Random Errors

The accuracy of the non-LTE line formation calculation depends on many factors, and an im- portant one is the accuracy of the atomic data. The inclusion of atomic data from different sources that have different accuracies represents a source of random errors in the estimated

78 Chapter5. Non-LTEforCalculationNii

Table 5.9: Rates of variation of the atomic data

Atomic parameter uncertainty

f -value ±10% f -value≥0.1

±50% f -value<0.1 Stark widths ±40%

Photoionization cross section ±20% Collision strength (Excitation)

R-Matrix ±10% Impact parameter approx. Factor of 2 Collision strength (Ionization) Factor of 5

equivalent widths. In order to quantify the errors, a series of Monte Carlo simulations were carried out following the procedure of Sigut (1996). In this simulation, two hundred random realizations of the nitrogen atom were generated where the various atomic data were changed within their estimated bounds shown in Table (5.9).

The oscillator strengths (f -values) of the bound-bound radiative transitions were allowed to change within ±10 % for f -value equal to or greater than 0.1, and within ±50 % for the transitions with smaller f -values. Such values were chosen because the OP length and velocity f-values differ by≈10 % for f-values equal to or larger than 0.1, while the difference can be as high as≈ 50 % for smaller f-values. The Stark widths were allowed to change within±40 %, which is in the order of the difference between our calculated Stark widths and experimental values for a number of Niilines, see Table (5.3). § The photoionization cross sections were

allowed to change within±20%. The photoionization cross sections have uncertainty of 10 % (Luo & Pradhan, 1989) , and an extra 10 % was added to include possible uncertainty in the photoionizing mean intensity, Jν (Sigut , 1996). The R-Matrix method represents the most accurate tool for the computation of the thermally-averaged collisional strengths, and Hudson & Bell (2004) show that their results agree with the the results of previous R-matrix calcula- tions to within ≈ 10 %. The impact approximation is not really as accurate as the R-Matrix §_{The assumed error bound of Stark widths is smaller than the differences which lie between≈ 50 %−60 %}

5.6. TheAccuracy of thePredicted non-LTE EquivalentWidths 79

procedure, and consequently, the collisional strengths of transitions computed with the impact approximation were allowed to vary by a factor of 2. Similarly, the collisional ionization rates computed using the procedure of Seaton (1962) are highly uncertain, and a factor of 5 was chosen as the error bound of their estimations.

A converged, non-LTE solution was found for each of the 200 randomly-realized atoms, and the distribution of each transition equivalent width was taken to estimate the uncertainty. The predicted distributions of equivalent widths (forλ3995 Å andλ6482 Å) are in shown in Figure (5.13) for Teff of 19,000 and 23,000 K, log g=4.0, andξt,=5 km s−1. The figure shows that the distributions are nearly Gaussian, and the average value and the standard deviation represent the predicted equivalent width and the associated uncertainty due to random errors. The predicted equivalent widths and the expected errors due to uncertainties in the atomic data of these two lines (at log g=4.0, andξt,=5 km s−1) with Teff’s between 15000 and 31000 K and

nitrogen abundances between 6.83 and 8.13, are given in Table (5.10).

Given in Table 5.11 are the correlations between the two hundred random realizations of each radiative and collision transition rate with the calculated equivalent widths of these two lines. The Pearson’s correlation coefficient, r, was calculated in order to investigate the cor- relation between the random sets of the atomic data and the predicted equivalent widths. The correlation coefficients are similar to the multi-multisimulations of the previous section; how-

ever, the rates are now varied within their expected errors and not changed by an arbitrary factor of 2 one at a time. These correlations allow one to deduce which transitions are driv- ing the uncertainty for a given line. Table (5.11) shows the correlation coefficients of the Nii

3995 and 6482 Å lines at effective temperatures of 19,000, 23,000 and 27,000 K, gravity of 4.0, and microturbulent velocity of 5 km s−1_{. For 200 random realization of the nitrogen atom,}

a correlation coefficient, r, of 0.18 is statistically significant at the 1 % level (Bevington, 1969). As shown in the Table, the error in the predicted equivalent widths is strongly affected by the uncertainties in the oscillator strength values of the radiative transition themselves and the ac- curacy of the collisional strength values. Tables C.1 to C.16 in Appendix C show the results of

80 Chapter5. Non-LTEforCalculationNii 60 65 70 75 80 0 5 10 15 20 25 30 35 Number

Equivalent width (mA)

NII 3995A : T

eff=19000K, log(g)=4 ,ξ=5km/s : µ= 70.05mA , σ =2.09mA

10 11 12 13 14 15 0 5 10 15 20 25 30 Number

Equivalent width (mA)

NII 6482A : T

eff=19000K, log(g)=4 ,ξ=5km/s : µ= 12.57mA , σ =0.62mA

110 115 120 125 130 0 5 10 15 20 25 30 35 Number

Equivalent width (mA)

NII 3995A : T

eff=23000K, log(g)=4 ,ξ=5km/s : µ= 119.83mA , σ =3.1mA

40 45 50 55 0 5 10 15 20 25 30 35 Number

Equivalent width (mA)

NII 6482A : T

eff=23000K, log(g)=4 ,ξ=5km/s : µ= 46.69mA , σ =2.1mA

Figure 5.13: The predicted equivalent width distribution forλ6482 Å andλ3995 Å for Teff=

19000 and 23000 K, log g equal to 4.0,ξt =5 km s−1, and the solar nitrogen abundance. Two hundred randomly-realized atoms were used, and the mean and the standard deviation of the equivalent width distributions (in mÅ) are shown at the title of each panel.

the Monte Carlo simulations for the other combinations of log g, (3.5, 4.0 and 4.5) andξ(0.0, 5.0 and 10.0 km s−1.

In order to test the accuracy of measuring the nitrogen abundances, given the above uncer- tainties, the equivalent widths of three singlet lines of Nii,λ3995 Å,λ4447 Å andλ6482 Å,

computed for selected nitrogen abundances, ǫN,MU LT I, and the unperturbed, reference atoms were used for abundance estimations using the results of the Monte Carlo multi simulations,

ǫN,MC, for Teff equal to 19,000 K, 23,000 K, 25,000 K and 31,000 K. In this test, nitrogen abun-

dances were obtained from each of the curves of growth of the 200 random realizations of the nitrogen atom for the three singlet lines. Figure (5.14) shows how the estimated abundances based on the 200 random atoms differ from the original abundance values used in the reference

5.6. TheAccuracy of thePredicted non-LTE EquivalentWidths 81

calculations. The computed abundances represent the mean of all abundance estimations using all Monte Carlo simulations. Also, the uncertainties of the estimated abundances due to inac- curacies in the atomic data are shown as error bars in the Figure which equal 2σ. The figure shows that abundances obtained using the results of the 200 Monte Carlo simulations match the original abundances, and that the expected errors in the estimated nitrogen abundances due to uncertain atomic data tend to increase with nitrogen abundance, e.g. for Teff = 23,000 K,

the uncertainty is ±0.02 dex for ǫN = 6.83 dex which rises to ±0.11 dex for ǫN = 8.13 dex. The figure also shows that the estimated errors vary with Teff. At the same nitrogen abundance

such as ǫN = 7.83 dex, the uncertainty is±0.05 dex for Teff = 19,000 K, rising to±0.07 dex

for Teff =23,000 K and then falling to±0.05 dex for Teff =29,000 K The highest error occurs

for Teff between 23,000 K and 25,000 K. This is in agreement with the observed behaviour of

the equivalent widths with Teff, where the maximum strengths of these lines occur at the same

temperatures (see Figure 5.6).