A VERY BRIGHT, VERY HOT, AND VERY LONG FLARING EVENT FROM THE M DWARF BINARY SYSTEM DG CVn

We fit the spectra obtained from the time intervals T0−30:T0+72 and T0+123:T0+328 from the BAT only (first interval) and XRT+BAT (second interval), using either a single temperature APEC (Astrophysical Plasma Emission Code) model or a nonthermal thick-target bremsstrahlung model. The amount of interstellar absorption was fixed to an N_H value of 4.7 × 10¹⁷ cm⁻², based on Mg ii and Fe ii column densities measured by Malamut et al. (2014) toward β Com, a star ∼45 away in angular extent and with a proximity of 9 pc (S. Redfield 2016, private communication). APEC describes a collisionally ionized plasma in coronal equilibrium, with line emission formed predominantly from a balance of collisional excitations populating excited ionic states, and radiative de-excitations; this is the usual assumption for stellar coronal plasmas. We use a custom version of ATOMDB (Smith et al. 2001) calculated out to 100 keV¹⁹ , since the standard ATOMDB energy grid delivered with XSPEC ends at 50 keV. The Volume Emission Measure (${\mathscr{VEM}}$) quantifies the amount of plasma emitting at the fitted temperature, and is equivalent to $\int {n}_{e}^{2}{dV}$. The nonthermal thick target model is often used to describe hard X-ray emission from solar flares. In this model, a power-law distribution of electrons with energy (described by the parameter δ_X) is modified by transport through a fully ionized plasma. From the observed spectrum, the index δ_X can be derived, as well as (for unresolved stellar observations) the power in the electron beam (see the discussion in Osten et al. 2007). In solar flare observations a broken power law is often observed, with low-energy cutoff (E₀) around 20 keV; in this case, we consider a single power law, with E₀ fixed to be 20 keV. The power required depends on the low-energy cutoff as ${(20/{E}_{0})}^{({\delta }_{X}-2)}$ as described in Osten et al. (2007), so decreasing the low-energy cutoff from 20 to 10 keV results in roughly a factor of two more power required for δ_X ∼ 3. The spectra are shown in Figure 2 and fit parameters are listed in Table 1. The 0.01–100 keV flux extrapolated from the best-fit model is also reported in Table 1. The two models are statistically indistinguishable for the same time interval.

Table 1.  Spectral Fit Parameters for Trigger and Initial Decay of Flares on DG CVn

Parameter	T0−30:T0+72	T0+123:T0+328a
	BAT only	XRT+BAT
Thermal Fit
TX (106 K)	${278}_{-92}^{+140}$	290 ± 31
${\mathscr{VEM}}$ (1054 cm−3)	${9}_{-2.7}^{+4.7}$	4.9 ± 0.1
χ2 (dof)	38.2 (36)	401.6 (432)
fX (14–100 keV) × 10−9	${3.6}_{-0.8}^{+0.2}$	${2.03}_{-0.15}^{+0.18}$
(erg cm−2 s−1)
fX (0.3–10 keV) × 10−9	(4.5)b	2.38 ± 0.03
(erg cm−2 s−1)
fX (0.01–100 keV) × 10−9	9.3	5.1
(erg cm−2 s−1)
Nonthermal Fit
δX	3.6 ± 0.4	${3.2}_{-0.1}^{+0.2}$
Power (1037 erg s−1)	${3}_{-1.1}^{+1.2}$	1.11 ± 0.03
χ2 (dof)	40.2 (36)	409.62 (432)
fX (14–100 keV)  × 10−9	${3.8}_{-0.4}^{+0.2}$	${2.24}_{-0.15}^{+0.19}$
(erg cm−2s−1)
fX (0.3–10 keV)  × 10−9	(5.8)b	${2.34}_{-0.02}^{+0.03}$
(erg cm−2 s−1)
fX (0.01–100 keV)  × 10−9	11.	5.3
(erg cm−2 s−1)

Notes.

^aN_H fixed at 4.7 × 10¹⁷ cm⁻². ^bFlux extrapolated from best-fit model in the 14–100 keV range.

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The thermal fit shows that the spectrum from 0.3 to 100 keV for each time interval is dominated by a single temperature. Attempts to use a more complicated model, like multiple temperature components, did not result in a statistically better fit, demonstrating the dominance of the hot temperature plasma. We confirmed that the spectrum is dominated by the continuum from such a hot plasma by redoing the fit using a bremsstrahlung model (brems) in XSPEC, and comparing the result to the APEC fit. The bremsstrahlung model only includes contributions from H and He, which confirms the high-temperature results from fitting with APEC. There is no evidence for any lower temperature plasma, as evidenced by the lack of He- or H-like iron at characteristic energies of 6.7 and 6.9 keV.

Previous reports of stellar superflares with Swift had reported the detection of the Fe Kα line at 6.4 keV (Osten et al. 2007, 2010). However, these appear to be due to a calibration artifact unrecognized at the time, namely charge trapping (Pagani et al. 2011). The spectral fits to XRT data shown in Figures 2 for BFF and 3 for F2 do not show any evidence of excess emission near 6.4 keV.

We extracted XRT spectra in four intervals during the peak and decay of the second large flare, F2. There was not enough signal in the BAT at this time to extract a spectrum, so we concentrate on the 0.3–10 keV range of the XRT. The four spectra were fit jointly, with a three temperature APEC model: the two lowest temperatures (corresponding to quiescent emission) were fixed to be the same for all four spectra, while the third temperature component corresponding to the flare emission was allowed to vary. We arrived at this after comparing the goodness-of-fit for models with different numbers of components. Values are given in Table 2. Columns labelled "Q1" and "Q2" list the results for the quiescent component; the metallicity was fixed to be unity (i.e., solar) for the spectral components. Spectral fits extracted for other times during the peak and decay of the flare are listed in separate columns. The temporal trends show a large temperature and volume emission measure at the earliest time interval, with generally decreasing plasma temperatures and volume emission measures during the decay of the flare. The abundance of the flaring plasma initially appears to be mildly sub-solar, but noise in the values fitted from spectra extracted at later times prevents the determination of a definitive trend. Since the spectra were fit together, a single value of the fit statistic was calculated: χ² of 749.81, with 787 degrees of freedom. Figure 3 shows the spectra along with the model fits. Table 2 also gives a flux extrapolated to the 0.01–100 keV range for intercomparison of the different time intervals and other flares.

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Table 2.  Spectral Fit Parameters During F2 Event on DG CVn

Parameter	Q1	Q2	T0+10860:T0+11849	T0+16496:T0+17605	T0+22437:T0+23367	T0+28019:T0+29125
TX (106 K)	${5.0}_{-1.3}^{+2.4}$	${14.7}_{-2.8}^{+2.7}$	${53.8}_{-3.8}^{+2.9}$	${41.5}_{-4.4}^{+4.9}$	55.7${}_{-13.0}^{+20.2}$	${36.2}_{-7.8}^{+11.3}$
${\mathscr{VEM}}$ (1052 cm−3)	${1.3}_{-0.6}^{+0.8}$	3.6${}_{-2.0}^{+2.3}$	${154.9}_{-6.3}^{+5.5}$	${105.3}_{-9.0}^{+8.6}$	${38.6}_{-11.2}^{+8.3}$	${43.0}_{-8.6}^{+8.2}$
Z	1 (fixed)	1 (fixed)	${0.4}_{-0.12}^{+0.12}$	${0.51}_{-0.20}^{+0.24}$	${1.18}_{-0.88}^{+2.12}$	${0.05}_{-0.05}^{+0.52}$
fXa (0.3–10 keV)  × 10−10	0.09b	0.20b	6.59${}_{-0.11}^{+0.07}$	${4.35}_{-0.11}^{+0.11}$	${2.26}_{-0.19}^{+0.10}$	${1.60}_{-0.06}^{+0.14}$
(erg cm−2 s−1)
fXc (0.01–100 keV) × 10−10	0.13	0.28	8.0	5.3	2.8	2.0
(erg cm−2 s−1)

Notes.

^aFlux calculated for flaring time intervals includes contribution from the quiescent plasma component. ^bQuiescent flux in this energy range calculated using best-fit model parameters. ^cFlux extrapolated into this energy range using best-fit model parameters.

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Calculation of the X-ray radiated energy is made difficult by data gaps arising from satellite occultations, as well as limitations of data from the BAT due to signal-to-noise constraints. We attempt to account for the energetics in a couple of ways, to estimate the total energy for BFF and F2. Table 1 lists the fluxes in two intervals of time for which flux measurements in the 14–100 keV bandpass can be made; for one of these time intervals, the flux in the 0.3–10 keV interval can be measured directly, and in the other it can be estimated by extrapolating the best-fit spectral model into this energy region. Because of the data gaps, we could not perform direct integration under the X-ray light curve. Instead, we assumed continuity of the flares across the data gaps and used exponential rises and decays to parameterize the light curve as a series of flare events; the parameters were not fit to the light curve, but rather were varied to approximate the shape of the light curve. Figure 5 shows the XRT light curve with these fits. Thus the energy estimate done this way is only approximate. We note that the radio light curve in Fender et al. (2015) shows the radio flare decay of BFF in the Swift data gap, showing its decay to be relatively simple. We used energy conversion factors from the spectral modeling described above to determine the integrated energy in the 0.3–10 keV bandpass. This gives an estimated X-ray radiated energy for BFF in the 0.3–10 keV range, from T0+123 to approximately T0+10⁴ s, of 4 × 10³⁵ erg. We can add in the radiated energy in the 14–100 keV bandpass from T0−30:T0+328, using the fluxes derived from spectral modeling in Table 1, and the 0.3–10 keV flux in the T0−30:T0+72 time range listed in Table 1 and extrapolated from the best-fit model to the 14–100 keV energy range. These last two contributions are ≈5 × 10³⁴ erg, putting a lower limit on the X-ray radiated energy in the 0.01–100 keV range for BFF of ≈4.5 × 10³⁵ erg. We also calculate an upper limit to the energy radiated in the 0.01–100 keV bandpass by using the ratio of flux in the 0.01–100 keV energy range to that in the 0.3–10 keV range, as listed in Table 1 for BFF. For BFF, the ratio for both time intervals in Table 1 is ∼2, so the radiated energy in the 0.01–100 keV bandpass could be as much as twice that deduced from the 0.3 to 10 keV energy range, or 8 × 10³⁵ erg. We take the duration of BFF to be the time from the optical rise preceding the trigger, occuring around T0–70 s as discussed in Caballero-García et al. (2015), until the transition from BFF to F2 in the optical light curves, at around T0+7750 s.

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The F2 event is considerably longer-lasting than BFF, extending from about T0+6800 to about T0+30800 s. The decay could be even longer; because of the decrease in count rate and shorter monitoring intervals it is difficult to see whether the behavior from about 2–5 × 10⁴ s is a continuing decay from the peak of F2 or whether there are subsequent smaller flares occurring. The spectral fitting for this event, in Section 3.1.2, does suggest that the plasma is cooling. The radiated energy in the 0.3–10 keV band for the F2 event is 9 × 10³⁵ erg, more than twice the already large energy of BFF; this is due primarily to the extended duration of F2, as its count rate is lower. Calculating an upper limit for the energy radiated in the 0.01–100 keV bandpass, using the ratio of flux in the 0.01–100 keV energy range to the 0.3–10 keV range, we find that the ratio for F2 peaks at 1.2 for the first time interval in the flare, and decreases thereafter. This implies a correction of at most 20% in the radiated energy determined for the 0.3–10 keV bandpass, or a 0.01–100 keV radiated energy of 10³⁶ erg. We take the duration of F2 to be approximately the interval T0+6800:T0+30800 s.

The radio data also reveal the presence of a radio flare at ∼1 day, which falls in a gap of the X-ray data. We accounted for this "missing" flare using an approximate rise and decay that would fit within the gap in the X-ray data (named F5). Accounting for the several smaller events that occurred afterward, we estimate the total radiated X-ray energy for the series of events spanning ∼19 days to be about 2 × 10³⁶ erg. We note that this is only an approximation, due to the data gaps, but it suggests that F2 was responsible for about half of the energy release during this extended period, with BFF responsible for half as much X-ray radiated energy.

The R-band light curve of F2 peaks at T0+9590 s and has a duration of 3.62 hr (from T0+8165 to T0+21,200 s). At the beginning of the R-band observations, DG CVn was already 0.78 mag brighter than quiescence, and was increasing in brightness. The observed duration of the rise phase is 1300 s, and at the peak of F2, the system is 1.49 mag brighter than quiescence. The flare energy is the quiescent luminosity multiplied by the equivalent duration of the flare (Gershberg 1972). The equivalent duration (of one star) for F2 is 2.84 × 10⁴ s, giving an R-band flare energy of E_F2,R = 8.5 × 10³⁴ erg. Figure 4 shows the V-band flare light curve, along with that of the "great flare" of AD Leo from HP91 for perspective.

V-band light curves are available for both BFF and F2: the former from Caballero-García et al. (2015), and the latter in this paper. From inspection of Figure 1 of Caballero-García et al. (2015), it is clear that the system was slightly elevated at the start of their optical data, before the burst beginning around t−T0 = −70 s, which corresponds to BFF. The disparity in timescales in the V band between BFF and F2 is apparent in Figure 4, with an FWHM duration of 20 s for BFF, and ∼3600 s for F2. We translated the light-curve data from Caballero-García et al.'s (2015) Figure 1, using GraphClick, to estimate what the integrated V-band energy of BFF might be, in combination with V-band measurements presented here, which reveal the light-curve behavior just prior to F2. This corresponds to an integrated energy in the V band of about 2.8 × 10³⁴ erg for BFF, from T0–146 to T0+7730.

We estimate the V-band radiated energy for F2 using direct integration from the light curve. From T0+7730 to T0+23754, the integrated energy is 5.2 × 10³⁴ erg. The light-curve behavior in Figure 1 for the V band appears to demonstrate that the system has returned to its quiescent value after F2, though the gaps in the photometry prevent a definitive statement. The impulsive phase initiation of F2 is obvious from the light curve, though the decay from BFF flattened out for a long time before this, being ∼0.6 mag above the quiescent system V magnitude. The fact that the V band photometry was also enhanced prior to the BFF burst at ${\rm{t-T0}}=-70$ s suggests that there may have been an even earlier event missed in all wavelengths.

Table 4 lists the X-ray and optical energies derived for both BFF and F2, as discussed in Sections 3.3.1–3.3.3. We have measurements of the X-ray and V-band energy for both events, as well as (for F2) radiated energy in the R band. F2 is the more energetic of the two events; there also appears to be a difference of about a factor of two in the energy of the two events in both the 0.3–10 keV X-ray band and V-band radiated energies. We estimate the U-band energy using E_U–E_V scalings from Hawley & Pettersen (1991). Although the X-ray radiated energy in the 0.3–10 keV band appears to differ only by about a factor of two for the two events, the high plasma temperatures derived for BFF increase the wavelength range over which significant emission is received. As discussed in Section 3.3.1, we can come up with a range of the likely coronal radiated energy by considering a wider photon energy range. Consideration of this wider wavelength range increases the amount of radiated energy from coronal plasma, up to the point where the coronal radiated energy from BFF is only slightly less than that from F2. This suggests that the energy partition in each event does not follow the same trend.

Osten & Wolk (2015) described energy partition in solar and stellar flares using smaller stellar flares, and demonstrated a rough agreement between solar and stellar flares in the relative fraction of radiated energy appearing coronal plasma and that in the hot blackbody emission, which dominates the U-band. From their Table 2, we estimate the bolometric radiated energy two ways: using the extrapolated U-band energy described above, as well as the X-ray energies calculated in Section 3.3.1. The estimation of bolometric energy using the coronal radiated energy is imprecise because the formulation of Osten & Wolk (2015) only considered the 0.01–10 keV energy range, whereas the BFF event clearly has significant contribution at higher photon energies. The two estimates of bolometric energy differ from each other by a factor of three or more. If the same relative contribution does hold for all of the coronal plasma, then accounting for the upper limit to the total coronal radiated energy in the 0.01–100 keV for BFF suggests that BFF and F2 may have been comparably energetic, at a few ×10³⁶ erg. The fact that the bolometric energy estimates generated each of two ways differ from each other by about a factor of three or so suggests that the energy partition is not constant from flare to flare. Considering the upper limits to X-ray radiated energy and the V-band radiated energy, the ratio varies from E_X/E_V ≈ 28 for BFF, and ≈20 for F2. The BFF event had significantly more coronal radiation than F2 when considered relative to the optical radiated energy. The data are too sparse to determine whether there is a relationship between the size of the flare or other parameters and the energy partition determined.

Fender et al. (2015) presented 15.7 GHz data obtained starting about 6 minutes after the Swift trigger of DG CVn. Since the radio flare traces the action of nonthermal particles, these measures constrain the amount of kinetic energy in the BFF. We follow the treatment of Smith et al. (2005) in estimating the kinetic energy from the radio light curve. By assuming a spectral energy distribution of the radio emission, and modeling the temporal evolution of the emission, we can estimate the total radio energy, and hence the kinetic energy. The free parameters are the magnetic field strength in the radio-emitting source and the distribution of the accelerated electrons. The radio light-curve data is taken from Fender et al. (2015), and the portion within ∼2 hr of the trigger is shown in Figure 6; we use these data to constrain the time profile of the radio flare. Since the radio data suggest there was a single decline from the peak, we model the time profile of BFF as a single exponential decay, with a peak at the start time suggested from the start of the V band burst reported in Caballero-García et al. (2015), namely T0−70 s. We assume a fast linear rise to the maximum flux. The decay is fit from the radio light curve, and is 3980 s; we assume there is a single exponential decay during the decline of the radio flare. We also assume that the radio spectrum has a spectral shape of the form

$$\begin{eqnarray}&&{S}_{\nu }=A{\nu }^{{\alpha }_{1}}{\rm{}}\,{\rm{for}}\,{\rm{}}\nu \leqslant {\nu }_{\mathrm{pk}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{S}_{\nu }=B{\nu }^{{\alpha }_{2}}{\rm{}}\,{\rm{for}}\,{\rm{}}\nu \geqslant {\nu }_{\mathrm{pk}}\end{eqnarray} \tag{ 3 }$$

where S_ν is the radio flux density, ν_pk is the peak frequency, separating optically thick (ν ≤ ν_pk) emission with spectral index α₁ from optically thin (ν ≥ ν_pk) emission with spectral index α₂, and A and B are prefactors describing the dependence of the radio emission on other parameters. We assume that the peak frequency does not change during the decay, though there is evidence from solar and stellar flares that the peak frequency does change during the impulsive phase (Lee & Gary 2000; Osten et al. 2005). The spectral indices for a homogeneous radio-emitting source, on either side of ν_pk are

$$\begin{eqnarray}&&{\alpha }_{1}=2.5+0.085{\delta }_{r}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\alpha }_{2}=1.22-0.90{\delta }_{r}\end{eqnarray} \tag{ 5 }$$

(Dulk 1985), where δ_r is the index of the distribution of nonthermal electrons. The dependence of the number density of nonthermal electrons with energy and time, n(E, t) is a separable function and has the form

$$\begin{eqnarray}&&n(E,t)=\displaystyle \frac{N(t)({\delta }_{r}-1)}{{E}_{0}}{(E/{E}_{0})}^{-{\delta }_{r}}\ \ \end{eqnarray} \tag{ 6 }$$

where E is the electron energy, N(t) describes the temporal behavior of the number of accelerated particles, and E₀ is a cutoff energy, usually taken to be 10 keV (Dulk 1985). Using this formalism, the time evolution seen at ν_AMI = 15.7 GHz, F(t), can be applied to all frequencies

$$\begin{eqnarray}&&S(\nu ,t)=F(t){\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{AMI}}}\right)}^{{\alpha }_{2}}{\rm{}}\,{\rm{for}}\,{\rm{}}\nu \geqslant {\nu }_{\mathrm{pk}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&S(\nu ,t)=\displaystyle \frac{(Ft){\nu }_{\mathrm{pk}}^{({\alpha }_{2}-{\alpha }_{1})}}{{\nu }_{\mathrm{AMI}}^{{\alpha }_{2}}}{\nu }^{{\alpha }_{1}}{\rm{}}\,{\rm{for}}\,{\rm{}}\nu \leqslant {\nu }_{\mathrm{pk}}\end{eqnarray} \tag{ 8 }$$

where we have assumed ν_pk < ν_AMI. Solar and stellar radio observations show that the peak frequency ν_pk is usually ∼10 GHz. Since DG CVn is at a known distance, we then convert this to L_r(ν, t) (erg s⁻¹ Hz⁻¹) to describe the temporal and spectral behavior of the flare.

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The kinetic energy at a given time is then determined by integrating over the energy dependence from the lower energy cutoff to infinity

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}(t)=N(t)V(t)\displaystyle \frac{{\delta }_{r}-1}{{\delta }_{r}-2}{E}_{0}\end{eqnarray} \tag{ 9 }$$

where V(t) is the source volume. For optically thin emission the flux density can be expressed as

$$\begin{eqnarray}&&S(\nu ,t)=k{{\nu }^{2}/c}^{2}\int {T}_{b}(\nu ,t)d{\rm{\Omega }}(t)\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{\eta (\nu ,t)V(t)}{{d}^{2}}\end{eqnarray} \tag{ 11 }$$

where k is the Boltzmann's constant, c is the speed of light, T_b is the brightness temperature, and dΩ is the solid angle subtended by the radio-emitting source. The equation can be rewritten using ${T}_{b}={c}^{2}/k/{\nu }^{2}\eta (\nu )L$ for τ_ν ≪1, η(ν) is the gyrosynchrotron emissivity, L the length scale of radio-emitting material, and d the stellar distance. The radio luminosity can then be expressed as

$$\begin{eqnarray}&&{L}_{r}(\nu ,t)=4\pi \eta (\nu ,t)V(t).\end{eqnarray} \tag{ 12 }$$

We use the analytic expressions for the emissivity η for X-mode emission from Dulk (1985)

$$\begin{eqnarray}\eta (\nu ,t) & = & 3.3\times {10}^{-24}{10}^{-0.52{\delta }_{r}}\\ & & \times {BN}(t){(\sin \theta )}^{-0.43+0.65{\delta }_{r}}\times {\left(\displaystyle \frac{\nu }{{\nu }_{{\rm{B}}}}\right)}^{1.22-0.9{\delta }_{r}}\end{eqnarray} \tag{ 13 }$$

where B is the magnetic field strength in the radio-emitting source, θ is the angle between the radio-emitting region and the line of sight, and ν_B is the electron gyrofrequency. We define A(ν) to contain the constant and frequency-dependent prefactors, and substitute this into η

$$\begin{eqnarray}&&\eta (\nu ,t)=A(\nu )N(t),\ \ \ \mathrm{with}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}A(\nu ) & = & 3.3\times {10}^{-24}{10}^{-0.52{\delta }_{r}}\\ & & \times B{(\sin \theta )}^{-0.43+0.65{\delta }_{r}}\times {\left(\displaystyle \frac{\nu }{{\nu }_{{\rm{B}}}}\right)}^{1.22-0.9{\delta }_{r}}\end{eqnarray} \tag{ 15 }$$

where we are assuming that the magnetic field strength in the radio-emitting source does not change appreciably with time. Then, L_r can be expressed as

$$\begin{eqnarray}&&{L}_{r}(\nu ,t)=4\pi A(\nu )N(t)V(t).\end{eqnarray} \tag{ 16 }$$

This can be rearranged so that

$$\begin{eqnarray}&&N(t)V(t)=\displaystyle \frac{\int {L}_{r}(t,\nu )d\nu }{\left.4\pi \int A(\nu d\nu \right)}\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&{E}_{\mathrm{kin},\mathrm{tot}}=\int N(t)V(t)\displaystyle \frac{{\delta }_{r}-1}{{\delta }_{r}-2}{E}_{0}{dt}.\end{eqnarray} \tag{ 18 }$$

Particles will be depleted and replenished during this time; this can be accounted for in the temporal variations. The incompleteness of the observational data, coupled with some of the assumptions made in the analysis, will not render this a precise estimate of the kinetic energy, but does allow for an order of magnitude estimation, given the magnetic field strength in the radio-emitting source and a constraint on the energy distribution of accelerated particles. Since stellar radio data are usually consistent with a relatively hard radio spectra, we examine δ_r in the range of 2.2 ≤ δ_r ≤ 3.9. The right panel of Figure 6 shows the parametric dependence of the kinetic energy on the unknown values of magnetic field strength and index of the nonthermal electron distribution. The implied kinetic energy ranges from very large values, of ∼10⁴⁰ erg for low values of magnetic field (tens of Gauss) and high values of δ_r, to 10³⁴ erg and less for kiloGauss fields and a range of δ_r. These constraints will be used in Section 4.2 to aid in the constraint on the thermal or nonthermal nature of the X-ray emission in BFF.

The X-ray decay analysis in Section 3.4 gives a loop semi-length of l = 2.0 ± 1.0 R_⋆. Based on the discussion in Section 3.2, we use constraints on the flare footpoint area from V- and R-band photometry for F2 obtained assuming a color temperature T_BB of 6000 K, or X_{F2, V}(T_BB = 6000 K) = 0.14. The total flaring area is ${A}_{{fl}}={X}_{{fl}}\pi {R}_{\star }^{2}$ = 2π${R}_{\mathrm{foot}}^{2}$. These two numbers constrain the value α = R_foot/(2l), assuming a single columnar loop with semi-length l as derived above and two footpoints contributing to the total optical area; the flare area becomes

$$\begin{eqnarray}&&{A}_{{fl}}={X}_{{\rm{F}}2,V}\pi {R}_{\star }^{2}=2\pi {R}_{\mathrm{foot}}^{2},\end{eqnarray} \tag{ 23 }$$

with R_foot the radius of one footpoint, and the value α = R_foot/(2l) can then be determined to be 0.07 using the vlaue of X_F2,V evaluated for T_BB = 6000 K. This is independent of the number of loops, as long as the loop length l does not change when generalized to N flaring loops. We can couple this with a simple picture of the emitting region as a loop (or N flaring loops), where the volume emission measure can be expressed as

$$\begin{eqnarray}&&{\mathscr{VEM}}={n}_{e}^{2}\pi {R}_{\mathrm{foot}}^{2}2l.\end{eqnarray} \tag{ 24 }$$

This can be rearranged to give

$$\begin{eqnarray}&&{n}_{e}={\left[\displaystyle \frac{{\mathscr{VEM}}\alpha }{\pi {R}_{\star }^{3}}{\left(\displaystyle \frac{{X}_{{\rm{F}}2,V}}{2}\right)}^{-3/2}\right]}^{1/2}\end{eqnarray} \tag{ 25 }$$

and with the ${\mathscr{VEM}}$ constrained from the spectrum at peak of F2, and α from the combination of the white-light event and coronal loop length, the coronal electron density is constrained to be 3 × 10¹¹ cm⁻³. The quantity α in Equation (25) is calculated for N = 1. Note that this approach can be applied to N flaring loops, and the electron density is unchanged. With n_e and T_X, the strength of the magnetic field required to confine the flaring coronal plasma is then

$$\begin{eqnarray}&&{B}_{\mathrm{conf}}=\sqrt{8\pi {n}_{e}{{kT}}_{X}}\end{eqnarray} \tag{ 26 }$$

with k Boltzmann's constant; evaluating this, we derive B_conf of ∼230 G for F2.

The V-band measurements at the peak of F2 as well as X-ray measurements constrain the flux ratio, which is useful for a comparison of the relative brightening of photospheric and coronal emissions, respectively. At T0+9709 ± 63 s, the Johnson V magnitude is 9.83. This gives a flux density of 4.3 × 10⁻¹³ erg cm⁻² s⁻¹ Å⁻¹, or integrated flux over the V filter bandpass of 3.6 × 10⁻¹⁰ erg cm⁻² s⁻¹, using a FWHM of 836 Å. From Table 2 the 0.3–10 keV flux from the nearest interval, T0+10860:T0+11849 s, is 6.59 × 10⁻¹⁰ erg cm⁻² s⁻¹. We use the flux estimated in the 0.01–100 keV energy range, 8 × 10⁻¹⁰ erg cm⁻² s⁻¹, for a flux ratio f_X/f_V of 2.2.

Is X ∼ 0.14 from a 6000 K blackbody reasonable? A preliminary multithread modeling approach to F2 uses the F13 beam heated atmosphere from K13, which were calculated with the RADYN code (Carlsson & Stein 1997). If we assume that the emission during F2 can be modeled by a superposition of impulsively heated loops (new kernels using the "average burst" spectrum (Table 1 of Kowalski et al. 2015; ${F}_{\mathrm{kernel}}$ here)) and decay phase emission from previously heated loops (F13 gradual decay spectrum at t = 4 s in Table 1 of K15; F_decay) with an area coverage that is 25 times the kernel emission, then we obtain a broadband spectrum that is generally consistent with the coarse Swift UV and optical colors (uvw2/V ∼ 1 compared to the observations ∼0.8–0.9). We can estimate an areal coverage using an actual RHD spectrum

$$\begin{eqnarray}&&{F}_{\mathrm{flare}}=25\ast {X}_{\mathrm{kernel}}\ast {F}_{\mathrm{decay}}+{X}_{\mathrm{kernel}}\ast {F}_{\mathrm{kernel}}\end{eqnarray} \tag{ 27 }$$

where F_kernel is the surface flux of the F13 model averaged over its evolution; F_decay is the surface flux of the F13 during the gradual phase.

For the peak of F2, X_kernel = 0.008 and 25 ∗ X_kernel = 0.2, which is similar to 0.14 for T = 6000 K. For this modeling, X(T = 10,000 K) = 0.023 is justified for a nonthermal interpretation and X(T = 6000 K) = 0.14 is justified for a thermal (decaying loop) interpretation. In Section 4.1.1, the best area to use is likely the T = 6000 K area, though the RHD model (at t = 4 s) that is used to represent the decay emission from previously heated loops is far shorter than the decay times obtained from the X-ray light curves in Section 3. In Figure 8, we show the flare specific luminosity (L_flare = $\pi {R}_{\mathrm{star}}^{2}$F_flare) for the multithread model from RADYN.

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Aschwanden et al. (2008), following on earlier work by Shibata & Yokoyama (1999), found a striking similarity between power-law dependences of flare peak volume emission measure against temperature for a sample of solar and stellar flares. The index of the scaling was the same for solar and stellar flares, even while the stellar flare temperatures were generally hotter. The trend, for stars, was ${{\mathscr{VEM}}}_{p}={10}^{50.8}{\left(\tfrac{{T}_{p}}{10\mathrm{MK}}\right)}^{4.5\pm 0.4}\ \ {\mathrm{cm}}^{-3}$ with T_p being the flare peak temperature and ${{\mathscr{VEM}}}_{p}$ being the peak volume emission measure of the flare. Using the peak X-ray temperature of F2 from spectroscopic fitting in Section 3.1.2, namely 48 MK, the emission measure expected from this scaling relation is 7 × 10⁵³ cm⁻³, so only a factor of 2.3 below the observed peak emission measure of 1.64 × 10⁵⁴ cm⁻³.

Table 1 lists the best-fit parameters for the trigger spectrum and ≈200 s where both XRT and BAT spectra were obtained, using a nonthermal thick-target bremsstrahlung model for the spectral fitting. The free parameters in the spectral fitting are the index δ_X of the accelerated electrons and the total power in the electron beam.

We have one constraint on the kinetic energy of the accelerated particles involved in the event by multiplying the power from each spectral segment by the integration time for that segment. From the parameters in Table 1, this is a staggering 5 × 10³⁹ erg, and is a lower limit, since there is no X-ray data for a large portion of the event, and the total power depends on the value of the low-energy cutoff, as described in Section 3.1.1. Estimation of the kinetic energy also proceeded from the radio light curve in Section 3.3.5; we use the value of δ_X from the X-ray spectral fits to determine plausible values of kinetic energy. Solar flare data shows that the nonthermal electrons producing nonthermal hard X-ray emission tend to be less energetic than those producing the radio gyrosynchrotron emission, but we assume δ_X = δ_r for simplicity. With that substitution, the kinetic energy then becomes a function of the magnetic field strength in the radio-emitting source, according to Figure 6. Matching the lower limit on kinetic energy implied by a nonthermal interpretation of the X-ray spectrum with the estimated kinetic energy inferred from the analysis of the radio flare requires a very low magnetic field strength, of the order of 20 G or less in the radio-emitting source.

The peak power of the electron beam, derived from spectral fitting, is 3 × 10³⁷ erg s⁻¹. The V-band measurement at the peak of BFF gives a constraint on the footpoints of the flaring loop to be X_BFF,V = 0.375, or an area of 9 × 10²⁰ cm². This implies a beam flux of 10¹⁶ erg cm⁻² s⁻¹, which is about four orders of magnitude larger than the largest beam fluxes investigated for solar flares (>5 × 10¹² erg s⁻¹ cm⁻² Krucker et al. 2011). For an F16 beam, the drift speed of the return current would have to be the speed of light and still the beam would not be neutralized, therefore, resulting in strong magnetic fields. Such a large beam flux seems physically implausible because it would require treatment of a return current and violation of fundamental assumptions, and we do not consider it to be a viable interpretation. Additionally, Caballero-García et al. (2015) argue that the time delay between the optical V band burst and the Swift trigger is evidence of the Neupert effect, which would be difficult to envisage if the X-ray emission was entirely nonthermal.

The standard flare decay analyses for the X-ray emission done for F2 will not work for BFF, because the temperature is not changing appreciably over the 200 s timescales over which we have X-ray data. Our insight into BFF is guided by an analysis of the F2 event, for which we see T(t), ${\mathscr{VEM}}(t)$, and from which we can infer coronal loop length, and a ratio of the radius of the loop (from white-light footpoints) to the coronal loop length. For the BFF event, V-band data give us the flare footpoint area, and by assuming that the same value of α applies to BFF as well as F2, we can estimate the loop length for BFF. For α = 0.07, and the value of X_BFF,V = 0.375 for a blackbody temperature of 10⁴ K, we infer a coronal loop semi-length of 3.2 R_⋆ or maximum height of 2.0 R_⋆. Tying these parameters together with the peak ${\mathscr{VEM}}$ for the thermal model from Table 1 using Equations (25) and (26), we derive n_e of 3 × 10¹¹ cm⁻³, and a B_conf of 580 G. These numbers are similar to what we derive for F2, and given that the energy partition between X-rays and V band appears to be similar, the likely case is a thermal plasma.

Table 5 compares key parameters of the two flares considered here on DG CVn, as well as the superflare on the nearby EV Lac described in Osten et al. (2010). The energy comparison is restricted to the energy bands with the most temporal coverage: in EV Lac and BFF there was significant HXR emission in the initial stages of the flare. Note that the estimation of f_V for EV Lac, proceeding using the same steps as described above for the F2 event of DG CVn, yields an integrated V filter flux of 6.15 × 10⁻⁹ erg cm⁻² s⁻¹, for a value of ${f}_{X}/{f}_{V}$ of 4.0. The peak X-ray luminosity relative to bolometric luminosity was calculated over the expanded energy range of 0.01–100 keV to account for the majority of the radiative losses of the hot coronal plasma; the value for EV Lac was taken using parameters in the first line of Table 2 in Osten et al. (2010) and calculated on this larger energy range. Using the scaling relationship between flare temperature and emission measure established for solar flares and a sample of stellar flares, for the DG CVn flare BFF, we would expect a ${\mathscr{VEM}}$ nearly 2400 times larger than observed, and, for the EV Lac peak flare, a factor of 88 larger than observed (see Table 5). This may be seen as problematic for the thermal interpretation; however, there have been previous suggestions of a departure from this behavior at the highest stellar flare temperatures previously observed. Getman et al. (2008) suggested that superhot flares may turn over in this relationship, based on inferring flare temperatures using a median energy analysis of flares from young stars, and suggested that even at temperatures in excess of 100 MK the ${\mathscr{VEM}}$ should be between 10⁵⁴ and 10⁵⁵ cm⁻³.

The high temperatures suggest that the plasma will lose its energy by conductive losses on a relatively fast timescale. The timescale on which the plasma will lose energy by radiative losses

$$\begin{eqnarray}&&{\tau }_{\mathrm{rad}}=\displaystyle \frac{3{k}_{{\rm{B}}}{T}_{e}}{{n}_{e}\psi ({T}_{e})}\end{eqnarray} \tag{ 28 }$$

depends on the electron density n_e, electron temperature T_e, Boltzmann's constant k_B, and the radiative loss function ψ(T_e). The radiative losses for a collisionally ionized plasma are evaluated by summing the contributions from line and continuum radiation at each temperature tabulated in the Astrophysical Plasma Emission Database (Smith et al. 2001), similar to what was done in Osten et al. (2006). The timescale on which the plasma will lose energy by conductive losses is

$$\begin{eqnarray}&&{\tau }_{\mathrm{cond}}=\displaystyle \frac{3{n}_{e}{k}_{{\rm{B}}}{l}^{2}}{\kappa {T}_{e}^{5/2}}\end{eqnarray} \tag{ 29 }$$

where l is the length scale and κ is the Spitzer conductivity coefficient (=8.8 × 10⁻⁷ erg cm⁻¹ s⁻¹ K^−7/2 Spitzer 1962). Using values for electron density and length scale derived from analyses above, we determine the dependence of the two timescales on electron temperature and compare with the duration and peak temperature of BFF. Figure 9 displays the results. It is curious that the location at which the two timescales are approximately equal is close to the peak temperature of BFF, and the value of the timescales are similar to the upper limit given to the event duration of BFF from the sparse data.

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Given the extreme parameters of BFF, we can also examine the ratio of the relaxation time of the plasma to the conductive cooling time. In order for such a hot temperature plasma to be observed, the relaxation time should not be larger than the timescale on which the plasma lose energy by conduction. The ratio of the thermal relaxation time of the plasma to the timescale for conductive cooling is

$$\begin{eqnarray}&&{{\tau }_{\mathrm{relax}}/\tau }_{\mathrm{cond}}=2\displaystyle \frac{{T}_{8}^{4}}{{n}_{11}^{2}{L}_{9}^{2}}\end{eqnarray} \tag{ 30 }$$

(see the discussion in Benz 2002), where T₈ is the temperature in units of 10⁸ K, n₁₁ is the electron density in units of 10¹¹ cm⁻³, and L₉ is the loop length in units of 10⁹ cm. Evaluating Equation (30) for the values appropriate for this flare, we find τ_relax/τ_cond to be 2 × 10⁻⁴. This demonstrates that the plasma does have time to relax to the observed thermal temperature.

The total kinetic energy in BFF cannot be constrained independently for a thermal interpretation of BFF; the results from Section 3.3.4 show it to depend on δ and B. However, an estimate for a lower limit to the total radiated X-ray energy for BFF is 4 × 10³⁵ erg. Studies of the global energetics of large solar flares (Emslie et al. 2012) show that the total radiation from soft X-ray emitting plasma is comparable to or slightly smaller than the energy in flare electrons accelerated to energies greater than 20 keV. If we assume that the energy partition is similar for BFF, then a rough equipartition between the radiated X-ray energy and the energy in accelerated electrons would suggest, via Figure 6, magnetic field strengths in the radio-emitting plasma of several hundred Gauss to about 1 kiloGauss. This is consistent with the field strengths derived above independently from equipartition between the gas pressure and magnetic pressure.

If we take the thermal interpretation as the more physically plausible explanation, given the constraints from the multi-wavelength observations, then we can still ask the question of what signature of nonthermal electrons might be expected to appear in the hard X-ray spectral range. Nonthermal particles propagate in a collisionless plasma. A lower limit to the particle energy required to cross a propagation path with length L across a density n_e is given by setting the propagation time of accelerated electrons equal to the collisional deflection time (Aschwanden 2002)

$$\begin{eqnarray}&&E\geqslant 20\sqrt{{L}_{9}{n}_{11}\left(\displaystyle \frac{0.7}{\cos \alpha }\right)}\ \ \ \mathrm{keV}\end{eqnarray} \tag{ 31 }$$

where L₉ is the loop length in units of 10⁹ cm, n₁₁ is the electron density in units of 10¹¹ cm⁻³, and α is the pitch angle. For the parameters in Table 5 the minimum energy is 580 keV; this analysis suggests that the accelerated particles filling the entire flare loop would produce nonthermal hard X-ray emission at energies above this to be potentially observable.