Analysis and modeling of the dynamics of the glow of calcium and hydrogen lines in solar and stellar flares


 We present intensity curves of solar flares obtained in the Hα hydrogen line and CaII H, CaIR 8542Å lines using multichannel spectrographs of Ondřejov Observatory (Czech Republic) for the period 2000–2012. The general behavior of observed intensity curves is practically the same for all flares and is consistent with temporal variations of X-ray emission. However, our results differ significantly from those obtained by other authors for selected flare stars, for example, AD Leo; EV Lac; YZ CMi. We tried to explain the difference in the behavior of Ca II and Hα radiation flux by appearance of a shock wave during a flare and slow heating of the plasma.


Introduction
The aim of this work was to study the behavior of the emission intensity curves in the Hα, H CaII and CaIR 8542 Å lines and to compare the results with the spectra of flare stars. The formation and temporal evolution of the Ca II line, which has a longer time for stellar flares, remains a mystery to this day. Perhaps this is due to the formation in the lower chromosphere and time-dependent, non-thermal model of electron heating at the moment of the impulsive phase, see Allred et al. (2006) or is the coronal X-ray reheat model described by Hawley et al. (1992). To determine the possible mechanisms of radiation intensity in the lines, the radiation flux was calculated under the assumption of a shock wave and, in the second case, a slow heating of the gas, which is stronger in this region than in neighboring regions.

Observations
All observational data on solar flares were obtained on two spectrographs of the Astronomical Institute of the Czech Academy of Sciences (Ondřejov). These are Multichannel-Flare-Spectrograph (MFS, 230 mm/13.5 m) and Horizontal-Sonnen-Forschungs-Anlage (HSFA-2, 500 mm/35 m), described in (Kotrč et al., 1992) and (Kotrč, 2009). Since the observations were carried out at two different instruments using analog and digital CCD matrices, the processing and calibration of the data differed. For the data obtained on HSFA-2 and MFS, we used the IDL software environment. We selected 25 class C, M flares with different positions on the solar disk for the period 2002-2012. The paper provides data for flares 2012-11-24 C3.3, 2002-08-20 M3.4, 2012-07-05 M1.8. a) The 2012-11-24, C3.3, NOAA 11618 flare was observed on the MFS spectrograph with digital cameras. Its position on the disk and obtained spectra are shown in Figure 1. After data calibration, the intensity plots were plotted in absolute units as a function of time. The graph in Figure 2 also presents an X-ray flux (GOES 15). b) The flare 2002-08-20 M3.4 in the active region NOAA 10069 was observed using an analog camera. We observed at the MFS simultaneously in three spectral lines. The position of the vertical line on the SJ fil-    Then we looked at the development of some stellar flares. (Kowalski et al., 2013) provide observational data from the Apache Point Observatory in southern New Mexico. As an example, note the evolution of the H CaII and Hα lines for the YZ CMi star flare ( Figure 7) observed in 2011 Feb 24. We also considered flares AD Leo observed in 2011 Feb 08 and EV Lac observed in 2010 Oct 11, see (Kowalski et al., 2013). For these three stars, the lag of the maximum for   the K CaII line relative to the Hα line is within the range of 12-24 min.

Calculation of the Plasma Heating Models
The theoretical analysis of radiation in the observed lines is carried out within the framework of a model of a gas layer localized in the chromosphere and heated to a higher temperature than the environment at the same height. The proposed work considers two complementary heating mechanisms: a quasi-static increase in the energy supplied and a non-stationary radiative cooling behind the shock wave. In the first variant, the flux of energy carried by MHD waves from some confined region located under the photosphere is increased. The additional energy is transferred to the chromosphere where dissipation of waves leads to stronger gas heating. In this paper we assume the layer to be homogeneous with single temperature T, density N and ionization state determined by balance equations. Due to the assumption of the layer homogeneity, the shock wave is a stationary one, that is, it moves on the undisturbed gas at a constant speed, designated V.
In the both variants, the stationary emission of the homogeneous layer and shock wave radiative cooling, the following physical processes are taken into account: freefree, free-bound and bound-bound collisional and radiative transitions in the diluted black body radiation which temperature is T * and dilution factor is 1/2. The layer is supposed to be transparent in the continuous spectrum but self-absorption in line frequencies is taken into account in the frame of the Biberman-Holstein escape probability method. Escape probability of photons in hydrogen lines was computed by integrating the Doppler and Holzmark convolution profile, and in metal lines is Voigt profile. In the case of hydrogen atom, all discrete levels allowed by the Inglis-Teller criterion are included (from 11 to 13, depending on the electron density), for the CaII ion -levels 4s 2 S, 5s 2 S, 3d 2 D, 4d 2 D, 5d 2 D, 4p 2 P o , 64p 2 P o , 4f 2 F o , 5f 2 F o .
In the stationary heating model for each chemical element at specified parameter values, we calculate the ionization state and the excitation degree of the discrete levels. Parameters of the stationary problem are the size of the layer L, the gas temperature, its density, and the black body temperature. The following parameters are taken into account: L = 1 km; N = 10 15 cm −3 ; T * = 5500 K; (1) 7000 K < T < 8700 K.
The results of the calculations are listed in the Table 1. The first column shows the gas temperature, and the first row represents spectral line designations. The table body contains radiation fluxes expressed in units erg/(cm 2 s). The third row of the table (T = 8000 K) gives the best approximation to the flux values in the Hα and the CaII lines for stellar flares. Analysis of the table data allows the following highlights to be noted in the line emission of the layer.

Steep Balmer decrement. A large value of ratio F(Hα)/F(Hβ) is typical for hydrogen impact excitation at intermediate temperatures. 2. Comparable fluxes in hydrogen and calcium lines.
Compensation for the large difference in the abundances of these chemical elements is due by the high excitation potential of the Balmer series and self absorption in the hydrogen lines.
3. Dependence r = F(3945 Å)/F(8542 Å) of the CaII ion lines on the gas temperature. The infrared line is stronger at low temperatures, the resonant one is stronger at high temperatures. Such dependence r(T) is determined mainly by change in the probability of resonant radiation due to ionization of calcium. 4. Ratio of fluxes in the Hydrogen Hα and CaII resonant line increases with temperature. Within the parameter range (1), it passes through one at T ≈ 8000 K.
In the frame of the shock wave model we solve the problem of the non-stationary cooling of the gas heated on the viscous jump. The equations describing radiative cooling downstream flux, and the methods of their solution are described by us in (Belova et al., 2014;Bychkov, 2018, 2019). The shocked gas becomes nonhomogenious because its temperature, density, excitation degree and the ionization state vary while it is cooling. The constant parameters are the density N 0 of the undisturbed gas and its speed V.

Discussion and conclusion
The results of solar flares analysis indicate the same behavior of Hα and CaII lines in time. Their impulse and gradual phases reach the same values at the same time. These observations contradict those of other stars, for example, of the spectral type dMe. These individual types of stars demonstrate a significant delay in the emission of CaII (H, K) lines, relative to Hα (Kowalski et al., 2013). Possibly, in stars the flare causes sharp ionization followed by slow recombination. These issues have not been adequately studied yet. The authors are grateful to the Ondřejov observatory team for the opportunity to participate in joint observations.

Funding information:
The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.