Can Flare Loops Contribute to the White-light Emission of Stellar Superflares?

After the recent discovery of stellar superflares in Kepler satellite data (Maehara et al. 2012), a series of papers rapidly emerged trying to explain this phenomenon mainly in terms of correlations between various stellar parameters. The total flare energy was estimated from the light curves, for statistical samples of both dMe red-dwarf and G-type solar-like stars, and was related to flare duration, magnetic activity (represented by large starspots), stellar rotation velocity, stellar age, etc. (Notsu et al. 2015; Shibata 2016; Namekata et al. 2017; Shulyak et al. 2017). However, only limited attention was devoted to understanding the mechanisms of the superflare emission. First of all, Kepler light curves represent the white-light (WL) emission of a flare, integrated in the broad optical passband from 400 to 900 nm. Therefore, all Kepler superflares are, according to standard solar terminology, the so-called White Light Flares (WLF). But superflares on cool stars seem to have much larger total power than solar flares, i.e., up to 10³⁸ erg compared with the maximum of 10³² erg in the solar cases (Shibata 2016). This follows from the time integrated fluxes measured by Kepler.

A plausible scenario of the energy release in superflares is the same as that for solar flares, i.e., the magnetic reconnection in the corona and the energy transport down to low atmospheric layers by particle beams (mainly electrons) and by thermal conduction. This then suggests that, on stars, we observe the surface structures analogical to solar-flare ribbons where all solar WLFs are normally detected. On the Sun, the WL ribbons usually disappear after the flare impulsive phase, while the decaying ribbons continue to be visible in various spectral lines during the gradual phase. This later phase is also characterized by the typical appearance of the so-called "post-flare loops," visible in different spectral lines. However, to our knowledge, these loops have never been detected in the WL against the solar disk and only recently, quite importantly, they have been rarely seen high above the limb during the off-limb observations of strong flares by SDO/HMI (Saint-Hilaire et al. 2014; Krucker et al. 2015). Therefore, on the Sun, these loops do not affect the WLF emission. If the same applies also for stellar superflares, we can think that their strong WL emission emerges entirely from ribbons and this is how the energy released during the reconnection is normally related to WL emission. However, with an increasing spatial resolution of solar-flare observations, we find that the solar ribbons are actually very narrow (although often long) features, occupying only a tiny fraction of the active-region area—see the latest GST movie in Jing et al. (2016), while the cool Hα loops cover a much larger space.

Extrapolating this to superflares where the magnetic loops are expected to be much larger and assuming that the WL visibility of flare loops during superflares can be quite different compared to the solar case, we investigate in this study whether such loops can contribute to WL emission of superflares. Based on our theoretical analysis, we propose that the flare loops can indeed significantly contribute to total superflare WL emission, if not dominate it totally, because they can be visible against the stellar disk (namely in case of very cool dMe stars) and they can occupy much larger areas than just the ribbons. This may then change our current picture of the WL emission of stellar superflares, but also the generally accepted paradigm of stellar-flare optical emission as being entirely due to flare ribbons. The radiation-hydrodynamical (RHD) models of stellar flares are one-dimensional models of a flare loop where all emissions in cooler lines and continua arise from the ribbons (Kowalski 2016). These models have not yet predicted the optical emission from overlying loops that are typically observed after an impulsive onset of solar flares. In particular, the line emission must be even stronger in the case of superflares and, moreover, we predict here the importance of WL emission of such loops.

The paper is organized as follows. In Section 2, we describe the loops typically observed during solar flares and discuss their visibility. Section 3 summarizes all relevant mechanisms of the WL emission produced by flare loops. In Section 4, we present our numerical results of the WL visibility of loops during stellar superflares. Since the critical parameter of such a visibility is the electron density in the loops, we make density estimates for the case of superflares in Section 5 using various physical arguments. In Section 6, we comment on spectral characteristics of stellar WL flares and Section 7 contains a discussion and conclusions.

In the majority of solar flares, including WLFs, we see dark Hα loops during the gradual phase—see, e.g., an excellent movie of the X1 flare taken at very high resolution with GST (Jing et al. 2016). These cool loops have been, somewhat misleadingly, called "post-flare" loops (Švestka 2007); in this paper, we will call them simply flare loops. To see cool flare loops against the solar disk in absorption in the Hα line, the electron density in the loop should not exceed 10¹² cm⁻³ (Heinzel & Karlicky 1987). The same loops, however, can be seen in emission in other lines like Mg ii h and k (Lacatus et al. 2017; Mikuła et al. 2017). At such densities or lower, the loops will not be visible in WL against the disk, but can be detected as faint WL loops high above the limb (Saint-Hilaire et al. 2014; Krucker et al. 2015). Those loops, however, should not be confused with the off-limb WL flare emission at chromospheric heights (Battaglia & Kontar 2011; Krucker et al. 2015; Heinzel et al. 2017). High-lying WL flare loops emit either in the Paschen recombination continuum with free–free contribution (depending on the temperature), or due to Thomson scattering of the photospheric radiation on loop electrons. For electron densities lower than 10¹² cm⁻³, the latter mechanism should dominate (Heinzel et al. 2017). Cool loops form from initially hot ones (10⁷ K), which have been formed by the evaporation during strong chromospheric and transition-region heating. Depending on the densities, the loops at different temperatures appear within typical cooling times, which depend on the conductive and radiative cooling processes. In the solar case, for electron densities of cool loops typically lower than 10¹² cm⁻³, the cooling time roughly takes minutes and this may explain why we see the cool loops later after the impulsive heating with electron beams is already over (HXR is also no longer detectable). However, in case of superflares, where we expect larger densities, cooler loops may appear practically immediately after the flare onset.

In the wavelength range of Kepler, the WL continuum emitted by dense superflare loops will be mainly due to the hydrogen recombination (namely the Paschen continuum) and due to the hydrogen free–free process. Below we detail these mechanisms (for general formulae, see also Hubeny & Mihalas 2015).

3.1. Hydrogen Recombination Continua

At low temperatures, the loops are partially ionized and assuming their temperature T and electron density n_e as free parameters, we can compute the WL radiation. The absorption coefficient for bound-free hydrogen transitions from atomic level i is

$\begin{eqnarray}&&{\kappa }_{\nu }^{\mathrm{bf}}={\alpha }_{\nu }({n}_{i}-{n}_{i}^{* }{e}^{-h\nu /{kT}}),\end{eqnarray} \tag{ 1 }$

where α_ν is the hydrogen photoionization cross-section

$\begin{eqnarray}&&{\alpha }_{\nu }=2.815\times {10}^{29}{g}_{\mathrm{bf}}(i,\nu )/{i}^{5}/{\nu }^{3}\end{eqnarray} \tag{ 2 }$

with g_bf being the Gaunt factor for bound-free opacity. n_i is the non-LTE population of the hydrogen level i from which the photoionization takes place and ${n}_{i}^{* }$ is its LTE counterpart. The second term represents the stimulated emission, which is normally treated as a negative absorption in the radiative-transfer equation. The departure coefficient from LTE is defined as ${b}_{i}={n}_{i}/{n}_{i}^{* }$. h and k are the Planck and Boltzmann constants, respectively. Since we are not solving here the full non-LTE radiative-transfer problem, we have to make some assumption about b_i. At relatively high densities that we will consider for the flare loops, detailed non-LTE modeling shows that the b-factors for third and higher hydrogen levels are close to unity. We thus assume here that b₃ (Paschen continuum) and b₄ (Brackett continuum) are equal to one; we neglect the opacity of higher continua in the WL range observed by Kepler. However, we will show later that even departures from unity of these factors do not affect our results substantially. With b_i = 1, we thus get

$\begin{eqnarray}&&{\kappa }_{\nu }^{\mathrm{bf}}={\alpha }_{\nu }{n}_{i}^{* }(1-{e}^{-h\nu /{kT}}),\end{eqnarray} \tag{ 3 }$

with

$\begin{eqnarray}&&{n}_{i}^{* }={n}_{p}{n}_{e}{{\rm{\Phi }}}_{i}(T),\end{eqnarray} \tag{ 4 }$

where ${{\rm{\Phi }}}_{i}(T)=2.0707\times {10}^{-16}2{i}^{2}{e}^{h{\nu }_{i}/{kT}}/{T}^{3/2}$ is the Boltzmann factor. In what follows, we assume a pure hydrogen plasma for which ${n}_{p}{n}_{e}={n}_{e}^{2}$. Finally, it can be shown that the source function of the Paschen and Brackett continuum is approximately equal to

$\begin{eqnarray}&&{S}_{\nu }\simeq \displaystyle \frac{1}{{b}_{i}}{B}_{\nu }(T)\simeq {B}_{\nu }(T).\end{eqnarray} \tag{ 5 }$

3.2. Hydrogen Free–Free Continuum

Hydrogen free–free opacity is expressed as

$\begin{eqnarray}&&{\kappa }_{\nu }^{\mathrm{ff}}=3.69\times {10}^{8}{n}_{p}{n}_{e}{g}_{\mathrm{ff}}(\nu ,T){T}^{-1/2}{\nu }^{-3}(1-{e}^{-h\nu /{kT}}),\end{eqnarray} \tag{ 6 }$

with the Gaunt factor g_ff. The source function is equal to the Planck function for the free–free process.

3.3. Radiative Transfer

Assuming a loop of diameter D, having a uniform continuum source function S_ν and observed against the stellar disk, we can express the emergent loop intensity as the formal solution of the radiative-transfer equation

$\begin{eqnarray}&&{I}_{\nu }={I}_{\mathrm{bg}}{e}^{-{\tau }_{\nu }}+{S}_{\nu }(1-{e}^{-{\tau }_{\nu }}),\end{eqnarray} \tag{ 7 }$

where the first term represents the background radiation from the stellar disk below the loop I_bg, attenuated by the continuum loop opacity with τ_ν being the optical thickness of the loop at a given continuum frequency. The second term is the radiation intensity of the loop itself. Note that for optically thin loops, the resulting intensity is a mixture of partially penetrating background intensity and the radiation intensity of the loop itself. Using the approximation b_i = 1, we set S_ν = B_ν(T). The total optical thickness is

$\begin{eqnarray}&&{\tau }_{\nu }=({\kappa }_{\nu }^{\mathrm{bf}}+{\kappa }_{\nu }^{\mathrm{ff}})D.\end{eqnarray} \tag{ 8 }$

3.4. Thomson Scattering on Loop Electrons

The Thomson scattering process is different from the free–bound and free–free emission and we include it here as an optically thin emission, simply added to the intensity as computed by Equation (7). More precise radiative-transfer calculations are not needed because (i) fb and ff opacity will be shown to be small as well, and (ii) the Thomson scattering is practically negligible for most conditions considered in this study. Contribution to the loop intensity is expressed as

$\begin{eqnarray}&&{I}_{\nu }^{\mathrm{Th}}={n}_{e}{\sigma }_{{\rm{T}}}{J}_{\nu }^{\mathrm{inc}}D,\end{eqnarray} \tag{ 9 }$

where ${\sigma }_{{\rm{T}}}=6.65\times {10}^{-25}$ cm² is the cross-section for Thomson scattering and the incident intensity ${J}_{\nu }^{\mathrm{inc}}$ is computed as the product of B_ν(T_eff) and geometrical dilution factor. T_eff is the effective temperature of the star under consideration. For the geometrical dilution factor, we take a value of 0.4, which corresponds to loop heights around 10,000 km. Note that both fb and ff emissivities are proportional to ${n}_{e}^{2}$, while the Thomson scattering scales only linearly with n_e. For M dwarfs, the incident radiation is much weaker than that for solar-type stars (low T_eff), and it will be even lower above large areas occupied by dark starspots.

4.1. Flare Areas

The flux of the entire preflare star is

$\begin{eqnarray}&&{F}_{s}=({A}_{s}-{A}_{f}){I}_{s}+{A}_{f}{I}_{\mathrm{bg}},\end{eqnarray} \tag{ 10 }$

where ${A}_{s}=\pi {R}^{2}$ is the stellar surface (R being the stellar radius), A_f is the flare area, and I_s represents the stellar surface intensity computed from the effective temperature T_eff of the star. During a flare, the stellar flux will be

$\begin{eqnarray}&&F={A}_{f}I+({A}_{s}-{A}_{f}){I}_{s}.\end{eqnarray} \tag{ 11 }$

The Kepler flux amplitude relative to the preflare flux is then defined as

$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}F}{{F}_{s}}=\displaystyle \frac{F-{F}_{s}}{{F}_{s}}=\displaystyle \frac{{A}_{f}}{{A}_{s}}\displaystyle \frac{I-{I}_{\mathrm{bg}}}{{I}_{s}-\tfrac{{A}_{f}}{{A}_{s}}({I}_{s}-{I}_{\mathrm{bg}})}.\end{eqnarray} \tag{ 12 }$

We consider two limiting cases for I_bg, I_bg = I_s (i.e., no starspots), and I_bg = I_spot meaning that the preflare area is covered by starspots having the radiation temperature T_spot. The former case is usually used in the literature as an approximation (Shibayama et al. 2013); in this study, we assume that I_bg = I_spot. For simplicity of exposition, we have omitted here the frequency indexes. In the following, we will show the results for a peak wavelength 600 nm of the Kepler WL passband. For a more precise comparison with Kepler observations, one has to integrate fluxes over the whole wavelength range of Kepler, weighted by its transmission profile. This is shown, e.g., in Shibayama et al. (2013), together with the procedure of computing the total flare energy, integrated over flare lifetime.

4.2. Theoretical Amplitudes

We have computed the flare intensity I for a variety of temperatures and electron densities expected in superflare loops, assuming different stellar T_eff, different I_bg, and varying D. Then the flux amplitudes ΔF/F_s were evaluated using parametric values of the filling factor A_f/A_s. From various models, we have selected four representative examples to show in this exploratory paper, and a more extensive analysis will be done in a future paper. Figures 1–4 show variations of theoretical amplitudes with the electron density, for five temperatures ranging from 10,000 K to 1 MK. In the same figures, we also plot the continuum optical thickness of the loops at 600 nm (dashed lines). In general, the amplitudes show a linear increase with increasing electron density in our log–log plots, which applies for the optically thin regime. This means that the flare intensity is a sum of partially attenuated background radiation and of the emission by the loop itself. However, when cool loops become optically thick at high electron densities (typically greater than 10¹⁴ cm⁻³), the intensity saturates to the blackbody value corresponding to the kinetic temperature of the loop and is no longer dependent on n_e. This is well visible in Figures 1 and 2 for T = 10,000 K. Both optically thin and saturated regimes also imply that the recombination component of the emergent radiation is independent of the value of b-factors so that our assumption that b₃ = 1 and b₄ = 1 is justified in these two regimes. Figures 1 and 2 show the situation for a solar-type star with T_eff = 6000 K, with a spot covering the flare region (T_spot = 4000 K). Figure 1 shows the case with D = 1000 km, a single loop having a typical diameter as known from solar observations, and the filling factor 0.1, which is a rather conservative estimate for stellar active regions. Figure 2 is for D = 5000 km and a larger flare area having the filling factor 0.2. In these two cases, we would need electron densities between 10¹³ and 10¹⁴ cm⁻³ to get the amplitude consistent with characteristic Kepler observations represented by the black horizontal line in all figures. However, the situation with M-type stars is different due to low luminosity of the stellar disk at T_eff = 3500 K and T_spot = 3000 K (we estimate the spot temperatures according to Berdyugina 2005). We immediately see that for cool M-type stars the flare loops can significantly contribute to WL amplitudes already at electron densities somewhat higher than 10¹² cm⁻³. As noted in Section 2, this density is a limiting one at which the Hα loops are still visible in absorption against the solar disk. Densities higher than 10¹² cm⁻³ require stronger heating and evaporation process, which we discuss in the next section.

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Another very important finding is that the range of electron densities needed to obtain the observed amplitude is rather narrow, for loop temperatures spanning two orders of magnitude. This means that both cool loops, as well as hot loops, may contribute significantly to the overall WL emission. The contribution of hot loops is somewhat smaller so that we need a slightly higher electron density to reach the observed amplitude. During solar flares, we actually see cool and hot loops at the same time (e.g., Mikuła et al. 2017), while for stars a mixture of such loops will contribute to the total stellar flux.

4.3. Relative Importance of WL Emissivities

Since we solve the transfer equation for three overlapping continua (i.e., Paschen, Brackett, and free–free), it is not directly evident which mechanism dominates the emergent intensity and thus the amplitudes for given values of T and n_e. However, we found that the most typical situation is that flare loops are optically thin for moderate electron densities considered in our grid of models. In such a case, the ratio of optically thin emissivities of the free–free and Paschen continuum takes the simple form (Heinzel et al. 2017)

$\begin{eqnarray}&&\displaystyle \frac{{I}^{\mathrm{ff}}}{{I}^{\mathrm{Pa}}}=8.55\times {10}^{-5}{{Te}}^{-h{\nu }_{i}/{kT}},\end{eqnarray} \tag{ 13 }$

assuming that both Gaunt factors are unity (note that in Heinzel et al. 2017 the exponent is positive by a misprint but the computed ratios are correct). This ratio is independent of the electron density and is only a function of temperature. We see that in the optically thin regime the free–free emission starts to dominate the Paschen recombination at temperatures higher than about 3 × 10⁴ K. At T = 10⁴ K the ratio is 0.15 but at 1 MK the free–free emission dominates the free–bound ones completely. Therefore, at high temperatures, the WL loop emission is dominated by the free–free process, especially at electron densities higher than 10¹² cm⁻³.

From the previous section, we can see under which conditions the flare loops can substantially contribute to the WL emission of superflares. In this section, we try to estimate the electron densities as a key parameter, by extrapolating our experience with solar flares to superflares.

According to the standard magnetic-reconnection model of solar flares (e.g., Shibata & Magara 2011), the energy flux F (erg cm⁻² s⁻¹), injected into the footpoint of the reconnected loop, can be written as

$\begin{eqnarray}&&F=\displaystyle \frac{{B}^{2}}{4\pi }{V}_{A}={10}^{12}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}{{\rm{s}}}^{-1}{\left(\displaystyle \frac{B}{100{\rm{G}}}\right)}^{3}{\left(\displaystyle \frac{{n}_{e}}{{10}^{9}{\mathrm{cm}}^{-3}}\right)}^{-1/2},\end{eqnarray} \tag{ 14 }$

where B(G) is the magnetic field strength in the inflow region of the flare current sheet, n_e (cm⁻³) is the electron density in the inflow region, and V_A is the Alfvén speed in the inflow region (Yokoyama & Shibata 1998, 2001). Let us assume that the fraction a of this energy flux is used to accelerate high energy electrons. Then, the energy flux F_e of nonthermal electron beams can be written as F_e = aF. When this electron beam collides with the dense chromospheric plasma with density n_ch, the beam energy is thermalized to heat the chromospheric plasma from 10⁴ K to flare temperatures T.

The radiation cooling time t_rad (s) for the chromospheric plasma with density n_ch (cm⁻³) can be expressed as

$\begin{eqnarray}&&{t}_{\mathrm{rad}}=\displaystyle \frac{3{n}_{\mathrm{ch}}{kT}}{{n}_{\mathrm{ch}}^{2}{\rm{\Lambda }}(T)}=100\,{\rm{s}}\,{\left(\displaystyle \frac{{n}_{\mathrm{ch}}}{{10}^{12}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{5/3},\end{eqnarray} \tag{ 15 }$

where we used Λ(T) = 10^−17.73T^−2/3 for 6.3 ≤ log T ≤ 7.0 In this case, radiation cooling time is too long to balance with the heating by electron beams. Hence the heated chromospheric plasma suddenly expands upward leading to chromospheric evaporation and downward leading to shock and chromospheric condensation.

Roughly speaking, the flare temperature is determined by the balance between reconnection heating and conduction cooling (Yokoyama & Shibata 1998, 2001) even if the main energy transport from the reconnection site to the chromosphere is due to nonthermal electron beams. This is because the nonthermal electrons quickly thermalize in the upper chromosphere, and then the thermal conduction finally transports heat throughout the flare loop. Then the flare temperature T (K) can be written (Yokoyama & Shibata 1998, 2001; Shibata & Yokoyama 1999, 2002)

$\begin{eqnarray}&&\displaystyle \frac{{\kappa }_{0}{T}^{7/2}}{L}={F}_{e}=a\displaystyle \frac{{B}^{2}}{4\pi }{V}_{A}.\end{eqnarray} \tag{ 16 }$

Namely, we get

$\begin{eqnarray}&&T\simeq 3\times {10}^{7}\,{\rm{K}}\,{\left(\displaystyle \frac{{F}_{e}}{{10}^{11}\mathrm{erg}{\mathrm{cm}}^{-2}{{\rm{s}}}^{-1}}\right)}^{2/7}{\left(\displaystyle \frac{L}{{10}^{9}\mathrm{cm}}\right)}^{2/7}.\end{eqnarray} \tag{ 17 }$

Note that a reference electron-beam flux of 10¹¹ erg cm⁻² s⁻¹ is obtained for B = 100 G, n_e = 10⁹ cm⁻³, and a = 0.1. In reality, this temperature is the maximum temperature in the flare region, and corresponds to the superhot component of flares (Shibata & Yokoyama 2002). The observed flare temperature is measured when the flare emission measure becomes maximum as a result of evaporation flow. At this time, the flare temperature is a bit smaller and comparable to 10⁷ K.

We can estimate the density of the evaporation flow n_ev (cm⁻³) from the balance between enthalpy flux carried by the evaporation flow and the electron-beam energy flux F_e

$\begin{eqnarray}&&5{n}_{\mathrm{ev}}{{kTC}}_{s}={F}_{e},\end{eqnarray} \tag{ 18 }$

where C_s (cm s⁻¹) is the sound speed of the heated flare plasma with temperature T estimated to be ${C}_{s}=5\times {10}^{7}\,\mathrm{cm}$ s⁻¹ (T/10⁷ K)^1/2. Then the density of the evaporation flow becomes

$\begin{eqnarray}&&{n}_{\mathrm{ev}}=3\times {10}^{11}\,{\mathrm{cm}}^{-3}\,{\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{-3/2}\displaystyle \frac{{F}_{e}}{{10}^{11}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}}.\end{eqnarray} \tag{ 19 }$

This is a lower limit of the flare-loop plasma density, because the evaporating mass accumulates in the loop so that the flare-loop density can increase in time as long as the evaporation flow continues. On the other hand, the upper limit of the flare-loop density is determined by the balance between the gas pressure of the flare-loop plasma and the magnetic pressure that confines the plasma, 2nkT = B²/8π. Hence, we obtain

$\begin{eqnarray}&&n={10}^{11}\,{\mathrm{cm}}^{-3}\,{\left(\displaystyle \frac{B}{100{\rm{G}}}\right)}^{2}{\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{-1}.\end{eqnarray} \tag{ 20 }$

The typical flare-loop density in solar flares derived from soft X-rays (for T = 10⁷ K) is 10¹¹ cm⁻³, and so the above order-of-magnitude theory seems consistent with observations assuming B ≃ 10² G.

Is it possible to have flare-loop densities larger than 10¹² cm⁻³, reaching 10¹³ cm⁻³ or even more ? If the coronal magnetic field in the reconnection region is 300 G, the electron-beam energy flux becomes 3 × 10¹² erg cm⁻² s⁻¹. In this case, n_ev reaches 10¹³ cm⁻³. However, such dense flare plasma cannot be confined by a 300 G magnetic loop. Instead, we need a 1 kG magnetic loop. Or, if the flare temperature becomes less than 10⁶ K, then such high density flare plasma can be confined.

In the case of M dwarfs, it is well known that huge starspots with a few kG are present in some active stars (e.g., Johns-Krull & Valenti 1996; Berdyugina 2005). Furthermore, the pressure scale height is shorter in the M dwarf photosphere and chromosphere so that the transition-region height is lower in the M dwarf atmosphere than in the solar atmosphere. As a result, the coronal field strength of M dwarfs can be stronger than that on the Sun even if the photospheric magnetic field distribution and strength of M dwarfs are the same as those of the Sun. Altogether, a 1 kG magnetic loop is quite likely on M dwarfs, so that the flare loops could be seen as a WLF on M dwarfs.

On the other hand, the momentum balance in a flare loop requires that the momentum of the downward moving chromospheric condensation (con) is balanced by the momentum of the evaporation flow (ev) in a hot loop above the condensation—this was first proven observationally by Canfield et al. (1987). Assuming that the hydrogen plasma density is roughly proportional to the electron density, we can write for the momentum balance

$\begin{eqnarray}&&{n}_{e}^{\mathrm{con}}{V}^{\mathrm{con}}={n}_{e}^{\mathrm{ev}}{V}^{\mathrm{ev}},\end{eqnarray} \tag{ 21 }$

where V represents the respective flow velocities. From various solar observations, we can roughly estimate the ratio V^ev/V^con to be around 10, which gives the ratio of electron densities of the same order of magnitude. Typical chromospheric densities derived from flare ribbon observations and modeling range between 10¹³ and 10¹⁵, where the latter value was obtained for an atmosphere strongly heated by electron beams having fluxes of the order of 10¹³ erg cm⁻² s⁻¹ (Kowalski et al. 2015). This then gives an estimate for electron densities in the hot loops in a range of 10¹² to 10¹⁴ cm⁻³. This is consistent with the above estimates based on the reconnection scenario.

Finally, using this range of electron densities, we can estimate the radiative cooling times for hot 10⁷ K loops, to be cooled down to temperatures at which we can see cool loops. Using Equation (15) for loop densities 10¹² and 10¹⁴ cm⁻³, we get the cooling times of the order of 100 to 1 s, respectively. This then means that cool loops can be detected quite early, before the flare maximum, and they will produce a significant portion of the WL emission during the gradual phase. In the solar case, the loop densities are lower and the cool loops appear in spectral lines later, namely at the onset of the gradual phase (see, e.g., Schmieder et al. 1995).

The mechanisms of the WL continuum formation in stellar flares, like free–bound (Balmer, Paschen), as well as free–free, have been considered by many other authors (see, e.g., Hawley & Fisher 1992 for older work and Kowalski 2016 for a recent review). However, those authors interpreted the WL continuum spectra, in analogy with solar flares, as arising from the flare ribbons, i.e., footpoints of the extended loops. In the optical range, the Paschen continuum formed at chromospheric levels, is usually optically thin unless the electron-beam fluxes are very strong (see F13 model of Kowalski et al. 2015). In the latter case, the Paschen continuum will saturate to blackbody spectrum with characteristic chromospheric temperatures. However, blackbody emission may also come from deeper photospheric layers presumably heated by backwarming, but that is detectable only if the overlying forming region of the Paschen continuum is optically thin. Altogether, such spectral distributions seem to be consistent with typically observed strong blackbody-like continua with best fits in the 8000–12,000 K range (e.g., Hawley & Fisher 1992; Kowalski et al. 2012). There is also the issue of the Balmer continuum, which we do not detail in this study.

A novel aspect of the present paper is to show under which conditions the flaring loops, overlying large areas, and spanning temperatures from cool to hot, can contribute to the total continuum radiation of stellar flares. As we already mentioned in the Introduction, the WL emission from solar loops was never detected against the disk and this is because the typical electron densities in solar-flare loops do not significantly exceed 10¹² cm⁻³. The same may apply for standard flares on cool stars, but the situation may differ in the case of superflares where we expect much larger densities (Section 5). On the Sun, we see both cool and hot loops simultaneously (namely during the gradual phase) and thus we have also considered a possible contribution of hot loops which mainly emit due to the free–free mechanism. However, we do not say that this must be dominant. If the cool loops occupy most of the arcade volume, the WL emission from them will be dominated by the free–bound Paschen continuum.

An important question is how can our models be consistent with WL spectral observations of stellar flares, which have been known for a long time to show the blackbody behavior at temperatures around 10,000 K. Here we focus just on the flare loops overlying the whole active region and we compute their WL emission, i.e., not the emission of narrow ribbons. In reality, however, the total stellar flux will be a mixture of both ribbon and loop components and thus a more sophisticated analysis of the observed spectra will be required to understand this behavior. We do not show the computed spectra in the present paper because they correspond only to the loop component. But we compare our models with typical amplitudes of Kepler superflares in order to set the limits on the loop detectability in WL. In fact, there exist no spectral observations yet of the Kepler stars during their superflares so that a comparison with models is impossible and any discussion would be rather premature. In this paper, we only say that in analogy with the Sun there must be present arcades of cool as well as hotter loops, presumably much more extended on superflare stars than on the Sun (much larger starspots) and this is generally accepted by the community. Then if such loops may reach electron densities around 10¹³ or higher (we estimate this based on the reconnection models in Section 5), such loops will substantially contribute to WL continuum emission of Kepler superflares. This contribution has to be added to emission arising from the flare ribbons, taking into account proper (but still largely unknown) filling factors for both types of structures and only then the model spectra can be compared with future spectral data on superflares.

Although no superflare from the Kepler sample was yet observed spectroscopically, there is one detection of a very strong flare (called megaflare) on dMe star YZ CMi (Kowalski et al. 2010, 2012). The flare spectra show a typical blackbody continuum with temperatures around 10,000 K. However, the flare is much more complex and, in fact, it also shows a strong Balmer (and perhaps Paschen in the visible) component, which, as the authors claim, is spatially much more extended. Since the spectra were taken during the gradual phase of this megaflare, one would expect large areas covered by loops, both cool and hot. This might be consistent with our model, which predicts the Paschen (and Balmer) continuum for cool loops rather than blackbody continuum, unless the Paschen continuum saturates to blackbody at very high loop densities (see our figures for T = 10,000 K).

As follows from our modeling, the WL amplitudes due to flare loops will critically depend on electron densities and relative flare areas (filling factors). In the preceding section, we have shown that required electron densities higher than 10¹²–10¹³ cm⁻³ are quite expectable as the result of strong evaporative processes during superflares; though, on the Sun, such densities are not common. However, the relatively large flare areas we used in Figures 1–4 deserve some discussion. From high-resolution observations of solar flares (e.g., Jing et al. 2016), we clearly see the ribbons as long but narrow features. Such ribbons appear even less extended in WL. On the other hand, the total area typically covered by the system of flare loops is much larger. Contrary to solar flares, in all current analyses of stellar superflares, it is assumed that the chromospheric flare condensation is optically thick in the Paschen continuum with a representative temperature 10⁴ K that leads to the blackbody WL spectrum (Shibayama et al. 2013; Katsova & Livshits 2015; Kowalski et al. 2015). Based on that, one gets relatively small areas that are typically between 0.01% and 1% of the whole stellar surface (H. Maehara 2018, private communication). For solar-type stars with an amplitude 0.01, the area of such ribbons is only 0.2%. However, the same amplitude can be reached by assuming that flare loops cover a much larger area, like 10%–20% of the stellar surface. This is also consistent with areas covered by expected large starspots (Berdyugina 2005; Aulanier et al. 2013; Maehara et al. 2017). In the case of the Sun, we see that WL ribbons are much smaller compared to spot areas (or active-region areas in general), M. Švanda et al. (2018, private communication) found a factor around 10 using a set of WL flare and sunspot observations by SDO/HMI. However, the areas covered by flare-loop arcades are much larger and comparable to the size of active regions, this is well documented on many images from SDO/AIA. For dMe stars modeled in Figure 3 and 4, the required electron densities are smaller, closer to solar ones, or the flare areas needed to reach the observed amplitudes can be smaller than those for solar-like stars—this is due to a higher contrast for stars having low T_eff. We thus conclude that the WL emission of superflares can be due to a mixture of WL ribbons and flare loops that are also strongly emitting in WL. The actual ratio will then depend on the assumed temperatures of ribbons and loops, on the electron density in the loops and on the areas covered by ribbons and loops. During the evolution of the whole flare structure where the loops grow up in time and can occupy large areas, the WL emission from stellar-flare loops can be dominant. On the other hand, we know from solar observations that the flare ribbons become less bright during the gradual phase (and mostly invisible in WL) and the whole active region is typically covered by a large system of loops.

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In summary, we propose in this paper that extended cool, as well as hotter, loops overlying the whole active region can significantly contribute to the total flux during flares. This is based on a close solar analogy with the so-called "post-flare loops." On the Sun, the two components—ribbons and loops—are spatially quite visible in spectral lines and we can study them separately. On the stars, however, they mix together and thus both may contribute to the total stellar flux in the optical range. Such a scenario is quite novel and certainly deserves further verification and development.

We acknowledge useful discussions with H. Maehara, Y. Notsu, K. Namekata, and M. Bárta and appreciate useful comments of the anonymous referee. This work was partially supported by grant No. 16-18495S of the Czech Funding Agency and by ASI ASCR project RVO:67985815. P.H. also acknowledges the support of the support by the International Research Unit of Advanced Future Studies of Kyoto University during his stay in Japan.