3.1. The Helium Abundance and Temperature Variations

The N(He⁺) / N(H⁺) ratios can be derived from equations of the type

(3.5)

where the effective recombination coefficients, , for hydrogen and helium have been computed by Hummer & Storey (1987) and Smits (1994), and I(_nm)_R is the pure recombination intensity that has to be obtained from the observed intensity, I(_nm). Radiative transfer effects and collisions from the 2³S level affect I(_nm) and have to be estimated.

The collisions to recombinations ratio of a helium line is given by

(3.6)

where is the effective collisional coefficient that depends strongly on T_e and

(3.7)

where ionizations from the 2³S level have been neglected (Kingdon & Ferland 1995).

The latest estimates of the I(_nm)_C / I(_nm)_R values for the different helium lines are those by Kingdon and Ferland (1995) based on the 29-state ab initio computation for collisions to He⁰ states with n 5 by Sawey & Berrington (1993) and the helium recombination coefficients by Smits (1994).

Robbins (1968) and Robbins & Bernat (1973) have computed the effect that atomic absorption has on the He I line intensity ratios. Robbins has used as a parameter the He I 3889 optical depth, (3889), for the triplet series. From the computations by Robbins and Cox & Daltabuit (1971) and the ratio of two He I lines it is possible to determine (3889) and consequently the effect of the radiation transfer on the triplet lines. A similar procedure can be followed for the singlet lines. It is found that the radiation transfer effect is almost negligible for 4472, 5876 and 6678; alternatively it is large for 3889, 7065 and 10830.

The He⁺ / H⁺ values derived from different helium lines, based on equations (3.5) and (3.6) for t² = 0.00, do not agree for a given object, particularly for those PN with high N_e and T_e values (e.g. Peimbert & Torres-Peimbert 1987a, b; Peña et al. 1995). The differences imply that the collisional effects have been overestimated. This problem has at least four solutions: a) the He line intensities have not been properly measured, b) there is an unknown process depopulating the 2³S level, c) the density has been overestimated (see equation 3.7), d) the temperature has been overestimated, i.e. t² 0.00.

Even if 10830 is affected by telluric absorption (Kingdon & Ferland 1991), I consider that possibility a) above plays a minor role in well observed objects. Possibility b) suggested by Peimbert & Torres-Peimbert (1987a, b) has been studied by Clegg and Harrington (1989) who find that photoionization can reduce the N(2³S) population by as much as 25% in compact optically-thick PN; alternatively for the vast majority of the observed PN and for giant extragalactic H II regions the effect is very small and can be neglected. Possibility c) could be important for objects with N_e 3000 cm^-3, but for PN with N_e >> 3000 cm^-3 is not important (see equation 3.7). Finally, possibility d) will be explored further.

In Figure 1 we present the He⁺ / H⁺ abundances for the type I PN Hu 1-2 (Peimbert, Luridiana & Torres-Peimbert 1995b) based on three He I lines that are almost unaffected by radiative transfer effects. The observations correspond to the average of three different regions of the nebula. The temperature at which the three lines reach the same He⁺ / H⁺ ratio is about 13 000 K, considerably smaller than that given by < T_e > (4363/5007) that amounts to 18 800 ± 600 K; this result implies a very large t² value. The density for the observed regions of Hu 1-2, < N_e > = 4 900 cm^-3, is higher than the critical density and errors in N_e possibly do not play a role in explaining the discrepancies in the He⁺ / H⁺ determinations. Peña et al. (1995) from a similar study of N66 also find that lower T_e and N_e values than those given by [O III], [O II] and [Ar IV] lines are needed to derive the same He⁺ / H⁺ abundances from the 4472, 5876 and 6678 lines.

Figure 1. N(He+) / N(H+) = y+() versus < Te > diagram for the type I PN Hu 1-2, where < Te > stands for the average of three different regions of the nebula (Peimbert et al. 1995b).

The I(3889) / I(4472), I(7065) / (4472) and I(10830) / (4472) ratios depend on (3889) and T_e. The T_e affects weakly the recombination coefficients but strongly the collisional excitation effects from the 2³S level. The relationship between (3889) and T_e for any line ratio is derived by comparing the observations with the computations by Robbins (1968). The three line ratios depend on different functions of (3889) and T_e, therefore the combination of two line ratios will provide us with a unique pair of (3889) and T_e values.

In Figure 2 we present a (3889) versus T_e diagram for NGC 7009 (Peimbert et al. 1995b) where we have adopted N_e = 6 000 cm^-3. From the I(3889) / I(7065), I(3889) / I(10830) and I(7065) / I(10 830) crossings we obtain T_e values of 8 000 K, 6 700 K and 6 300 K respectively. For this object T_e (4363)/(5007) is equal to 10 000 K, the differences between T_e(4363)/(5007) and the crossing temperatures are mainly due to the t² value (which is similar to that derived by Liu et al. 1994); while the smaller T_e values derived from the two I(10830) crossings relative to that derived from the I(3889) / I(7065) crossing probably is due to telluric absorption and dust destruction inside NGC 7009 of 10830 photons (Clegg & Harrington 1989; Kingdon & Ferland 1991, 1993).

Figure 2. (3889) versus Te diagram for NGC 7009. The solid lines stand for the I() / I(4472) ratio, the dotted lines to the right and to the left at a given (3889) correspond to ratios 10% higher and 10% lower than observed, respectively (Peimbert et al. 1995b).