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Astron. Astrophys. 321, 549-556 (1997)

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2. The response of a stellar chromosphere to impulsive heating

Let's consider the processes in the region adjacent to the footpoint of a loop. The particles and heat fluxes propagate along the magnetic lines of force, allowing a one-dimensional approach. The initial structure of the stellar atmosphere contains a corona, a transition region, a chromosphere and a photosphere. We consider the loop to be like a vertical tube with a constant cross-section with height. We suppose that the flux of the accelerated electrons is injected into the loop from above. The gas-dynamic processes which develop in this structure are analyzed.

The main problem confronting the modelling of impulsive processes is the computation of the characteristics of a region of very high temperature gradient. Additional assumptions were adopted such as the continuity of the pressure [FORMULA] in the region of the temperature jump (thermal conduction front), a different dependence of the degree of ionization on the temperature in front of and behind this jump. Taking into account our previous experience of numerical simulation, we have changed the statement of the problem in the following way: namely we abandon any attempt to describe the region of the temperature jump, and use a unified dependence of [FORMULA] both for the low and high temperature layers.

Previous experience of gas-dynamic modelling of solar and stellar flares, leads us to use a single-fluid approach to a description of gas-dynamic processes in a collision dominated chromospheric and coronal plasma. Similar time dependent equations are used by Cheng in all his computations (Cheng et al. 1983). In principle, we could also restrict the model to a single-temperature approximation (Katsova et al. 1981); a two-temperature approach gives qualitatively the same solution, but allows us to determine more exact quantitative values of physical parameters which are characteristic of the flare process, such as the density, the temperature, and the thermal conduction flux. For a comparison between two- and single-temperature computations for a solar flare see in Boiko & Livshits (1995).

The set of equations for a two-temperature, single-fluid plasma is as follows:





Here the ratio of the specific heats for hydrogen plasma is [FORMULA] ; the internal energy of the electron component is [FORMULA] ; the internal energy of heavy particles is [FORMULA] (here it is assumed that the temperature of neutral atoms is the same as that of protons), where [FORMULA] is the ionization potential of hydrogen, n is the density of equivalent hydrogen atoms (i.e. [FORMULA]), [FORMULA] are densities of neutral hydrogen, protons and electrons respectively; [FORMULA] is degree of ionization; [FORMULA] is the viscosity; [FORMULA] is viscosity coefficient; [FORMULA] is thermal conduction flux, where [FORMULA] is the thermal conduction coefficient; [FORMULA] is heating function; L is radiative cooling function; [FORMULA] is heating balancing the radiative losses at the initial time, and Q is the energy exchange function between the electron and ion components.

The following boundary conditions apply: static upper and lower boundaries, an absence of thermal conduction fluxes;




Here the number of the hydrogen nuclei in the column, determined above, [FORMULA] is the Lagrangian coordinate.

As the initial model we use the Hawley-Fisher's QT3 model (1992), where the photosphere, the chromosphere and the transition region are included. The densities are small enough in the upper layers of the QT3 model (column mass above the transition region [FORMULA] in [FORMULA] is [FORMULA]). We add the coronal layers to this model up to heights of 30 000 km in accordance with Katsova et al. (1987). The distribution of the parameters of the initial model are shown in Fig. 1.

[FIGURE] Fig. 1. The distribution of temperature (solid line) and the density (dashed line) shown as a function of the column number for the initial model (see references in the text). The horizontal axis at the top of the figure represents heights above the level where [FORMULA] for this initial model.

We suppose that the chromosphere is heated by an electron beam of energy [FORMULA] and the time profile of the heating flux has a rectangular form. The value of the heating flux [FORMULA] is set to agree with the soft X-ray flux for stellar flares. We choose the spectrum of the electrons hard enough as compared with typical one, detected from the hard X-ray bursts on the Sun. Namely, we assume [FORMULA] where v is the velocity of the electrons, E is their energy, and [FORMULA] is the differential spectrum. For hard solar bursts, [FORMULA] is proportional to [FORMULA] - [FORMULA] An injection of this electron beam into the chromosphere leads to a heating distributed over the Lagrangian coordinate. The level of heating is evaluated according to Syrovatskii & Shmeleva (1972). The results of our gas-dynamic modelling remain valid for any mechanisms of powerful impulsive heating not only for electron beams, but also others, such as an absorption of impulsive X-ray radiation, etc, which affect the heating distributed over the Lagrangian coordinate.

It is necessary to consider the gas-dynamic processes in a stellar atmosphere taking into account the radiation of the plasma. At lower temperatures, radiative losses and the degree of ionization are computed using the results from a consideration of the kinetics of the hydrogen atom in layers with a given optical depth at the centre of the [FORMULA] - line (Bruevich & Livshits (1993)). The same function [FORMULA] as the one calculated for the layer with [FORMULA] is used now in the gas-dynamic modelling for all layers with different optical depths. Although it leads to a small overestimate of the temperature of the layers below the transition region, the assumption that the degree of ionisation is independent of optical depth over a range in column number allows us to avoid consideration of a complete set of radiative gas-dynamic equations, and to restrict our approach to a typical set of equations, where the radiation has only dissipative terms in the energy equation.

The radiative cooling function for the temperatures [FORMULA] is taken from Raymond & Smith (1977), (see the details in Boiko & Livshits, 1995). The heating of the initial atmosphere is calculated as


The energy exchange between the electron and the ion components of the plasma is


The final expression for the heating flux and radiative loss function can be found in Boiko & Livshits (1995). The degree of the ionization [FORMULA] is taken from Bruevich & Livshits (1993) for the layer with [FORMULA]. The values of [FORMULA] are distinguished from given ones in Boiko & Livshits (1995) by a factor of 0.9 for [FORMULA].

Numerical solution of the set of equations (1)-(4) is troublesome. This problem cannot be solved solely in Lagrangian coordinates. Therefore, for a more exact computation of the motions of the shock wave and thermal conduction front, we compute the motions of the knots of a numerical grid. The numerical method uses the Langangian approach with a re-interpolation of all the functions at each time-step to form a new grid with a new set of physical parameters for the knots (Boiko & Livshits, 1995). Of course, this gas-dynamic problem is a multi-parameteric one; its final solution is sensitive to the degree of heating, the kind of heating agent (accelerated electrons or thermal flux), the hardness of the electron beam, and to the choice of the initial model. The dependence of the solution on the parameters adopted have been investigated for the solar case (see a review by Fisher (1986); Kosovichev et al. 1980). In particular, the density in the source of the optical continuum emission is proportional to the heating flux. Effects in the chromosphere are more remarkable in the case of heating by a hard electron beam, while those in the corona are more pronounced for the case of heating by the softer electron beam. Solutions for small heating fluxes and/or soft electron beams lead to low values of the density and temperature in the source of the optical radiation; therefore they are less useful for stellar flare modelling. For a variant that is close to the computation by Cheng & Pallavicini (1991) (where the initial heating expands to coronal layers above the transition region) we obtain similar results. This paper contains results of the alternative model with a powerful initial heating expanding up to the upper chromosphere. The latter is realized by a choice of the initial model with [FORMULA]

For stellar flare modelling, it would be of interest to carry out the computation with higher values of heating fluxes; however, the thermal flux saturates, and return electric currents arise that restrict the penetration of accelerated electrons into the chromosphere. Boiko & Livshits (in prep) tried to take into account these new effects for solar flares. We enlarge here on the model with a heating flux [FORMULA] where these effects can be neglected.

2.1. Explosive evaporation of an elementary burst

The conditions for explosive evaporation (Fisher et al. 1985), when the heating time scale in the evaporating region is less than the time scale of hydrodynamic expansion, are assumed to be satisfied in our case of powerful impulsive heating. This process consists of two parts: at the flare onset, in the first 0.1 - 0.2 s, a plasma is heated strongly in the upper chromospheric layers. In this phase there is no motion, the density is constant and the pressure increases sharply. This situation is presented in Fig. 2 for the time [FORMULA] after the commencement of heating. The region of high pressure is seen at heights 610 - 670 km above the stellar photosphere. Then, 0.3 s after the flare onset, two disturbances subsequently propagate upward and downward from this high-pressure region.

[FIGURE] Fig. 2. Distributions of the density, velocity, total pressure and temperatures (electrons - solid line, ions - dashed line) versus height for the time [FORMULA] after the beginning of the heating. Negative values of velocities correspond to motions toward the photosphere.

An important feature of this part of the process is the formation of a region with [FORMULA] at heights 550 - 610 km and [FORMULA]. The downward shock wave arises at [FORMULA], just below the temperature jump, and travels through all the quasi-equilibrium regions with [FORMULA].

The second part of the process of explosive evaporation is characterized by the slow downward motion of a temperature jump, ahead of which a shock wave travels. Hot gas moves outwards from the temperature jump. Our solution is shown in Fig. 3 for [FORMULA]. The gas behind the downward radiative shock front is compressed by a factor of about 100, and the geometrically thin low-temperature condensation arises between the shock wave and thermal conduction front. It is identified as the source of the low-temperature radiation of flares. The temperature is distributed non-uniformly inside the downward moving condensation: at 4 s it reaches about 8000 K behind the shock front (at the height 242 km) and increases up to 10000 - 12000 K at the top of the condensation, at 262 km, where the thermal flux is able to penetrate. The temperature in the middle of the condensation can be as low as 5000 K. Therefore the flare optical continuum and the Balmer line emission can arise only in the upper and lower layers of the condensation.

[FIGURE] Fig. 3. The same parameters as in Fig. 2, but for the time [FORMULA] after the beginning of the heating at the height 242 km above the photosphere.

Such a distribution of the temperature with a rise at the top and at the bottom of the optical emission source is typical, particularly for a source forming during the gas-dynamic process, when heating is switched on. If this powerful heating is stopped, as in the gradual phase of the flare, it becomes necessary to take into account absorption of the outer X-ray emission. If this effect is considered (Hawley & Fisher, 1992), the temperature stays high enough ([FORMULA]) over all the condensation, and the gas begins to emit effectively in the optical, not only in outer parts, but also in the inner regions.

The propagation of the low-temperature condensation downward through the AD Leo atmosphere is presented in Fig. 4. The velocities of the cool gas vary gradually from [FORMULA] at 0.4 s to [FORMULA] at 8 s. Note that in all former computations maximal densities didn't exceed [FORMULA]. Their increase here at [FORMULA] is due to choosing of a variant of the solution, in which weak disturbances from the downward moving temperature jump maintain the shock wave. In many real flares the shock wave can attenuate at [FORMULA] faster that it is shown in Fig. 4.

[FIGURE] Fig. 4. The density profiles versus the height. The time in seconds is denoted in the upper part of the figure. The straight line refers to the initial model.

The physical parameters in the upward flux at a height of 5000 km are presented in Fig. 5. The electron temperature [FORMULA] rises to [FORMULA] during the first 0.5 s. The ion temperature remains low until the shock wave arrives at this height. The shock front velocities change from [FORMULA] for [FORMULA] to about of [FORMULA] for [FORMULA]. When the shock front reaches the given height of 5000 km, the ions heat up strongly (up to [FORMULA]), and the ion temperature becomes close to the electron temperature (see Fig. 5). At this height the velocities of the gas exceed [FORMULA] upwards. The evaporated hot gas moving upwards gradually fills the coronal part of the loop. The flux of the particles is about [FORMULA]. This flux of evaporated material increases quickly ([FORMULA]) and then remains constant; thus, the mass loss from unit area ([FORMULA]) during the entire elementary burst is about [FORMULA]. The ion temperature of the flare plasma remains lower than the electron temperature in the average over the loop. If we consider the gas-dynamic process, even in a diverging coronal part of the loop, then values of velocities, temperature and mass loss from a unit area ([FORMULA]) will be little changed (Getman & Livshits, 1996).

[FIGURE] Fig. 5. The time dependence of the electron temperature (solid line), the ion temperature (dashed line) and velocity of the upwards evaporated plasma (dotted line) in the upper part of the flow (at the height 5000 km). The shock front reaches this level 6 s after the flare onset.

2.2. The properties of the flare radiation accompanying the elementary gas-dynamic process

The low-temperature condensation arises in a few tenths of a second after the flare onset. Physical conditions therein are such that the condensation can emit both in the optical continuum and in the Balmer lines (Livshits et al. 1981; Katsova et al. 1981).

The densities inside the low-temperature condensation are high enough (more than [FORMULA], Figs. 3 and 4), and physical conditions therein are close to LTE. This is favourable to the production of optical continuum. Gas-dynamic modelling is carried out for a heating flux of 3 times less than in our previous computation. Nontheless, the optical depth of the condensation, in optical continuum at [FORMULA], remains close to 1. Optical continuum emission arises 2 s after the flare onset and lasts a few seconds. The absorption of the soft X-rays in the condensation should lead to an increase in the duration of the optical continuum emission. Thus, this condensation is a source of radiation over a wide spectral range from the optical to the UV ([FORMULA]).

Theoretical modelling of a flare with optical and UV ([FORMULA]) observational data confirms our ideas about the thermal origin of flaring optical continuum. This allows us to estimate with confidence the flare area. For the flare of [FORMULA] on the AD Leo, we compute its area as [FORMULA], if we adopt the physical conditions of the emission source as in Fig. 3.

The first lines of the Balmer series should be broadened due to the Stark effect. The calculations show that typical FWHM of these lines are close to [FORMULA], 2 s after the flare onset. Sometimes the upper part of the condensation can emit an intensive narrow [FORMULA]) [FORMULA] -line. The red shift of the Balmer lines should be small in accordance with small velocities of the downward gas motions. The necessary consequence of the model is that the flare optical continuum emission appears at the time when the slope of the Balmer decrement becomes less steep, independently of the parameters of the emission source. The first consideration of this problem based on the gas-dynamic modelling (Katsova et al. 1991) is given by Katsova (1990), and these results are confirmed by new the solution.

In response to the heating of the chromosphere an emitting region should be formed with a temperature of about [FORMULA]. The source of the UV lines like C IV, He II, etc. appears at the flare onset and should exist for several tenths of a second (Fisher, 1987; observations of stellar flare with the ASTRON satellite by Burnasheva et al. 1989 and their interpretation by Katsova & Livshits, 1989), see Fig. 2. According to our computation, the column density of this source is [FORMULA] for [FORMULA]. The optical depth at the centre of the resonant C IV line [FORMULA] is [FORMULA] For these conditions, the source function is close to the Planck function, and the flux at the Earth in both lines of the C IV doublet can be estimated as


where d is the distance to the star (for the AD Leo d = 4.9 pc), the width of the C IV line is [FORMULA] [FORMULA] is the Planck function. If we suppose that the area of the UV-line emission is the same as in the optical continuum emission the C IV doublet line flux is [FORMULA] for AD Leo. Such a flux should accompany the beginning of this process in each elementary burst. A fast burst like this should be observed in the He II [FORMULA] and possible in some other lines.

The UV emission doesn't end after the formation of the shock wave. The plasma with a temperature of [FORMULA] in the lower part of the evaporated flow is the source of the EUV lines in the first few seconds after the beginning of the process. These lines have to be broadened by the Doppler effect, but they shouldn't have significant blueshift. The blueshifted component of the high-temperature lines observable during flares are those emitted by ions such as Fe XXIV and Ca XIX. The intensities of the EUV lines depend on the flare area, which is close to the area of the optical continuum source. The rise of the intensity of these lines occurs 1-2 s ahead of the appearance of the optical continuum and about 3 - 10 s ahead of the soft X-ray radiation during powerful events with duration no longer than a few minutes (i.e. such events can be considered as a response to a single act of primary energy release). The soft X-ray emission is due to hot gas evaporated upwards. For a given particle flux of [FORMULA] and the area of one footpoint of the loop, mentioned above, the emission measure of hot gas per one elementary burst is estimated as [FORMULA] It is clear that in order to provide the observed value of the emission measure we need a set of elementary bursts. These results concerning the evaporation of hot flaring plasma agree well with the computations by Cheng & Pallavicini (1991) for red dwarf stars.

2.3. The result of a set of elementary events

The injection of an electron beam into a given point of the stellar chromosphere lasts, probably, not more than several seconds. This can be repeated, both in the second footpoint of the same low-lying loop, and in some adjacent loops. The hard X-ray data on solar flares, observed with high temporal and spatial resolution, show that all impulsive flares consist of a number of elementary events, and that these spikes are distributed randomly during the impulsive flare.

The above discussed gas-dynamic processes are subsequently developed in several low-lying loops of active regions on the star. The total radiation of an impulsive flare can be the sum of the emission accompanying several single, elementary acts of primary energy release. It includes few additional components. First, each elementary gas-dynamic process relaxes during tens of seconds due to the propagation of the disturbance into the photosphere. In single bursts, the intensities of optical continuum and broad lines decrease slowly after the end of heating. Usually, the total duration of powerful impulsive flare is about of 100 s, which includes several elementary events and their decline. Second, the observable soft X-ray emission should arise in tens of seconds after the beginning of the flare as the plasma is evaporated from the chromospheric footpoint to the corona during 1-3 elementary bursts.

The soft X-ray emission can last longer. This is determined by the radiative loss rate and the efficiency of the process of heat conduction along the loop from coronal levels downwards. The characteristic times of both processes are close to each other, and are equal to


where [FORMULA] and T are the electron density and the temperature of flaring plasma, and [FORMULA] is radiative loss per [FORMULA] (see also Pallavicini et al., 1990).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998