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The only way to test big bang nucleosynthesis (BBN) and therefore cosmology at an age of order seconds to minutes, is through the observational abundances of the light elements D, 3 He, 4 He, and 7 Li (see. e.g., Olive, Steigman, & Walker 2000). Because there are no measurements of 3 He at very low metallicity (i.e., significantly below solar) at this time, a higher burden is placed on the remaining three elements. The measurements of D/H in quasar absorption systems are very promising (Burles & Tytler 1998a; 1998b), although not all data agree (Webb et al., 1997; Levshakov, Tytler, & Burles 1998; Tytler et al. 1999). Similarly, 7 Li measurements are continually improving (Ryan, Norris & Beers 1999) and systematic uncertainties are being reduced (Ryan et al. 2000), but the accuracy of the primordial abundance determinations for 7 Li are not probably not much less than a factor of 2. Testing the theory of BBN requires reliable abundances of at least two isotopes. Unlike the other light element abundances, in order to be a useful cosmological constraint, 4 He needs to be measured with a precision at the few percent level. Thus, the determination of the 4 He abundance with improved accuracy continues to be of prime importance to cosmology.

To date, the most useful 4 He abundance determinations are made by observing helium emission lines in HII regions of metal-poor dwarf galaxies. These measurements, which span metallicities ranging down to 1/50th of the solar oxygen abundance, all show 4 He abundances, Y, between 22 and 26% by mass. This is one of the strongest indications that the majority of the 4 He observed in these systems is in fact primordial and that BBN occurred. At the next level of precision, however, it is necessary to be able to extract a primordial abundance, Yp, from these data (e.g., Pagel et al. 1992, hereafter PSTE). The most common method to determine Yp is by means of a linear regression with respect to a tracer element (Peimbert & Torres-Peimbert 1974; 1976) such as oxygen or nitrogen (other methods such as a Bayesian analysis gives similar results, Hogan et al. 1997). To first order, we expect that along with the stellar production of heavy elements, there is a component of stellar contamination of primordial He. The uncertainty in the primordial abundance of 4 He due to this contamination and its exact relationship to the production of heavy elements is reduced by observing the lowest metallicity objects. Currently there is some controversy concerning the best estimate of Yp. Izotov & Thuan (1998b, hereafter IT98) assembled a sample of 45 low metallicity HII regions, observed and analyzed in a uniform manner, and derived a value of Yp = 0.244 ± 0.002 and 0.245 ± 0.001 (with regressions against O/H and N/H respectively). This value is significantly higher than the value of Yp = 0.228 ± 0.005 derived by PSTE. Analysis based on the combined available data (Olive & Steigman 1995; Olive, Skillman, & Steigman 1997; Fields & Olive 1998) yield an intermediate value of 0.238 ± 0.002 with an estimated systematic uncertainty of 0.005. Peimbert, Peimbert, & Ruiz (2000, hereafter PPR) have derived a very accurate helium abundance for the HII region NGC 346 in the Small Magellanic Cloud, and from this they infer a value of Yp = 0.2345 ± 0.0026. These different results depend, in part, on differences in the analyses of the observations. Thus, it is important to better understand any systematic effects that may result due to different analyses methods. Furthermore, as one can plainly see, the differences in the various determinations of the 4 He abundance appears to be many sigma. Thus it is clear that present systematic errors have been underestimated, and the main goal of this paper is to specify methods to better quantify and reduce the systematic uncertainties in 4 He abundance determinations.

Of course, the degree to which we can make an accurate determination of the primordial He abundance ultimately depends on our ability to extract accurate 4 He abundances from individual extragalactic HII regions. All of the information comes from the relative strengths of the emission lines of He I and H I, and the emission lines of heavier elements such as oxygen, nitrogen, and sulfur. To determine a 4 He abundance from the emission line intensities, it is necessary to determine the physical characteristics of the HII region. The electron temperature of the HII region is usually determined from the temperature sensitive ratio of [O III] emission lines (but see PPR). Electron densities can be derived from the ratios of [S II] and [O II] lines, although these may not be favored (see Izotov et al. 1999 and also section 2.4). While the relative H I and He I emissivities have very small dependencies on the electron densities, certain He I emission lines have an enhanced density dependence due to the collisional excitation of electrons out of the metastable 2S state. Additionally, some He I emission lines are subject to enhancement or diminuation through the radiative transfer effects of absorption or florescence.

One also needs to ascertain whether or not neutral helium (or neutral hydrogen) corrections are important (e.g., Shields & Searle 1978; Dinerstein & Shields 1986; Viegas, Gruenwald, & Steigman 2000). Finally, corrections due to possible effects of underlying He I stellar absorption in the spectra must be considered, though in the past, these have usually been neglected. Underlying He I absorption was shown to be an important effect for the NW region of I Zw 18 (Izotov & Thuan 1998a). Skillman, Terlevich, & Terlevich (1998) have demonstrated that the effects of underlying He I absorption may be more important than claimed by IT98 and may explain some of the "anomalous" line ratios observed by them (which led to the rejection of certain objects from the linear regressions used to determine Yp).

In some studies (e.g., Skillman & Kennicutt 1993; Skillman et al. 1994), 4 He abundances determinations were based on a single emission line at lambda6678. This line was deemed preferable as it is less subject to the effects of collisional enhancement relative to the stronger He I lines at lambda4471 and lambda5876 (cf. Pagel & Simonson 1989). Its proximity to Halpha also means that the ratio lambda6678 / Halpha is practically unaffected by a reddening correction. However, there is always a danger relying on a single emission line. Fortunately, other He I emission lines are available. The three lines lambda4471, lambda5876, and lambda6678 are all relatively insensitive to density and optical depth effects. This means that, on the one hand, the conversion of these line strengths to a helium abundance can be done with greater certainty. On the other hand, they do not provide a reliable estimate of the either the optical depth or density. The latter is known to make a correction of order 1% to 5% (depending on both the density and the temperature) due to collisional excitations. Nevertheless, one could make a case for using only these lines to determine the helium abundance.

Recently, a "self-consistent" approach to determining the 4 He abundance was proposed by Izotov, Thuan, & Lipovetsky (1994, 1997, hereafter ITL94, ITL97) 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 very important for lambda7065). This is the method used by IT98 in their most recent estimate of Yp. 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 (see Section 5).

In this paper, we will attempt to better quantify the true uncertainties in the individual helium determinations in extragalactic HII regions. After a brief discussion of the available observables needed in the determination of y+ = He+ / H+, we present methods for determining the reddening correction, C(Hbeta), the degree of underlying absorption in H I and He I, and ultimately y+. In addition we will show specifically how various uncertainties in measured quantities affect y+. In section 4, we will describe several alternative methods for deriving the helium abundance based on 3 to 6 emission lines. Here we will investigate the utility of adding a sixth He I line, lambda4026, which may be used to make a quantitative correction for the presence of underlying stellar He I absorption. For various cases based on several emission lines, we will test the minimization procedure (with respect to the recombination values) by calculating Monte Carlo realizations of the input data. This enables us to check the stability of any given solution of the minimization and better estimate the uncertainties in our result. In section 5, we present some examples of synthetic data to demonstrate the power of the method and the dangers of systematic uncertainties in the observed He I line strengths. Our conclusions and prospects for accurate helium determinations will be given in section 6.

The goal of this paper is to explore different analysis methodologies and to promote particular observational and analysis techniques. In the future, we will apply the recommended methods to both new observations and other observations reported in the literature.

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