TIME-RESOLVED PROPERTIES AND GLOBAL TRENDS IN dMe FLARES FROM SIMULTANEOUS PHOTOMETRY AND SPECTRA^*

Magnetically active M dwarfs are those with a persistent chromosphere often diagnosed by Hα or Ca ii H and K line emission even outside of flares. Chromospheric line emission is attributed to strong magnetic fields (∼few thousand Gauss) covering sometimes ∼50% or more of the stellar surface (Saar & Linsky 1985; Johns-Krull & Valenti 1996). These active M dwarfs regularly produce flare emission across the electromagnetic spectrum, from soft X-rays (∼0.4–50 keV) to the radio (∼10 MHz–10 GHz). Due to the low photospheric background at blue and NUV wavelengths, white-light flares on M dwarfs produce a large contrast which facilitates flare detection and reduces the contribution of quiescent (non-flare) emission. The contrast of the flare emission against the quiescent background is known as the "flare visibility" (Gershberg 1972) and makes the Johnson U-band filter preferred for flare studies (Moffett 1974). The U-band flare energy comprises ∼1/6 of the white-light energy (Hawley & Pettersen 1991), which in turn dominates the energy observed at shorter wavelengths, such as in the EUV and soft X-ray (Hawley et al. 1995; Fuhrmeister et al. 2011). Active M dwarfs with spectral subtypes dM3e–dM6e⁸ have been found to flare frequently with good visibility (Lacy et al. 1976) and thus are the main targets for flare monitoring.

Flare light curves are typically divided into an impulsive phase and a gradual decay phase (Moffett 1974; Moffett & Bopp 1976). The impulsive phase consists of a fast rise lasting tens of seconds or more, a peak, and a fast decay. The quasi-exponential, gradual decay phase begins with a transition from fast to slow decay (Hawley & Pettersen 1991) and can last from minutes to hours. These two phases comprise the classical flare light curve morphology, although much more complex light curves are observed (e.g., Moffett 1974; Kowalski et al. 2010).

Stellar flares produce greatly enhanced emission in chromospheric lines, such as the hydrogen Balmer (HB) series, Ca ii H and K, and He i. These lines are typically associated with chromospheric temperatures ranging from ∼6000–20, 000 K. The hydrogen lines have a fast rise phase but typically peak several minutes after the peak of the (U-band) continuum emission (Kahler et al. 1982; Hawley & Pettersen 1991; García–Alvarez et al. 2002). The energy in the emission lines is only a small percentage (∼4%) of the total FUV to optical flare energy in the impulsive phase but larger (∼17%) in the gradual decay phase (Hawley & Pettersen 1991), indicating that the major channel of atmospheric cooling occurs through continuum radiation for the flare duration. Some flares produce line emission that contributes a larger percentage, ∼30%–50%, of the total energy (Hawley et al. 2007). Broadening of the Balmer line profiles has been observed, with full widths at the continuum level that approach ∼20 Å for large flares (Hawley & Pettersen 1991; Fuhrmeister et al. 2008). The broadening of hydrogen (and helium) lines has been interpreted as an indication of turbulent or directed mass motions of tens to several hundred km s⁻¹ (Doyle et al. 1988; Eason et al. 1992; Fuhrmeister et al. 2008) or Stark broadening due to the electric fields from increased electron densities on the order of 10¹³–10¹⁴ cm⁻³ (Švestka 1972; Worden et al. 1984; see also Kurochka & Maslennikova 1970). Broadening of the Ca ii lines is not observed (Paulson et al. 2006), but the total Ca ii K flux exhibits a characteristic late peak after the Balmer lines at the beginning of the continuum gradual phase (Hawley & Pettersen 1991). In almost all previous studies, the entire Balmer series has not been captured simultaneously in order to achieve wavelength coverage in the blue, usually at the expense of Hα. Eason et al. (1992), Crespo-Chacón et al. (2006), Fuhrmeister et al. (2011) have provided data covering most of the Balmer series (the Hβ line was not included in the studies of Eason et al. 1992 and Fuhrmeister et al. 2011), but with the red and blue data obtained at significantly different cadence. The relative flux in each hydrogen line (the Balmer decrement) over the duration of the flare is an important diagnostic for the evolution of electron densities (Drake & Ulrich 1980).

Candidates for the emission mechanism that produces the white-light continuum have been described by Cram & Woods (1982), Giampapa (1983), and Nelson et al. (1986), and include blackbody (BB), hydrogen free-free (ff), hydrogen bound-free (bf), and H⁻ bound-free radiation processes. Inverse Compton scattering of quiescent infrared radiation from relativistic electrons has also been proposed (Gurzadian 1988), but this "fast-electron hypothesis" has not been confirmed with X-ray observations (Mullan 1990). The emission types have been constrained using multicolor Johnson broadband photometry (hereafter, "colorimetry") of flares with energies ranging from E_U ∼ 10³¹ erg to E_U ∼ 10³⁴ erg. Hawley & Pettersen (1991) and Hawley & Fisher (1992) used UBVR photometry and IUE SWP/LWP data to study the continuum shape evolution during a larger (10³⁴ erg) flare on the dM3e star AD Leo, finding that the peak flux occurs in the U band and a significant amount of flux (27% of the total) is observed in the NUV. The broadband distribution was fit very well by a blackbody with T = 9000–9500 K in the impulsive (rise, peak) phase and T = 8400–8800 K in the gradual decay phase. The data from this flare also indicated a reddening of the continuum in the gradual decay phase, and this was suggested to be the result of the presence of two (or more) competing emission mechanisms, including a contribution from hydrogen (Paschen continuum) recombination radiation.

A two-component model was first proposed using simultaneous colorimetry and spectra of dMe flares by Kunkel (1970)—see also Moffett & Bopp (1976)—who concluded that a single, isothermal (T_e = 3000–30, 000 K) optically thin hydrogen emission (bf+ff) model was too blue to explain the observed flare colors, nor could it account for the spread of colors among a sample of flares. Instead, they proposed a model consisting of a dominant component of hydrogen bf (recombination) radiation with a secondary contribution from a heated photosphere, which increases in relative contribution over time during the flare decay. Other studies have similarly concluded from colorimetry that flare radiation consists of a combination of hot blackbody emission (with temperatures as high as 13, 000–18, 000 K) and optically thin Balmer continuum (BaC) recombination radiation (de Jager et al. 1989; Zhilyaev et al. 2007). It has been speculated that the blackbody is short-lived and the BaC becomes more dominant in the gradual decay phase (de Jager et al. 1989; Abdul-Aziz et al. 1995; see also Abranin et al. 1997; Zhilyaev et al. 2007), but the direct characterization of both components (i.e., with spectra) has thus far not been possible. Furthermore, Allred et al. (2006) recently showed that continuum constraints using colorimetry are fraught with degeneracies; a model spectrum that has a large Balmer jump (due to hydrogen bf radiation) may exhibit the shape of a hot, blackbody with T_BB ∼ 9000 K when convolved with broadband filters.

Previously, spectra of the Balmer jump (covering ∼1000 Å near 3646 Å) have been obtained during several large flares on dMe stars: AD Leo (Hawley & Pettersen 1991), UV Ceti (Eason et al. 1992), Gl 866 (Jevremovic et al. 1998), AT Mic (García–Alvarez et al. 2002), CN Leo (Fuhrmeister et al. 2008), and YZ CMi (Doyle et al. 1988). None of these studies showed conclusive evidence of a component that could be attributed to hydrogen (Balmer) recombination radiation, in contrast to the findings of Kunkel (1970). Interestingly, Eason et al. (1992) noted that the BaC appeared in absorption,⁹ a property that we study in this paper.

Flux-calibrated spectra at wavelengths redder than the Balmer jump has revealed blackbody temperatures consistent with those subsequently inferred from colorimetry. Mochnacki & Zirin (1980) used a multichannel spectrophotometer to map the evolution of the hot blackbody component, which they speculated may be dominant at flare maximum. They found that the rise phase could be caused by increasing area coverage of the hot component while the decay phase is explained by both rapidly decreasing temperature (from 9500 K at peak to 5500–7000 K in the decay) and relatively constant area. They were unable to accurately observe the BaC due to spectral vignetting, but they did note a smaller Balmer jump than predicted by Kunkel (1970) and found the decay of NUV emission was slower than in the optical, perhaps indicating two components in action. Similar temperatures of 8000–11, 000 K have been directly measured from blue/optical spectra (at λ > 4000 Å) (Kahler et al. 1982; Katsova et al. 1991; Paulson et al. 2006; Kowalski et al. 2010). de Jager et al. (1989) determined a temperature of 16, 000 K from (non-flux calibrated) spectra of a ΔB  ∼ 5 mag flare on UV Ceti. During a different ΔU  ∼ 5 mag event on UV Ceti, Eason et al. (1992) concluded that an optically thick thermal-bremsstrahlung (hydrogen ff) model with T  ∼  13, 000 K gave the best fit to their spectrum.

Direct spectra of the blackbody peak are not yet available due to the difficulty of observing in the NUV at λ < 3200 Å, but spectra around the Balmer jump have provided several important clues. The spectra from Hawley & Pettersen (1991) of AD Leo showed that the flux distribution increases toward the blue with little (if any) observed Balmer jump; Hawley & Fisher (1992) suggested that the peak lies somewhere between λ = 3000–3500 Å. Other spectra indicate the possibility that the continuum peaks outside the atmospheric window, λ < 3250 Å (Fuhrmeister et al. 2011). The largest observed flux is emitted within the U band (Hawley & Fisher 1992; Hawley et al. 2003), and the blackbody fits indicate a peak flux at λ ∼ 3000 Å. Fuhrmeister et al. (2008) and Schmitt et al. (2008) measured the NUV shape of the flare continuum down to the atmospheric limit at λ ∼ 3250 Å with VLT/UVES and found a temperature of 11,300 K; the temperature fits have large uncertainties (∼4000 K), possibly due to the long integration times, narrow wavelength range (λ = 3250–3800 Å), and lack of a robust flux calibration (without standard stars) for these echelle data. The same authors also measured the continuum shape in red, higher cadence spectra and found temperatures that were different than from the NUV: 20, 000–27, 000 (± 5000) K at peak and 3200–5600 K during the decay. Fuhrmeister et al. (2011) observed a flare on Proxima Centauri with a similar VLT/UVES setup and did not find a good blackbody fit to the flare spectrum.

In Kowalski et al. (2010), we reported initial results from a "Megaflare" on the dM4.5e star YZ CMi. In particular, we found evidence for both BaC and hot blackbody emission which varied in strength during secondary flare heating events. See Section 6 for additional discussion of this flare.

The origin of the T  ∼  10, 000 K blackbody component is currently unknown, and indeed its existence has been contested by van den Oord et al. (1996) and Nelson et al. (1986) on grounds that it requires very high heating fluxes from a solar-type flare heating beam of ∼5 × 10¹¹ erg s⁻¹ cm⁻². However, we know today that fluxes larger than this are possible even on the Sun (Neidig et al. 1993; Krucker et al. 2011). The inferred areal coverages of the blackbody component are small, ⩽0.5%, for even the largest stellar flares (e.g., Hawley & Fisher 1992), supporting the idea that this hot, blackbody emission component originates from a compact source at locations of intense and focused heating, perhaps at the footpoints of magnetic loops.

The blackbody has been reproduced in static phenomenological models. In Cram & Woods (1982), their model atmosphere #5 features extreme heating from the chromosphere through the deep photosphere and results in a T_BB ∼ 14, 000 K emission component and an Hα line with a central absorption; they note that this atmosphere most closely matches the continuum observations of stellar flares and could represent the stellar analog of solar flare kernels where there is deep and concentrated atmospheric heating. Houdebine (1992) also produced model spectra that rise into the NUV, similar to a hot blackbody but with a sizeable Balmer jump; their models employ very large electron densities of 10¹⁶ cm⁻³. They suggest that hydrogen recombination radiation and blackbody continua contribute in varying proportions depending on the various parameters of the flare atmospheres, but provided few details. Phenomenological models of an energetic flare observed in the extreme-ultraviolet on a dM4e star gave a very small Balmer jump and continuum peak in the NUV (2400 Å; Christian et al. 2003). Kowalski et al. (2011a) used the static, NLTE RH code (Uitenbroek 2001) to model the secondary heating events during the "Megaflare" in Kowalski et al. (2010). They found that a Gaussian temperature "hot spot" placed below the temperature minimum in a quiescent M dwarf atmosphere produces an optical spectrum with T_BB ∼ 11, 000–18, 000 K and strong absorption in the HB features, as observed during the secondary events (see Section 6.3).

Self-consistent models that use realistic flare heating mechanisms typically result in a white-light continuum that is dominated by a strong hydrogen recombination component (Hawley & Fisher 1992). The sophisticated one-dimensional RHD models of Abbett & Hawley (1999) and Allred et al. (2005, 2006) used the RADYN code (Carlsson & Stein 1994, 1995, 1997) to simulate flares on an M dwarf and on the Sun using moderate (10¹⁰ erg cm⁻² s⁻¹, F10) and large (10¹¹ erg cm⁻² s⁻¹, F11) fluxes of mildly relativistic electrons injected at the top of a semi-circular flare loop. The RADYN models employ the thick-target formulae of Emslie (1978), Ricchiazzi & Canfield (1983), and Hawley & Fisher (1994) that describe how the nonthermal electron beam deposits energy throughout the atmosphere. The accelerated electron distribution in the Allred et al. (2006) models employs a double power-law energy distribution of beam electrons with a minimum cutoff energy, E_c, which is assumed to be 37 keV (inferred from solar flare hard X-ray observations with RHESSI; Holman et al. 2003). The energy from the electrons is deposited in the chromosphere, which explosively evaporates into the corona, illustrating the chromospheric evaporation scenario developed by Fisher et al. (1985). The Allred et al. (2006) F10 and F11 models represent impulsive heating, and they were run with constant beam fluxes for 230 s and 16 s, respectively.

The Allred et al. (2006) RHD predictions of HB line emission, such as line broadening and flux decrements, are generally consistent with observations. However, an important shortcoming of the model predictions is the lack of hot blackbody emission which is clearly present, and in fact dominant, in the spectra of stellar flares. Instead, the dominant continuum components at λ > 2000 Å are a large spectral discontinuity at the Balmer jump and prominent Balmer (bf) continuum emission. The optical flare emission is due to Paschen (bf) continuum emission and radiation from the moderately heated photosphere. The photospheric heating is at most ∼1000 K and results from incident NUV backwarming radiation; direct beam heating contributes relatively little to the heating of deep layers, and cannot reproduce the heating or densities implied by phenomenological models. The ultimate problem in these physically self-consistent models is therefore not enough heating at high densities. However, as mentioned previously, the broadband colors of the model flare spectrum produce a continuum distribution with the general shape of a blackbody with T ∼ 9000 K and also appear to match the observed broadband UV, UBVR fluxes of a moderate sized flare quite well (flare 8 of Hawley et al. 2003). A recent observation of the flare decay phase for the first time directly detected a BaC in emission which matched the shape of the F11 RHD model BaC (Kowalski et al. 2010; see also Section 6).

The last study of a large sample of flares with simultaneous spectra and photometry was that of Bopp & Moffett (1973) and Moffett & Bopp (1976), who revealed several global trends in the spectroscopic characteristics of flares. Their sample consisted of five flares with low-resolution spectral coverage from λ = 3700–5700 Å and high cadence U-band photometry. The exposure times of the spectra were between 30 s and 3 minutes, with most >1 minute. Their main result was the demarcation of flares into "spike" and "slow" phases according to the relative contribution of line and continuum emission; they speculated that a two-component model with hydrogen recombination could explain these phases. Other results were shown for the He i λ4026, He i λ4471, and Mg ib lines, indicating no apparent relation between flare properties and their detection/non-detection. They found a longer time delay between continuum and emission line maxima for higher luminosity stars, but their use of equivalent widths for line diagnostics makes interpretation difficult. The continuum shapes were not analyzed and the lack of blue wavelength coverage did not allow for an assessment of possible BaC radiation.

As a modern day extension of the Moffett & Bopp (1976) study, we have obtained high signal-to-noise spectral observations of flares in the blue/NUV, including the Balmer jump wavelength, for a homogenous analysis of the line and continuum properties. These data are necessary to break the degeneracy of fitting emission types to broadband photometry and to determine which continuum processes contribute (and how much) to the white light. In the past, colorimetry was preferred in order to achieve a high signal-to-noise at good time resolution, but the U band is difficult to interpret because it straddles the Balmer jump. Modern, 4 m class telescopes now provide good blue/NUV sensitivity and allow time-resolved spectra to be used to characterize faint levels of flare flux varying on short timescales. A large, systematic study of blue/NUV flare emission will reveal if hydrogen recombination (BaC) radiation is present in flares and how it relates to the blackbody component. Including a range of flare types (e.g., fast versus slow) is useful to assess why this continuum component does not obviously appear in the most recent, high quality observations (e.g., Hawley & Pettersen 1991; García–Alvarez et al. 2002; Fuhrmeister et al. 2008). For example, perhaps the disappearance of the BaC is a phenomenon that only occurs during large impulsive flares (IFs), which coincidentally are the only ones that have yet to be studied in detail with blue/NUV spectra.

We have obtained broadband photometry simultaneously with the spectral observations to connect with decades of single-filter and colorimetric flare studies. Most importantly, simultaneous observations allow us to relate the spectral continuum characteristics to the diverse types of light curves. Perhaps coincidentally, most of the largest flares which have been studied with NUV spectra have time profiles in the U band that deviate from the classical model and include a secondary, (usually lower) amplitude continuum enhancement following the fast decay phase of the first peak. Some flares have three or more continuum peaks in the impulsive phase (Kahler et al. 1982; Eason et al. 1992), while other flares have low amplitudes but gradually emit a large amount of energy over a longer time period (Hawley et al. 1995).

A basic phenomenological question therefore is, "how do the continuum properties evolve through the different phases of the U-band evolution, and can we ultimately use these properties to create a flare continuum model that can explain the gamut of flare light curves that are observed?" For example, Osten et al. (2005) studied two U-band flares on the dM3.5e star, EV Lac: their durations were 4.5 and 7 minutes, yet their peak amplitudes were a factor of 20 different: how is the physics (i.e., flare heating and subsequent radiation) different between these two flares? With a large sample of flares that have simultaneous time-resolved spectra and photometry, we can constrain the detailed continuum parameters over a range of flare characteristics to provide constraints for future models.

The most general aspect of this question is how the continuum components compare between the impulsive and gradual decay phases, and therefore how their respective "fast" and "slow" evolution are related to the dominant flare heating mechanism, atmospheric cooling response, and radiation at those times. In addition to identifying the differences between the individual phases of flares, we also seek to identify similarities between the same phases of different flares to clarify important phenomena that must be reproduced by RHD models.

This paper is organized as follows. In Section 2 we discuss the observations and data reduction. In Section 3, we introduce several parameters used to analyze the data. A general overview of the flare sample is given in Section 4, and Section 5 contains the detailed emission line analysis. We apply the two-component blue continuum analysis of Kowalski et al. (2010) to all flares in the sample in Section 6. The gradual decay phase is analyzed in Section 7, and we introduce a third continuum component in Section 8. We compare to the Allred et al. (2006) RHD model predictions in Section 9. Finally, we analyze filling factors and flare speeds in Section 10. In Section 11, we summarize our findings and discuss their physical significance for flare processes. Finally, in Section 12 we discuss future observations that are needed. There are several appendices where we discuss detailed aspects of the data and analysis algorithms. Throughout this paper, we refer the reader to relevant sections and appendices of Kowalski (2012).

Over the course of three years, we obtained high-cadence simultaneous photometric and spectroscopic observations of nearby, dMe flare stars. We monitored the dMe flare stars that had the highest measured optical/NUV flare rates (∼1 hr⁻¹, Lacy et al. 1976; Pettersen et al. 1984), and our initial sample consisted of four bright, nearby stars (EV Lac, YZ CMi, AD Leo, EQ Peg) to allow for short exposure times. The multiwavelength properties of the flares on these stars have been studied extensively (e.g., Kahler et al. 1982; Hawley et al. 1995; van den Oord et al. 1996; Osten et al. 2005, 2010). The targets and basic stellar parameters are given in Table 1.

A fifth target star, GJ 1243, is a star that has not had its flare rate previously measured. It is known to be an active star (Gizis et al. 2002) of spectral type dM4e, and its long baseline photometric starspot activity was recently measured by Irwin et al. (2011). It is particularly important to examine GJ 1243 due to its inclusion in our Kepler GO program with observations at 1 minute cadence (the Kepler flare properties will be presented in subsequent papers).

Spectra were obtained with the Dual-Imaging Spectrograph (DIS) on the ARC 3.5 m telescope at the Apache Point Observatory (APO). We employed the low-resolution B400/R300 gratings, which provided continuous wavelength coverage from λ ∼ 3400–9200 Å, except for a dichroic feature that affected the flux calibration from λ ∼ 5200–5900 Å. The CCD was binned by 2 and windowed to ∼130 pixels along the spatial axis (∼100''), thereby reducing the readout time from 40 s to ∼10 s. Integration times ranged from 1 s to 45 s (most were between 10 and 20 s). Short cadence (1–8 s) spectra were occasionally interspersed in the observing sequence in order to avoid non-linearity and saturation in the red during the longer integration times which were necessary to obtain visible counts on the two-dimensional image at λ ∼ 3450 Å. This typically provided an adequate signal-to-noise of ∼10 at 3600 Å in quiescence. We obtained data with the 15 slit for the first two years and primarily with the 5'' slit for the last year (if the conditions were clear). The wide slit facilitated absolute flux calibration (so that we could check the scaled spectra against the original flux-calibrated spectra; see below), mitigated the effects from miscentering the star on the slit and from seeing variations, and allowed the exposure times to be reduced. The slit was automatically rotated to the parallactic angle in order to account for atmospheric differential refraction (Filippenko 1982).¹⁰ Care was taken to ensure the star stayed centered on the slit through the course of the observations, but in some instances small deviations may have affected the observations. The observations were taken under moderate to good weather conditions; the seeing was estimated with each spectrum but rarely exceeded the narrow (15) slit width by more than ∼05.

The spectra were reduced using standard IRAF¹¹ procedures via a customized PyRAF¹² wrapper, developed from the reduction software of Covey et al. (2008). Initial processing included bias, overscan, and flat-field corrections. Aperture extraction and background subtraction were performed and a wavelength solution was applied using HeNeArHg and HeNeAr lamps. The resulting dispersions were 1.82 Å pixel⁻¹ for the blue and 2.3 Å pixel⁻¹ for the red. The spectral resolutions were determined from the He i λ4471 arc line taken at the beginning of the night. For the 15 slit, the resolution at this wavelength was 5.5–7.3 Å (R ∼600–800) and ∼18 Å (R ∼250) for the 5'' slit. For the very wide (5'') slit width, the profiles of arc lines are wider and less Gaussian than for point sources, and measuring the quiescent emission line profiles of the M dwarfs reveals an actual resolution closer to 13–15 Å (R ∼320). Although broad, the line profiles for the spectra of M dwarfs taken with the wide slit were in general nearly Gaussian and allowed line fluxes to be measured.

Observations of spectrophotometric standard stars (white dwarf or sdO stars) were obtained every night and were used to convert from counts to an energy scale. An airmass correction was applied to the spectra using the atmospheric extinction curve for APO, published by the Sloan Digital Sky Survey (SDSS). The spectral shape accuracy across the λ = 3400–5200 Å range was usually better than 10%, as calculated from observations of multiple spectrophotometric standard stars. Synthetic U, u, and g absolute magnitudes obtained from the spectra were accurate to within 20%–25% during good conditions; the large uncertainties likely result from not knowing the precise blue response of all optical elements. The spectrophotometric standard star fluxes were obtained from Oke (1990); second order corrections to the APO atmospheric extinction curve were not applied to the data. The wavelength ranges that we use for continuum analysis in this paper are λ = 3420–5200 Å and λ = 5900–7500 Å. The He i λ5876 line may be used in future analysis but is marginally affected by the far end of the dichroic. The Ca ii IR triplet can also be analyzed (Kowalski 2012), but spectral fringing affects the red flux at λ > 8350 Å (with amplitude of ∼2%). For the observations between 2010 February and 2011 July, the wavelength region at λ < 3600 Å was affected by a ∼2% "ripple" variation in flux, which is apparent in very high flux levels (such as at flare peak and in the standard stars). A similar effect has been attributed to inter-pixel variations of quantum efficiency (Rutten et al. 1994), but we have attributed it as a transient artifact in instrument performance. The ripple is visible in flat-fields but is not removed in the reduction process. Further details about the data reduction and flux accuracy are given in Appendix A of Kowalski (2012).

The observing log for each target star is given in Table 2. The monitoring time each night, number of exposures, exposure times, spectral resolutions, and available simultaneous photometry are also provided. For the number of exposures given, the three values indicate the number used for blue wavelength analysis, the total number of spectra recorded, and the number of spectra obtained at a shorter exposure time and lower cadence for red wavelength analysis: (nB, nT, nR). For a spectrum to be considered in the blue wavelength analysis (and be counted in nB), the standard deviation of flux divided by the average flux just blueward of the Balmer jump was required to be less than 15%. This allowed us to select spectra that were (mostly) unaffected by weather and cosmic rays. Note that some short-exposure spectra (a fraction of nR) are included by this requirement and some were excluded explicitly (see caption for Table 2); the remaining short exposure spectra are analyzed independently.

Photometry was obtained from the NMSU 1 m telescope and the ARCSAT 0.5 m telescope at the APO. The 1 m is operated robotically (Holtzman et al. 2010), and provided continuous Johnson U-band photometry. The U-band data were taken with exposure times of 10 s for YZ CMi, 4 s for AD Leo, 4 s for EV Lac, and 15 s for EQ Peg AB (the A and B components were both visible but not completely resolved). With a readout of ∼10 s,¹³ the cadence was therefore several observations per minute. Observations were reduced as part of the standard 1 m pipeline (and in some cases by hand using standard IRAF procedures). We used the Flarecam instrument (Hilton et al. 2011) on the 0.5 m, which was remotely operated. Flarecam has enhanced UV sensitivity, a fast readout (∼1 s), and rapid filter wheel rotation among the available SDSS ugri filters. The exposure times varied depending on conditions, but gri band exposures were typically 1–2 s. A variety of imaging sequences were employed during the campaign in order to determine the optimal balance of duty cycle and wavelength coverage. The data were reduced using standard IRAF procedures with the ejhphot reduction wrapper (Hilton et al. 2011).

Differential aperture photometry was performed using a nearby bright star, and a quiescent window was chosen to normalize the count flux for the night.¹⁴ By comparing the times in the spectra and photometry headers, we determined that the timing of the Flarecam images was not always synched with UTC. This offset ranged between 7–30 s, and we adjusted the center times of the measurements accordingly. This timing precision is adequate for our study, since the spectra have a cadence ⩾11 s.

Table 2 contains information about the photometry used for each night. Note that the photometry and spectra for 2009 January 16 are the same as those presented in Kowalski et al. (2010).

From the U(ug)-band and Hγ equivalent width light curves of 31 nights of observations, we selected 18 flares from these five stars to analyze in detail. In order to facilitate analysis of the NUV continuum flux (which Earth 's atmosphere efficiently scatters or absorbs), the largest Uug amplitude events were chosen. The flares occurred on 14 separate nights which comprised 75 hr of spectral monitoring with 7780 spectra.¹⁵ In addition, we consider the data obtained during the large flare of 2009 October 27 on EV Lac, which was discussed in Schmidt et al. (2012). The blue spectra were obtained at the Dominion Astrophysical Observatory (DAO) and have much lower time resolution (200–300 s) and only cover the wavelength range λ = 3550–4700 Å with R ∼ 750, but these data encompass an unusually fast and large amplitude flare. We also calculate relevant quantities from the Great Flare on AD Leo of 1985 April 12 (Hawley & Pettersen 1991, hereafter HP91) for comparison. The total number of flares in our sample is therefore 20.

Line fluxes were calculated for hydrogen Balmer α, β, γ, δ, + Ca ii H, Ca ii K, several helium i lines (λ4026, λ4386, λ4471), He ii λ4686, and several prominent lines with ambiguous identifications that possibly represent a combination of helium i, Fe ii, or Mg ib lines (λ ∼ 4924 Å, λ ∼ 5018 Å, λ5171 Å).¹⁶ A line flux can be calculated by measuring the equivalent width of the line and multiplying by the nearby continuum (Reid et al. 1995). If the continuum is normalized to absolute photometry or if the observations are spectrophotometric, this method accounts for the effects of weather and slit-loss. However, the continuum changes dramatically during flares, especially in the region of the blue lines. Alternatively, one could look for a region of the continuum with an accurate calibration whose value does not vary during flares and use this to calculate the equivalent width and flux normalization. After close inspection, we found that the entire optical continuum (3400–9200 Å) experiences significant flux variations during the largest flares. The red continuum near λ = 8650 Å is most nearly constant, but the flux calibration here is not always reliable due to spectral fringing in the far red.

The calculation of line fluxes during flares is further complicated by the dramatically changing widths of the line wings during flares (Doyle et al. 1988; HP91). The integration limits and continuum ranges are given in Table 3, and were chosen to be wide enough to account for the maximum amount of broadening observed in the flare sample; the same windows were used for all spectra of all flares in the DIS sample in order to be consistent. An example peak flare spectrum with significant broadening is shown with the integration windows in Figure 1.

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Table 3. Line and Continuum Windows

Line	Line Integration Window	Continuum Region
	(Å)	(Å)
Hα	6520–6610	6465–6484, 6610–6665
Hβ	4828–4898	4813–4825, 4895–4912
Hγ	4310–4370	4260–4305, 4375–4420
Hγ (DAO)	4321–4361	4260–4305, 4375–4420
Hδ	4065–4135	4040–4064, 4137–4167
Hδ (DAO)	4084–4122	4040–4064, 4137–4167
H + Ca ii H	3944–3997	3914–3923, 4000–4017
Ca ii K	3923–3945	3914–3923, 3946–3955
He i λ4471 (15 slit)	4464.6–4481	4380–4461, 4488–4554
He i λ4471 (5'' slit)	4460–4495	4380–4461, 4488–4554
Continuum Measure	Spectral Region
	(Å)
C3615	3600–3630
C4170	4155–4185
C4500	4490–4520
C6010	5990–6030

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The measurement of line fluxes employed in this paper is as follows. Starting with the total (flare+quiescent) flux in each spectrum, we define local continuum regions and determine a linear fit between regions on both sides of each emission line. The linear fit allows us to estimate a first-order change in the continuum beneath the line (as is important during flares). We then compute the flux in the line region (Table 3). Measurements of the line fluxes in Hγ, Ca ii K, and He i λ4471 can be reliably calculated before subtracting a preflare spectrum; flare-only emission is obtained after calculating the total line fluxes by subtracting the quiescent or pre-flare line fluxes. However, for the other lines, it was necessary to subtract the quiescent spectrum before calculating the line flux because the lines were either at very low-level and were not readily visible (e.g., the other He i lines) or the surrounding continuum is poorly modeled by a linear function due to "jagged" quiescent molecular features (as is the case for Hα, Hβ, Hδ). For these lines, the line flux was calculated after subtracting a quiescent spectrum (see Section 2.6), allowing for a more precise fit of the line to the local continuum. For Hα (and for flare-emission in faint lines) this method resulted in negative features away from line-center if the quiescent lines were not aligned precisely with the flare features (e.g., due to occasional single pixel jumps from wavelength instabilities); therefore, we summed the positive and negative flux values over the Hα line.

The local continuum near important features such as Hδ, Hγ, Hβ, and Hα also contains numerous photospheric molecular features which present an additional complication for defining line and continuum regions. Because we integrate over a large wavelength window, we include some molecular features in the line flux (for Hγ, He i λ4471; see above); however, the molecular flux is assumed to be removed by subtracting the quiescent spectrum. The line calculations are done with an automatic routine, and they were examined by eye to ensure that we accounted for all of the excess flare flux. The line windows given in Table 3 were adjusted by a small amount for every spectrum depending on the centroid of the line, which was determined initially for the Balmer Hα, Hβ, Hγ, and Hδ lines; the wavelength shifts for the Ca ii K and He i λ4471 lines were forced to be the same as for the Hγ line. The wavelength centroid stability is typically <1 Å  but can vary up to a pixel (Δλ = 1.83 Å pixel⁻¹) from one spectrum to the next.

An additional step in flux calibration was necessary to correct for exposure-to-exposure gray variations in the level of flux due to variable seeing, variable transparency, and imperfect centering of the star in the slit. In Kowalski et al. (2010), we used simultaneous U-band photometry to apply a single scaling factor to each spectrum. Since then, we have developed an improved method to scale the spectra which minimizes the subtraction residuals in the molecular features in the red continuum. Importantly, this new technique allows us to independently compare the spectra and photometry, as the integration times for the photometry (especially during the fast impulsive phase of a flare) may differ from the spectra.

For each night, we determined a master quiescent or pre-flare spectrum by identifying non-variable times from the photometry and the Hγ line. We scaled the spectra during the quiescent or pre-flare interval to a common flux at λ = 4500 Å in order to account for weather or slit-loss variations over the course of the spectra within this time window. Synthetic Johnson B and V fluxes were compared to the accepted magnitudes (in Table 1, obtained from Reid & Hawley 2005 who compiled magnitude data from Bessell 1990; Koen et al. 2002; Leggett 1992, GJ 1243 measurements were obtained from Reid et al. 2004), using the Johnson (1966) flux zeropoints. The observed fluxes were then scaled so that the synthetic fluxes were equal to the accepted fluxes, which is important for placing all nights (for a given star) on the same baseline flux level.

A scaling for each flare spectrum relative to the master quiescent spectrum is then performed as follows. For each spectrum during the flare, we multiplied by a large range of possible scale factors (0.2–4.0), subtracted the quiescent spectrum, and calculated the sum of the standard deviation of the subtraction residuals in three spectral regions (outside of features from Earth's atmosphere which can change over time) from λ = 6600–6800 Å (excluding the region around He i 6678 Å), λ = 7000–7100 Å, and λ = 7350–7550 Å.¹⁷ These regions correspond to strong flux changes in the quiescent spectrum due to the presence of molecular bandheads; therefore, errors in flux scaling appear as significant over- or under-subtractions at these wavelengths. The best scale factor minimized the sum of the standard deviation of the subtraction residuals. We tested the accuracy of this procedure, which we describe in Appendix A. Essentially, we generated a model flare spectrum and multiplied by an arbitrary scale factor to simulate flux loss (from weather or slit-loss). We found that our simple algorithm determines the correct scaling factor for all but extremely large amplitude flares that increase the U-band flux by a factor of ∼100 or more, which are not in our sample. A similar scaling method was employed by Abdul-Aziz et al. (1995). The principle behind the scaling method is analogous to PSF subtraction in imagery of protoplanetary disks, where the optimal subtraction is found by minimizing the subtraction residuals in the background (e.g., Wisniewski et al. 2008).

Ultimately, a single scaling factor was determined for each spectrum. The final stage in flux calibration was to multiply the flux density, line fluxes, continuum fluxes, and synthetic filter fluxes by the scaling factor during the flare times. The flare-only fluxes were then obtained by subtracting the quiescent values. Figure 35 in Appendix A demonstrates the recovery of flare variations during times of variable cloud cover.

In Figure 2, we show a light curve from 2008 October 1 of a large flare on EQ Peg A. We use this figure to illustrate several of the following empirical values that can be directly measured from the photometry.

• 1.  
  t_1/2. To describe the time evolution of a light curve, we define t_1/2, the full width of the light curve at half-maximum. This measures the "timescale" of the impulsive phase of the flare, including both fast rise and fast decay times. The measure of t_1/2 does not assume a functional form for the decay, which can be complex, as seen in Figure 2. We also measure t_1/2 for the light curves of spectral components (Section 3.2). In some cases, the rise time is fast compared to the photometric or spectral cadence; in these cases we must interpolate between data points to obtain t_1/2. For the example flare in Figure 2, we illustrate the t_1/2 value.
• 2.  
  I_f, I_f + 1. The measure I_f is the familiar quantity in flare studies (Gershberg 1972). It is the ratio of flare-only count flux (photons cm⁻² s⁻¹) in a given band to the quiescent count flux in that band. I_f + 1 is the flux enhancement, or the total count flux relative to quiescence. In solar physics, I_f is used to express the intensity contrast.¹⁸ If C(t) is the total count flux ratio (counts_target/counts_comp) in the differential photometry, normalized to 1 during quiescence, then

  $$\begin{equation} I_f(t) = C(t) - 1; I_f(t) + 1 = C(t). \end{equation} \tag{ 1 }$$

  For the example flare in Figure 2, I_{f, U, peak} = 20.4.
• 3.  
  ED. The equivalent duration (ED) in a given bandpass is the integral of I_f over the duration of a flare (Gershberg 1972). The units are in seconds and multiplying by the quiescent luminosity in the band gives the energy of the flare.
• 4.  
  $\mathcal {I}$. To characterize the shape of the light curve, we use an "impulsiveness index," $\mathcal {I}$, which is defined as

  $$\begin{equation} \mathcal {I} = I_{f,{\rm peak}} / t_{1/2}. \end{equation} \tag{ 2 }$$

  The quantity $\mathcal {I}$ is a measure of the peak relative flux of a flare weighted by how fast it rises to peak and decays. Both a more luminous-at-peak flare and a smaller t_1/2 (faster timescale) can give rise to larger values of $\mathcal {I}$. We find that this measure provides a way to quantitatively sort the flares according to their light curve evolution, while only using observables measured directly from the light curve.
• 5.  
  $\mathcal {L.}$ The specific luminosity ($\mathcal {L}$, in units of erg s⁻¹ Å⁻¹) is useful for characterizing the luminosity without the ambiguity of using a spectral window of width, Δλ. For example, U-band luminosities (and energies) assume Δλ = 700 Å whereas B-band luminosities (and energies) assume Δλ = 900 Å, making luminosities in the two bands not directly comparable. In some cases, we present the integrated (over wavelength) quantities L, E in typical bandpasses for comparison to previous studies. However, when comparing spectral continuum measurements, we use $\mathcal {L}$. All measures (L, E, $\mathcal {L}$) assume an isotropically emitting source and employ the distances in Table 1.

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We now refer to Figure 3 to describe the measured parameters from the spectra. This spectrum illustrates the flare-only emission at the peak of a simple, moderate-amplitude flare on AD Leo from 2010 April 3.

• 1.  
  Spectral Zones. We divide the spectrum into four zones: the NUV zone (λ = 3420–3646 Å), the intermediate zone (λ = 3646–4000 Å), the blue-optical zone (λ = 4000–5200 Å), and the red-optical zone (λ = 5800–7550 Å). These are useful for our analysis of continuum variations during flares.
• 2.  
  C3615 and C4170. In each spectrum, we measured the average flux in several ∼30 Å-wide continuum windows across the DIS spectral range. The continuum windows were chosen to correspond to spectral windows free of major (and most minor) emission lines that appear during flares. The two continuum measures that we use to characterize the blue are the average flare-only flux in the wavelength region from 3600–3630 Å (denoted C3615) and the average flux in the wavelength region from 4155–4185 Å (denoted C4170). The C3615 region was chosen to be blue of the Balmer jump at 3646 Å, while also red enough to obtain a reasonable signal-to-noise in small to moderate-size flares. This measure covers the approximate central wavelength of the U band, which is much broader. The C4170 region was chosen to emulate the NBF4170 continuum filter, which is a custom continuum filter that is also employed on the solar camera ROSA (Jess et al. 2010) and stellar camera ULTRACAM (Dhillon et al. 2007). C4170 provides a measure of the continuum flux redward of the Balmer jump and unaffected by blending of high order Balmer lines. The continuum windows are summarized in Table 3 (including two other continuum windows, C4500 and C6010, used in Section 8).
• 3.  
  χ_flare. To describe the flare color across the blue and NUV wavelengths in (mostly) line-free continuum bands, we use the quantity:

  $$\begin{equation} \chi _{\mathrm{flare}} = \mathrm{C3615} / \mathrm{C4170.} \end{equation} \tag{ 3 }$$

  The error in this quantity is obtained by propagating the standard deviation of the fluxes in C3615 and C4170,

  $$\begin{equation} \sigma _{\chi , \mathrm{flare}} = \chi _{\mathrm{flare}} \sqrt{ \left(\frac{\sigma _{\mathrm{C3615}}}{ \mathrm{C3615}}\right)^2 + \left(\frac{\sigma _{\mathrm{C4170}}}{\mathrm{C4170}}\right)^2 }. \end{equation} \tag{ 4 }$$

  Formally, the uncertainties of C3615 and C4170 are the standard errors of the mean values, but some weak emission line features (e.g., Fe i, Fe ii) appear in this spectral region; therefore, a better estimate of the uncertainty in the continuum level in each window is given by the standard deviation of the flux.χ_flare is similar to the Balmer jump, J, that has been measured from T Tauri star spectra (Valenti et al. 1993; Herczeg & Hillenbrand 2008) and also in previous M dwarf flare studies in the blue to derive electron temperatures assuming an isothermal, isodensity slab of hydrogen (Kunkel 1970).
• 4.  
  BaC3615. The quantity C3615 (see #2 above) is the average flare-only continuum flux from λ = 3600–3630 Å  consisting of BaC emission from hydrogen recombination and other possible components that contribute toward the continuous emission throughout these wavelengths (such as Paschen continuum and blackbody continuum). Our estimate for only the flare BaC emission at 3615 Å, BaC3615, is obtained by extrapolating and subtracting a continuum that is fit to the blue-optical zone. In particular, we fit a straight line to the wavelength windows listed in Table 4 from λ = 4000–4800 Å (BW1–BW6), extrapolate to λ = 3600 Å, and subtract an average of these extrapolated values at λ = 3600–3630 Å from the flare-only flux average (C3615) to obtain BaC3615. In Section 6, we find that a T_BB ∼ 10, 000 K blackbody fits well the shape from λ = 4000–4800 Å and that a 10, 000 K blackbody is approximately linear in this wavelength range. This procedure is shown for an example flare spectrum in Figure 3. Note that by definition, BaC3615 ⩽ C3615.
• 5.  
  BaC_tot. The estimate for the wavelength-integrated BaC energy from λ = 3420–3646 Å. The BaC is estimated using the same fitting, extrapolation, and subtraction procedure as for BaC3615. Instead of averaging the extrapolated line value, the line extrapolation was subtracted at every wavelength in this region.
• 6.  
  PseudoC. The intermediate zone (between blue-optical and NUV zones) contains the higher order Balmer lines (H7 and greater) in addition to the Ca ii H and K lines (λ ∼ 3934, 3968 Å, respectively). Although Ca ii H is blended with H (H7) in these low resolution data, Ca ii K is resolved. Within this zone, we integrate the flare flux from λ = 3646–3914 Å (from the Balmer jump through H8) and refer to it as the "PseudoC" because many of the hydrogen lines blend together (are partially or completely unresolved) at these wavelengths to form a pseudo-continuum. Again, as with the BaC_tot, we use an extrapolation of the line-fit to the blue-optical to estimate the underlying continuum.
• 7.  
  S#. We refer to the time-sequential spectrum number (starting at 0 at the beginning of each night) as S#. These numbers refer to the nB subcategory of spectra used for blue continuum analysis as described in Section 2.2 (see also Table 2).
• 8.  
  Times. The times on the light curves indicate the number of hours elapsed on the respective MJD from Table 2. Times always refer to midtimes of the exposure. The times for the flare data from 2009 January 16 are given in "elapsed hours from flare start," as used in Kowalski et al. (2010); to obtain the number of hours elapsed on MJD 54847, add 4.2483 hr to the number of elapsed hours from flare start.

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Additional terminology used to describe the photometry and spectra are the following:

• 1.  
  Impulsive Phase. An impulsive phase consists of a fast rise, peak, and fast decay of the light curve.
• 2.  
  "Peak" or "maximum continuum emission." "Peak" or "maximum continuum emission" always refer to the maximum value of C3615 during a flare. The peak times are given for each flare in Table 5.
• 3.  
  Gradual Decay Phase. Gradual phase emission is observed during times of slowly rising or slowly decreasing emission. Following HP91, the "gradual decay phase" is defined as the turnover from fast to slow decay. The gradual decay phase spectra are chosen from the section of the light curve as near to the break from fast decay to slow decay as possible. We select three spectra around a time when C3615 is not changing rapidly so that the spectra can be co-added to increase the signal-to-noise without largely affecting the interpretation of atmospheric parameters. The red vertical dashed lines in Figures 36–54 in Appendix B indicate the times that we selected to represent the gradual decay phases for the flares in our sample. The gradual decay phase times analyzed for each flare are given in Table 5.
• 4.  
  "blackbody continuum component." This term refers to a continuum slope that matches the slope of a Planck function with temperature T_BB. A "hot blackbody" is used to designate a blackbody continuum component with T_BB ≳ 8500 K.

The impulsiveness index ($\mathcal {I}$) provides the main light curve morphological classification scheme employed in the remainder of the paper. For our flare sample, $\mathcal {I}$ ranges from 0.02–100. The impulsive flare (IF) events are those that have $\mathcal {I} > 1$, whereas the gradual flare (GF) events have $\mathcal {I} < 1$. For flares that are close to this dividing line (∼0.6–1.8), we assign the classification hybrid flare (HF) events, as these flares have a prominent impulsive phase (or several impulsive phases) but also share properties with the GFs. We also considered the fast and slow flare classification scheme from Dal & Evren (2010), but this grouping employs a total decay time measurement; in some cases, poor weather, a standard star sequence, or secondary flares interrupted the decay measurements. Using t_1/2 (in the definition of impulsiveness) bypasses the ambiguities with measuring precise start and stop times.

The bluest available photometry (SDSS u, Johnson U, or SDSS g) for the flares in our sample are shown in Figure 4 (nine impulsive IF events), Figure 5 (two impulsive IF events—IF0 and IF10—with less data), Figure 6 (four hybrid HF events), and Figure 7 (five gradual GF events). In Appendix B, we show figures of each flare with the integration times of the spectra (Figures 36–54) and the spectrum numbers (S#'s) indicated. IF0 on AD Leo from HP91 is known as the "the Great Flare," and IF1 on YZ CMi (Kowalski et al. 2010) is known as "the Megaflare." In the decay phase of IF1, we refer to the sub-peak at t = 2.1441 hr as the "Megaflare decay secondary peak #2" or "MDSF2."¹⁹

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From the light curves, it is apparent that our sample contains a diverse set of peak amplitudes, total durations, and light curve morphologies. The naming convention (IF, HF, and GF) and the ordering of the flares within this classification scheme is based²⁰ on the value of $\mathcal {I}$ as detailed in Section 3.1. Table 6 summarizes the key properties of the U-band photometry: flare ID (Column 1), star name (Column 2), date (Column 3), time of peak C3615 (Column 4), I_{f, U} + 1 at peak photometry (Column 5), equivalent duration in U (Column 6), U-band energy (Column 7), U-band luminosity at peak photometry (Column 8), t_{1/2, U} (Column 9), and $\mathcal {I}$ (Column 10). The time-integrated photometric quantities are calculated only during the time period when spectra were obtained. Note that the time of peak photometry and time of peak C3615 may not precisely coincide due to the different cadences and integration times.

Table 6. Flare Summary Table (1): Key Photometry Properties

ID	Star	Date	tpeaka	If, U, peak + 1	ED	EU	LU, peak	t1/2, U	$\mathcal {I}$
					(s)	(1032 erg)	(1029 erg s−1)	(minutes)
IF1b	YZ CMi	2009 Jan 16	2.1441	208	93 690	38	84.63	7.73	27.0
IF2	YZ CMi	2011 Feb 8	8.6540	12.2	690	6.2	4.50	0.48	23.3
IF3	YZ CMi	2011 Feb 24	2.1500	78.2	45 810	18.5	29.98	3.82	20.2
IF4	EQ Peg A	2008 Oct 1	10.4686	21.4	4710	3.7	16.04	2.46	8.3
IF5	EV Lac	2010 Oct 11	6.1929	5.4	460	0.28	2.7	0.84	5.2
IF6	EV Lac	2009 Oct 10	4.7934	2.6	70	0.04	0.97	0.32	5
IF7	YZ CMi	2010 Feb 14	2.6245	6.4	810	0.33	2.19	1.18	4.6
IF8	YZ CMi	2010 Dec 11	11.6274	2.9	130	0.05	0.78	0.69	2.8
IF9c	AD Leo	2010 Apr 3	3.9120	4.4	1300	2–4	5.11	1.47	2.3
HF1	YZ CMi	2011 Feb 8	8.4909	5.3	920	0.37	1.72	2.43	1.8
HF2c	GJ 1243	2011 Oct 20	4.6452	5.4	500	0.86	1.29	4.19	0.93
HF3	YZ CMi	2010 Dec 13	9.3138	3.2	440	0.18	0.88	2.84	0.77
HF4	EQ Peg A	2008 Oct 1	8.7889	2.4	530	0.41	1.11	2.40	0.58
GF1	EV Lac	2010 Oct 11	2.1265	8.04	10 180	6.20	4.29	13.02	0.54
GF2	YZ CMi	2011 Mar 2	4.8504	2.1	2020	0.81	0.43	3.68	0.3
GF3	EV Lac	2009 Oct 10	2.4744	1.87	570	0.35	0.38	2.88	0.3
GF4	AD Leo	2011 Feb 8	12.0746	1.32	80	0.12	0.48	2.13	0.15
GF5c	YZ CMi	2009 Jan 26	6.3106	1.3–1.5	170	0.36	0.13	21	0.02
IF0	AD Leo	1985 Apr 12	4.8172, 4.9550	70.2 (43d)	50000	79	88.06	3.53	19.6
IF10	EV Lac	2009 Oct 27	4.9069	45.4 (9d)	8720	5.3	27.0	0.46	96

Notes. ^aTime in hours from the beginning of the MJD obtained from spectra using the mid exposure of maximum C3615, except for IF1 (see note on times in Section 3.2). ^bAll properties pertain to spectral observation window except peak amplitude of U, t_{1/2, U}, and $\mathcal {I}$. ^cU-band properties were estimated from spectra because no U-band data were obtained. The t_1/2 for GF5 estimated by smoothing the light curve over three spectra. The energies for HF2 are lower limits because the observations ended before the end of the gradual phase. ^dEstimated from spectra with longer integration time than the photometry.

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In Appendix C, we present the Flare Atlas with a time-sequence of flare-only spectra from λ = 3400–7500 Å for each flare event. In Figures 8(a)–(c) and 9, the flare-only spectra at maximum continuum emission (i.e., maximum C3615) are presented in the same order as the photometry in Figures 4–7. The quiescent levels are shown as dotted lines for comparison. Table 7 gives the I_{f, C4170, peak}, t_{1/2, C4170}, χ_{flare, peak}, and χ_{flare, decay} for each flare as a reference; these values are important constraints for flare models (Section 9). A detailed analysis of the continuum will be discussed in Section 6; here, a simple Planck function (light blue line) has been fit to the windows in the blue optical zone (λ = 4000–4800 Å; BW1–BW6 in Table 4) to parameterize the slope of the continuum. The best-fit temperatures and χ_{flare, peak} values are shown in parentheses. Except for GF1 (and possibly GF3 and GF5), the GF events are generally too faint for an accurate continuum fit in the blue-optical zone.

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There are varying amounts of the excess continuum at λ ≲ 3640 Å above the extrapolation of the blackbody (blue line), especially among the IF events. The HF and GF events have large amounts of excess continuum at λ ≲ 3640 Å. The spectral trends in the NUV zone for the IF, HF, and GF events are similar to the underlying blackbody curves. To illustrate this, we scale the light blue blackbody curves to the flux at λ = 3600–3630 Å and show these in yellow, which basically have the same slopes as the light blue fits.

The flares in our sample represent relatively large amplitude and high energy events on dMe stars. For example, the average U-band flare energy on YZ CMi is ∼0.03 × 10³² erg (Lacy et al. 1976), which is slightly smaller than the lowest energy flare (IF8) on this star in Table 6. The IF events have a large spread of amplitudes from low (I_{f, peak} + 1 ∼ 2.5–3) to very large (I_{f, peak} + 1 > 10). The IF events also generally have a classical, simple shape: a fast-rise, a fast decay, and a more gradual decay beginning at a low level, ∼20% or less, relative to the peak. Durations range from several minutes (e.g., IF6, IF8) to several hours (e.g., IF1, IF3). Some IF events have secondary flares but they are usually dominated by a single, large-amplitude peak. The HF events also have fast rise components, but they exhibit marked deviations from the classic flare shape, such as multiple continuum peaks of comparable amplitude during the impulsive phase (e.g., HF1) and an elevated or prolonged decay phase (e.g., HF2). These flares are low to moderate amplitude (2 < I_{f, peak} + 1 < 5) and usually longer lasting (>1 hr) than the IF events of comparable amplitude. The GF events are low-amplitude (I_{f, peak} + 1 < 2.2) except for GF1, which has I_{f, peak} + 1 ∼ 8. The rise phases are notably slower, although they can have distinct periods of faster and slower emission; and may be accompanied by intermittent peaks (e.g., GF1 and GF3). However, these continuum peaks do not significantly contribute to the overall timescales, which can be several hours for even the low amplitude flares.

There is often another local maximum (i.e., a secondary flare) just after the first peak but before the gradual decay phase. Secondary flares are especially evident in IF0, IF1, IF3, IF10, HF2, and HF4. Secondary flares usually occur at about half the peak flux level or less. Although the four largest amplitude events all have secondary flares (with ∼15 or more occurring during IF1), lower amplitude events also show them. IF4 is a large-amplitude event that shows a stall in the fast decay resulting in a nearly constant flux level before continuing a fast decay. This also may be interpreted as a relatively low-amplitude secondary flare.

In addition to the quantitative "impulsive," "hybrid," and "gradual" classification schemes, we find the following descriptive groupings using the data in Tables 6 and 7 in addition to the light curve data (not shown here; see Chapter 3 of Kowalski 2012) of the spectral components from Section 3.2.

• 1.  
  Simple, classical flares (IF2, IF5, IF7, IF9). These flares have moderately large amplitudes (I_{f, U, peak} + 1 ∼ 5–12) and energies (E_U ∼ 2 × 10³¹–2 × 10³² erg). A defining characteristic is that the U band (or bluest photometry) follows the evolution of C4170, which in fact, holds for most of the flares. The different timescales of decay between the spectral components are evident: although C4170, BaC3615, and Hγ vary from fastest to slowest, there is a spread of relative t_1/2 values, with IF9 having the smallest ratio of t_{1/2, C4170}/t_{1/2, BaC3615}. Except for IF7, these flares do not have obvious secondary flares in the decay.
• 2.  
  Low amplitude flares (IF6, IF8, GF2, GF3). The low amplitude flares have I_{f, U, peak} + 1 ∼2 (GF) and ∼3 (IF). There are both complex and simple events in this group, and the durations range from minutes to hours. These flares have lower energies: the low-amplitude short-duration flares have E_U ∼ 5 × 10³⁰ erg and the low-amplitude long-duration flares have E_U ∼ 10³¹ erg in their first main peaks; the complex flares are more than 10 times as energetic as the simple flares, even though the simple flares have larger peak amplitude in U (or u).
• 3.  
  Multiple-peaked, medium amplitude flares (HF1, HF2, HF3, HF4). These are medium amplitude flares with multiple peaks in the impulsive phase, having I_{f, Upeak} + 1 ∼ 2.5–5.5. These flares are all HF events. Generally, in the HF flares, the evolution of the U band closely matches the evolution of the BaC3615, whereas typically in the IF flares, the evolution in the U band matches the evolution in C4170. In these flares, χ_{flare, peak} is 2.3–3.
• 4.  
  High energy flares (IF0, IF1, IF3, IF4, IF9, IF10, GF1). The high energy flares have E_U > 3 × 10³² erg. The most impulsive flares (except IF2) tend to be the most energetic. C4170 and U(ug) photometry track each other well in the high energy flares, whereas the BaC3615 is more similar in evolution to the Hγ line. Secondary flaring to varying degrees is observed in all cases.
• 5.  
  Low amplitude, simple gradual flares (MDSF2, GF4, GF5). These flares have a moderate amount of energy (E_U ∼ 10³¹–10³² erg) for their low peak amplitudes. Interestingly, they lack a prominent fast decay phase after the peak emission. In GF5, short impulsive events are observed later in the flare decay. MDSF2 during the IF1 gradual decay phase is included in this group; its values of E_U ∼ 10³² erg and $\mathcal {I_U}\sim 0.6$ qualify it a high energy HF/GF.

4.3.1. General Relationships between Spectral and Photometric Properties

The general relationships between spectral properties (χ_{flare, peak}, Hγ/C4170) and photometric Uug light curve properties ($\mathcal {I}$) of the Flare Atlas are summarized with three figures, Figures 10–12. Figure 10 (χ_{flare, peak} versus the impulsive index, $\mathcal {I}$) indicates that the instantaneous continuum shape at maximum amplitude in the broadband light curve is linked to the overall evolution of the flare. From this figure (see also values in Table 7), it is evident that the most impulsive flares have the lowest χ_{flare, peak}, with most ≲ 1.8 and all ≲ 2.2. The HF events have intermediate values, χ_{flare, peak} ∼2.3–3, and the GF events have larger but more uncertain values (besides GF1), χ_{flare, peak} ∼3 +. The errors on χ_{flare, peak} are typically 0.01–0.12 (corresponding to relative errors of 3%–10%), but some flares have significantly larger errors with the GF events generally having the largest uncertainties. We do not consider the χ_flare values with $\sigma _{\chi _{\mathrm{flare}}}/$χ_flare  >  0.2. This excludes GF4 from χ_{flare, peak} analysis and IF5, IF6, IF8, HF4, GF2, GF3, GF4, and GF5 from χ_{flare, decay} analysis. Using standard error propagation, we determine the confidence levels by which the IF, HF, and GF sequence is ordered according to the χ_{flare, peak} parameter. We find that IF9 and HF1 are separated by 4.5σ, HF4 and GF1 are separated by <1σ, HF1 and GF1 are separated by 5σ, and IF5 and IF9 separated by almost 4σ. The differences in χ_{flare, peak} are generally more significant between the IF and HF events than for the HF and GF events; this is not surprising given that the IF events tend to have the larger signal-to-noise due to their larger amplitudes. We conclude that the differences in χ_{flare, peak} are significant between IF, HF, and GF events, and even between certain IF events.

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In Figures 11 and 12, we show the Hγ line flux divided by the continuum C4170 flux (both taken at peak C4170; essentially this is the equivalent width of the C4170 flare continuum in units of Å). The flares are color-coded by the IF/HF/GF designation. Figure 12 shows a narrower range of values than Figure 11 where IF2, IF3, IF7, IF8, IF9, and IF10 cluster together at χ_{flare, peak} ∼1.6–1.8. The first and second peaks of the Great Flare (IF0) are also included in Figure 12. Note that IF8 and IF3 are the smallest and largest amplitude IF events (with full spectral and time coverage) respectively on YZ CMi, yet they show very similar peak characteristics. IF4 has the lowest χ_{flare, peak} (∼1.3) and also line-to-continuum ratio (∼6)

In Figures 11 and 12, we see that the light curve morphology and χ_{flare, peak} are related to the ratio of Hγ/C4170. The IF events have Hγ/C4170 ≲50 (most IF events fall below 30), the HF events show 38≲Hγ/C4170≲75, and the GF events show 85≲Hγ/C4170≲165. We find a very strong relationship between χ_{flare, peak} (recall, χ_{flare, peak} = C3615/C4170) and Hγ/C4170:

$$\begin{equation} \chi _{\mathrm{flare,peak}} \approx 0.020(\pm 0.002){\rm H}\gamma /\mathrm{C4170} + 1.28(\pm 0.05). \end{equation} \tag{ 5 }$$

Among the IF events, IF3, IF5, IF7, IF8, and IF9 fall closest to the best fit line. Although IF5 and IF6 are the IF events (see Figure 4) with the largest values of χ_{flare, peak} ∼ 2.2, they also have relatively large values of Hγ/C4170 ∼40–50. IF1 is an outlier (with much more relative Hγ radiation for the χ_{flare, peak} value predicted by the red line), as the peak data correspond to the peak of a secondary flare (MDSF2) during the gradual decay phase. As χ_flare is effectively a measure of the Balmer jump height, this relation implies that flares with larger Balmer jumps relative to the C4170 flux have larger Balmer line fluxes relative to C4170 flux. This implies a connection between the relative amount of Balmer line radiation and BaC radiation. χ_flare is thus a very important quantity because it can be measured without spectra (Kowalski et al. 2011b), yet apparently it can be used as a diagnostic of the Balmer line and continuum radiation.

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As was discussed in Section 4.3.1 (Figures 11 and 12), a larger ratio of Hγ flux to C4170 generally results from flares with larger χ_{flare, peak}. Therefore, the height of the Balmer jump is related to the relative amount of Balmer line radiation produced at peak emission. Hγ is a useful diagnostic because it is a strong, easily measured line, in both high and low flaring states. This line is also the highest order hydrogen line calculated in most RHD models to date. Furthermore, its properties have been studied extensively in the past for dMe flares. The U band is widely used for flare monitoring (Moffett 1974), and it is a diagnostic of the continuum flux and energy, since colorimetry studies have shown the peak of the white light occurs in the U band or at shorter wavelengths (Hawley & Fisher 1992).

The U band contains higher order Balmer lines and Ca ii H and K, but ≳90% of the U-band energy is due to continuum radiation (Doyle et al. 1988; see also HP91). It is a well-established property that the Balmer lines evolve more slowly than the continuum, staying elevated longer (Kahler et al. 1982) and sometimes peaking as late as the end of the impulsive phase (HP91, García–Alvarez et al. 2002; Gurzadian 1984). The time-integrated energies scale over approximately 4.5 orders of magnitude with E_U ∼ 25 × E_Hγ (HP91). Using a larger sample in this study, we find that the BaC3615 component of the U band is even better correlated with the properties—including peak luminosity, total energy, and time evolution.

Here, we analyze the timing in detail. Figure 13 shows the relation between t_{1/2, Hγ} and t_{1/2, BaC3615}. We find a linear relation among the IF and HF events without multiple peaks (crosses). The fit to these flares is shown as a light blue line, given by

$$\begin{equation} t_{1/2, \mathrm{BaC3615}} \approx 0.54(\pm 0.01) t_{1/2, {\rm H}\gamma } + 0.4(\pm 0.1). \end{equation} \tag{ 6 }$$

In other words, t_1/2 is twice as fast in the BaC as in Hγ.

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The GF events (shown in open circles) follow a different trend with nearly equal timescales in Hγ and BaC3615. GF1 is a multiple-peaked flare, and it produces copious BaC3615 with a long timescale. The IF and HF events with 2–3 peaks (IF0, IF4, HF1, HF2, and HF4) spaced relatively close in time²¹ are shown with red squares and are labeled. IF4 and HF2 are double-peaked flares and are apparently outliers (the red squares with t_{1/2, Hγ} ∼ 16 minutes). IF4 falls closer to the GF distribution, and HF2 (also possibly IF0) have much faster BaC3615 timescales given the Hγ timescale predicted by the light blue line. HF1 and HF4 fall near the single-peak distribution and the GF distribution, respectively. Note that t_1/2 depends on which peak dominates and also how it is measured (e.g., the time delay between multiple peaks). A larger sample of double-peaked IF and HF events would constrain their different timing behavior in Hγ and BaC3615.

The IF0, IF4, and HF2 events exhibit large delays of ∼10, 4, and 3 minutes, respectively, between the times of maximum continuum (C3615) and maximum line (Hγ) emission, suggesting a difference in the heating and cooling timescales of the continuum and line radiation during the impulsive phase. We find that these relatively common, large lags result when secondary continuum peaks lag the primary events, as discussed above for Figure 13. For most flares without a relatively large secondary event following closely after the first peak, however, there is a difference of less than one minute (and in most flares, no lag within the time resolution of the spectra) between the peak times of the continuum and Hγ line. The short delay in peak times of <1 minute suggests that a common heating mechanism produces the impulsive phase of line and continuum radiation during high energy flare events with relatively simple morphology, such as IF3 and IF9. The largest (apparent) lags of 18 and 15 minutes result in the GF2 and GF3 events, which have two (or more) primary peaks in C3615 of similar amplitude separated by large amounts of time (≳ 15 minutes). However, considering the time only around the first impulsive phase in the GF2 and GF3 events, the lags between the local maxima of Hγ and C3615 are 0–0.5 minutes. We speculate that lags are the result of superimposed flare events, and the emission components with different decay timescales (e.g., Hγ and C3615) add to produce different timings of the peaks. We plan to investigate this further in a future paper.

We connect the timing properties of the HB components with a simple new relation, the time decrement. We use the exceptionally high-quality data covering the time evolution of the hydrogen lines in flares IF3, IF9, HF2, and GF1 (Figures 14, 15, 16, and 17, respectively) to show this relationship and how it varies among flare type. In addition to Hα, Hβ, Hγ, and Hδ, we show the continuum evolution of PseudoC, BaC3615, C4170, Ca ii K, and He i λ4471. The fluxes are normalized to their peaks to illustrate the different decay trends. Figure 14 (bottom panel) also shows the rise phase in detail for the absolute fluxes of Hα, Hβ, Hγ, and Hδ.

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From the data in Figures 14–17, it is evident that the higher order Balmer lines have a faster decay time compared to the lower order lines, declining to a lower relative flux by the end of the impulsive phase. This effect was noted by Doyle et al. (1988) and HP91. According to t_1/2, the ordering of the components from fastest to slowest is C4170, He i λ4471, BaC3615, PseudoC, Hδ, Hγ, Hβ, Hα, and Ca ii K. Hγ and Hδ have rather similar decay rates, but Hγ is apparently slower.²² Ca ii K will be discussed in Section 5.3.

To connect the timescales across the Balmer series, we plot the t_1/2 value of each transition as a function of the wavelength of the transition in Figure 18 for IF3 (red asterisks), IF9 (black diamonds), GF1 (black circles), and HF2 (black squares). An estimate of t_1/2 for the H10 line (using an extrapolation from the blue-optical as the underlying continuum) from the PseudoC component is included as well as the value of t_1/2 for BaC3615. Remarkably, this "time-decrement" relationship among the Balmer spectral components appears nearly linear in wavelength space for the two classical, impulsive (IF) events in the figure. The time decrement for IF3 is fit with a linear relation (red dashes) to show the trend

$$\begin{equation} t_{1/2}=0.014\lambda - 36.96. \end{equation} \tag{ 7 }$$

IF9 has a similar time decrement as IF3 but with a factor of ∼1.7 shorter timescales. The linear relation for IF9 (purple dashes) is

$$\begin{equation} t_{1/2}=0.0077\lambda - 21.83. \end{equation} \tag{ 8 }$$

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The scaling between the time decrement relationships of these two flares indicates a fundamental similarity between the heating/cooling processes of Balmer emission produced in medium and large classical flares. A simple discussion of the physical parameters that produce the linear time decrement of the IF events is given in Section 11.1, but this phenomenon should be investigated with detailed RHD models.

The gradual flare GF1 has the same t_1/2 for Hγ and Hδ as IF3, but the lower order line evolution is faster and the higher order line evolution is slower: in other words, the time decrement for GF1 is flatter compared to the IF events.

The time decrement of HF2 is flat for the lower order lines and steepens for the higher order lines and BaC3615. HF2 has two continuum peaks with a highly elevated gradual decay phase, and its overall time evolution is a result of the combined (i.e., spatially unresolved) heating during the two emission peaks. Recall that in Figure 13 (Section 5.1) we compared the t_1/2 of Hγ and the BaC3615 between all flares, and found a general relation for the simple events while complex events, such as HF2, behaved differently. Figure 18 elucidates this difference, consistent with its hybrid classification. The time evolution of HF2 is studied further in Appendix D of Kowalski (2012). To understand the time decrement of HF2, it will be necessary to superpose two simple events (each with a linear time decrement) with the spacing of the two peaks in HF2. A study that explores the results of superposing emission of simple events will be presented in a future paper.

The t_{1/2, C4170} values (Table 7) for these four events are also shown in Figure 18 as light blue symbols. The very fast evolution (t_1/2 = 1–15 minutes) of C4170 does not follow the time-decrement relationship among the Balmer emission components. The ratio of the t_{1/2, C4170} values (not used in the linear fits) between IF3 and IF9 is ∼3.8 which is larger than the scaling factor of ∼1.7 between the time decrements of the respective Balmer components. This indicates that several heating/cooling processes are simultaneously present during the flare—a dominant process for the Balmer emission component, which approximately scales (e.g., similar heating over larger area) between classical flares, and a dominant process for the C4170 emission component, which does not scale in the same way between classical flares. Therefore, it is possible that these different heating processes are present for different lengths of time (e.g., the C4170 heating process only in the impulsive phase). If the C4170 originates from the same (or similar) heating process that produces the Balmer lines—as was concluded from the similar timing of the peaks of Hγ and C3615 for these two flares (Section 5.1—the fast timescale of C4170 either implies formation in a denser region of the atmosphere where the cooling is more efficient or a threshold in the strength of the heating process that can produce C4170.

Additionally, the time evolution of C4170 gives important insight into how the heating processes (and hence time decrement) vary between the types of flares (IF, HF, GF). The t_{1/2, C4170} is relatively large for GF1, and this may be related to the general flatness of the Balmer time-decrement relation. In other words, the GF1 event has similar timescales for its spectral components, implying that they are more closely related in their formation and persistence over time. The t_{1/2, C4170} for HF2 is slightly large compared to the Balmer time decrement, and this flare consists of two temporally resolved, yet spatially unresolved, peaks superposed. The different heating properties in the two peaks of HF2 may generate the mixed time-decrement behavior. The IF events show a simple, linear relationship among the Balmer emission components, yet they also exhibit the largest difference between Balmer emission and C4170 timescales. The timing of the C4170 and Balmer series are important constraints for consistently modeling these spectral components together.

The formation and time evolution of the Ca ii K line has long been a mystery, including why it responds slowly in flares and peaks after the Balmer lines. It has been associated with formation in the lower chromosphere in time-dependent, nonthermal electron heating models of the impulsive phase (Abbett & Hawley 1999; Allred et al. 2006), and phenomenological modeling has shown that Ca ii K emission can also be associated with a hotter, higher region in the flaring chromosphere (Schmidt et al. 2012). Models of coronal X-ray backwarming have shown that relatively large amounts of Ca ii emission originate from a range of heights in the flare chromosphere for T ∼ 5000–7600 K (Hawley & Fisher 1992). The timing properties have been interpreted with a scenario in which hot flare loops cool down to the temperature of Ca ii K formation (Gurzadian 1984; Houdebine 2003; Crespo-Chacón et al. 2006), but this remains to be tested with RHD models of the gradual decay phase.

The Neupert effect (Neupert 1968) is an observed relation between the signatures of impulsive phase nonthermal particles and gradual phase coronal heating. The Neupert effect is usually reported as the proportionality between the integral of nonthermal microwave, hard X-ray, or white-light emission and the luminosity of thermal soft X-rays. The Neupert effect has been observed in solar (Dennis & Zarro 1993) and stellar (Hawley et al. 1995; Guedel et al. 1996; Osten et al. 2004; Fuhrmeister et al. 2011) flares, and is a fundamental aspect of the standard flare model. It is usually interpreted in terms of the chromospheric evaporation process (Fisher et al. 1985), whereby the nonthermal particles impact chromospheric material, which then ablates into the corona and emits thermal radiation at millions of degrees well into the gradual phase of the nonthermal particle emission. The models of Hawley & Fisher (1992) showed that the Ca ii K line flux evolution during the Great Flare could be produced from X-ray backwarming from a flare corona at T ∼ 10 MK. Here we investigate whether Ca ii K follows a Neupert effect representing the gradual phase emission (like soft X-rays) using C4170 to represent the impulsive phase emission.

Ca ii K is the most gradual line in the intermediate and blue-optical spectral zones (Figures 14–17); well-known characteristics are a longer rise time than the Balmer lines, a late peak in the beginning of the gradual decay phase, and a slow return to quiescence (e.g., HP91). In Figures 19 and 20, we show the flux of Ca ii K (blue diamonds), the flux of C4170 (crosses) and the cumulative integral of C4170 (red lines) for flares with ARC 3.5 m/DIS data. We calculate the cumulative integral of C4170 until the time of Ca ii K maximum, which in some cases occurs between the end of the impulsive phase and beginning of the gradual decay phase (IF0, HF1, HF2, GF1) and in some cases occurs deep into the gradual decay phase (IF3, IF9). This variation in the timing of the Ca ii K peak has been seen in the literature (Houdebine 2003). There is a varying degree of similarity in the evolution between the Ca ii K line and the cumulative integral of the continuum. The closest similarities are found for the flares IF6, IF7, IF9, HF1, HF2, GF1, and GF2; the largest differences are found for IF2, IF3, IF4, and GF3. For the fastest flares, it would be useful to test this relation with higher cadence data. For the flare IF1, we show the cumulative integral of the U-band light curve scaled to match the maximum of the Ca ii K light curve. Despite not knowing if the maximum of Ca ii K within this time window (at t ∼ 1.75 hr) is the absolute maximum for the flare, the cumulative integral of U follows the Ca ii K from t = 1.3 hr to t = 1.75 hr. In some flares, the Ca ii K line evolution does not obviously respond during the first part of the impulsive phase, showing evidence of a delay compared to C4170; this is seen in IF3, IF4, HF1, and HF3. Ca ii K shows a decrease (although by a small amount) relative to the previous spectrum in the impulsive phase of the flares IF2, IF4, IF5, IF9, HF1, and HF3. In the flare IF2, a decrease in line flux is most noticeable, disappearing completely from the decay phase value of HF1 before increasing again in the beginning of the gradual decay phase of IF2. We speculate that the lack of response in Ca ii K is a result of the formation of strong hot, blackbody emission, resulting in Ca ii K absorption (Section 6.3) effectively canceling the amount of emission from a different spatial location. Of course, this explanation requires confirmation from detailed models.

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If the Neupert effect underlies the relation between the cumulative integral of C4170 and the flux of Ca ii K, then the gradual evolution of Ca ii K could be related to the chromospheric evaporation process. Under this interpretation, C4170 represents impulsive phase heating of the lower atmosphere and material is evaporated into coronal loops; regions of the chromosphere (perhaps even in distant regions away from the main flare loops) are then heated to ∼6000 K from, e.g., incident X-ray and EUV (XEUV) backwarming radiation originating from coronal plasma. A connection between the fastest component, C4170, and the slowest component, Ca ii K, in our stellar flare observations provides important constraints for RHD models that seek to produce a consistent picture of the flare process whereby the C4170 can be incorporated into the standard solar flare model of, e.g., Martens & Kuin (1989). Because we observe this relation in flares relatively independent of morphological type, the degree to which the Ca ii K Neupert relation holds gives additional constraints for how the heating process varies between flares of a given type. For example, of the four most implusive flares in the DIS sample, the Ca ii K Neupert relation clearly does not hold for IF2 and IF4. Note that Osten et al. (2005) found a violation of the Neupert effect from soft X-ray observations during a (rather impulsive) U-band flare, suggesting that the relation may break down in some heating scenarios.

The amount of flux that we can attribute to Balmer emission compared to the total flare emission constrains the amount of unexplained energy (e.g., white-light continuum) for model predictions. In this section, we characterize the amount of non-HB radiation as a function of morphological flare type. We also investigate the relationship between the HB flux and χ_flare. Current models fail to reproduce χ_{flare, peak} (see Section 9); therefore, the relative amount of HB emission present at times of peak flux can guide modeling efforts.

For quantitative purposes, we denote the HB component as the sum of the fluxes in Hδ, Hγ, Hβ, the PseudoC, and the BaC_tot (see Section 3.2).²³ The HB component does not include H (H7) in the blue optical because it is blended with Ca ii H,²⁴ and it does not include Hα because it is not available for some flares due to lack of wavelength coverage or saturated flux values (IF0, IF1, IF10, IF4, IF6, HF4, and GF3; see Kowalski 2012 for an analysis of the Hα line evolution in the Flare Atlas). We investigate the coarse time evolution of the percentage of HB flux ("%HB" or "HB flux ratio") to total (λ = 3420–5200 Å) flare flux at the time of maximum continuum emission and near the beginning of the gradual decay phase (these times and spectrum numbers are given in Table 5 and are indicated by red vertical lines in Appendix B, Figures 36–54). The gradual decay phase measurements are averaged over three spectra, and the peak measurements are averaged in several of the GF events to increase signal-to-noise. The results are shown in Figure 21 (left panel) and given in Table 8 for the IF, HF, and GF events. The spectra S#23–25 (gradual decay) and #113 (just before peak of MDSF2) measurements are shown for IF1 as light blue star symbols and the IF0 peak and decay spectra are shown as dark purple star symbols.

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Nearly every flare shows an increasing percentage of HB flux from the peak to the gradual decay phase,²⁵ indicating that the gradual decay phase emission is marked by an increased relative importance of HB emission. The percentages increase by ∼20% for most flares; larger changes of ∼30% occur during IF0, IF3, IF4, IF5. Except for IF3, these flares also have very sudden breaks from impulsive to gradual phases. Significantly smaller changes, <20%, occur during HF3, GF1, and GF5 because the gradual decay phase of these flares begins at a relatively high amplitude compared to the peak, and the break from impulsive to gradual phases is much less defined.

There is also an increasing percentage of HB emission during both the peak and decay phases according to the IF/HF/GF sequence, in agreement with the flux of Hγ divided by C4170 at peak emission in Figure 11. The IF events show a range of 3%–24% at peak, with most between 11%–17%, but changing to ∼30%–52% during the gradual decay phase. The HF events are scattered around 25%–35% at peak and 40%–50% in the gradual decay phase, while the GF events are scattered around 35%–50% at peak and 40%–65% in the gradual decay phase. The GF events show more scatter and less uniformity in their behavior compared to the IF and HF events. Note the light curve morphology of GF1 (Figure 50 of Appendix B); there are impulsive phases at t ∼ 1.92 hr, t ∼ 1.94 hr, and t ∼ 2.02 hr prior to the major, broad emission peak at t ∼ 2.13 hr. The values in these peaks are ∼28%, 25%, and 36%, compared to 38% in the main peak.

The individual flux contributions from Hδ, Hγ, Hβ  and Hα are also given in Table 8 for the peak phases (maximum C3615) of each flare. The values for Hδ, Hγ, and Hβ also trace the morphological classification of a flare: IF events produce <1%–2%, HF events produce ∼2%–3%, and GF events produce 3% or more of the λ = 3420–5200 Å flux in each of these hydrogen lines. When available, the flux contribution from Hα is generally similar to the other hydrogen lines at maximum continuum emission. The individual contributions are important constraints for RHD models that calculate the detailed radiative transfer using a relatively small hydrogen atom (six levels is typically used) and because blending from neighboring hydrogen lines does not affect the flux measurements of the lower order transitions.

The evolution of χ_flare (C3615/C4170) from the peak to the gradual phase times is shown in the right panel of Figure 21 and the values are given in Table 7. Indeed, both χ_flare and the percentage of HB emission increase during the flare gradual decay phases, implying a connection between these measures. The IF events show rather similar trends: they have a change in the percentage of HB emission of ∼20%–30% and Δχ_flare∼1 from the peak to the gradual decay phase times.²⁶ The IF5 and IF6 events stand out as IF events that have a large amount of HB (>20%) and large χ_flare (∼2.2) at peak (see also Section 4.3.1). These two flares have percentages of HB emission that rise to large values (42%–52%) and values of χ_flare that increase to ∼4 in the gradual decay phase, further strengthening the connection between these two measures (χ_flare, HB flux/total flux) even for the seemingly peculiar IF events that are not as similar as the other flares in the IF type.

The peak phase value of the percentage of HB emission for IF1/MDSF2 coincidentally falls within the same range as the other IF events; however, the percentage of the newly formed Balmer emission is uncertain due to Balmer absorption during this secondary flare (Section 6.3). The χ_flare of IF1 for peak of MDSF2 and decay at the start of the spectral observations (light blue star symbols) are also consistent with the other IF events, even though a large amount of decay emission is present at the start of MDSF2.

A time-resolved analysis of IF1 reveals the strong connection between the detailed evolution of χ_flare and the percentage of HB emission during successive impulsive (e.g., MDSF2) and gradual decay phases. In Figure 22 (top), the χ_flare evolution is compared to the U-band evolution over 1.3 hr of the decay phase of IF1. The trends are anti-correlated during the secondary flares with U-band peaks at t ≈ 1.6 hr and t ≈ 2.16 hr (MDSF2), as shown in Kowalski et al. (2011b). In Figure 22 (bottom panel), the percentage of HB emission is shown, and it varies similarly with χ_flare. The anti-correlated behavior between the percentage of HB emission and the U band (blue) is very similar to that between the U band and χ_flare. The maximum percentage of HB emission is 35%, but then drops to a minimum of 15% at t ≈ 2.17 hr just after the U-band peak of MDSF2.²⁷ The vertical line in both panels denotes the time of minimum percentage of HB emission (at S#116), which occurs just before the minimum χ_flare and just after the maximum U-band enhancement. The time differences between maxima and minima in χ_flare, percentage of HB emission, and the U band are intriguing and should be confirmed with higher cadence data covering the peak and initial fast decay phase.

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The percentages of HB flux in flares (Table 8, Columns 2 and 3) should be directly compared to RHD models. The non-HB flux [1−HB flux/total flux] is largely due to flux from the underlying white-light continuum that extends from the NUV through the optical. This accounts for the majority (≳ 50%) of the total flux in all flares at peak, and as much as 83%–89% of the total flux at peak in most IF events. For the first time, we have included the BaC at λ > 3420 Å in the energy budget with the other Balmer features, and we have shown that the Balmer component is not the dominant source of flux in flare emission in the blue. At most, the Balmer contribution accounts for ∼50% of the blue-optical flux, and only in GF events and the decay phase of other flares. The continuum component that accounts for the non-HB flux is represented by C4170. In the next section (Section 6), we will model this non-HB emission as hot, blackbody emission.

In Section 4 (Figures 8(a)–(c) and 9), we showed the flare spectra at peak times for all flares. The flare peak spectra exhibit a steeply rising continuum toward NUV wavelengths, which is nearly ubiquitous and continues into the NUV beyond the spectral range of DIS. In Section 5.4, we found that typically only 11%–17% of the flux from λ = 3400–5200 Å in the flare peak spectra of the IF events is accounted for by HB (line and continuum) emission (Figure 21). We find that the spectral shape of non-HB emission matches that of hot blackbody emission which pervades the entire NUV, blue-optical, and red-optical zones; C4170 is dominated by the emission from the blackbody continuum component.

The blackbody fitting is performed as follows. We calculate a color temperature, T_BB, of the blue-optical continuum by fitting a Planck function to the flux in the continuum windows from λ = 4000–4800 Å (BW1–BW6, similar to those used in Kowalski et al. (2010)) given in Table 4. We used the peak flare spectrum from 2011 February 24 (IF3) and a decay spectrum from 2009 January 16 (IF1) to guide our selection of line-free regions. The same windows were used to fit a straight line to the continuum in order to calculate BaC3615 (Section 3.2). We do not include the flux at shorter wavelengths due to the difficulty in determining bona-fide continuum from blended hydrogen lines (PseudoC) and due to the contribution from BaC emission. We do not include flux from redder wavelengths, because we find evidence for complicated behavior in the red-optical (Section 8). We also fit a filling factor ($X_{\mathrm{BB}} = R_{\mathrm{fl}}^2 / R_{\mathrm{star}}^2$; as in Hawley et al. 2003 and Kowalski et al. 2010) using the IDL routine mpcurvefit, where X is the fraction of the visible hemisphere covered by the projected area²⁸ of the flare (discussed in Section 10.1). The two parameters are fit simultaneously, which is an important distinction to the analysis from Kowalski et al. (2010) where T_BB was fixed to 9000 K, 10, 000 K, and 11, 000 K. The fits for IF1 are redone here using the new flux scaling (Section 2.6), and some quantities change compared to those in Kowalski et al. (2010, 2012). Good quality fits are obtained for the IF events (except for IF6²⁹), the HF events, GF1, GF3³⁰ and GF5.

The blackbody fits are indicated by light blue lines in the flare peak spectra panels (Figures 8(a)–(c) and 9) and are given in parentheses. In Figure 23, we show the distribution of T_BB at maximum continuum emission for the 17 flares with well-determined temperatures (values are given in Table 7). The flares are colored by the morphological type, but there is no discernible trend with type or peak $\mathcal {L}_U$. We find that T_BB = 10, 000–14, 000 K for the IF events (Figure 8(a)). By comparing to the dotted (pre-flare) spectra we see that this hot, blackbody component is present for a variety of flare amplitudes, appearing in flares with peak U-band enhancements greater than I_{f, U} + 1 ∼3, and as large as I_{f, U} + 1 ∼80. The high signal-to-noise ratio of our spectra makes the characterization of this component possible even with relatively small contrast in the blue-optical at λ > 4500 Å. The blackbody component contributes ∼76%–97%, and on average 84%, of the λ = 3420–5200 Å flux during the peak times for the IF events (Section 5.4).

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The flare peak spectra of the HF events (Figure 8(b)) also show the hot, blackbody component, with temperatures between T_BB ∼9500–12,000 K. The lower amplitudes of these flares are noticeable, e.g., by comparing to the pre-flare spectra. The HF events also have a conspicuous amount of excess emission above the extrapolated light blue curve at λ < 3800 Å. Of the GF events in Figure 8(c), the large-amplitude, high energy gradual flare GF1 shows the most convincing evidence of a moderately hot blackbody, with T_BB ∼9000 K. A significant NUV excess is also present during this flare. GF1 has a small-amplitude impulsive phase with a strong response in C4170 that occurs before the main flare peak (see Figure 17); the newly formed flare emission in this pre-cursor flare peak has T_BB ∼ 14, 500 and is included in Figure 23 as GF1'. GF3 shows evidence of a high temperature, but these data have a more uncertain calibration³¹; however, we averaged three spectra near the peak and derived a similar value of T_BB for this flare. We have averaged four spectra at the flare peak of GF5, which results in a relatively good fit, but with a lower temperature of T_BB ∼ 6700 K. In GF2 and GF4, we cannot accurately ascertain the characteristics of the slope of the continuum; nonetheless, the very low-amplitude flares (I_f + 1 ∼ 1.3–2.2, |ΔU| < 1 mag; GF2–GF5) do show excess C4170 emission.

The ∼5% blue-continuum shape uncertainty coupled with the uncertainty in the scale factor, R (see Section 2 here and Appendix A of Kowalski (2012)), and also the contamination of continuum windows from emission lines (HB line wings and low-level Fe and Ti lines) lead to systematic uncertainties in the blackbody temperatures of about 500–1200 K. The statistical uncertainties are only ∼200 K for most events, and are indicated by the error bars in Figure 23. Kowalski et al. (2010) found a ±1000 K range in temperatures that well-represented the blue continuum shape. In Appendix F of Kowalski (2012), the detailed systematic errors involved with temperature fitting are examined. The important result is that for the IF2 event, the extreme amount of line broadening causes the BW1 and BW2 fitting windows to have possible contamination from Balmer line wing emission; the acceptable range of T_BB for this flare is therefore 11,700–14,100 K, though the temperature of 14,100 K better accounts for the shape of the total emission at λ ≲ 3646 Å.

In summary, we observe a continuum at flare peak times of the impulsive phase with a spectral slope matching that of a hot, blackbody with T_BB ∼ 9000–14, 000 K (most within 10,000–12,000 K) which is relatively independent of flare amplitude over a large range of peak U-band specific luminosities, total energies, and light curve morphologies. For this range of temperatures, the spectrum would not be expected to strictly follow a λ⁻⁴ slope according to the Rayleigh–Jeans law. Instead, a linearly decreasing fit to the blue optical zone gives a reasonably good fit and looks very similar to a Planck function for these temperatures: in the formula for the Planck function, the steeply decreasing λ⁻⁵ part multiplies with the increasing exponential part, giving a linear decrease toward the red for λ = 4000–5000 Å.

The interflare variation in T_BB at flare peak is likely significant despite the large systematic uncertainties of ∼1000 K. We show convincing evidence that interflare variations are significant between two flares, IF2 and HF1, in Appendix G, and we discuss the implications for flare heating differences in Section 11. Additional high signal-to-noise data would be useful to determine if the color temperature decreases to ∼7000 K at flare peak during very small amplitude events (I_f + 1 ≲ 1.5), as implied by the observations of GF5. NUV data covering the flare peak of the continuum (likely near λ = 2500–3000 Å) would help to more precisely determine the blackbody temperature at flare peak and its evolution during the impulsive phase.

This hot, blackbody continuum component has been detected previously using colorimetry and spectra (see references in the Introduction). Self-consistent flare heating models of the impulsive phase have thus far not reproduced this (dominant) emission component; we compare to current models in Section 9. In summary, we have found that a T_BB ∼10⁴ K blackbody function and linear function are useful mathematical representations for this continuum component; however, we will shortly show (Section 6.3) that a single blackbody or line does not account for important absorption features in the spectrum.

We find that there is always excess flux in the NUV zone, above an extrapolation of the Planck function at these wavelengths. The slope of the total NUV spectral zone emission at flare peak generally follows the same slope as the underlying Planck function. In Figures 8(a)–(c) and 9, we scaled the Planck function to the flux at C3615, and showed in yellow that it matched the flux at λ < 3646 Å, illustrating that the overall shape of the continuum at λ < 3646 Å for the IF, HF, and some GF events can be represented by about the same temperature fit as for the blue-optical. We measure the slope of the NUV flare emission at 3420 Å ⩽ λ ⩽ 3630 Å by fitting a line to three wavelength bins in this region. These values are given in Table 7 (Columns 9 and 10 are the values for the flare peak and gradual decay phase; Columns 11 and 12 are the values for the flare peak and gradual decay phase with the underlying blackbody emission subtracted) are consistent with the blue spectral shape indicated by the yellow curves, and will be important for comparing to detailed BaC model predictions in Section 9. Note, the flux calibration errors in the NUV are larger (∼5%–10%), and the statistical uncertainties in the fits can also become comparatively large; higher signal-to-noise measurements should be obtained in the NUV.

Although the total flux at λ < 3646 Å follows the same general slope as the underlying blackbody, we attribute the excess flux in the NUV to chromospheric BaC radiation from recombination to the n = 2 level of hydrogen, as in Kowalski et al. (2010). We find that BaC emission is ubiquitous at flare peak for all flares in our sample, consistent with our conclusion that BaC emission affects the value of χ_{flare, peak} and is correlated with the relative amount of Hγ emission (Section 4.3.1, Figure 11). As explained in Section 3.2, we use both the BaC3615 and BaC_tot quantities as measures of the BaC emission.

In Figure 24, we show BaC3615/C3615 (the fraction of the NUV flux emitted in BaC emission) and find a general ordering between the IF, HF, and GF events. There is a decreasing trend with peak specific U-band luminosity with a range of values for each morphology class, especially for the IF events, which have a standard deviation that is twice the standard deviation of the percentage of HB emission (Section 5.4). The IF events are least dominated by BaC3615, with most having values of 0.2–0.35. These flares are dominated by the hot blackbody emission component which also contributes to the flux of C3615. While the IF5 and IF6 events have larger values, ∼0.4, IF4 has a very low value ∼0.05 (but also required additional calibration—see Appendix A of Kowalski 2012). The HF events have about equal contributions (∼0.5–0.6) and the GF events have the majority of their peak NUV flux (0.55–0.8) in BaC emission. In Column 8 of Table 8, we present these values. We conclude that the general relationship is that the overall light curve evolution (flare morphology) is physically connected to the relative importance of the BaC flux at the time of maximum continuum emission.

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Several flares in our sample have impulsive phase durations that are significantly longer than the spectral cadence, thereby allowing detailed time-resolved analyses of the two continuum components. In particular, IF1, IF3, and GF1 have the greatest number of spectra during their respective impulsive phases (see figures in Appendix B). Note that the impulsive phase of IF1 refers to the subpeak MDSF2, not the main flare peak which did not have spectral coverage.

In this section, we analyze the rise phase and peak spectra of MDSF2, to show that the hot, blackbody continuum component is not a featureless Planck function across our wavelength range. Here, we establish the presence of absorption features in impulsive phase flare emission, and thereby redefine how we understand the "blackbody" continuum properties (e.g., Figure 23).

An indication for the presence of absorption in our spatially unresolved flare data is an anti-correlation between the time evolution of BaC3615 and the time evolution of the total amount of the blue/NUV continuum, as diagnosed by the U-band flux or C4170 flux. This effect has been observed during the secondary flares in the decay phase of IF1 (Figure 1(d) of Kowalski et al. 2010) and presented graphically through a sequence of rise-phase spectra (Figure 1 of Kowalski et al. 2012). Those figures illustrate that as the U band increases during the secondary flare MDSF2, the entire blue/NUV continuum becomes dominated by the blackbody component whereas the absolute (and relative) amount of BaC3615 decreases.

In previous papers on IF1/MDSF2 (Kowalski et al. 2011a, 2012), we have denoted the sum of all (spatially unresolved) decaying and impulsive phase emission as $F_{\lambda }^\prime$ (obtained by subtracting the preflare quiescent spectrum), and newly formed flare emission isolated from an impulsive event as $F_{\lambda }^{\prime \prime }$ (obtained from subtracting the preflare decaying emission).³² During MDSF2, we find direct evidence that the newly formed flare emission ($F_{\lambda }^{\prime \prime }$) during the rise phase resembles the spectrum of a hot star. This was first shown in Kowalski et al. (2011a); in this section, we present an analysis of the time evolution.

Figure 25 shows the newly formed flare spectra ($F_{\lambda }^{\prime \prime }$) during the rise (black; same spectrum as presented in Kowalski et al. (2011a)) and just prior to the sub-peak (gray) at λ < 6500 Å. At both times, the blue-optical shapes are well-matched by the spectrum of Vega (from Bohlin 2007) scaled and plotted in red to show the striking similarities in the spectral slope and absorption features. The difference between the flux at λ < 3600 Å in the gray spectrum and the flux in Vega could be due to the spatially unresolved chromospheric BaC emission forming during the event or due to the difference in gravity between the hot star and an M dwarf (log g = 4 and log g = 5, respectively). Models of deep heating (e.g., Kowalski et al. 2011a) in M dwarfs are required to understand these detailed spectral differences.

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Mihalas (1970) discusses the detailed formation of hot-star spectra: the lower emission at λ ∼ 3615 Å relative to the emission at λ ∼ 4170 Å (in other words, the BaC is in absorption) is a result of the non-uniform temperature structure of the atmosphere coupled with wavelength dependent hydrogen Balmer and Paschen continua opacities (and flux redistribution) in an optically thick medium. In other words, the optically thick emitting material producing the λ ∼ 3615 Å emission is at a lower temperature (e.g., from lower column mass or higher height for a radiative equilibrium stratification) than the optically thick emitting material at λ ∼ 4170 Å. We have therefore revealed the multithermal contributions that comprise the hot blackbody emission component (see Section 6.1 and Figure 23). The use of the blackbody function (and T_BB) is a useful parameterization of the λ = 4000–4800 Å slope, but it is important to realize that it is an approximation of the detailed radiation field originating from a flare atmosphere with an opacity dependence as a function of wavelength and a temperature dependence as a function of height.

Temperature evolution during the rise of MDSF2 is qualitatively evident in Figure 25 by comparing the blue-optical spectral shape to Vega from flare rise (black) to flare peak (gray). The spectrum becomes steeper (corresponding to a hotter color temperature), and there are three ways to measure this for any given time during MDSF2. For example, we consider the spectrum near maximum continuum emission (S#113, the gray spectrum in Figure 25):

• 1.  
  Method 1: Color temperature of total flare-only emission. The color temperature of the total emission with quiescent level subtracted ($F_{\lambda }^\prime$) is ∼11, 000 K (Figure 23). This represents the color temperature averaged over the entire flaring area over the duration of the exposure (Kowalski et al. 2012).
• 2.  
  Method 2: Color temperature of newly formed flare-only emission. The temperature of the newly formed flare emission ($F_{\lambda }^{\prime \prime }$) is ∼18, 000 K just prior to the U-band peak. This temperature is obtained by subtracting the previously quiescent+decay flare emission. This technique provides a better estimate for the shape of the continuum that is produced during the times when there is an impulsive phase. Relating the continuum shape to a color temperature with this technique assumes that the (1) the previously decaying emission is small or that it does not decay significantly and (2) the newly formed flare emission originates from a spatially distinct flare area that was not previously heated. Otherwise, the non-linear dependence between T and intensity (e.g., for the Planck function) results in an incorrect association between T_BB and $F_{\lambda }^{\prime \prime}$.
• 3.  
  Method 3: Adjusted color temperature of newly formed flare-only emission. In Appendix D, we estimate the amount by which the flux of the previously decaying emission has decreased during the secondary flare. We apply these estimates to the times of the spectra during MDSF2 which allows the best derivation of the properties of the newly formed flare emission. We find that the color temperatures are ∼1800–2700 K lower than method #2 (above). The value at flare peak of MDSF2 (T_BB ∼ 15, 000 K) falls just beyond the high end of the sample's distribution of T_BB in Figure 23.

We summarize the derived color temperature values at three times in Table 9 using these three methods. The times correspond to the time just before MDSF2, halfway through the rise phase of MDSF2, and just prior to flare peak of MDSF2. The observed apparent anti-correlation between the time evolution of BaC emission (BaC3615) and blackbody emission (Kowalski et al. 2010, 2011b, 2012) is now explained by the presence of an emission component with "absorption" features forming during the flare. If there are significant absorption components in other flares, then the value of the derived BaC3615 value for these flares (obtained from a linear extrapolation—see Section 3.2) is an underestimate of the intrinsic amount of BaC emission originating from the chromosphere.

We briefly examine whether BaC absorption has an obvious effect on the distribution of the peak properties of the Flare Atlas. We calculate the peak specific luminosities of C4170 and BaC3615 (see Sections 3.1 and 3.2). We find that in general, the IF events (with lower χ_{flare, peak} values; Section 4.3.1) are more luminous in C4170 whether the BaC3615 is emission or absorption (on average, $\mathcal {L}_{\mathrm{C}4170}/\mathcal {L}_{\mathrm{BaC}3615} = 2.4$), the GF events are more luminous in BaC3615 (on average, $\mathcal {L}_{\mathrm{BaC}3615}/\mathcal {L}_{\mathrm{C}4170} = 2.4$), and the HF events are slightly more luminous in BaC3615 (on average, $\mathcal {L}_{\mathrm{BaC}3615}/\mathcal {L}_{\mathrm{C}4170} = 1.5$). The trend between morphological type and the relative amounts of BaC3615 and C4170 could be related to the amount of BaC absorption produced by the blackbody (C4170) component. Since the amount of BaC3615 that we measure is probably less than the intrinsic BaC3615 emission (due to this absorption component) the emission and absorption may add together to produce smaller values of BaC3615—and χ_{flare, peak}—for the IF events. The HF and GF events may produce intrinsically more BaC3615 than the IF events, they may produce less BaC3615 absorption than the IF events, or they may have both effects. However, we suggest the possibility that absorption can also lead to both low values of χ_{flare, peak} and large values of C4170/BaC3615. The BaC emission produced in large amounts in the early flare phases, and persisting over a relatively long time compared to the blackbody component, effectively veils the absorption signatures except in the flares with the most extreme heating. Detailed modeling of the blackbody component over a large parameter space is needed to understand the role of absorption in flares of all morphological types.

We now present cases of line broadening in larger flares where there is evidence of absorption in the wings. In Figure 26, we show Hδ, Hγ, Hβ, and He i λ4471 profiles ($F_{\lambda }^{\prime }$) for IF4, at maximum continuum emission (black, S#665) and at maximum Balmer line emission (S#672, turquoise). The profiles are plotted with the local continuum subtracted with a straight line and then are normalized to the peak of the line. The striking effect here is a "depression" (deficit) in the line wings in the black profile (maximum continuum) between ±10 Å to ±30 Å from line center. Furthermore, the amount of wing depression increases from Hβ to Hδ. The differences in the widths (measured at the 10% line flux level) between the times of maximum continuum flux and maximum line flux are ∼6 Å for Hδ and Hγ and ∼2 Å for Hβ. Although the change in the line widths for Hδ and Hγ are just ∼2σ (from 3 Å uncertainty in widths), we observe a flux depression (deficit) over at least 10 contiguous pixels in the wings on both sides of the line center, which increases the significance of the flux depression. During the rise phase spectrum (S#664, corresponding to a flux of ∼40% of the peak continuum emission), the Balmer lines are narrower than at both maximum line and continuum emission, but there is less evidence for flux depressions in the line wings. In the spectrum following flare peak, the Balmer lines have begun to broaden significantly and no wing depressions are seen.

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A similar wing depression effect is seen in the IF1 spectra during the peak continuum emission of the subpeak MDSF2. In Figure 27, we show the profiles for the IF1 flare near the time of maximum Balmer line emission within the spectral observation window (turquoise) and at the peak continuum emission (black) during MDSF2. The wing depression is present, but relatively less than in IF4 (Figure 26), and does not appear as far from line center as in IF4. The wing depression increases for the higher order Balmer lines as in IF4. The line widths (measured at the 10% line flux level) at the time of maximum continuum emission are 2–3 Å narrower than at the time of maximum line emission.

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We find similar evidence for wing depressions during flare peak of IF3 (see Appendix E), given that the rise phase emission is subtracted from the flare peak.

For IF1, the newly formed flare emission during MDSF2 was found to have a spectrum like a hot star, with the Balmer lines in absorption (see Section 6.3). In particular, note the similarities in the widths of the Balmer absorption lines between the Vega spectrum and the flare spectra in Figure 25. We therefore interpret the wing depressions in IF1 and IF4 as direct signatures of Balmer line absorption³³ forming during these flares. The absorption wings in model hot star spectra with log g = 4–5 are very broad (Castelli & Kurucz 2004) due to Stark broadening (Peterson 1969). The superposition of an emission line component and an absorption line component leads to the net decrease that we observe in the wings in the flare spectra. As a result of the superposition of emission and absorption, the measured line widths (at the 10% flux level) also decrease. The Balmer absorption line flux decrement for Vega is Hδ: Hγ: Hβ ≈1.09: 1.00: 0.75 (using the spectrum from Bohlin 2007); therefore, one would expect the line widths and wing fluxes of the higher order Balmer lines in flare spectra to be more strongly affected in the presence of hot star-like absorption. The Balmer emission line component originates from the flaring chromosphere (Hawley & Fisher 1992; Allred et al. 2006) and veils the Balmer absorption component in the total, spatially unresolved flare spectrum. See Johns-Krull et al. (1997) for a phenomenological illustration using quiet-Sun Hα absorption and superimposed chromospheric Hα emission.

Wing absorption signifies heating in deep layers of the atmosphere, which is an important constraint for flare heating models. The heating differences between flares—e.g., between the primary flare of IF4 and the secondary flare MDSF2—should be studied with RHD models that successfully produce the range of blackbody continuum properties. These flares have two of the three lowest χ_{flare, peak} values in the sample (χ_{flare, peak} ∼ 1.3–1.5; Table 7) and also show evidence of Balmer line wing absorption, implying a common heating scenario. Further, the data are suggestive of differences in χ_{flare, peak} and the amount of Balmer line wing absorption (IF4 having a smaller χ_{flare, peak} and also more wing absorption, as is most evident for Hδ). Also, the peak U-band amplitude is much larger in IF4 (I_{f, U} ∼ 20) than in MDSF2 (I_{f, U} ∼ 7). Assuming that IF4 produced the same type of "blackbody" emission as MDSF2 (Section 6.3) with T_BB = 15,000 K (Table 9), the inferred area of this emission would be over three times as large in IF4 and formed over a timescale nine times as fast (t_{1/2, C4170} ∼ 1 minute for IF4, t_{1/2, C4170} ∼ 9 minutes for MDSF2; Table 7). As a result, more "blackbody" emission formed over less time leads to more obvious absorption wings at peak. On the other hand, it is possible that the "blackbody" emission is intrinsically different between the two flares, related to the speed at which the light curve develops and ultimately to the depth of primary heating (IF4) compared to the depth at which primary heating (IF1) may cause secondary flare heating (MDSF2; see Section 10.2). It may also be important to consider that I_{f, U} is much higher—therefore with more blackbody emission present—in the maximum line emission spectrum for IF1 (I_{f, U} ∼35 at S#24) than for IF4 (I_{f, U} ∼4 at S#671).

Searching for wing depressions is a method for directly detecting hot star signatures in $F_{\lambda }^{\prime }$ without needing to isolate newly formed emission, a technique that relies on several assumptions (Section 6.3). This analysis can be employed with other flare data that cannot be accurately flux-calibrated, or for spectral data with limited wavelength range. A larger sample of primary and secondary peaks during high energy flares with high time resolution would be useful for constraining the timing of the absorption wings during the flare and also the relationship between a very small value of χ_{flare, peak} ∼1.5 and the presence of absorption wings.

To study the gradual phase,³⁵ we average three spectra at the times given in Table 5 in order to increase the signal-to-noise in the blue-optical zone for determining a blackbody temperature fit. Figures 28(a)–(f) show the averaged spectra from the beginning of the gradual phase (Table 5 and indicated by vertical red lines in Appendix B, Figures 36–54) for flares with sufficient signal-to-noise for analysis. Note that IF4, IF5, and HF4 have only a single spectrum shown because the data quality was variable during the low levels of gradual phase emission. Two component fits, allowing T_BB, X_BB, and X_BaCF11 to vary, are shown in color with the quiescent spectrum as a dotted line. The gradual phase spectrum for IF10 is given as S#32 even though it contains contributions from (relatively low-level) secondary flares. In parentheses, we have listed χ_{flare, decay} and T_BB; the values are excluded when they are not well-determined (i.e., very little or no continuum emission in the blue-optical zone). We remind the reader that T_BB is a parameterization of the continuum slope; detailed modeling is required to deduce the multithermal structure of the atmosphere (see discussion in Section 6.3).

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A striking similarity among the gradual phase spectra is the value of T_BB, ∼7300–8500 K (typically around 8000 K) in the flares IF1, IF2, IF3, IF7, IF9, IF5, GF1, and HF2. However, some flares have unusual characteristics. The gradual phase of IF4 was taken near the peak of the line emission (due to the uncertainty in flux calibration later in the flare at very high airmass), and this flare has a larger color temperature of 9500 K. The gradual phases of HF1 and HF3 are also hotter with temperatures exceeding 10, 000 K. We have averaged the spectra just following the peak (synthetic U-band) emission of GF5 to show the remarkable detection of white-light continuum (and all the typical flare emission lines) in an event with only I_f + 1 < 1.5 and I_C4170 + 1  ∼ 1.03. This flare has a gradual phase T_BB ∼ 4900 K, the lowest detected temperature in our sample, and is about 2000 K cooler than at peak.

Note that the gradual phase spectrum of IF1 (Figure 28(a)) is that shown in Figure 1(b) in Kowalski et al. (2010), where we modeled this spectrum with a T = 9000–11, 000 K blackbody. T_BB ∼ 8250 K results with our current, better determined fit, with a larger inferred areal coverage, X_BB ∼ 0.4% of the visible hemisphere (∼2 times the value given in Kowalski et al. 2010). Among the three spectra averaged to form the gradual phase spectrum of IF1 in Figure 28(a), the inferred temperature range is between 8000–8500 K. The difference between 8250 K and 10, 000 K is significant, as our estimate of the systematic uncertainty in the temperature determinations is ∼1000 K (Kowalski 2012, Appendix F).

We calculate the specific luminosities of BaC3615 and C4170 (see Sections 3.1 and 3.2) and find a trend among the IF events showing T_BB ∼ 8000 K emission in the gradual decay phase and χ_{flare, peak}< 2: IF1, IF2, IF3, IF7, and IF9. These flares are described by the relation,

$$\begin{eqnarray} &&\mathrm{log_{10}} \mathcal {L}_{\mathrm{BaC3615}} = -0.63(\pm 0.86) + 1.03(\pm 0.03) \times \mathrm{log_{10}} \mathcal {L}_{\mathrm{C4170}}. \nonumber \\ \end{eqnarray} \tag{ 9 }$$

A slope of ∼1 indicates that the BaC3615 and C4170 specific luminosities increase in equal percentages during the gradual decay phase according to the size of the flare. The flares with χ_{flare, peak} >2 and which form a T_BB ∼ 8000 K component at the beginning of the gradual decay phase—IF5, HF2, and GF1—may comprise a second distribution which is offset from those with χ_{flare, peak} <2 and T_BB ∼ 8000 K in the gradual decay phase. However, it is not clear whether the distribution is offset horizontally (signifying that they are "deficient" in C4170) or vertically (signifying that they have excess BaC3615). In any case, the data suggest that the persistence of C4170 with T_BB ∼ 8000 K formation is related to the formation of BaC3615 in the gradual decay phase.

The χ_flare value is approximately constant in the gradual decay phases of IF3 and GF1 (χ_{flare, decay} ∼ 2.7 and 3.5, respectively). What can we learn from this? This may indicate that the hydrogen BaC and the C4170—and therefore the T_BB ∼ 8000 K blackbody component—vary together during the gradual decay phase. However, the interpretation of χ_flare is complicated because variations in both the blackbody and BaC components can lead to variations in C3615, and therefore the value of χ_flare. For IF3, we investigate the ratio of BaC3615 to C4170. This ratio is similar to the evolution of χ_flare for the duration of the flare, with both measures being relatively constant over ∼0.6 hr of the gradual phase. Because BaC3615/C4170 is nearly constant with values of ∼1.5–1.6 over this time period, we conclude that the local flux in the BaC at λ = 3615 Å does follow the local flux in the blackbody at λ = 4170 Å during the gradual phase.

In summary, we find evidence for a T_BB ∼ 8000 K component at the beginning of the gradual decay phase during eight flares, most of which are IF events. It is not known if the emission is formed in the same areal region as the impulsive phase emission (with T_BB ∼ 9000–14, 000). We find a strong correlation among different flares between the C4170 (T_BB ∼ 8000 K) specific luminosity and BaC3615 specific luminosity during the gradual decay phase, suggesting that the late-phase persistence of C4170 and BaC3615 are driven by the same process. Modeling results indicate that the gradual phase C4170 may be a combination of Paschen continuum from hydrogen recombination and increased photospheric emission (e.g., H⁻ recombination) induced by BaC (Allred et al. 2006) or X-ray backwarming (Hawley & Fisher 1992). However, color temperatures as high as 8000 K have not yet been produced in these models. We explore the gradual decay phase continuum further in the analysis of the red continuum (Section 8).

Three flares—IF1, IF7, and GF5—are particularly interesting to compare because they occurred on the same star and the data were obtained at the same spectral resolution (1.83 Å pixel⁻¹, using the 15 slit). The gradual decay phase spectra of these three flares on YZ CMi are presented in Figure 29 (from the times given in Table 5). These comprised the smallest amplitude flare (GF5, blue), a medium-amplitude flare (IF7, red), and the largest amplitude flare (IF1, black) in the sample. At the times of these spectra, the U band was emitting at 1.4, 2, and 25 times the quiescent level, respectively. The best-fit T_BB values are 7600 K for IF7 and 8300 K for IF1. GF5 has very low amounts of emission, but is fit nonetheless with T_BB ∼ 4900 K.

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Despite these differences, there are striking similarities across the NUV and blue-optical zones, including (1) small-scale features at the base of the hydrogen lines; (2) the appearance of the prominent helium i lines at λ4471 and λ4026, He ii at λ4686, Ca i at λ4227+Fe ii at λ4233, and Fe ii (or Mg ib) at λ4924 +λ5018+λ5169; (3) two plateaus in the underlying level of the pseudo-continuum (one from λ =3780–3950 Å and one from λ =3650–3760 Å); (4) hydrogen lines resolved through H14 at λ3722 Å (H15 at λ3712 Å, H16 at λ3704 Å, and He i at λ3704 Å are blended); and (5) similar small-scale features, which may be blended Fe and Ti lines in emission in addition to the BaC emission at these wavelengths (e.g., λ ∼ 3580 Å, 3680 Å). These qualitative similarities and the scaling relationship given in Section 7.1 illustrate that the gradual emission is nearly identical, but "scaled up" in flares according to their amplitudes and energies. Clearly there are important similarities in the gradual decay phases of flares covering a large range of impulsive phase peak amplitudes and energies.

The time evolution of the Conundruum flux compared to other spectral features provide constraints on its origin. For IF3, we show the evolution of the ratio of HB (see Section 5.4) to total λ = 3420–5200 Å flux in the top panel of Figure 30. The evolution is nearly flat in the gradual decay phase. If we consider not just the ratio of HB to total flux, but instead the ratio of HB to blackbody flux (not shown), the ratio increases steeply in the gradual decay phase, similar to the trend in Hγ flux/C4170, which is shown in black asterisks in Figure 30 (bottom). Therefore, there must be an extra amount of flux in addition to the blackbody flux that contributes to the gradual phase emission to make the HB/total flux ratio in Figure 30 (top panel) approximately constant. From the excess flux (non-HB, non-blackbody) we also subtract Ca ii K, Ca ii H, He i lines, and the Fe ii (λ4924 +λ5018+λ5169) triplet. The excess emission, which represents the Conundruum flux, is shown in Figure 30 (top panel) as green squares. The relative contribution steadily rises into the decay phase contributing a maximum of ∼9% of the total λ = 3420–5200 Å flare flux.³⁷ We note that there is also a decreasing value of T_BB from ∼8000 K at the beginning of the gradual phase to less than 7000 K after t ≈ 2.7 hr during this flare (see Figure 33, Section 10.2), which is probably due to the increasing contribution from the Conundruum at the redder end of the blackbody fit near λ ∼ 4800 Å, thereby affecting the blackbody fit. The Ca ii K line flux reaches its maximum at this time also (Figure 14).

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In addition to becoming more apparent in the gradual decay phase, the Conundruum flux also has a larger relative contribution at redder wavelengths (λ ∼ 6000 Å). Here, we use a new continuum measure (C6010) in the red-optical zone (Table 3). In Figure 30 (bottom panel), several continuum fluxes are measured relative to C4170 for IF3 throughout its entire evolution: C6010, C4500, and C3615 (Table 3). In addition, we show the Hγ/C4170 as a function of time (peak values for the sample were analyzed in Figure 11). Figure 30 (bottom panel) illustrates that, strikingly, C6010/C4170 has a similar time evolution to χ_flare and the Hγ/C4170 ratio. The C4500/C4170 ratio, on the other hand, is relatively flat showing small variations that reflect the small changes in T_BB through the impulsive phase and into the gradual phase (see Section 10.2). The similar behavior seen between C6010/C4170 and Hγ/C4170 suggests a time evolution of red-optical emission that more closely resembles the blue HB emission. In the decay phase, the C6010/C4170 ratio is more similar to the Hγ/C4170 evolution than to the χ_flare evolution. We conclude that the Conundruum has a relatively larger contribution at red wavelengths (C6010) and that its time evolution is closer to the lower order Balmer lines (Hγ, Hβ, and Hα) and Ca ii K than to BaC3615 (which contributes to the value of χ_flare).

For several flares, we can directly measure the color temperature (spectral shape) of the Conundruum. In Figure 31 (left), we show for the first time a complete (gradual phase) flare spectral energy distribution (SED) from λ = 3400–9200 Å. This is an average of the spectra S#23–25 from IF1.³⁸ The T_BB fit to the blue-optical zone is shown in blue and clearly does not account for all of the red continuum emission. The color temperature of the red optical zone (using windows RW1–RW3, Table 4) for this flare ranges between T_BB ∼4500–5500 K (fit shown in red). In Figure 31(right), we show the complete SED during the decay phase of IF3 (time listed in Table 5). This flare also exhibits a red color, with a best fit blackbody temperature of ∼3700 K.

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At the peak of IF3, the Conundruum is also apparent but with a smaller relative contribution to the total flux of C6010 compared to the gradual decay phase. In Figure 32, we show the extrapolation of a blackbody curve fit with T_BB =12,100 K to the blue-optical at the peak emission during IF3 (S#31) in order to illustrate the excess flux of C6010 unaccounted for by a simple isothermal Planck function that matches the blue-optical spectral zone. We tried various combinations of fitting Planck functions to the entire spectrum to account for the Conundruum. First, a single blackbody fit to both the blue and red optical zones reveals a temperature of T_BB ∼ 10,200 K; it fits the overall shape well, but misses the detailed shape of the blue-optical zone. If we fit a blackbody to the red optical zone only (using the continuum windows RW1–RW3 in Table 4), we find a temperature of T_BB ∼ 7700 K, but this fails to account for much of the blue-optical flux. Clearly, an isothermal fit cannot explain the full flare SED at peak.

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Using a two-component blackbody to model the peak of IF3, we find that a combination of a T ∼ 6000–6400 K (X_BB ∼1.5%) blackbody and a superhot blackbody T ∼ 10⁵ K–3 × 10⁶ K (X_BB ∼0.0042%, ∼0.00012%, respectively) fit the λ = 4000–6800 Å flux distribution the best using the continuum windows BW1–BW6 and RW1–RW3 (Table 4). The optical signature of a very hot temperature (T ≳ 10⁵ K) component can only be confirmed with bluer observations. Hawley & Fisher (1992) discuss that free-free emission from a superhot 1 MK source reproduces the optical (UBVR) colors of the Great Flare well but that it cannot account for the observed turnover of the broadband flux in the FUV and NUV.

It is possible that the large number of free parameters for two blackbodies allow an excellent, yet unphysical, fit to the IF3 peak spectrum. In particular, the two bluemost fitting windows (BW1, BW2; Table 3) may be contaminated from hydrogen wing emission in such a large flare, therefore giving unrealistically large temperatures to fit the rise of the spectrum from λ = 4000–4150 Å. If we exclude the two bluemost continuum windows for the fit, we find that a combination of temperatures of 4700 ± 500 K (X_BB =2.3%) and 13,700 ± 1000 K (X_BB =0.17%) and scaled by 0.98 fit the underlying λ = 4150–6800 Å continuum flux distribution the best (overplotted in red dashes in Figure 32). This fitting gives a hot blackbody consistent with a single temperature fit to BW1–BW6 (T_BB ∼ 12,100 K; Table 7) and a single temperature fit using only BW3–BW6 (T_BB ∼ 11,000 K; for further discussion on systematic temperature fitting errors, see Appendix F of Kowalski 2012).

Possible explanations of the Conundruum are the following:

• 1.  
  Calibration artifact. We first considered whether the Conundruum was a calibration issue, as the DIS dichroic affects the flux calibration in the Johnson V-band range. The largest effect is near 5500 Å (e.g., in flat-field lamp exposures). We took a conservative range and did not consider the flux between λ =5200–5800 Å due to this issue.
• 2.  
  Minor species emission lines. It is well known that spectra of QSOs and Seyfert 1 galaxies have a forest of Fe ii lines between λ =5000–5500 Å (Osterbrock 1977; Puetter et al. 1981). There have been detections of strong Fe ii lines in stellar flares (Abdul-Aziz et al. 1995) and Fe i and Fe ii lines from solar flares (Severnyi et al. 1960; Johns-Krull et al. 1997) at these wavelengths. Other authors have also seen the presence of Fe lines in this range (Eason et al. 1992); many small scale features are visible in the high signal-to-noise spectra of IF1, and the Conundruum may be a blend of these lines.
• 3.  
  Flux redistribution from minor species emission lines. Due to the appearance of the iron forest at these wavelengths, line blanketing (Rutten 2003) may cause the nearby continuum to increase, similar to flux redistribution generating a higher-than-expected effective temperature for the Sun (Böhm-Vitense 1989). Higher spectral resolution data of flares from 4500–6000 Å would help characterize bona-fide continuum regions. Line-blanketing-induced continuum enhancements also need to be understood throughout the entire blue optical and NUV zones.
• 4.  
  Non-hydrogen continua. There is a He ii continuum edge (n = ∞ to n = 5) at λ = 5694 Å. He ii continuum has been detected in solar flares in the EUV (Linsky et al. 1976; Milligan et al. 2012).
• 5.  
  Continuum fitting artifact. By including the bluemost windows in the blackbody fitting (BW1, BW2), it could be a concern that we misfit the continua at longer wavelengths. These continuum windows have low-lying features within their narrow range. In extreme cases, the wings of Hδ and H could contaminate the continuum windows. Appendix F of Kowalski (2012) tested whether removing BW1 and BW2 before fitting the blackbody function can explain the apparent Conundruum flux. In three of the four flares tested, various amounts of Conundruum remained after excluding these bluemost continuum windows.³⁹
• 6.  
  Multithermal flare emission. The sum of all flare emission at any given time from a flare is very likely not isothermal; therefore, emission from regions (along all spatial directions in the atmosphere) with different temperatures would superpose to produce a spectrum that is inconsistent with a fit of a single blackbody over a large wavelength range. In Section 8.1, we were able to explain the blue-optical hot blackbody+Conundruum contribution at the peak of IF3 using a double-blackbody fit (after minimizing the continuum fitting artifacts described in possibility #5).
• 7.  
  Higher order hydrogen continua. In Figure 2 of Kowalski et al. (2012), they accounted for the Conundruum through a superposition of model spectra using hot spot models and the RHDF11 model. The RHDF11 model ascribes the flux at redder wavelengths as due to a combination of hydrogen Paschen continuum from the chromosphere and increased photospheric continuum that results from chromospheric backwarming. If the Conundruum has a large contribution from Paschen and possibly Brackett continua (hydrogen recombination to n = 3 and n = 4, respectively), a subsequent transition to n = 2 from either of these two levels would result in emission of an Hα or Hβ photon, respectively. This scenario could possibly explain the relation between the Conundruum and the lower order Balmer lines, which we discussed for Hγ in Figure 30 (Section 8.1). Higher order hydrogen continua have been suggested to be a possible contribution to white-light radiation in the infrared (Tofflemire et al. 2012).In Section 9, we will discuss the interpretation of the Conundruum further and show that items #6 and #7 are the leading possibilities.

To model the flare BaC, we used the impulsive-phase RHD model spectrum at t = 15.9 s from the F11 simulation of Allred et al. (2006), hereafter A06, following Kowalski et al. (2010). We subtracted a linear extrapolation of the blue-optical zone in the model to estimate the amount of flux originating from the BaC. We refer to this model spectrum of the BaC as BaCF11 to distinguish from the published RHDF11 model.⁴⁰ We repeated this calculation for the F10 model at t = 230 s from A06, giving an F10 BaC model spectrum BaCF10. These times (15.9 s and 230 s) correspond to the maximum amount of emission produced at λ = 4170 Å in the models. We calculated the spectral slopes, m_NUV, of the BaCF11 and BaCF10 model spectra and give these values in Table 10 (Columns 4 and 5) to compare to the observations in Table 7. The value of m_NUV ∼ 5.1–5.5 indicates that the model flux at 3420 Å is ∼90% of the model flux at 3615 Å. At the peak of the flares IF2, IF3, and IF9, the slope of the excess flux is blue (m_NUV < 0; Column 11 of Table 7, Section 6.2) and is not consistent with the red slope of BaCF11 or BaCF10 (m_NUV > 5). Whereas the peak phases of the IF and HF events often show negative values of m_NUV (see Columns 9 and 11 of Table 7), the GF events are more similar to the RHD predictions. This finding (determined from above) implies that our understanding of the impulsive phase BaC (and thus the flare chromosphere) is not complete with the current suite of RHD models. In the flare gradual decay phases, the IF and HF events tend to be more positive so the models may be more accurately portraying gradual decay phase (and GF) flare conditions.

To investigate the BaC in the gradual decay phase spectra in more detail, we added the BaCF11 model spectrum to the blackbody fit at λ < 3646 Å. The results are shown in red in Figures 28(a)–(e). We quantitatively compare the slopes of the observed BaC emission (after subtraction of the blackbody) to the slope of the BaCF11 (∼5.1) and BaCF10 (∼5.5) in Column 5 of Table 10. In all of the gradual decay spectra, there is a relatively good match to the positive slopes (flux increasing toward redder wavelengths) of the model spectra, as we found for IF1 (Kowalski et al. 2010; see also Figure 28(a)). The statistical fitting errors on the slopes (given in parentheses) are quite large, likely due to the difficulty in modeling the "true continuum" underlying the forest of low-excitation metallic lines that are likely present in the NUV during the decay phase (Figure 29). There are also systematic errors due to flux calibration of ∼5%–10% at these wavelengths (Appendix A of Kowalski 2012); this uncertainty corresponds to about 2–4 in the units given in the table. The best matches to the BaCF11 model are obtained during IF9 and GF1, which have BaC slopes of 4.9 ± 2.4 and 5.9 ± 1.3, respectively. We conclude that the observed BaC are reflecting flare chromospheric conditions from the models of A06, which apparently predict the gradual decay phase emission using (impulsive phase) RHD beam models having a non-thermal electron energy flux of 10¹⁰–10¹¹ erg cm⁻² s⁻¹, a low energy cutoff of E_c = 37 keV, and a power law distribution of nonthermal electrons with δ = 3–4. More accurately flux-calibrated spectra with higher spectral resolution (to resolve the faint metallic lines) extending further into the NUV will provide better constraints for the models.

The persistence of the BaC during late stages and the relatively good match to the shape of impulsive phase RHD model BaC, implies continued heating from accelerated particles during the gradual decay phase of these flares. Gyrosynchrotron microwave radiation has been detected far into the gradual phase of white-light flares on M dwarfs, suggesting that nonthermal particles are accelerated after the impulsive phase (van den Oord et al. 1996; Osten et al. 2005). Simultaneous, high cadence optical spectra, photometry, and radio data (e.g., from the EVLA) would better constrain the timing of gyrosynchrotron emitting particles and white-light radiation.

The impulsive phase RHDF11 model also accounts for the shape of the red optical Conundruum emission component rather well, with a best-fit blackbody temperature in the red of T_red ∼ 4600 K (RHDF10 shows about the same color in the red). This color temperature is similar to the color temperature of the total emission in the red-optical zone in the gradual decay phase of IF1 and to the color temperature of the cooler blackbody using a double blackbody fit in the peak phase of IF3. However, none of the RHD models (F10 or F11) show evidence for a color temperature of ≲ 4000 K in the red part of the spectrum, as is needed to explain the red emission in the IF3 gradual decay phase. The RHDF11 spectrum predicts increased photospheric radiation from chromospheric UV backwarming and Paschen continuum from hydrogen recombination in the flare chromosphere. The opacity description in the RADYN model does not include molecular features that are important in M dwarfs (e.g., TiO) and which would likely have a strong influence on the energy balance in the photosphere, and hence the backwarming physics. Modeling the backwarming is further complicated due to the large area over which it may originate (Fisher et al. 2012; Isobe et al. 2007). Obtaining separate models for the Paschen continuum and backwarming radiation is not currently possible with RADYN.

A reddening continuum shape during the gradual decay phase of IF0 in the V and R bands (covering λ ∼ 5000–7000 Å) was reported in HP91, who speculated about the presence of a Paschen continuum at these wavelengths. The best-fit blackbody temperature in the gradual decay phase to the UV, UBVR photometry was 8400–8800 K, similar to the lower temperatures we find in the gradual decay phase of the DIS sample using only the blue-optical continuum fitting. The mismatch to the blackbody continuum implied by these temperatures in Figure 11 of Hawley & Fisher (1992; bottom two panels) is due to the addition of this red component. Excess flux in the R band is also apparent in the moderate size flare on AD Leo described in Hawley et al. (2003). We speculate that the Conundruum rises into the red and accounts for the excess R-band flux observed in these studies.

Paschen continuum emission originating from typical temperatures (6000–8000 K) of the flare chromosphere show a characteristic red color (Hawley & Fisher 1992), and we consider the possibility that the red continuum emission (and hence Conundruum) is primarily due to Paschen continuum. From the RHDF11 spectrum, we estimated the height of the Paschen jump (λ ∼ 8204 Å) as F_{λ = 8200} − F_{λ = 8300}, which is 1/16.5 times the Balmer jump height (F_{λ = 3640} − F_{λ = 3655}). The predicted flux of the Paschen jump is shown in Figure 31 as red error bars indicating the height of the Paschen jump (using 1/16.5 times the flux of BaC3615). We do not find convincing evidence of a Paschen jump to indicate the presence of Paschen continuum emission, but the spectral region around the Paschen jump is contaminated by features from Earth's atmosphere and instrumental fringing which make the quiescent subtraction uncertain and the characterization of flare-only continuum difficult at these wavelengths. The Paschen jump has been (tentatively) detected in a solar flare (Neidig & Wiborg 1984).

Using simple, isothermal, isodensity, optically thin, hydrogen bf+ff slabs from Kunkel (1970) as a guide, we can explore the shapes of Paschen (+Brackett) continua from λ = 3646–8205 Å. If we fit only to the red continuum, λ = 6000–6800 Å, T_e =12 500–25,000 K (where T_e is the electron temperature) is required to achieve a blackbody temperature of T_BB ∼ 4500–5500 K (which corresponds to the color of the RHDF11 impulsive phase spectrum and the IF1 gradual phase red spectrum). If we consider the red gradual phase spectrum of IF3, which has a redder T_BB compared to IF1 (T_BB ∼ 3500–4000 K), its shape is reproduced by an isothermal model with T_e = 7000–9500 K. Clearly, a large range of electron temperatures (7000–25, 000 K) can give rise to a relatively small range of red continuum colors (T_BB ∼ 3500–5500 K). More advanced, multithermal, self-consistent atmospheric modeling including photospheric opacity calculations is needed to make further progress in understanding this enigmatic continuum component.

The largest discrepancies between the current RHD models and the observations are the shape of the continuum emission in the blue-optical zone and the height of the Balmer jump. We measured the same observational parameters, χ_flare and T_BB, for the F10 and F11 continuum predictions from A06. The times analyzed are t = 230 s (F10) and t = 15.9 s (F11), respectively. We find that for the flare-only flux, χ_{flare, peak} ≈ 4.8 and T_BB ≈ 5000 K for F10 and χ_{flare, peak} ≈ 8.2 and T_BB ≈ 5300 K for F11 (given in Table 10). The values of χ_flare are much higher than the observations (χ_{flare, peak} between 1.5–4.5; Figure 11), indicating that the Balmer jump is too large. The color temperatures are far too low compared to the observations, which show a range at their peak times of T_BB ∼9000–14, 000 K (Sections 6 and 6.3). Using state-of-the-art observations and new diagnostics, we have revisited the long-outstanding problem that too much flare energy is deposited in the mid to upper chromosphere, where a large Balmer jump in emission is produced. In the RHD models, there is not enough energy deposited at high column mass (in the low chromosphere and photosphere) where a Balmer jump in absorption and/or blue-optical zone continuum with a shape similar to a blackbody having temperatures T_BB ∼ 9000–14, 000 K can be generated (based on the phenomenological models of Cram & Woods 1982; Houdebine 1992; Kowalski et al. 2011a). Only when the impulsive phase "hot, blackbody" (white-light) emission is successfully reproduced by the models can we begin to understand the detailed multithermal origin of the BaC, the Paschen continuum, the Conundruum, and the gradual phase 8000 K blackbody component; furthermore, we can then begin to understand how physical processes, such as flare heating, differ between GF and IF events and even between individual IF events. Thus, RHD models incorporating additional physics—such as higher energy electron beams and proton beams—are urgently needed.

A physically significant quantity that can be derived from the spectra is the filling factor, X, which is the fraction of the area of the projected visible stellar disk that is emitting flare continuum emission. Values of X for the white-light emission are often calculated in the literature (van den Oord et al. 1996; Hawley et al. 2003; Kowalski et al. 2010; Osten et al. 2010; Walkowicz et al. 2011; Davenport et al. 2012). Flare areas allow us to determine the degree to which individual stellar flares are "scaled-up" versions of one another (i.e., same spectral properties with larger area). Filling factors allow us to understand what type of heating distribution is responsible for the observed phenomena—i.e., using solar flare terminology, whether the flare heating is diffuse over a large area such as in extended ribbons or prominences, or whether the heating is focused into a small area such as in compact kernels. Finally, we find that filling factors are important for deriving the speeds of stellar flare development.

The area of a stellar flare is strongly dependent on the emission type used to model the spectrum. An important concern is that real stellar atmospheres do not produce featureless blackbody emission spectra. In Kowalski (2012), we investigated the corrections obtained using more realistic Castelli & Kurucz (2004) models of hot star atmospheres (summarized in Appendix F), and we found that the corrected, inferred areal coverages (compared to those inferred from a Planck function) increase by a factor of 1.4–2.3 for the peak temperatures of T_BB ∼ 9000–14, 000 K in our sample. Moreover, the effective temperatures of the hot "blackbody" emission⁴¹ component can be estimated this way, giving T_eff ∼ 7700–9400 K for this range of T_BB (for log g = 5 model atmospheres). The bolometric heating flux necessary to sustain the hot "blackbody" emission component is therefore ≈2–4 × 10¹¹ erg cm⁻² s⁻¹. Starting in Section 10.2, we correct X_BB using the algorithm in Appendix F.

We now discuss important systematic uncertainties when calculating filling factors of flare emission. First, there may be more than one source of continuum emission contributing to the total blue-optical (e.g., C4170) flare continuum emission at λ > 3646 Å at peak times during flares. For example, Conundruum flux likely extends into the blue (see Figure 31). The RHDF11 model predicts continuum flux at λ > 3646 Å, and this was used in the modeling of the continuum in Kowalski et al. (2012). Therefore, a single-component hot "blackbody" representation places only an upper limit on the areal coverage. We suspect that if the hot "blackbody" was the only source of blue-optical continuum, the absorption signatures seen in the Castelli & Kurucz (2004) models would be more prominent in flare spectra; even in giant flares, such as IF3 and IF4, the absorption phenomena only become measurable at peak emission when the hot "blackbody" has reached an areal coverage of ∼0.2% of the visible stellar hemisphere, corresponding to a physical area of ∼3 × 10¹⁸ cm².

To calculate a filling factor for the (chromospheric) BaC emitting region, we use the BaCF11 model from A06, as was presented previously for IF1 (Kowalski et al. 2010). Despite the discrepancy between the observed slopes and model slopes in the NUV spectral zone (Tables 7 and 10), we use the RHD BaC flux to model the observed BaC as it is the best model we have currently available. A major source of uncertainty in the area of the BaC-emitting region is absorption predicted in the blackbody component (Castelli & Kurucz 2004) at λ < 3646 Å: the BaC emission and absorption components are spatially unresolved in our stellar flare spectra. Interpreting the peak phase emission of IF2 using these hot star spectra, the measured amount of BaC flux (BaC3615) would be three times smaller than the intrinsic amount of BaC emission originating from the flare chromosphere if we also account for the absorption. In other words, a simple linear extrapolation from the blue-optical (as is used to calculate BaC3615 and therefore the filling factor of BaC emission) may vastly underpredict the true amount of chromospheric BaC emission. However, we note that preliminary modeling of "hot spots" in M dwarf atmospheres (Kowalski et al. 2011a) indicates that the amount of "missing" BaC flux is only ∼1.5 times the measured BaC3615.

Another important systematic uncertainty in the area of the BaC emitting region is our selection of the F11 model to represent the observed BaC flux. For example, using the BaC from the F10 RHD model of A06, we find that the inferred areal coverages of the BaC-emitting region would then be 8 times larger than inferred from the F11 model (Kowalski et al. 2010). The model BaC shape is not constrained from the observations due to the very small differences in the shape of the BaCF11 and BaCF10 between λ = 3420–3630 Å (Table 10). Given the discrepancy between NUV slopes observed at flare peak (for IF and HF events) and for these RHD models discussed earlier (Section 9), a method for discriminating between models would only be applicable during the gradual phase and peak phase of GF events. We are seeking to determine a method for constraining the appropriate model of the BaC.

Third, a more accurate modeling of the energy deposition by nonthermal electrons would take into account the pitch-angle distribution of electrons via the Fokker–Planck beam formulae (e.g., McTiernan & Petrosian 1990). As the energy deposition is not as localized in the chromosphere, employing these formulae for a given beam flux would likely require the areas of the RHD BaC to be larger to account for the observed emission (Mauas & Gomez 1997, A06).

Given these considerations, we calculate X_BB, X_BaCF11, and X_BaCF11/X_BB for all of the flares in the Flare Atlas. The values of X_BB (from fitting a blackbody, before applying the corrections in Appendix F) are given in Table 7 (Column 7). At peak, the blackbody component exhibits an areal coverage on the order of ∼0.3% for the large amplitude flares, ∼0.03% for the medium amplitude flares, and ∼0.005% for the small amplitude flares, a range of ∼60× in area coverage in the DIS sample. We find that larger U-band emission at peak is due to a combination of both increasing blackbody and BaC areal coverage. The BaC emitting region is also compact, though larger than the blackbody component, originating from ∼0.05% to 1.7% of the visible stellar hemisphere. The ratio of the inferred surface area coverages (X_BaCF11/X_BB) is 1–26 at peak, with the apparent amount of BaC originating from a larger area than the blackbody. However, the range of 3 ⩽ X_BaCF11/X_BB ⩽15 is found to represent most flares at peak emission; this range was also found during the decay phase of IF1 in Kowalski et al. (2010), as the flare went through successive impulsive and gradual phases of secondary flares. The largest luminosity flares have areal ratios of ∼1–10. The ratio of areal coverages, X_BaCF11/X_BB, after applying the corrections in Appendix F to the blackbody emission component, ranges between 2–8 for 12 out of the 17 flares with well-determined blue-optical temperatures. We predict that accounting for all of the aforementioned uncertainties in our area calculation would ultimately increase the BaC emitting area relative to blackbody emitting area. It would be instructive to recalculate the ratios of filling factors using more accurate RHD models of the BaC and blackbody emission components.

The IF3 event produced a rise phase lasting 2.7 minutes and a fast decay phase lasting 5 minutes, allowing a unique time-resolved study of the impulsive phase of a large-amplitude flare. Following the continuum analysis of Section 6.1, we fit a blackbody function to each spectrum from λ = 4000–4800 Å, and the filling factor, X_BB, is corrected using the algorithm in Appendix F. The evolution of T_BB and X_BB is shown in Figure 33. The temperature evolution is as follows: almost immediately (S#25–26) in the fast rise phase, we measure a color temperature of ∼10, 000 K, which increases to ∼11, 500 K by S#27 when C3615 is one-third of the peak value. T_BB reaches a maximum value of ∼12,100 K at the peak of C3615 at S#31. Whereas T_BB increases by ∼2000 K in the rise phase, X_BB experiences a large increase by a factor of 20 and stays elevated near its maximum value (possibly increasing very slightly) for three spectra after the peak. In subsequent spectra after the peak, the color temperature drops and then decreases monotonically at an average rate of ∼800 K minute⁻¹ during the fast decay phase. At the end of the fast decay phase, the temperature reaches the characteristic gradual decay phase value of ∼8000 K. For equal C3615 flux levels during the fast rise and fast decay phases, the fast decay phase is more than 1000 K cooler, which implies that the area (X_BB) must be greater during the fast decay phase. In other words, the fast rise is hotter and smaller than the fast decay. Mochnacki & Zirin (1980) observed similar impulsive phase trends for X_BB and T_BB using continuum data during several dMe flares from λ = 4200–6900 Å (e.g., see their Figure 3). The larger temperatures inferred from our data likely result from fitting a narrower spectral window (λ = 4000–4800 Å) to avoid contamination from the Conundruum component.

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We now use the rate of areal increase (X_BB) during the rise phase of IF3 to estimate the speed at which the flare area is changing, which can be used to constrain different scenarios of flare heating.

We use a simple flare areal model, whereby the white-light emitting region is circular with radius r, and the inferred filling factor, X, is changing at a rate dX/dt. The speed of the leading edge (perimeter) of the expanding circular flare area is given by

$$\begin{equation} v(t)_{\mathrm{flare}} = \frac{dr(t)_{\mathrm{flare}}}{dt} = 0.5 \frac{dX(t)}{dt} \frac{1}{r(t)_{\mathrm{flare}}} R_{\mathrm{Star}}^2, \end{equation} \tag{ 10 }$$

where $r(t_i)_{\mathrm{flare}} = (R_{\mathrm{Star}}^2 X(t_i))^{1/2}$ and $\frac{dX_i}{dt} = (X_i - X_{i-1}) / (t_i - t_{i-1})$ where i is the spectrum number with midtime t. If instead we assume a more complex geometry having two expanding flare areas to represent the two footpoint structures observed during solar white-light flares (Hudson et al. 2006; Maurya & Ambastha 2009), the inferred speeds decrease by a factor of two. Note, we use the filling factors X_BB adjusted using model hot star atmospheres of Castelli & Kurucz (2004) described in Appendix F. The uncertainties in the speed calculations are estimated to be ∼40%, primarily due to the systematic uncertainties in the temperatures (see Appendix F of Kowalski 2012). An additional uncertainty results from possible projection effects (see footnote ²⁸ in Section 6.1) such that a flare near the limb would show a smaller expansion velocity (by an amount proportional to (cos  θ)^1/2, where θ is the foreshortening angle) compared to a flare at disk center.

We show the results for the speeds (v_flare) during IF3 in Figure 34. According to this simple, circular flare model, the leading edge of the flare is moving at a high speed (∼100 km s⁻¹) during the rise phase and decreases to small speed (<10 km s⁻¹) after the peak. The average speed during the flare rise is ∼50 km s⁻¹, and the maximum speed (∼110 km s⁻¹) is attained in the mid rise phase. The rise-phase speeds are supersonic for the photosphere and chromosphere of an M dwarf (c_s ∼ 5–10 km s⁻¹) and we suggest several mechanisms below that could be responsible for the expansion rates observed. van den Oord et al. (1996) also considered the expansion speed⁴² of an erupting filament during a white-light flare on YZ CMi, and found ∼100–500 km s⁻¹, but they did not have spectra which are needed to determine the speed of the blackbody continuum component.

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IF9 is another (IF) event in our sample where the blackbody has a rise and peak phase detection. Applying the same analysis, we find velocities of ∼130 km s⁻¹, ∼90 km s⁻¹, and ∼0 km s⁻¹ in the rise, peak, and fast decay phases, respectively. Although the uncertainties are again ∼40%, IF9 and IF3 show a similar pattern: fast speed in the mid-rise, decreasing speed at peak, and very slow speed in the immediate post-peak phase.

We repeat these calculations for the rise phase of MDSF2 because it is a flare (albeit a secondary flare during IF1) with a simple, time-resolved rise phase. Only the average speed can be calculated for this flare. We find that during the rise phase, the area is expanding with an average speed of ∼8(±3) km s⁻¹, much smaller than during IF3. Although the flux calibration is less certain, IF4 is apparently a very quickly evolving flare exhibiting speeds of ∼65–150 km s⁻¹ through the rise and peak phases. We note in passing that both the "slowest" and "fastest" flares (MDSF2 and IF4, respectively) show signatures of Balmer wing depressions (Section 6.4) indicative of extreme heating scenarios, discussed in the next section. In Appendix G, we also consider the speeds during the HF1 and IF2 events.

There are several possible scenarios which could drive the expansion of a flare area, and we discuss the two most extreme here.

• 1.  
  Scenario 1 (steady bombardment of a single region): v_flare ∼ c_{s, chrom}, dA_beam/dt = 0. For this scenario, we assume that the beam⁴³ bombardment occurs at a fixed location and over a constant area (dA_beam/dt = 0). The sound speed is an important parameter for the expansion of a flare area occurring on the dynamical timescale in response to persistent beam heating from above (e.g., from the particles driven into the lower atmosphere during reconnection). As pressure equilibrium is achieved with the ambient atmosphere at the speed of sound, radiative and conductive heating lead to a temperature increase in the surrounding region. The rate at which the flare area expands is therefore related to the sound speed, and the rate at which the temperature increases is a combination of the rates from these heating processes which depend on the detailed physics and chemistry (e.g., specific heat) of the heated atmosphere. As the flare area expands, the average temperature may eventually drop because the beam heating (over constant area) can no longer sustain the temperatures against cooling from expansion. It is known from one-dimensional RHD models that constant heating of a given region increases the column mass in the corona so as to inhibit further beam penetration (A06); future studies of this effect with (three-dimensional) RHD models would illustrate how relevant Scenario 1 may be during flares.
• 2.  
  Scenario 1b (expanding bombardment of a single region): v_flare ∼ v_reconnection ∼ c_{s, chrom}, dA_beam/dt > 0. Scenario 1b is a variation on Scenario 1. The flare speed would also equal the sound speed if a sound wave can trigger magnetic reconnection and beam heating in nearby regions. Given that the sound wave disturbance persists for an extended duration, dA_beam/dt > 0.
• 3.  
  Scenario 2 (sequential bombardment of multiple regions): v_flare ∼ v_reconnection > c_{s, chrom}, dA_beam/dt > 0. A second scenario which may be important during the rise phase of flares is transient, fast heating (sequential bombardment) of many small neighboring regions. In this scenario, the total area of all beams increase. This scenario allows for speeds larger than c_s in the lower atmosphere without having to invoke shock heating. In fact, this scenario could occur at a speed which is driven by the speed of reconnection at heights (i.e., in the corona) above the site of flare heating. The emission we see at any given time is a superposition of all flare areas, and the increase of flare area during the rise phase is then related to the increasing number of these individually heated regions—not to the expansion of a single region. However, persistent heating and expansion could also be taking place simultaneously as each area that is heated by a beam expands according to the physics described in Scenario 1. An alternative explanation for Scenario 2 would be supersonic expansion of a flare area, resulting in heating contributions in neighboring regions from an expanding shock wave.Of the many important physical parameters that would inform this picture, we consider the depth of continuum formation and the decay timescale of the continuum to be essential information. Given the depth of continuum formation, we would know the relevant sound speeds and could test the relative roles of shock heating, conduction, radiative heating, and dynamical expansion. The timescale of continuum decay sets how long the beam needs to persist in individual flare areas in order to explain the observed amounts of emission.

We now discuss these scenarios given the derived speeds during IF3 and MDSF2. The average speed during MDSF2 is perhaps coincidentally, consistent with the speed (5–10 km s⁻¹) of the expanding disturbance from the initial |ΔU| ∼ 6 mag at peak event (IF1) that was assumed to trigger MDSF2 in Kowalski et al. (2012). The slow speeds during the rise phase of MDSF2 are therefore consistent with Scenario 1b, giving a connection between the speed of a wave disturbance in the lower M dwarf atmosphere and the rate at which flare beam heating occurs at a new site. We speculate that the slower flare speed in MDSF2 led to more prolonged heating in a given area. In order to be consistent with the lack of Balmer line emission produced in MDSF2, it is also possible that the heating source was focused deep in the atmosphere (similar to the heights at which reconnection is thought to occur in order to produce Ellerman bombs, mentioned by Kowalski et al. 2012). These effects produced the higher color temperatures (T_BB ∼ 13, 000–15, 000 K; Section 6.3 and Table 9) and more easily detectable absorption signatures.

Comparing the speeds for IF3 and MDSF2 signifies a physical connection between the flare speed and the type of blackbody spectrum (i.e., type of flare heating) observed. The average rise phase flare speed during IF3 is more than five times faster than the speed during MDSF2. IF3 produced large amounts of chromospheric (e.g., Hγ, BaC3615) radiation, especially during the first half of the rise phase when the velocities were higher. If IF3 is described by Scenario 1, then the flare heating would have to be higher than the chromosphere (c_{s, chrom} ∼ 5–10 km s⁻¹) to be consistent with flare expansion occurring at large sound speeds. The rise and peak phases of IF3 are therefore most likely described by Scenario 2, implying that the areal increase would be related to the rate of individual areas being heated. Understanding the depth of formation of the IF3 continuum would allow us to place constraints on the heating of individual flare bursts so that the emission superposes correctly to form the observed light curve. At the beginning of the fast decay phase of IF3, the speeds decrease to <10 km s⁻¹, implying that Scenario 1 (or Scenario 1b) takes over after the peak. Unlike in the rise phase of MDSF2 (Scenario 1b) when the color temperatures are relatively high (13, 000–15, 000 K), the color temperatures in the fast decay phase of IF3 are decreasing and relatively lower (≲ 11, 000 K). Slow expansion is observed for a range of color temperatures, which may be related to the difference in heating mechanisms during the rise of MDSF2 and initial fast decay phase of IF3.

Additional higher time resolution data during the rise and fast decay phases of other types of flares would help us understand the relative importance of these scenarios and provide invaluable constraints on future three-dimensional models that attempt to reproduce the formation of the hot blackbody continuum. In Section 11.4, we discuss the implications of flare speeds for the solar–stellar connection.

How are the flare spectral properties related to the U-band light curve "morphology"? We began our study by separating the flares according to morphological characteristics from the broadband light curves, and we showed that the peak properties of the blue/NUV continuum are connected to the overall time evolution of the light curve. We differentiate between impulsive and gradual flares according to a new value, the impulsiveness index, $\mathcal {I}$. Our designation is not to be confused with the impulsive and gradual phases of an individual flare: for example, a flare may be gradual if it has a low peak amplitude and/or large t_1/2 even though it may have distinct impulsive and gradual phases. The impulsiveness designation instead identifies the emission type (i.e., fast or slowly evolving emission) that contributes most to the overall evolution. The flares that had the largest value of $\mathcal {I}$ are generally the classical, impulsive flares; the gradual flares are typically more complex featuring a slowly changing light curve near peak but also intermittent faster continuum variations. Some flares exhibited properties of both categories and are classified as hybrid flares (HFs); these typically had several closely spaced (in time), fast yet resolved, continuum variations. Houdebine (2003) devised a similar classification scheme between "impulsive," "gradual," and "combined" flares. However, the Houdebine (2003) classification does not include white-light continuum emitting flares in the gradual flare category and our respective definitions for combined and hybrid flares differ: our definition includes the timescales of primarily the impulsive phase. Also, some of their combined flares would likely be included as implusive flares according to our criteria.

BaC in emission is ubiquitous during stellar flares, although it is present in varying amounts, contributing as little as ∼20% of the NUV flux (C3615) during IF events at peak and as much as ∼80% at peak of the GF events (Section 6.2, Figure 24), with the relative amount also decreasing as a function of flare peak amplitude. Though it dominates the relative flux in the GF events, the BaC is seen in large absolute amounts in some IF events (e.g., IF1, IF2, IF5, IF9). For the first time, we provide detailed light curves estimating the BaC emission from flare start to flare finish, and we find that it is generally similar to the Balmer lines, and evolves quickly, like the higher order Balmer transitions (as suggested in Doyle et al. 1988). The t_1/2 values of the BaC light curves for the well-measured flares were ∼5–20 minutes, and future models should seek to reproduce this property.

We established a temporal relationship—the "time decrement" (Section 5.2, Figure 18)—between six of the Balmer features for several flares. The time decrement is a linear relationship between the wavelength of the Balmer transition and the t_1/2 value of the light curve for that transition. The time decrement is similar for the classical flares IF3 and IF9, indicating a possible scaling relationship—similar heating parameters but over a larger area—between moderately large energy flares (E_U ∼ 10³² erg) and high energy (E_U ∼ 10³³ erg) flares for the HB emitting region. The time decrement for representative HF and GF events behaved differently from the IF events. For the HF event, the time-decrement behavior is probably affected by the superposition of two distinct flare impulsive phases. For the GF event, the flatness of the time decrement is related to the larger ratio of t_{1/2, C4170} to the t_1/2 of the Balmer series; for the IF events, this ratio is normally very small.

To begin to understand the time decrement, we look to Drake & Ulrich (1980), who related the larger energy difference to smaller collisional frequency between the upper level and any given lower level in the higher order lines compared to lower order lines. The time decrement is therefore a result of each transition's individual sensitivity to the decreasing densities in the flare chromosphere during the flare decay (see, e.g., Figures 7(b) and 10(c) of Drake & Ulrich (1980)) in addition to the different oscillator strengths in the formula for C_lu (Rutten 2003). Due to the larger optical thickness, the lower order lines are less sensitive to the electron density of the flare chromosphere. The higher order lines and BaC emission are less optically thick and therefore have contributions from higher column mass (higher densities) where there are, generally, shorter cooling timescales, resulting in stronger sensitivity to n_e and thus shorter t_1/2. In addition to density, the time decrement is likely dependent on several other important flare parameters in the decay phase: the change in chromospheric temperature structure, the sustained level of beam heating in the gradual phase, and the incident XEUV radiation field and overpressure from the superheated corona. Given all of these effects, which are changing as a function of atmospheric column mass and time during the flare, it is critically important to investigate the causes of the time decrement with RHD models. Higher spectral resolution data of Balmer lines with high time cadence would help constrain electron densities (via the widths of the Balmer wings) in order to relate to the time-decrement behavior.

The continuum measured redward of the Balmer jump (C4170) is the fastest component in the blue/NUV and does not follow the time decrement of the HB series. This continuum component contributes a large fraction of the emission throughout the NUV, blue-optical, and red-optical spectral zones, and we relate this to the "blackbody" emission continuum that has been detected in previous λ > 4000 Å spectral (Mochnacki & Zirin 1980; Kahler et al. 1982) and UBVR broadband color (Hawley & Fisher 1992; Hawley et al. 2003; Zhilyaev et al. 2007) studies of flares. The U-band amplitude and time evolution are largely determined by the blackbody emission component although the U band is also affected by the BaC, which adds to the flare emission at λ < 3646 Å. The BaC contributes relatively more flux in the HF and GF events and is the reason for their smaller amplitudes and more gradual time evolution as observed in the U band. At peak, the percentages of C3615 (total NUV flux) due to BaC3615 are ∼20%–44% (IF), 50%–60% (HF), and 55%–80% (GF; Section 6.2, Figure 24). The four flares with the smallest amount of BaC at peak relative to the total flux are the largest amplitude events: IF0, IF1, IF3, and IF4. We therefore conclude that the slower evolution of the GF events is due to a larger influence from the BaC, which is slower than the blackbody component (as characterized by t_1/2); conversely, the faster evolution of the IF events is due to the dominance of the blackbody emission component (most dominant in the largest amplitude flares), which evolves the quickest as diagnosed by C4170. The HF events are intermediate because of the approximately equal contributions of BaC and blackbody to the U band at peak. Due to the largely different timescales among the spectral components, the relative amounts present at peak are therefore related to the overall light curve morphology. Clearly, a large sample was necessary for adequately understanding that not all stellar flares are identical, "scaled-up" versions of one another and that there are important spectral differences (presumably related to underlying flare heating mechanisms) to consider when modeling the gamut of flare behavior.

The IF events with the largest relative amount of BaC emission at peak are IF5 and IF6. These flares are also unusual for their IF designation with large χ_{flare, peak} (∼2.2 ± (0.1–0.2)), large Hγ to continuum ratios (∼40–50), and a large fraction of HB flux (∼40%) from λ = 3420–5200 Å. Coincidentally, these two flares both occurred on EV Lac. As illustrated by these two flares, the relative amount of one continuum component compared to the other is indeed variable even during the IF events. Nonetheless, the relative amount of BaC at peak generally determines the evolution of the flare as either impulsive (dominated by blackbody emission), gradual (dominated by BaC emission), or hybrid (both present, but generally more BaC emission). Also, the BaC does not always have a slow decay but is sometimes quite impulsive like the blackbody. This occurs typically with the HF events (see the time-decrement relation for HF2; Figure 18), or for the IF events with large χ_{flare, peak}.

Successful modeling of the exceptions within each classification scheme (e.g., IF5 and IF6 for the IF events; MDSF2 for the HF/GF events) undoubtedly will provide invaluable insight into the flare heating mechanism; this study awaits self-consistent modeling of the blackbody component which has not yet been attained.

The main parameter we use to describe the blue/NUV continuum is χ_flare, the ratio of flux values to the blue (C3615) and red (C4170) of the Balmer jump. The IF events have small χ_flare (and hence small Balmer jumps) at peak continuum emission whereas the GF events have larger Balmer jumps. We also measure the ratio of Hγ emission to the nearby continuum: Hγ/C4170. We connected the size of the Balmer jump to the Hγ/C4170, and find a suggestive linear trend such that flares with larger Hγ/C4170 ratios have larger Balmer jumps (Section 4.3.1, Figure 12). Most of the IF events have strikingly similar χ_{flare, peak} values of 1.6–1.8 over several orders of magnitude in flare peak amplitudes and energies. They generally also have Hγ/C4170 ratios of 15–25 and only 11%–17% of the emission in the HB component at peak. A narrow range of peak flare properties suggests a common impulsive heating mechanism among IF events. The χ_flare and Hγ/C4170 measures indicate the relative importance of each emission component—HB versus blackbody emission—while also giving information about the flare morphology. χ_flare is a measurement that can be made without spectra, for example at very high time resolution and high signal-to-noise with the custom continuum filters used on the stellar instrument ULTRACAM (Dhillon et al. 2007; Kowalski et al. 2011b) and solar instrument ROSA (Jess et al. 2010).

For the IF events, the Balmer jumps increase by the beginning of the gradual decay phase (Δχ_flare ∼ 1) and also show 20%–30% more blue and NUV emission in the HB (continuum and line) component compared to peak (Section 5.4, Figure 21). These trends indicate that the white-light heating/cooling mechanism in the impulsive and gradual phases of flares changes, as flares become more dominated by HB emission in the (fast and gradual) decay phases, which has been a phenomenon suggested by several works such as Moffett & Bopp (1976) and Abdul-Aziz et al. (1995).

The Balmer jump has not been detected in our low-resolution spectra (although we note that several instances in different flares show suggestive features near λ ∼ 3646 Å, especially in the gradual phase). It remains an open question what causes the large amount of apparent blending (which we have referred to as the "pseudo-continuum" or PseudoC) between λ = 3646–3914 Å, as the Balmer jump is not detected even in high spectral resolution, R ∼ 40,000 spectra (Schmitt et al. 2008; Fuhrmeister et al. 2008). We raise the question, what is the "BaC" that we observe in our low-resolution spectra? We suggest that the amount that we measure may be affected by the Stark broadening of higher order Balmer line wings and the continuum edge (Donati-Falchi et al. 1985). High spectral resolution, high time resolution data near λ = 3646 Å (while also covering some continuum regions at λ > 4000 Å in order to characterize the blackbody component) together with better modeling of the Stark broadening in the higher order Balmer lines (following the work of Donati-Falchi et al. (1985) for solar flares) are needed to understand the detailed pseudo-continuum properties from λ = 3646–3920 Å.

The large percentage (∼40%–90%) of the wavelength-integrated blue+NUV (3420–5200 Å) flux in the peak and gradual decay phases is emitted in continuum emission other than HB radiation for all flares (Table 8, Section 5.4). We attribute this continuum emission to a "blackbody" (or blackbody-like) source. We measured the slope of the blue continuum from λ = 4000–4800 Å and found it to be linearly decreasing with wavelength, thus allowing us to match it well to a blackbody with a moderately hot color temperature, T_BB ∼ 10, 000 K (Section 6.1). The range of peak temperatures for the 17 IF, HF, and GF events with well-measured slopes is between T_BB ∼ 9000–14, 000 K (with systematic errors of about 1000 K; Figure 23). The secondary flare (MDSF2) during IF1 has the highest color temperature of ⩾15, 000 K at peak. The lowest temperature was measured as T_BB ∼ 6700 K for the low-amplitude flare GF5. The flares for which we measured blackbody temperatures have a range of ∼2.5 orders of magnitude in flare peak specific luminosity in the U band. Why such a narrow range of temperatures? We speculate that the origin and transient nature of the blackbody radiation may be due to a thermal regulation process, when a temperature threshold is reached in the deep atmosphere prior to runaway heating whereby material heats to higher temperatures and lower densities (i.e., "explosive evaporation;" see Fisher et al. 1985). The observed Neupert relationship between Ca ii K and C4170 (the blackbody) may be indicative of an evaporative process occurring at high column mass (where C4170 is produced) if there is a connection between the formation of Ca ii K emission and the evaporated plasma through, e.g., X-ray (from T > 20 MK) radiative backwarming (Hawley & Fisher 1992), condensations (Abbett & Hawley 1999), or heat conduction channels (Canfield et al. 1984) from the corona.

We also explored interflare and intraflare T_BB variations. In Appendix G, we combined spectra from multiple peaks in a hybrid flare (HF1) to obtain a spectrum with the same total flux level of a very fast flare (IF2) and found bona-fide differences in the value of T_BB, the fraction of HB to total flux, and the evolution of HB to total flux. Of the many parameters that could give rise to interflare T_BB variations, we suggested that the speed of the flare areal increase is an important parameter, which is summarized below. However, it is imperative that RHD models be used to model temperature differences between flares. With these state-of-the art spectral observations and new analysis techniques, we also attempted to constrain the intraflare variation of the blackbody continuum component. For two flares with time-resolved rise phases (MDSF2 and IF3; Table 9 and Figure 33), we found an apparent increase of ∼2000 K in the color temperature of the blackbody component during the rise phases.

The impulsive phase "blackbody" continuum component is not a featureless emission spectrum from λ = 3400–6500 Å. We detected an A-type star spectrum during the rise and peak phases of a secondary flare (MDSF2) that occurred during the Megaflare (IF1; K10). This finding gives important insight into the nature of the blackbody component: the absorption features and continuum shapes (i.e., T_BB) encode information about the multithermal stratification of the flaring atmosphere at high densities. We hypothesize that the multithermal structure of the atmosphere is a byproduct of the thermal regulatory process, described above, such that the atmosphere responds in such a way that energy is most efficiently (and quickly) removed from the atmosphere through the continuum. We will explore this aspect of flare atmospheres in detail with future RHD models (A. F. Kowalski et al. 2013, in preparation).

Absorption signatures were also detected in the wings of the Balmer lines during MDSF2 and IF4. Note that hot star-like absorption during flares also affects the Balmer flux and time decrements (Kowalski 2012). Our analysis techniques can be applied to other flare observations to understand absorption phenomena in spectral data that lack broad wavelength coverage or robust flux calibration.

We extended the simple blackbody modeling to include the effects of flux redistribution and wavelength dependent opacities, as in a real stellar atmosphere, resulting in "Castelli–Kurucz" corrections which are described in detail in Kowalski (2012) and Appendix F. In summary, more realistic filling factors (X_BB) of the "blackbody" emission component can be obtained, which are about a factor of 1.5–2.3 times larger, than those found using the Planck intensity to estimate areas. Also, effective temperatures can be estimated from the measured color temperatures of T_BB (9000–14, 000 K). The corresponding range of T_eff is ∼7700–9400 K (with the largest being ∼10, 500 K during the MDSF2 peak) indicating that at least 2 × 10¹¹ erg cm⁻² s⁻¹ is required for the heating flux to sustain the "blackbody" component. Detailed modeling of these "hot spots" in the M dwarf atmosphere would constrain the density and temperature stratification of the lower flaring atmosphere.

Most flares do not show obvious absorption features in the Balmer continuum or Balmer lines, likely due to the large amount of (chromospheric) HB flux in emission that generates an effective veiling of the photospheric absorption. The veiling leads to an anti-correlation in the apparent amount of BaC emission and the amount of blackbody-like continuum emission. As the absorption becomes greater—either through a larger filling factor (areal coverage) or because of higher blackbody temperature—the measured BaC emission estimated using a linear extrapolation from the blue-optical zone decreases. Again, accurate models (e.g., phenomenological "hot spot" models) would elucidate how much BaC emission is being missed.

An additional benefit of our large sample of flares and homogeneous analysis is evident. Not all flares have the same proportion of each emission component and therefore some flares reveal more about a given emission property than others. For example, MDSF2 exhibited a spectrum that had an unusually strong contribution from the "blackbody" component alone, thereby allowing unambiguous detection of the absorption features. The IF0 event from HP91 also had such a strong contribution from the "blackbody" component that the relative amount of BaC emission was among the lowest in the sample. In contrast, the IF1 decay phase showed an excessive amount of BaC emission, thereby allowing a thorough characterization of its flux and spectral shape. We predict that more observations will reveal additional, important variations in the flare continuum components that will contribute new constraints on flare models.

We measured the speed at which the inferred "blackbody" flare area grows during the rise phase of IF3. We discussed ways in which the inferred flare areas are rather uncertain, and a more realistic determination was obtained by modeling the "blackbody" emission as a hot-star spectrum. During IF3, the speed is large in the early and mid-rise phase, with v_flare ∼100 km s⁻¹, decreasing to ∼50 km s⁻¹ at the peak and <10 km s⁻¹ in the post-maximum phase.

These speeds are strikingly similar to the speeds of the development of two-ribbon flares parallel and perpendicular to the magnetic neutral line in solar active regions. In particular, motions of white-light, hard X-ray, Hα  and C iv kernels are observed on the Sun (Kosovichev & Zharkova 2001; Wang 2009; Nishizuka et al. 2009; Krucker et al. 2011; Inglis & Dennis 2012), and are observed to propagate at similar speeds of between 50–130 km s⁻¹ during the development of two-ribbon structures parallel to the magnetic polarity inversion line (PIL; Fletcher & Hudson 2001; Schrijver et al. 2006). There is also slower motion perpendicular to the flare ribbons at the locations of the kernels, outward from the PIL with velocities of ∼15 km s⁻¹ (Fletcher et al. 2004; Keys et al. 2011) or as small as a few km s⁻¹ (Qiu et al. 2010), presumably the result of reconnection at progressively higher heights. Perhaps, the higher speeds in the rise phase are analogous to the formation of flare ribbons, and the smaller speed after flare maximum represents the perpendicular motions (e.g., "spreading moss," Berger et al. 1999). This scenario has been deduced by UV/HXR observations to explain solar flare light curve morphology (Qiu et al. 2010).

The heating of new flare footpoints, resulting in increasing flare area, was a conclusion from the models of Hawley & Fisher (1992, 1994) to explain solar and stellar flare evolution. During solar flares, it is an open question what triggers particle acceleration (a possible source of flare heating) in neighboring flare regions (Inglis & Dennis 2012). Possible scenarios are an unstable flux rope that erupts sequentially along a flare arcade (the "tether-cutting scenario" Moore et al. 2001) and magnetoacoustic slow waves that propagate away from the initial flare site (Nakariakov & Zimovets 2011). Three-dimensional models of reconnection predict motions of the reconnection site, which undergo changes in velocity to slower speeds due to mass loading (Linton & Longcope 2006).

The evolution of v_flare during IF3 (Figure 34) suggests that a similar (two-ribbon) flare development process involving sequentially heated footpoints occurs during solar and stellar flares, despite our likely over-simplification of the geometrical morphology of stellar flare regions (i.e., as an expanding circle). The quantity v_flare is therefore an exciting new measure that can be used to further address aspects of the solar–stellar connection. Broad wavelength coverage solar flare spectra (e.g., Neidig 1983), which are not feasible with current standard instrumentation, would be invaluable for providing detailed insight into the spatially resolved continuum properties during ribbon and footpoint formation.

We also found a possible relation between the average speed during the rise phase and the temperature of the blackbody component at peak, with a very slow speed (∼8 km s⁻¹) observed for hotter temperatures (T_BB ∼ 15, 000 K) and prominent Balmer absorption features, e.g., in MDSF2. Perhaps the heating occurs by a process similar to solar Ellerman bombs on the Sun, where reconnection is thought to occur in the low chromosphere or photosphere. The connection to Ellerman bombs and their (possible) formation at high densities and low heights has implications for the heating properties of M dwarf flares that show Balmer absorption signatures. However, we do not exclude the possibility that the heating during the secondary flare MDSF2 (which is classified as a HF event; Figure 10) is connected to the Megaflare decay and may exhibit different heating properties than classical flares.

Determining the fundamental connection between speed and the type of white-light continuum emission will require further modeling work on the spatial development of flare regions. However, we began by comparing the speeds between flares and relating to two possible heating scenarios. We present two extreme scenarios: the expansion of a single hot spot in the lower atmosphere (Scenario 1, v_flare ∼ c_{s, chrom}) and the formation of several hot spots (Scenario 2, v_flare > c_{s, chrom}). Insight into the detailed physics of these scenarios requires three-dimensional modeling, but we can gain initial insight by comparing the speed of the flare to the speed of sound in the level of the atmosphere where the (blackbody) emission is formed. Higher speeds (50–130 km s⁻¹) are observed during the formation of a hot blackbody, T_BB ∼ 10, 000–14, 000 K (Figure 23) and values of BaC3615/C3615 of 0.25–0.35 (Figure 24) for fast rise and peak phases of the classical flares IF9, IF3, and IF2 with moderate to large peak U-band amplitudes and energies. In the impulsive phase of these events, we find evidence of Scenario 2-type speeds during the rise and peak phases followed by Scenario 1-type speeds during the initial fast decay phase. Low speed (≲ 10 km s⁻¹) and high temperature were derived from the rise phase of MDSF2, whereas low speeds and relatively lower temperatures were derived in the initial fast decay phases of IF2, IF3, IF9, and HF1. Therefore, there is a range of heating that is observed at times of low flare speeds. The rise phase of MDSF2 is consistent with a variation on Scenario 1 (Scenario 1b), whereby the beam heating (and reconnection?) is triggered by a sound wave generated in the lower atmosphere by the initial flare peak of IF1.

The white-light continuum persists well into the gradual decay phase of IF, HF, and GF events as the flare spectrum becomes increasingly dominated by HB (line and continuum) radiation. At the beginning of the gradual decay phase, the blackbody component exhibits a cooler temperature of T_BB ∼ 8000 K. The change in color temperature from peak to gradual decay phase was first discovered by Mochnacki & Zirin (1980). We find that the evolution of C4170 (blackbody flux) begins to follow the BaC flux in the gradual decay phase. Another continuum component is detected redward of Hβ. This is observed as an increasing contribution of red flux during the flare gradual decay phase, and may be responsible for the redder colors seen in broadband photometry during the gradual decay phases of flares in the past (e.g., Kunkel 1970; Hawley & Pettersen 1991; Hawley et al. 2003). This "Conundruum" flux rises into the red spectral zone and has a color temperature of ≲5500 K during the peak and gradual decay phases. The colors in the red-optical are similar to those (albeit with large errors) reported during the gradual phase of a large flare on CN Leo (Fuhrmeister et al. 2008). This study found large temperatures (∼20,000–30,000 K) in the red during the impulsive phase, whereas we find a peak phase color temperature of T_BB ∼ 7700 K (using a single blackbody fit; Section 8). The Conundruum is present in some flares at peak also, but the hotter (blackbody) component dominates the white-light continuum shape at peak emission in the blue and, to a lesser extent, in the red. According to our analysis in Sections 6–8, the red continuum (λ ≳ 6000 Å) during dMe flares is a superposition of the hot, blackbody emission component and Conundruum emission component. Impulsive phase red emission (e.g., in the R band) is largely due to the action of the hot blackbody component, and late phase gradual emission is largely due to the Conundruum.

The red spectral shapes are similar to RHD model predictions (Allred et al. 2006), from which we infer the origin of the Conundruum to be Paschen and Brackett recombination radiation from the flare chromosphere, and there may also be a contribution from H⁻ recombination radiation originating from a larger backwarmed area of the star (similar to the geometry in Fisher et al. 2012). We explored the Conundruum flux in detail for IF3, and suggest that it has a physical connection to the lower order (Hγ, Hβ, and Hα) Balmer lines.

In summary, we find three continuum components that are important at different stages of the flare evolution: (1) BaC emission (most important in the mid rise phase and during the gradual decay phase); (2) hot "blackbody" emission (most important at the peak phase, but present with cooler temperatures in the fast decay and gradual decay phases); and (3) Conundruum emission (can be present in all stages, but most important in the late gradual decay phase).

The emission at flare peak cannot be explained by simply scaling up small amplitude (e.g., in U band) flares to get large amplitude flares, thereby implying important differences in heating mechanisms between IF, HF, and GF events. The emission components originate from different atmospheric parameters among the different types of flares, and even within a given flare type. We have shown several ways that IF3 and IF9 have scaled-up Balmer components (via the time decrement; Section 5.2). However, the C4170 components appear to differ and do not follow a scaled up relation in their time evolution, suggesting that C4170 may be formed differently even among the IF events. We examined the scaling relationship between two flares in detail in Appendix G—HF1 and IF2 on YZ CMi—and concluded that one large peak phase is not a simple superposition of several smaller peak phases. Nonetheless, the $\mathrm{log}_{10} \mathcal {L}_{\mathrm{BaC3615}}$ versus $\mathrm{log}_{10} \mathcal {L}_{\mathrm{C4170}}$ relation among the IF events with χ_{flare, peak}< 2 and T_BB ∼ 8000 K (IF2, IF3, IF7, IF8, and IF9) at the beginning of the gradual decay phase follows a linear relation, possibly indicating that these flares are scaled versions of one another after the fast decay of the C4170 (Section 7.1). Further evidence that gradual phase emission follows a scaling relation were deduced from a qualitative analysis (Figure 29) of the fine spectral features and continuum shapes spanning the gradual decay phases of IF1 (a large flare), IF7 (a medium amplitude flare), and GF5 (a small amplitude flare). One must independently consider the degree to which the HB (line and continuum) emitting region and blackbody continuum emitting regions are scaled between two flares and between individual flare phases. Furthermore, the production of the emission components is not mutually exclusive, as indicated by the short (<1 min) delay between the peaks of hydrogen line emission and C3615 such as during IF3 and IF9 (Section 5.1). In fact, the sources of the emission components are intricately tied to one another via the flare heating mechanism and must be self-consistently accounted for in flare models that attempt to reproduce the evolution of these components and their variation during IF, HF, and GF events and during the impulsive and gradual phases of a given flare.

Transient astrophysical phenomena across the universe may be related to the underlying physics in the dMe flares, and we plan to explore these connections in a future paper. Here, we briefly note several interesting similarities to gamma-ray bursts and accretion phenomena. Gamma-ray burst (GRB) light curves have been divided into two classes based on the duration of the hard X-ray light curves: Class I GRBs with two to several hundred second durations and Class II GRBs with <2 s durations (Kouveliotou et al. 1993). Kouveliotou et al. (1993) further found that the Class II GRBs exhibited harder spectral slopes in X-ray emission. Similarly, we found that impulsive flares have different optical timescales and spectral slopes (given by χ_flare) compared to gradual flares. Hurley et al. (2005) proposed that a Class II GRB originated from a magnetar flare (MF). The MFs may be a subclass called soft gamma ray repeaters that occur relatively close to the Milky Way and account for only a few percent of the short/hard GRBs (Palmer et al. 2005); the current leading model is the merging of two compact objects, one of which is a neutron star (Blinnikov et al. 1984; Nakar 2007). Interestingly though, the impulsive phase of a MF has been fit well with a blackbody. In contrast to flare impulsive phase emission, the blackbody was found to have a much higher temperature ∼2 × 10⁵ K. An impulsive phase in the light curve during times of prominent optically thick (blackbody or blackbody-like) emission is a similarity between dMe flares and MFs and implies a common cooling (and heating?) process during the most luminous events in the solar neighborhood and the most luminous events in the universe.

A second astrophysical setting in which we find similarities to dMe flares is accretion, such as occurs in T Tauri stars and dwarf novae. The Balmer jump is a common diagnostic in T Tauri spectra (e.g., Valenti et al. 1993; Herczeg & Hillenbrand 2008). In fact, some T Tauri spectra show small Balmer jump ratios that are rather similar to the χ_flare values for flare spectra. Optical veiling from continua is often seen at λ > 4000 Å, which can inform our understanding of the "blackbody" continuum in flare spectra. Therefore, the physics of accretion models for T Tauri spectra can be used to inform our understanding of the white-light continuum. We also note that dwarf nova accretion events produce superposed Balmer line emission and Balmer line absorption features in their spectra, with most conspicuous absorption at times of maximum continuum emission that resembles an A or B type star (Hessman et al. 1984).