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.
The formal solution of radiative transfer equation (RTE) can be written down
as the lambda operator acting on the source function,
, where is
directional cosine and 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 ), i.e.
By choosing an appropriate approximate lambda operator we obtain desired accelerated lambda iteration scheme.
It was shown that the approximate lambda operator
working with an optimal convergence rate is the diagonal part of the true
lambda operator (Olson et al. 1986). To construct the
diagonal 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
For line radiative transfer in a moving media the Feautrier variables are defined as (Mihalas 1978)
where is a distance from the line center. The line source function has to satisfy the condition , 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
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.
The general form of equations of statistical equilibrium is (Mihalas
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
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
is the specific intensity for frequency and directional cosine . 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 is the line profile function. Its dependence on for a vertical macroscopic velocity v takes the form
where is the line center frequency.
For continuum transitions the Ull' and Vll' are given by
where all' is the photo-ionization cross section and is the Saha-Boltzmann function (see Mihalas 1978).
The total emissivity can be expressed as
where is the background emissivity.
Substituting for in Eq. (4 (click here)) we obtain
the resulting formula for intensity
Note that has cancelled in the second and third term.
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 () leads to the stationary form of the momentum equation. Thus
where m is the column mass, p is the total pressure, is the density and 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.
Substituting for 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 . 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 to be the mean
nucleon mass, we can rewrite Eq. (13 (click here))
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
( is the total amount of levels) as
where . 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 ). 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 appear only in its last column.
The system of linear algebraic Eqs. (18 (click here)) is solved for corrections . The new variables are obtained as and iterations proceed until the changes are small enough, i.e. .