2.2.1. The Direct Method

Following Dinerstein (1990), the direct method can be broken down into 5 (easy) steps:

This step may seem trivial and self-explanatory. It assumes that you have taken proper care to reduce and calibrate your data. At the telescope, it is important to observe several standard stars (preferably from the HST spectrophotometric standards of Oke 1990). From my experience you can do a fairly good job just by following the standard procedures (with the addition of testing each step before moving on). I would recommend to the reader two guides available from NOAO (ftp: ftp.noao.edu; cd iraf/iraf/docs) "A User's Guide to CCD Reductions with IRAF" (Massey 1997) and "A User's Guide to Reducing Slit Spectra with IRAF" (Massey, Valdes, & Barnes 1992). Typically, one-dimensional spectra are extracted from long-slit (2-D) observations. Special care needs to be taken setting the extraction aperture width. The aperture should be sufficiently wide that small alignment errors do not give rise to systematic errors (this comes at a cost in signal/noise, but its worth the price). Additionally, one should avoid attempting "optimal" extraction. This works for stellar spectra that are inherently point sources and will be normalized, but not for extended nebulosities! And, finally, one should preferably integrate under the emission line profile (as opposed to fitting the line with a Gaussian profile). Fitting procedures can introduce systematic differences between high signal/noise and low signal/noise lines (and you will generally need to measure both accurately).

I'd like to insert yet another editorial comment here. I have had some experience trying to produce accurate emission line ratios from multi-order (echelle) spectrographs. Since the efficiencies change rapidly across the orders and since calibrating the relative efficiencies between orders is non-trivial, my experience tells me that emission line ratios from echelle spectrographs will never be as accurate as those derived from simple, long-slit spectrographs. I think that this is important, especially in the context of the design of spectrographs for the next generation of large telescopes - most of the planned spectrographs are echelle types (because of the ability to achieve high resolution and large wavelength coverage simultaneously). Thus, it is important to plan for long-slit spectrographs at (at least) a few of the large telescopes.

Because we know the theoretical emissivities of the recombination lines of H, and because there are a number of H recombination lines spread through the optical spectrum, it is possible to use the observed line ratios to solve for the reddening of the spectrum. The most recent H emissivity calculations are those provided by Hummer & Storey (1987), and, for the Balmer lines, these are very similar to the values calculated by Brocklehurst (1971). If one assumes a reddening law (f(), e.g., Seaton 1979), in principle, it is possible to solve for the extinction as a function of wavelength by measuring a single pair of H recombination lines. Values of C(H), the logarithmic reddening correction at H, can be derived from:

(2.1)

where I() is the intrinsic line intensity and F() is the observed line flux corrected for atmospheric extinction. Assuming a reddening law introduces a degree of uncertainty. Studies in our Galaxy show that the reddening law shows large variations between different lines of sight, but these variations are most important in the ultraviolet (Cardelli, Clayton, & Mathis 1989). Note also that the total measured extinction can have both Galactic and extragalactic components.

In practice it is best to measure several H recombination lines as this will provide both an estimate of the effect of stellar absorption underlying the emission lines and an estimate of the error in the measurement of the reddening. Due to the lack of H recombination lines in the near ultraviolet, it is possible to use He recombination lines in the same way. Here one can use the emissivities calculated by Smits (1991), which, in general, agree well with those calculated by Brocklehurst (1972), but do differ significantly in some cases (and thus, the Smits values are preferable).

There is a subtle problem with the reddening correction related to the distribution of the dust. The above method assumes a uniform "screen" of dust in front of the HII region. This is probably reasonable for reddening by foreground Galactic dust, but will not work so well if there is dust mixed within the HII region. Israel & Kennicutt (1980) pointed out that the extinction derived from Balmer line ratios tended to give lower values when compared to values derived from comparisons between a Balmer emission line and the thermal radio continuum emission. This implies clumpy extinction, internal extinction, or both. This was supported by detailed observations of Large Magellanic Cloud HII regions by Caplan & Deharveng (1986) and Balmer line - Brackett line - radio continuum emission comparisons of M101 HII regions by Skillman & Israel (1988). Mathis (1983) suggested that albedo variations may also be partially responsible for the optical - radio differences. While this effect is intrinsically interesting, it appears to have only a minor effect on abundance determinations; this is probably because the regions with the lowest amount of reddening will dominate the integral spectrum, so the lower values of reddening are probably appropriate. For a recent discussion of this problem, see Calzetti (1997).

Corrected emission line ratios should carry realistic errors. With CCD detectors, which can be configured to be linear to within measurable limits, it is possible to do a very accurate job of accounting for the different sources of error. Standard errors (L) for the line strengths (L) can be calculated from the following formula (where all numbers are in accumulated electrons):

(2.2)

where C₁ is the total number of counts (line plus continuum and sky), C₂ is the continuum, S is the sky background in a single row and n_o and n_s are the number of rows summed over for the object and the number of rows averaged for the sky, nAN² accounts for the readout noise, where n is the number of integrations, A is the area summed over in pixels, and N is the r.m.s. readout noise. The fifth term represents the reddening error, where C(H) is the error in C(H), and the sixth term accounts for errors in the flat fielding (estimated at 1%). The seventh term represents errors in the flux calibration, where F is the r.m.s. relative error for the fit to the flux calibration points for the standard stars (typically 2%). Leaving out the terms accounting for the errors in reddening or flux calibration will result in unrealistically small errors.

By "local physical conditions" one means the temperature and density of the gas. In fact, since the temperature is governed by the balance between the heating and cooling processes, and since the cooling is governed by different ionic species in different radial zones, one expects different ions to have different mean temperatures (cf. Stasinska 1990; Garnett 1992). While this is best treated with a complete photoionization model, a reasonable compromise is to treat the spectrum as if it arose in two different temperature zones, roughly corresponding to the O⁺ and O⁺⁺ zones (since the oxygen ions play a dominant role in the cooling, this is a reasonable thing to do).

The average density can be derived by measuring the relative intensities of two lines which arise from a split upper level. In the "low density regime" collisional de-excitation is unimportant and all excitations are followed by emission of a photon. The ratio of the fluxes then simply reflects the ratio of the statistical weights of the two levels. In the "high density regime'', where the level populations are held at the ratio of their statistical weights, the emission ratio becomes the ratio of the product of the statistical weights and the radiation transition probabilities. In the intermediate regime, near the "critical density" the line ratios are excellent density diagnostics. The best known is that of [S II] 6717 / 6731 which is sensitive in the range from 10² to 10⁴ cm^-3 and can be observed at moderate spectral resolution. At higher spectral resolution, one can use several other line pairs (e.g., [O II] 3726 / 3729).

In order to convert these line ratios into densities, one needs to know the energy level separations, the statistical weights of the levels, and the radiative and collisional excitation and de-excitation rates. Fortunately, one can use the five-level atom program originally written by De Robertis, Dufour, & Hunt (1987) which has been made generally available within IRAF by Shaw & Dufour (1995; SD95). This program has the additional great advantage that the authors have promised to keep the input atomic data updated. I recommend the SD95 article in PASP to any reader interested in abundances from emission lines as it covers much more detail than is allowed here. It is a well written and instructive article.

Figure 3. A diagram of the temperature sensitive emission line ratios for five ionic species observable in the optical/near-IR. These line ratios were calculated with the five-level atom program of Shaw & Dufour (1995) at an assumed density of 100 cm-3.

Deriving temperatures in HII regions is pleasantly simple. A glance at Figure 2 shows that the ratio of the emission from an upper level (an "auroral line", e.g., 4363) relative to the emission from a lower level (a "nebular line", e.g., 4959,5007) will be highly temperature sensitive. In Figure 3, I have used the program of SD95 to produce a comparison of five different temperature sensitive emission line ratios. Note that I have used observer axes (linear) as opposed to theoretician axes (logarithmic). This rather clumsy representation emphasizes the ranges of utility for the different line ratios (it can be difficult to accurately measure relative emission line ratios with differences exceeding 200!). Clearly the [O III] ratio is most useful at high temperatures, although the 4363 line will always be relatively weak in any HII region. On the other extreme, the [S II] ratio enjoys a wider temperature range of applicability, but actually loses much of its sensitivity to temperature above 15,000K (although there is still a 30% change from 15,000K to 20,000K - maybe the choice of linear axes isn't that great!). Of course, the ionization fraction will affect the absolute line strengths, so [S II] temperatures are rarely available for high temperature HII regions.

All of these ratios share the common problem that they are increasingly difficult to measure at low temperatures (high metallicities). For a sort of record in this game I direct the reader to Kinkel & Rosa (1994). They obtained a measurement of the [N II] 6584 / 5755 ratio of 200, implying a temperature of 6820K and an almost solar oxygen abundance in the HII region Searle 5 in M101.

Ionic abundances can be derived by comparing the strength of an ionic emission line to the strength of an H recombination line and then comparing that ratio to the ratio of theoretical emissivities. Stated as an equation:

(2.3)

where is the emissivity of the respective lines.

The theoretical emissivities can be calculated from the program of SD95. The emissivity for collisionally excited ions in the low-density limit is:

(2.4)

where N_i is the number density of the ion of interest, N_e is the electron density, h is Planck's constant, the frequency of the transition, the average electron collision strength from the lower level, g the statistical weight of the lower level, k Boltzman's constant, T the electron temperature, and the energy difference between the two electronic levels. For recombination, the emissivities are significantly more complex, but, in the end, their temperature and density dependences can be fitted with suitable power laws.

Normally, for observations of extragalactic HII regions, the observed recombination emission is restricted to H and He and the observed emission lines from the heavier elements are all collisionally excited. Because of this, absolute abundances (e.g., O⁺⁺/H⁺) will have a strong temperature dependence (and therefore the uncertainty can be dominated by the uncertainty in the electron temperature). Absolute abundances derived from recombination lines (e.g., He⁺/H⁺) will have very small temperature dependences. Similarly, relative abundances derived from two collisionally excited lines (e.g., N⁺/O⁺) will have relatively small temperature dependences. Peimbert, Storey, & Torres-Peimbert (1993) have discussed the utility of measuring absolute oxygen abundances using OII recombination lines; unfortunately, given the relative weakness of these lines, this technique will be limited to only the brightness Galactic nebulae.

When there are emission lines from all of the relevant ionic states, the total elemental abundance is just the sum of the ionic abundances. Unfortunately, this is rarely the case. This problem has given rise to a number of empirical formulae for converting ionic abundances into elemental abundances. Historically, some of these corrections were based on the near coincidences of ionization potentials. For example, consider oxygen:

(2.5)

In this case, O⁺ and O⁺⁺ are both observable in the optical (3727 and 4959, 5007). The unobserved O³ state is corrected for by assuming that the O³ zone is coincident with the He⁺⁺ zone (based on the coincidence of the second ionization potential of He of 54.4 eV with the third ionization potential of O of 54.9 eV) which is observable via the 4686 line. This is probably relatively safe (but this can be checked with far-IR measurements of [O IV] emission at 26 µm). On the other hand, similar ionization correction prescriptions have failed. This is due, in part, to the fact that photoionization cross-sections are complex functions and therefore ionization potentials can only serve as approximate guides. Thus, modern calculations of ionization correction factors rely on detailed photoionization models (e.g., Mathis 1985; Mathis & Rosa 1991). For very detailed discussions I direct the reader to the cases of sulfur (Garnett 1989), nitrogen (Garnett 1990), carbon (Garnett et al. 1995a) and silicon (Garnett et al. 1995b).

I close this discussion of the "direct method" by attempting to answer one of the questions which came from the class, "Just how accurately can one measure HII region abundances?" My answer: if one has a sufficiently high quality spectrum, it is possible to measure oxygen abundances in HII regions with accuracies of roughly 10 - 20%! (even smaller uncertainties are routinely encountered in the literature). This is certainly adequate for most astrophysical applications. Such a tremendously valuable tool can and should be used in a wide range of applications.