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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 tex2html_wrap_inline1564. 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.
 equation453

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 tex2html_wrap_inline1608 for the tex2html_wrap_inline1398 line center was subsequently used to define the velocity structure. We adopt two different approaches to define velocities.

 

layer models
V0 ( tex2html_wrap_inline1616 , tex2html_wrap_inline1618 )
(km s-1) (0.01,0.1) (0.1,1.0) (1.0,10.0)

10

u10 m10 l10
30 u30 m30 l30
50 u50 m50 l50
gradient models
V0 tex2html_wrap_inline1624
(km s-1) 0.032 0.32 3.2

10

u10 m10 l10
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

 

  figure488
Figure 1: Velocities of layer and gradient models with the same parameter V0 as a function of tex2html_wrap_inline1398 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 tex2html_wrap_inline1618 and tex2html_wrap_inline1616, respectively.
  2. Models with a velocity gradient. The velocity of gradient models has the form
    displaymath1606
    Note that the velocity goes to zero for tex2html_wrap_inline1644, to 2V0 for tex2html_wrap_inline1648 and is equal V0 for tex2html_wrap_inline1652. 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 tex2html_wrap_inline1398 line profiles and population departures for layer models with F1 temperature structure. The population departures are plotted for the models with velocity parameter tex2html_wrap_inline1656

 
  Figure 3: The tex2html_wrap_inline1398 line profiles and population departures for gradient models with F1 temperature structure. The population departures are plotted for the models with velocity parameter tex2html_wrap_inline1656

 
  Figure 4: The tex2html_wrap_inline1398 line profiles and population departures for layer models with F2 temperature structure. The population departures are plotted for the models with velocity parameter tex2html_wrap_inline1656

 
  Figure 5: The tex2html_wrap_inline1398 line profiles and population departures for gradient models with F2 temperature structure. The population departures are plotted for the models with velocity parameter tex2html_wrap_inline1656

For layer models we chose three regions where we let the material move. These are: the upper part of the flare chromosphere with tex2html_wrap_inline1670, the middle part with tex2html_wrap_inline1672 and the lower part with tex2html_wrap_inline1674. These values of tex2html_wrap_inline1676 were substituted for layer-model parameters tex2html_wrap_inline1618 and tex2html_wrap_inline1616. The gradient model parameter tex2html_wrap_inline1624, 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 tex2html_wrap_inline1624 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 tex2html_wrap_inline1700 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)) tex2html_wrap_inline1704 below 10-4 was reached within about 100 iterations for layer models and within about 150 for gradient models.


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