Validez

The numerical solution of the non-LTE line formation problem starts with initial estimates of the atomic level populations, such as their LTE values, and these estimates are substituted in the statistical equilibrium equations. Also, initial estimates of the mean intensities are obtained

4.2. TheRadiativeTransferEquation 43

from the Planck function. However, these values will not satisfy the statistical equilibrium equations. To overcome this problem, the atomic level populations are perturbed byδnm_i , rep- resenting the required corrections for the atomic level populations to satisfy the statistical equi- librium equations. The superscript m represents the mth_{iteration of the numerical solution. The} initial statistical equilibrium equation can be written as

nm_i Nmax X j,_i Pm_{i j}−nm_i Nmax X j,_i Pm_{i j}−Em_i = 0, (4.20) where Em

i is the error in the net rate, which should be zero. Introducing the correctionsδn m i and linearizing, the rate equation becomes

−Em_i =δnm_i Nmax X j,_i Pm_{i j}+nm_i Nmax X j,_i δPm_{i j}− Nmax X j,_i δnm_jPm_ji− Nmax X j,_i nm_jδPm_ji . (4.21)

The corresponding corrections for the rates,δPi j, are given by

δPm_{i j} = δRm_{i j} = Bi j 2 Z +1 −1 Z ∞ 0 φνδIµνm dνdµ , (4.22)

in the case where the rate i to j is a bound-bound radiative transition with Einstein absorption coefficient BPi j. δIνµis the corresponding change of the specific intensity of the radiative field. This change in the intensity can be related to the source function through

δIνµ(τνµ)= Λνµ[δSν(τνµ)], (4.23)

whereΛ_νµis known the “lambda operator” which represents as an integral over frequency and optical depth of the formal solution of the radiative transfer equation. As the source function perturbationδSν(τνµ) can be related to the perturbations to the upper and lower levels of the transition involved,

44 Chapter4. TheNon-LTE RadiativeTransferProblem

where cm_i and cm_j are coefficients determined numerically, Equations 4.21 can be cast as a system of linear equations for the population correctionsδnm_i that can be solved by standard techniques. This is only a brief sketch of the numerical solution to the line transfer problem. The reader is referred to Carlsson (1992) for the specific implementation in the case of the multi

code used in the current work. An entire field of radiative transfer was started by Scharmer (1981) who first suggested that great computational advantage could be gained by using not the full Λ-operator occurring in Equation 4.23 but a well-constructed approximate operator that nevertheless retains the essential physics. ApproximateΛ-operators are further discussed by Hubeny (1992).

Chapter 5

Non-LTE for Calculation N

ii

∗

5.1

Introduction

The departure from LTE is a well known occurrence in the atmospheres of B-type stars (Kurucz 1979). Consequently, it is important to solve the non-LTE radiative transfer problem of the singly ionized nitrogen atom in order to obtain accurate nitrogen abundances for the B stars. This was done using themulticode (v.2) of Carlsson (1992). The goal of such a calculation is to

compute grids of the non-LTE equivalent widths of the strongest optical Niilines commonly

used in abundance analysis. Also, an important additional goal is to obtain estimates of the error bounds in the computed equivalent widths due to errors in the atomic data. This was done via Monte Carlo simulation following the procedure of Sigut (1996). This method is useful for cases where the traditional way of estimating uncertainties from the dispersion of the measured elemental abundance using many weak spectral lines in not applicable.

The structure of this chapter is as follows: In Section (5.2), I give a brief summary of previous, non-LTE calculations for Niiand in Section (5.3) I discuss the atomic data used to

construct the nitrogen atom. In Section (5.4), the results of the current, non-LTE nitrogen cal- culations are given, and the error bounds on the predicted equivalent widths due to random errors and systematic errors are discussed. The results of the multi-multi analysis are pre-

sented in Section (5.5). Section (5.6) discusses the achievable accuracy of nitrogen abundance ∗_{A version of this chapter is a published article in the Monthly Notices of Royal Astronomical Society; ”Non-}

LTE equivalent widths for Niiwith error estimates” Ahmed, A. & Sigut, T.A.A. 2016, MNRAS, 455, 1099A

46 Chapter5. Non-LTEforCalculationNii

determinations in B-type stars.