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We present here only a brief overview of abundance determining methods relevant to emission-line objects. A list of prominent nebular emission lines is presented in Table A1, where we provide the ion, wavelength, and dominant excitation mechanism for each line. Readers interested in greater detail are urged to consult information in Spitzer (1978), Aller (1984), Osterbrock (1988, 1989), and Williams & Livio (1995). The atomic data used for the abundance calculations are reviewed by Butler (1993).

Table A1. Prominent Emission Lines In Nebulae

Ion Wavelength (Å) Excitation a Ion Wavelength (Å) Excitation

C IV 1549 C H I 4861 R
He II 1640 R [O III] 4959,5007 C
C III] 1909 C [N II] 5199 C
[OII] 3727 C HeI 5876 R
[NeIII] 3869, 3968 C [OI] 6300, 6360 C
HeI 3889 R [SIII] 6312 C
[SII] 4072 C [NII] 6548, 6584 C
HI 4101 R HI 6563 R
HI 4340 R [SII] 6716, 6731 C
[OIII] 4363 C [ArIII] 7135 C
HeI 4471 R [OII] 7325 C
HeII 4686 R [SIII] 9069, 9532 C

a Excitation mechanism, where C = collisional, R = recombination.

The basic method for obtaining the abundance of an element in emission-line objects comprises two steps: (1) determine abundances of the ions of that element whose emission lines are directly observable, and (2) adjust the total of the ionic abundances by a factor which accounts for ions of the same element which are unobservable.

Consider step 1. An ionic abundance relative to H+ is related to the observed strength of an emission line integrated over wavelength, corrected for interstellar reddening, and expressed relative to the Hbeta strength, Ilambda / IHbeta, through the respective reaction rate coefficients, epsilonlambda and epsilonHbeta in ergs cm3 s-1 sr-1, such that

Equation A1 (A1)

where Ni, NH++, and Ne are local number densities of the ion giving rise to the line lambda, H+, and electrons, respectively. The integrals arise because local products are integrated along the line of sight. Note that the rates are functions of the local electron temperature (Te) and density (Ne), although a good simplifying assumption is that these are constant within the nebular regions actually dominated by the ions in question. Te is usually determined using the line strength ratio of two lines such as [O III] lambda4363 and lambda5007 whose upper energy levels are relatively far apart. Ne is derived from the ratio of two lines such as [S II] lambda lambda6716, 6731 whose upper energy levels are closely spaced but the transitions differ significantly in their sensitivities to collisional deexcitation.

Adding the observed ionic abundances for an element together gives us a subtotal which differs from the desired total by the abundances of the ions whose emission lines are not observed. Thus, in step 2 above we determine an ionization correction factor, ICF(X), for element X by which we multiply the subtotal to produce the total elemental abundance relative to H+. Mathematically, the number density of an element NX / NH can be expressed as

Equation A2 (A2)

Ionization correction factors may be inferred from model simulations of nebulae or estimated by assuming that ions with similar ionization potentials are present in the gas in similar ratios to their total abundances. Thus, because the ionization potentials of O+2 and He+ are 54.9 and 54.4 ev, respectively, the total abundance of unobservable (in the optical) higher ionization stages such as O+3 and beyond with respect to total O is similar to relative amounts of He+2, an optically observable ion, with respect to total He. A good compilation and discussion of a broad range of ICFs may be found in the appendix of Kingsburgh & Barlow (1994). Additionally, model grids such as those by Stasinska & Schaerer (1997) may be used to derive ICFs for a nebula assuming the central star temperature is known.

The above method breaks down most frequently for metal-rich nebulae with low equilibrium temperatures, in which case auroral lines such as [O III] lambda4363 are too weak to measure, and thus the temperature cannot be determined. One way around the problem is to calculate a detailed photoionization model of the nebula using input abundances and other physical parameters which produce an output set of line strengths closely matching the observed ones. Actual abundances are then inferred from the model input. A simpler solution is the "strong-line method," which uses a composite of strong, observable emission line strengths whose value tracks an abundance ratio. The most significant example is R23 ident ([O III] + [O II]) / Hbeta, first introduced by Pagel et al. (1979), which comprises the sum of [O III] and [O II] nebular line strengths relative to the strength of Hbeta and is related in a complicated but understandable way to the total oxygen abundance O/H. This method has been refined and discussed more recently by Edmunds & Pagel (1984), Edmunds (1989), and McGaugh (1991), and of course it is not problem-free. Because the metal-rich portion of this relation must currently be calibrated with models, uncertainties arise from parameter choices such as relative depletion (Henry 1993; Shields & Kennicutt 1995) and gas density (Oey & Kennicutt 1993). Finally, analogous methods for obtaining N/O and S/O are presented by Thurston, Edmunds, & Henry (1996) and Díaz (1999), respectively.

Finally, the accuracy of abundances in emission-line systems is threatened by the proposed existence of small-scale temperature fluctuations along the line of sight, first described by Peimbert (1967). In this picture, an electron temperature measured with forbidden lines is actually overestimated when fluctuations are present but ignored. This in turn causes an underestimation of an abundance ratio such as O+2/H+ when it's based upon a forbidden/permitted line ratio such as [O III] lambda5007 / Hbeta. When ratios of permitted lines are used the effect is minimal and so abundances inferred from permitted/permitted line ratios are unaffected and systematically higher than abundances from forbidden/permitted ratios. Temperature fluctuations have been used to explain, among many other things, the significant discrepancy in planetary nebula carbon abundances (Peimbert, Torres-Peimbert, & Luridiana 1995), where those determined using C II lambda4267 / Hbeta, say, are often several times greater than abundances inferred from C III] lambda1909 / Hbeta. Esteban et al. (1998) found the effect to be small in the Orion Nebula, while Liu (1998) found a large effect in the planetary nebula NGC 4361, although it was insufficient for explaining the discrepancy between carbon abundances from recombination and collisionally excited lines. The issue of temperature fluctuations is an important one, albeit unresolved. Further details can be found in Peimbert (1995), Mathis, Torres-Peimbert, & Peimbert (1998), and Stasinska (1998).

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