2. TEMPERATURE STRUCTURE

To a second order approximation we can characterize the temperature structure of a gaseous nebula by two parameters: the average temperature, T₀, and the root mean square temperature fluctuation, t, given by

(2.1)

and

(2.2)

respectively, where N_e and N_i are the electron and the ion densities of the observed emission line and V is the observed volume (Peimbert 1967; Osterbrock 1970).

To determine T₀ and t² we need two different methods to derive T_e: one that weights preferentially the high temperature regions and one that weights preferentially the low temperature regions. For example the temperature derived from the ratio of the [O III] 4363, 5007 lines, T_e(4363/5007), and the temperature derived from the ratio of the Balmer continuum to I(H ), T_e(Bac / H ), that are given by

(2.3)

and

(2.4)

respectively. It is also possible to use the intensity ratio of a collisionally excited line of an element p + 1 times ionized to a recombination line of the same element p times ionized, this ratio is independent of the element abundance and depends only on the electron temperature. By combining this ratio with a temperature determined from the ratio of two collisionally excited lines like T_e(4363/5007) it is also possible to derive T₀ and t².

To determine abundance ratios, T_e(4363/5007) has been used very often under the assumption that t² = 0.00. In the presence of temperature variations however, the use of T_e(4363/5007) yields O abundances that are smaller than the real ones (e.g. Peimbert 1967; Peimbert & Costero 1969). In general, abundance ratios derived from the ratio of a collisionally excited line to a recombination line are underestimated, while those derived from the ratio of two collisionally excited lines with similar excitation energies or from the ratio of two recombination lines, are almost independent of t². Nevertheless for some applications like the determination of the primordial helium abundance, which is based on the ratio of recombination lines, the errors introduced by adopting a t² = 0.00 value are very small but non-negligible.

From observations it has been found that 0.01 t² 0.09, with typical values around 0.04; while from photoionization models of chemically and density homogeneous nebulae it has been found that 0.00 t² 0.02, with typical values around 0.005. An explanation for the differences in the t² values predicted by photoionization models and those found by observations has to be sought.