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Having discussed many of the potential pitfalls in determining the 4 He abundance in individual extra-galactic HII regions, we can now discuss the methodology for making such a determination. As we noted earlier, a He abundance can be inferred for each He I emission line observed by comparing the ratio of its observed intensity to Hbeta with the theoretical ratio and correcting for the effects of collisional excitation, florescence and underlying He I absorption. Thus, as per the discussion of the previous sections, we need to determine three physical parameters, the density, n, the optical depth, tau, and the equivalent width for underlying helium absorption, aHeI. As argued by ITL94 and ITL97, a self-consistent determination of the parameters, if possible, is preferable. Below we describe a few possible methods for such a determination and stress the need for a careful accounting of the resulting errors, which we deem requires a Monte Carlo simulation of the data.

As we noted above and discuss in detail below, different He I lines are more or less sensitive to the different physical parameters. In principle, it is possible to fix these parameters by minimizing chi2 using only the three best determined line strengths, lambda4471, lambda5876 and lambda6678. However, because these line strengths are not very sensitive to any of the physical parameters of interest, it may be preferable to consider two or even more additional wavelengths. We describe these various possibilities below. Once the parameters and their associated uncertainties have been fixed, the He abundance may be determined by averaging over all of the He I lines used in the determination of the physical parameters.

We note that we are adopting a different philosophical approach here compare to that in IT98. In the final calculation of y+, IT98 use only the main three lines to obtain the final He abundance. Additionally, they adopt and report a minimum density of 10 cm-3 (reduced from the minimum density of 50 cm-3 adopted in ITL97) and not lower densities which may be derived from their minimizations. To be truly "self-consistent" would imply that the helium abundance is derived from all observed lines and the physical parameters are those resulting from the minimization. An inspection of the IT98 data reveals that often the He/H ratios derived from the lambda7065 and lambda3889 lines are significantly different from the He/H ratios derived from the main three lines. Additionally, when their minimization routine is applied, one often finds unrealistically small values of the density. We take these as warning signs that in some cases either the minimization is not finding the best possible solution due to a degeneracy in the chi2 minimization, or there are problems with the input data. In such cases, it makes sense to either reject the object from derivations of the primordial helium abundance or to attribute a larger uncertainty to account for the lack of self-consistency in the minimization solution.

We begin our discussion of the merits of various minimization routines by examining the dependence of the line strengths (for the six He I lines of interest) on the physical parameters, n, tau, and aHeI. Figure 4 shows six He I emission lines and their relative dependences on the different effects discussed in the last sections. We show the relative effects for a baseline model of T = 15,000K, n = 10 cm-3, tau = 0, and no underlying stellar He I absorption (aHeI = 0). The top panel of Figure 4 shows the effects of an error of 500 K. This is of order or larger than the errors typically quoted for electron temperatures for high quality spectra. It can be seen that reasonable errors in electron temperature (or temperature fluctuations) will have a relatively small effect on the derived He/H abundances (note, however, that Peimbert, Peimbert, & Ruiz 2000 have found that a coupling between temperature and density allows solutions with small differences in temperature to result in significant differences in density resulting in larger than expected changes in the derived helium abundance).

Figure 4

Figure 4. Histograms showing the effects on helium emission lines due to changes in physical parameters. The baseline model is a photoionized gas at 15,000 K with a density of 10 cm-3, a negligible optical depth in the lambda3889 line, and no underlying absorption in the helium lines (the underlying spectrum has a lambda-2 slope and the equivalent width of the Hbeta emission line is 100 Å). The top panel shows that the helium lines are very insensitive to variations in temperature, with differences of order 1% or less for a 500 K variations. The second panel shows that all of the helium lines are sensitive to an increase in density, but that the lambda7065 line is far more sensitive than the other lines. The third panel shows that only the lambda3889 and lambda7065 lines have significant sensitivity to optical depth effects. The bottom panel shows that all lines are very sensitive to small increases in the underlying absorption.

The second panel from the top in Figure 4 shows the effect of increasing the density from 10 to 100 and the subsequent collisional enhancement of the He I lines. Clearly, of the six lines, lambda7065 is most sensitive to this effect. Of the three lines normally used to calculate He/H abundances, lambda5876 is the most sensitive and lambda6678 is the least sensitive. lambda7065 would be an ideal density diagnostic if not for the sensitivity to optical depth shown in the third panel.

The third panel from the top in Figure 4 shows the effect of increasing the optical depth tau(3889) from zero to one. lambda7065 has a strong sensitivity to optical depth effects. lambda3889 is also sensitive to tau(3889), and in the opposite sense, so that in combination these two lines could act to constrain both density and optical depth. Unfortunately, lambda3889 is blended with H8 (lambda3890). Thus, in order to derive an accurate F(lambda3889) / F(Hbeta) ratio, the F(lambda3890) must be subtracted off and underlying stellar H I (and He I) absorption must be corrected for. This generally implies a relatively large uncertainty for F(lambda3889), and thus, a larger uncertainty in the density and optical depth measures than one would hope for.

Finally, the bottom panel in Figure 4 shows the effects of 0.2 Å of underlying stellar absorption. The difference of a factor of three between the effect on lambda5876 and lambda4471 and lambda6678 means that there is some sensitivity to underlying absorption through the analysis of just those three lines. However, the effect is very strong for the weaker lambda4026 line. Thus, we will explore the possibility of adding lambda4026 as a diagnostic line.

It is very important to note from Figure 4 the strong trade-off between density and underlying He I absorption. All six He I line strengths are increased by increasing the density, while all six He I line strengths are decreased by increasing the underlying He I absorption. While the relative effects vary from line to line, the main result is a basic trade-off between density and underlying absorption when both are included in a minimization routine. This means that adding underlying absorption as a free parameter in a minimization routine will open up a larger range of parameter space for good solutions. On the other hand, it means that if absorption is not included in minimizations, its effects may be masked by driving the solutions to lower He abundances or densities.

4.1. Using 3 Lines

In principle, under the assumption of small values for the optical depth tau(3889), it is possible to use only the three bright lines lambda4471, lambda5876, and lambda6678 and still solve self-consistently for He/H, density, and aHeI. Of course, because these lines have relatively low sensitivities to collisional enhancement, the derived uncertainties in density will be large. However, as we will show, if there is some reason to suspect a problem with any of the additional lines, the three line method can actually lead to a more accurate result, and hence should be used as a diagnostic if nothing else. Using a minimization routine, as opposed to a direct solution, it is not even necessary to assume that tau(3889) = 0 in order to derive a helium abundance from just three lines.

The detailed procedure we use to determine the physical parameters along with the He abundance is given in Appendix C. The procedure is actually independent of the number of lines used, though when using fewer lines (as in the present case of 3 lines) the results are likely to be less robust.

4.2. Using 5 Lines

A self-consistent approach to determining the 4 He abundance was proposed by Izotov, Thuan, & Lipovetsky (1994, 1997) by considering the addition of other He lines. First the addition of lambda7065 was proposed as a density diagnostic and then, lambda3889 was later added to estimate the radiative transfer effects (since these are important for lambda7065). By minimizing the difference between the ratios of lambda3889 / lambda4471, lambda5876 / lambda4471, lambda6678 / lambda4471, and lambda7065 / lambda4471 and their recombination values, the density, optical depth, and helium abundance can be determined. The latter is determined by a weighted mean of the helium abundance based on lambda4471, lambda5876, lambda6678 once the values of n and tau(3889) are fixed. This is the method used by IT98 in their most recent estimate of Yp. Underlying He I absorption is assumed to be negligible in their method.

While this method, in principle, represents an improvement over helium determinations using a single emission line, systematic effects become very important if the helium abundances derived from either lambda3889 or lambda7065 deviate significantly from those derived from the other three lines. In addition, working with the ratios of all of the He I lines to a single He I line puts undue weight on that single line (in this case lambda4471). This is especially vulnerable to systematic errors in the presence of undetected underlying stellar absorption.

Here, we also consider using these five lines for determining the He abundance along with the physical parameters. However, as described in the appendix B, our minimization procedure is based on the weighted average of the He abundance as determined from the five lines independently. We allow for the presence of underlying He I absorption through the assumption that it will be identical (in terms of equivalent width) for all of the He I lines. In addition, once the physical parameters have been determined by the minimization, all five values of y+(lambda) are used in a weighted mean to determine the final Y+.

4.3. Using 6 Lines

Adding lambda4026 as a diagnostic line increases the leverage on detecting underlying stellar absorption. This is because the lambda4026 line is a relatively weak line. However, this also requires that the input spectrum is a very high quality one. lambda4026 is also provides exceptional leverage to underlying stellar absorption because it is a singlet line and therefore has very low sensitivity to collisional enhancement (i.e., n) and optical depth (i.e., tau(3889)) effects.

Our procedure for this case is identical to the one above with the addition of the sixth line. By adding lambda4026 as a diagnostic line, we increase our dependence on the assumption of equal equivalent width of underlying absorption for all of the He I lines. Our philosophy is that it is most important to discover underlying absorption when it is present. If underlying absorption is important in an individual spectrum, conservatively, it may be better to reject the object from consideration from studies constraining the primordial helium abundance. If a solution implies significant underlying absorption, and all of the helium lines give the same abundance within errors, it may be taken as an endorsement of the assumption of equal EW of underlying He I absorption.

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