are two processes, namely ther-mal conduction or radiation losses, and domination of one of theprocess over the other would result in different behaviours of tem-perature. The temperature evolution during radiative cooling can bedescribed as 𝑇 ( 𝑠, 𝑡 ) = 𝑇 ( 𝑠 )[ − ( − 𝛼 ) 𝑡 / 𝜏 𝑟 ] /( − 𝛼 ) , where 𝜏 𝑟 is the radiative cooling time at the start of the radiative phase, 𝛼 isthe coefficient of radiative losses function and 𝑠 is the coordinatealong the magnetic field (here and after, Cargill et al. 1995). Thecoefficient 𝛼 is assumed to equal to -0.5 for log(T)>5. As the tem-perature range of the data analysed here is less this value, we cannotuse this assumption. The dependence 𝑇 ( 𝑡 ) = 𝑇 ( + 𝑡 / 𝜏 𝑐 ) − / de-scribes the temperature evolution for the case of static conductivecooling (where 𝜏 𝑐 is the conductive cooling at the beginning). We Figure 10.
Average (median) time profiles (black) fitted by functions oftemperature evolution depended on radiative( blue) and conductive (red)losses.MNRAS , 1–8 (2020)
Kashapova et al. performed fitting using these two functions for the analysed spectralbands, using 𝜏 𝑟 , 𝜏 𝑐 and ( − 𝛼 ) as variables. The results can beseen in Figure 10. The averaged time profiles for all bands are bet-ter fitted by the function corresponding to radiative cooling. Onlythe 304Å band time profile demonstrates good agreement with thefitting based on conductive cooling for a short time interval when 𝑡 < . 𝑡 / . Based on the results of reconstruction and analysis of averagedtime profiles of solar flare decay phases obtained within 304 Å,1600 Å and 1700 Å spectral bands, we can conclude the following: • The processes of cooling during the decay phase of solar flaresare the most closely described by fitting of a combination of twoindependent exponents for t1 = [0, 0.5] 𝑡 / and t2 = [3, 10] 𝑡 / . • The parameters obtained for the 1700 Å solar flare time profile,which is theoretically closest in temperature to the M dwarf flares,are consistent with models that imply that the white light M dwarfflare emission is formed in a higher density of layer. • Fitting a broken power-law model allows the contribution ofchromospheric evaporation to be taken into account. However, it isnot sufficient for a full description of the second part of the decayphase.
ACKNOWLEDGEMENTS
This research was partly supported by the grant of the RussianFoundation for Basic Research No. 17-52-10001. A.M.B. acknowl-edges the support of the Royal Society International Exchangesgrant IEC/R2/170056. This research was partly supported by thebudgetary funding of Basic Research programs No. II.16 (LKK).
REFERENCES
Allred J. C., Hawley S. L., Abbett W. P., Carlsson M., 2006, ApJ, 644, 484Aschwanden M. J., 2004, Physics of the Solar Corona. An IntroductionAschwanden M. J., Tsiklauri D., 2009, ApJS, 185, 171Bentely R. D., Freeland S. L., 1998, in Crossroads for European Solarand Heliospheric Physics. Recent Achievements and Future MissionPossibilities. p. 225Borucki W. J., et al., 2010, Science, 327, 977Broomhall A.-M., et al., 2019, ApJS, 244, 44Cargill P. J., Mariska J. T., Antiochos S. K., 1995, ApJ, 439, 1034Davenport J. R. A., et al., 2014, ApJ, 797, 122Dominique M., Zhukov A. N., Dolla L., Inglis A., Lapenta G., 2018,Sol. Phys., 293, 61Emslie A. G., et al., 2012, ApJ, 759, 71Fang C., Ding M. D., 1995, A&AS, 110, 99Fletcher L., et al., 2011, Space Sci. Rev., 159, 19Freeland S. L., Handy B. N., 1998, Sol. Phys., 182, 497Gryciuk M., Siarkowski M., Sylwester J., Gburek S., Podgorski P., Kepa A.,Sylwester B., Mrozek T., 2017, Sol. Phys., 292, 77Hawley S. L., Fisher G. H., 1992, ApJS, 78, 565Hudson H. S., Wolfson C. J., Metcalf T. R., 2006, Sol. Phys., 234, 79Jess D. B., Mathioudakis M., Crockett P. J., Keenan F. P., 2008, ApJ,688, L119Kawate T., Ishii T. T., Nakatani Y., Ichimoto K., Asai A., Morita S., MasudaS., 2016, ApJ, 833, 50Kleint L., Heinzel P., Judge P., Krucker S., 2016, ApJ, 816, 88Kretzschmar M., 2011, A&A, 530, A84 Kupriyanova E., Kolotkov D., Nakariakov V., Kaufman A., 2020,Solar-Terrestrial Physics, 6, 3Lemen J. R., et al., 2012, Sol. Phys., 275, 17Machado M. E., Avrett E. H., Vernazza J. E., Noyes R. W., 1980, ApJ,242, 336Maehara H., Shibayama T., Notsu Y., Notsu S., Honda S., Nogami D.,Shibata K., 2015, Earth, Planets, and Space, 67, 59Matthews S. A., van Driel-Gesztelyi L., Hudson H. S., Nitta N. V., 2003,A&A, 409, 1107Namekata K., et al., 2017, ApJ, 851, 91Notsu Y., et al., 2019, ApJ, 876, 58O’Dwyer B., Del Zanna G., Mason H. E., Weber M. A., Tripathi D., 2010,A&A, 521, A21Pascoe D. J., Anfinogentov S., Nisticò G., Goddard C. R., Nakariakov V. M.,2017, A&A, 600, A78Reznikova V. E., Shibasaki K., Sych R. A., Nakariakov V. M., 2012, ApJ,746, 119Shibata K., Magara T., 2011, Living Reviews in Solar Physics, 8, 6Simões P. J. A., Reid H. A. S., Milligan R. O., Fletcher L., 2019, ApJ,870, 114Van Doorsselaere T., Kupriyanova E. G., Yuan D., 2016, Sol. Phys.,291, 3143Xu Y., Cao W., Liu C., Yang G., Jing J., Denker C., Emslie A. G., Wang H.,2006, ApJ, 641, 1210
APPENDIX A: DETAILS OF FLARES IN SAMPLE
Table A1 contains details of the flares in our sample.
DATA AVAILABILITY
The datasets were derived from sources in the public do-main: SDO/AIA data was obtained from Joint ScienceOperations Center (JSOC) ( http://jsoc.stanford.edu );GOES data were obtained from the GOES Flare Catalogue( https://hesperia.gsfc.nasa.gov/goes/goes_event_listings/ ). This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000
The datasets were derived from sources in the public do-main: SDO/AIA data was obtained from Joint ScienceOperations Center (JSOC) ( http://jsoc.stanford.edu );GOES data were obtained from the GOES Flare Catalogue( https://hesperia.gsfc.nasa.gov/goes/goes_event_listings/ ). This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000 , 1–8 (2020) verage solar flare time profile Table A1.
List of analysed events. 𝑇 𝑚𝑎𝑥 - UT time, 𝑡 / - minutes.N Date 𝑇 𝑚𝑎𝑥 GOES Location 304 Å 1600 Å 1700 Åyy/mm/dd GOES class 𝑇 𝑚𝑎𝑥 𝑡 / 𝑇 𝑚𝑎𝑥 𝑡 / 𝑇 𝑚𝑎𝑥 𝑡 /
001 11/02/14 17:26 M2.2 N56W18 17:25:23 4.8 17:25:06 2.4 17:25:20 4.4002 11/02/16 14:25 M1.6 S20W32 14:24:11 2.4 14:23:54 2.8 14:23:44 4.4003 11/03/08 22:23 M1.0 N31W76 22:20:11 5.0 22:19:54 4.8004 11/03/14 19:52 M4.2 N18W48 19:51:23 1.2 19:51:30 0.8 19:51:20 1.2005 11/04/15 13:51 C2.8 N13W24 13:50:21 1.0 13:50:18 0.8006 11/07/30 02:09 M9.3 N14E35 02:09:23 2.4 02:07:54 2.4 02:08:08 2.4007 11/08/04 03:57 M9.3 N19W36 03:54:11 12.6 03:52:42 6.0 03:54:08 5.6008 11/08/09 08:05 X6.9 N17W69 08:06:35 3.0 08:02:18 3.2 08:02:08 3.6009 11/09/04 01:07 C8.3 N18W79 01:08:11 3.6 01:05:30 5.2010 11/09/23 23:56 M1.9 N11E52 23:50:42 2.0011 11/09/24 09:40 X1.9 N12E60 09:36:21 1.8 09:36:18 0.8 09:36:08 1.6012 11/09/25 15:33 M3.7 N16E43 15:31:47 3.4 15:31:06 2.8 15:31:44 3.6013 11/09/26 05:08 M4.0 N13E34 05:07:23 2.2 05:06:42 2.4 05:06:32 2.4014 11/09/26 14:46 M2.6 N14E30 14:44:10 11.4 14:42:42 6.0015 11/11/15 22:35 M1.1 N20W80 22:32:59 2.0 22:32:42 3.2016 11/11/16 18:54 C2.9 S18E16 18:54:11 6.6 18:53:54 3.6017 11/11/16 21:43 C1.6 S19E13 21:40:45 3.2 21:40:42 0.8018 11/11/20 11:55 C3.0 S14E41 11:48:59 8.8 11:49:06 1.6019 11/11/22 04:04 C4.9 N13E50 04:02:59 1.4 04:03:06 0.8020 11/11/29 03:32 C2.1 N19E20 03:30:09 3.6 03:30:18 2.4021 11/12/05 15:18 C4.7 S20E00 15:17:47 1.0 15:17:06 0.8022 11/12/05 19:07 C2.5 S20W02 19:06:35 1.2 19:06:18 0.4023 11/12/18 02:05 C1.9 N19W04 02:05:23 1.8 02:04:18 2.8024 11/12/24 08:45 C5.2 S21W20 08:32:57 8.2 08:33:06 8.8 08:34:08 10.0025 11/12/25 18:16 M4.0 S22W26 18:15:23 1.0 18:14:42 1.2 18:14:56 0.8026 11/12/27 04:22 C8.9 S17E32 04:16:35 14.8 04:16:18 6.8027 11/12/28 14:25 C7.2 S23E85 14:23:23 2.2 14:23:30 3.2028 11/12/30 08:25 C3.4 S26E62 08:22:59 2.6 08:23:06 3.6029 11/12/31 13:15 M2.4 S25E46 13:13:23 2.8 13:13:06 1.6 13:13:20 2.4030 12/03/08 02:53 C7.2 S18W03 02:52:59 1.8 02:52:42 1.2 02:52:32 1.6031 12/03/14 15:21 M2.8 N14E05 15:19:23 12.4 15:17:06 6.8 15:16:56 8.8032 12/05/09 14:08 M1.8 N06E22 14:06:35 0.6 14:06:18 0.8 14:06:32 0.8033 12/05/10 04:18 M5.7 N13E22 04:16:58 0.6 04:16:42 1.2 04:16:32 2.0034 12/05/15 22:16 C3.0 N13W61 22:15:33 6.2 22:15:30 2.4035 12/06/07 15:43 C9.1 N13W06 15:40:35 7.6 15:37:30 5.6036 12/06/08 07:17 C4.8 N13W40 07:16:11 1.6 07:15:30 1.6037 12/06/29 06:48 C6.2 S16E60 06:47:47 2.0 06:46:42 3.2038 12/06/29 09:20 M2.2 N17E37 09:20:11 1.8 09:19:54 1.6 09:20:08 1.6039 12/07/01 19:18 M2.8 N14E04 19:15:46 5.4 19:15:53 4.0040 12/07/02 10:52 M5.6 S17E08 10:49:47 2.8 10:48:41 3.2 10:48:55 4.0041 12/07/05 11:44 M6.1 S22E68 11:43:46 3.4 11:43:53 1.6 11:43:43 2.8042 12/07/09 23:07 M1.1 S19E39 23:06:58 1.8 23:06:41 1.2 23:06:55 0.8043 12/07/13 12:19 C1.5 S22W27 12:18:58 1.4 12:17:53 0.4044 12/07/27 04:02 C5.0 N17E22 04:01:05 2.0045 12/10/23 03:17 X1.8 S15E59 03:15:46 0.8 03:15:53 0.4046 12/10/23 13:15 C2.3 S15E51 13:14:20 3.2 13:13:53 2.8 13:13:19 5.6047 12/11/13 02:04 M6.0 S24E47 02:03:22 2.6 02:02:41 1.2 02:02:31 1.6048 12/11/13 05:50 M2.5 S29E43 05:47:46 2.0 05:47:29 1.6 05:47:43 3.2049 12/11/17 18:10 C2.8 S24W18 18:09:22 4.4 18:09:29 3.2 18:11:19 3.6050 12/11/18 13:07 C4.3 N09E11 13:06:58 1.8 13:06:17 2.0051 12/11/20 12:41 M1.7 N11W90 12:38:44 2.0 12:39:29 0.8052 12/11/24 13:40 C3.3 N09W26 13:36:44 4.4 13:36:41 0.8053 13/01/13 00:50 M1.0 N18W18 00:50:10 0.4 00:48:41 1.6 00:48:31 1.6054 13/04/05 17:48 M2.2 N07E88 17:42:34 7.0 17:40:41 5.2 17:40:55 6.0055 13/05/02 05:10 M1.1 N10W26 05:04:58 1.8 05:04:41 1.2056 13/05/19 15:15 C6.3 S09W63 15:15:22 10.4 15:15:05 1.2057 13/05/20 15:06 M1.7 N13W08 15:03:46 2.6 15:03:29 2.4058 13/05/20 16:26 C6.0 N13W09 16:24:58 2.0 16:25:29 1.2059 13/06/24 11:32 C9.9 S17E54 11:31:22 1.6 11:31:29 2.0 11:31:19 3.2060 13/07/07 00:58 C6.1 S14E07 00:58:34 2.2 00:58:17 2.0 00:58:31 2.4MNRAS , 1–8 (2020) Kashapova et al.
N Date 𝑇 𝑚𝑎𝑥 GOES Location 304 Å 1600 Å 1700 Åyy/mm/dd GOES class 𝑇 𝑚𝑎𝑥 𝑡 / 𝑇 𝑚𝑎𝑥 𝑡 / 𝑇 𝑚𝑎𝑥 𝑡 /
061 13/10/28 14:05 M2.8 N07W85 14:02:58 1.8 14:03:05 0.8 14:02:55 1.2062 13/11/02 04:46 C8.2 S23W04 04:44:34 1.8 04:44:17 1.6063 13/11/05 22:12 X3.3 S12E46 22:11:47 0.2 22:11:29 0.8 22:11:43 0.4064 13/12/29 07:56 M3.1 S18E01 07:53:22 2.2 07:53:29 1.6 07:53:19 4.0065 14/01/01 18:52 M9.9 S14W47 18:46:58 12.2 18:47:05 6.0 18:47:19 6.8066 14/01/03 12:50 M1.0 S04E52 12:48:58 5.8 12:49:05 4.4067 14/01/07 10:13 M7.2 S13E11 10:11:46 7.6 10:11:29 0.8 10:11:43 1.2068 14/01/28 04:09 M1.5 S15E88 04:08:10 1.0 04:07:53 1.2069 14/01/28 07:31 M3.6 S10E75 07:30:58 1.0 07:30:41 1.2070 14/02/01 01:25 M1.0 S11E26 01:23:46 5.4 01:23:29 3.2071 14/02/02 18:11 M3.1 S10E08 18:09:22 2.2 18:10:17 1.2 18:10:07 2.8072 14/02/07 04:56 M2.0 S15W50 04:54:58 5.4 04:54:41 4.0073 14/02/13 01:40 M1.7 S12W09 01:37:45 4.0074 14/02/13 06:07 M1.4 S11W11 06:06:10 2.8 06:05:53 3.2075 14/02/25 00:49 X4.9 S12E82 00:45:29 4.0 00:45:43 5.6076 14/03/11 03:50 M3.5 N13W55 03:49:46 3.8 03:49:29 2.0 03:49:19 2.8077 14/03/20 01:57 C8.3 S11E75 01:56:58 2.4 01:57:05 1.6078 14/03/30 11:55 M2.1 N08W43 11:49:46 13.6 11:48:41 4.4079 14/05/06 04:32 C7.1 S13W86 04:32:56 10.8 04:30:17 4.8080 14/06/11 08:09 M3.0 S14E68 08:06:10 1.8 08:05:53 2.8 08:05:43 5.6081 14/06/12 09:37 M1.8 S25W53 09:35:46 5.2 09:35:29 1.2 09:35:43 0.8082 14/08/01 14:48 M2.0 S09E35 14:47:47 0.8 14:47:29 0.4 14:47:19 0.8083 14/08/25 11:18 B7.8 S12E41 11:18:20 3.0 11:17:53 1.6084 14/10/16 13:03 M4.3 S15E84 13:02:34 0.6 13:02:41 0.8 13:02:31 1.2085 14/10/22 05:17 M2.7 S15E14 05:14:32 0.2 05:14:41 0.4 05:14:31 0.4086 14/10/23 09:50 M1.1 S16E03 09:47:46 2.6 09:47:29 0.8 09:47:19 2.0087 14/10/23 19:15 C3.3 S21E05 19:14:44 2.4 19:14:41 2.0088 14/10/30 17:58 C3.4 S05E70 17:56:56 4.0 17:57:05 3.2089 14/11/09 07:20 C4.4 N18E19 07:19:56 2.0 07:17:53 4.0090 15/01/29 18:15 C5.0 N08E63 18:14:20 2.6 18:14:17 1.2091 15/01/30 12:16 M2.4 N07E52 12:14:34 1.4 12:14:17 1.6 12:14:31 1.6092 15/03/09 14:33 M4.5 S15E49 14:30:58 1.0 14:31:05 3.2093 15/03/12 04:46 M3.2 S15E11 04:43:22 2.2 04:43:29 0.8 04:43:19 2.0094 15/05/05 22:11 X2.7 N15E79 22:09:22 1.4 22:08:41 2.0 22:08:31 2.4095 15/08/22 21:24 M3.5 S15E15 21:22:33 1.8 21:22:16 2.4 21:22:06 3.6096 15/08/28 19:03 M2.1 S13W70 19:03:21 1.0 19:03:04 1.2 19:02:54 1.6097 15/09/29 11:15 M1.6 S21W37 11:13:21 4.2 11:13:28 2.4 11:13:18 1.6098 15/09/29 19:24 M1.1 S20W36 19:24:09 2.2 19:23:28 2.0 19:23:18 2.4099 17/04/02 02:46 C8.0 S12W08 02:45:18 2.4 02:45:03 1.6100 17/04/02 13:00 M2.3 N13W61 12:56:56 3.2 12:56:39 3.2 12:56:29 6.4101 17/04/03 14:29 M5.8 N16W78 14:23:42 3.6 14:23:51 2.8 14:23:41 4.0102 17/09/05 01:08 M4.2 S09W14 01:06:56 1.4 01:06:39 2.0 01:06:53 4.0103 17/09/05 17:43 M2.3 S09W24 17:42:30 5.4 17:41:51 4.0104 17/09/07 10:15 M7.3 S07W46 10:15:44 1.0 10:15:27 1.2 10:15:17 1.6105 17/09/08 02:24 M1.3 S09W54 02:22:56 1.8 02:22:39 1.6 MNRAS000