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2. Method

Some authors used static NLTE models to determine flare atmosphere parameters and then performed the formal solution of the radiative transfer equation with a prescribed run of the velocity to evaluate the emergent intensities. For macroscopic velocities comparable with the thermal velocity this can be a valid simplification. However, one can imagine that this approach must fail for velocities that exceed a certain limit. Thus we need a method that consistently includes the velocities. We will show that dynamic NLTE models based on ALI-approach can be satisfactorily used. Here we briefly summarize the multilevel ALI method of Rybicky & Hummer (1992) (so-called MALI) and point out the modifications needed for flare modeling.

2.1. MALI

The formal solution of radiative transfer equation (RTE) can be written down as the lambda operator acting on the source function, tex2html_wrap_inline1454, where tex2html_wrap_inline1456 is directional cosine and tex2html_wrap_inline1458 means frequency. The idea of ALI method is to split the lambda operator in two parts and to take the source function in the second term as known from previous iteration (denoted by tex2html_wrap_inline1460), i.e.
By choosing an appropriate approximate lambda operator tex2html_wrap_inline1462 we obtain desired accelerated lambda iteration scheme.

It was shown that the approximate lambda operator tex2html_wrap_inline1462 working with an optimal convergence rate is the diagonal part of the true lambda operator (Olson et al. 1986). To construct the diagonal tex2html_wrap_inline1462 we use the method based on Feautrier variables as proposed by Rybicky & Hummer (1991). Static Feautrier variables can be used in the continuum radiative transfer equation, because it is not affected by the moving plasma.
For line radiative transfer in a moving media the Feautrier variables are defined as (Mihalas 1978)
where tex2html_wrap_inline1468 is a distance from the line center. The line source function has to satisfy the condition tex2html_wrap_inline1470, so these Feautrier variables can be used only for non-overlapping lines.

Instead of the lambda operator we use the psi operator that acts on total emissivity tex2html_wrap_inline1472
This choice appears to be the most useful for our problem formulation (for details see Rybicky & Hummer 1992) and thus the lambda operator will be substituted in final formulas.

2.2. ESE

The general form of equations of statistical equilibrium is (Mihalas 1978)
nl is the atomic level population of the level l and Cll' and Rll' are, respectively, the collisional and radiative transition rates from level l to l'. Levels are ordered according to their energy (i.e. El>El' for l>l'). In this exploratory work we don't consider the non-thermal collisional rates.

If we assume the relaxation times of collisional and radiative processes to be much shorter than hydrodynamic ones, the statistical equilibrium is not affected by the hydrodynamics of the plasma. Thus we can write the equations of statistical equilibrium in their standard static form

Using the formulation of Rybicky & Hummer (1992) which takes into account the angle dependence of the absorption coefficient, we express the radiative rates as
tex2html_wrap_inline1496 is the specific intensity for frequency tex2html_wrap_inline1458 and directional cosine tex2html_wrap_inline1456. The quantities Ull' and Vll' are defined for each transition between levels l and l'. For line transitions there are given by
where All' and Bll' are the Einstein coefficients and tex2html_wrap_inline1514 is the line profile function. Its dependence on tex2html_wrap_inline1456 for a vertical macroscopic velocity v takes the form
where tex2html_wrap_inline1520 is the line center frequency.

For continuum transitions the Ull' and Vll' are given by
where all' is the photo-ionization cross section and tex2html_wrap_inline1528 is the Saha-Boltzmann function (see Mihalas 1978).

The total emissivity can be expressed as
where tex2html_wrap_inline1530 is the background emissivity.

Substituting for tex2html_wrap_inline1472 in Eq. (4 (click here)) we obtain the resulting formula for intensity
Note that tex2html_wrap_inline1530 has cancelled in the second and third term.

2.3. Momentum equation

  The momentum equation reduces to the equation of hydrostatic equilibrium in the case of a static atmosphere. Assumption that the time changes of the velocity are small (tex2html_wrap_inline1544) leads to the stationary form of the momentum equation. Thus

where m is the column mass, p is the total pressure, tex2html_wrap_inline1550 is the density and tex2html_wrap_inline1552 is the surface gravity. The second term on the lefthand side will decrease the total particle density N in the region with velocity as p=NkT. In the cases when the atmosphere is not stationary the stationary form of the momentum equation can be less accurate than the hydrostatic equilibrium equation. We have therefore the following possibilities to overcome this difficulty: 1) to use hydrostatic equilibrium equation, 2) to use the stationary form of the momentum equation or 3) to use the density as the input parameter from hydrodynamic simulations.

2.4. Preconditioned ESE and constraint equations


Substituting for tex2html_wrap_inline1496 in (7 (click here)) from Eq. (12 (click here)), we obtain an iterative system of non-linear equations for the new populations nl. The preconditioning strategy assures this system to be linear in nl provided we know a priori tex2html_wrap_inline1564. We use here preconditioning within the same transition only (see Rybicky & Hummer 1992).

ESE are closed using the charge conservation equation
where ql is the charge of the ion the level l belongs to.

The electron density is related by particle conservation equation

Using the state equation p=NkT and defining tex2html_wrap_inline1572 to be the mean nucleon mass, we can rewrite Eq. (13 (click here)) as

The set of preconditioned ESE, constraint Eqs. (14 (click here)) and (15 (click here)), together with momentum balance (16 (click here)) can be rewritten in a matrix form for the vector of unknown variables tex2html_wrap_inline1574 (tex2html_wrap_inline1576 is the total amount of levels) as
where tex2html_wrap_inline1578. Although the preconditioned ESE are linear in level populations nl, this system remains non-linear in the unknown variables due to products of electron density with level populations (the matrix A in Eq. (17 (click here)) depends on tex2html_wrap_inline1564). To correctly solve it we followed Heinzel (1995) and linearized system (17 (click here))
As the matrix A depends only on the electron density, the non-zero elements of the matrix tex2html_wrap_inline1588 appear only in its last column.

The system of linear algebraic Eqs. (18 (click here)) is solved for corrections tex2html_wrap_inline1590. The new variables are obtained as tex2html_wrap_inline1592 and iterations proceed until the changes are small enough, i.e. tex2html_wrap_inline1594.

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