3. Calculations

3.1. Numerical code

We wrote the numerical code in C-language. The code uses a 12 levels plus continuum hydrogen atom model. All transition, included continua, are treated explicitly, which means that radiative rates of all transitions are computed in each iteration. We assume all lines to be in CRD which is reasonable in flares due to high . The code was tested using a five-levels-plus-continuum model of hydrogen for the temperature structures of the quiet solar atmosphere model VAL3C (Vernazza et al. 1981) and solar flare atmosphere models F1, F2 (Machado et al. 1980). We used the approximate formula for the charge conservation equation that includes ionization from other elements.
 

The differences for hydrogen level populations and electron densities didn't exceed 15%.

3.2. Models

To investigate the influence of the macroscopic velocity field on emergent intensities we proceed as follows. We took the temperature structure of the chromospheric flare model F1 and F2 (Machado et al. 1980) and computed the five-level-plus-continuum models of hydrogen with no velocities. The optical depth scale for the line center was subsequently used to define the velocity structure. We adopt two different approaches to define velocities.

 

(km s^-1) (0.01,0.1) (0.1,1.0) (1.0,10.0)

30 u30 m30 l30

50 u50 m50 l50 gradient models V₀

(km s^-1) 0.032 0.32 3.2

30 u30 m30 l30

50 u50 m50 l50

Table 1: Velocity parameters for layer and gradient models. The positive values mean downward velocity. The letter means a position of the moving material (l - lower, m - middle, u - upper part of the chromosphere) and the number is the velocity parameter V₀

 

  
Figure 1: Velocities of layer and gradient models with the same parameter V₀ as a function of line center optical depth

1. A layer moving with a constant velocity. These layer models are described by three parameters: the velocity of the layer V₀ and the heights of the lower and upper edge of the layer and , respectively.
2. Models with a velocity gradient. The velocity of gradient models has the form
   
   Note that the velocity goes to zero for , to 2V₀ for and is equal V₀ for . This formula was used by Mihalas (1976) to describe an expanding atmosphere. Here the formula describes a velocity field with downward velocity increasing with height and we assume that positive values of velocity mean downward motions.

 
  Figure 2: The line profiles and population departures for layer models with F1 temperature structure. The population departures are plotted for the models with velocity parameter

 
  Figure 3: The line profiles and population departures for gradient models with F1 temperature structure. The population departures are plotted for the models with velocity parameter

 
  Figure 4: The line profiles and population departures for layer models with F2 temperature structure. The population departures are plotted for the models with velocity parameter

 
  Figure 5: The line profiles and population departures for gradient models with F2 temperature structure. The population departures are plotted for the models with velocity parameter

For layer models we chose three regions where we let the material move. These are: the upper part of the flare chromosphere with , the middle part with and the lower part with . These values of were substituted for layer-model parameters and . The gradient model parameter , which corresponds to the height where the velocity is equal to V₀, was set to be in the center of each region of the layer models with respect to the logarithmic scale. So the value of was 0.032, 0.32 and 3.2, respectively. To summarize, we have chosen three regions for both types of models denoted upper, middle and lower, with a uniform velocity for layer models and with a velocity increase with height for gradient models.

Once having defined the position of the moving material we chose the velocity V₀ to be 10, 30 and for both types of models. Note that in the center of each selected region the velocities for both types of models with the same parameter V₀ are identical. The relationship is shown in Fig. 1 (click here).

Values for each particular model are summarized in Table 1 (click here).

Hydrodynamic simulations have shown that the density of the moving material is higher than its surrounding (Fisher et al. 1985). As the static form of the momentum equation leads to lower densities in the region with velocities, we used the hydrostatic equilibrium equation instead (see Sect. 2.3 (click here) for details).

We started to compute models with the lowest velocity and proceeded to higher velocities. As a starting guess (the level populations and electron density) for models with high velocities we used the results from the models of the same type with a lower velocity. This appeared to be very useful for faster convergence of some models. The convergence (see Sect. 2.4 (click here)) below 10^-4 was reached within about 100 iterations for layer models and within about 150 for gradient models.