For 4 He, we follow an analogous procedure to that described above. We again start with a set of observed quantities: line intensities I() which include the reddening correction previously determined and its associated uncertainty which also includes the uncertainty in C(H); the equivalent width W(); and temperature t. The Helium line intensities are scaled to H and the singly ionized helium abundance is given by
where E() / E(H) is the theoretical emissivity scaled to H. The expression (C1), also contains a correction factor for underlying stellar absorption, parameterized now by aHeI, a density dependent collisional correction factor, (1 + )-1, and a florescence correction which depends on the optical depth . Thus y+ implicitly depends on three unknowns, the electron density, n, aHeI, and .
To be definite, we list here the necessary components in expression (C1). The theoretical emissivities scaled to H are taken from Smits (1996):
Our expressions for the collisional correction , are taken from Kingdon & Ferland (1995). We list them here for completeness. They are:
where D = 1 + 3130n-1 T4-0.50. The corrections for florescence are given in terms of the optical depth for the He I 3389 line. We use the IT98 fit of the Robbins (1968) enhancement factors:
f (4026) is not given by IT98, but is assumed to be 1 because it is a singlet line (as is the case for 6678).
Once the individual values for y+() are determined, we can begin the process for self-consistently determining the physical parameters. As described in the text, we may wish to consider 3, 5, or 6 different 4 He emission lines. Depending on the number of lines used, we next determine the average helium abundance.
This is a weighted average, where the uncertainty () is found by propagating the uncertainties in the observational quantities stemming from the observed line fluxes (which already contains the uncertainty due to C(H), the equivalent widths, and input temperature. Since the average, , depends on the parameters, n, and aHeI, we must make an initial estimate for these.
From , we can define a 2 as the deviation of the individual He abundances y+() from the average,
We then minimize 2, to determine n, aHeI, and . Uncertainties in the output parameters are determined as in the case for aHI and C(H), that is by varying the outputs until 2 = 1. Propagation in the latter uncertainties give us a reasonable handle on the systematic uncertainties in our final result for y+.
This procedure differs somewhat from that proposed by IT98, in that the 2 above (C6) is a straight weighted average, whereas IT98 minimize the differences between ratios of He abundances from pairs of He I lines (referenced to one wavelength, typically 4471). When the reference line is particularly sensitive to a systematic effect such as underlying stellar absorption, the uncertainty propagates to all lines this way. In our case, the individual uncertainties in the line strengths are kept separate.
Finally, as in the case for the hydrogen lines, we have performed a Monte-Carlo simulation of the data to test the robustness of the solution for n, aHeI, and from the 2 minimization and the true uncertainty in these quantities. As before, starting with the observational inputs and their stated uncertainties, we have generated a data set which is Gaussian distributed for the 6 observed He emission lines (plus the temperature). From each distribution, we randomly select a set of input values and run the 2 minimization. The selection of data is repeated 1000 times. We thus obtain a distribution of solutions for n, aHeI, and , and we compare the mean and dispersion of these distributions with the initial solution for these quantities.