6.2.2 Temperature structure

Table 8 represents mean electron temperatures weighted by ionic abundances for Models 1 and 2, as well as the ring region and the inside region of the PN. We also see each ionic temperature corresponding to the temperature-sensitive line ratio of a specified ion. The definition for the mean temperatures was given in Ercolano et al. (2003b); and in detail by Harrington et al. (1982). Our model results for $T_{\rm e}[$O III$]$ compare well with the value obtained from the empirical analysis in §4. Fig. 11 (top left) shows $T_{\rm e}$ obtained for Model 2 (adopted best-fitting model) constructed in $45 \times 45 \times 7$ cubic grids, and with the ionizing source being placed in the corner. It replicates the situation where the inner region has much higher $T_{\rm e}$ in comparison to the ring $T_{\rm e}$ as previously found by plasma diagnostics in §4. In particular the mean values of $T_{\rm e}[$O III$]$ for the ring (torus of the actual nebula) and the inside (spheroid) regions are around $12\,000$ and $15\,000$K in all two models, respectively. They can be compared to the values of Table 4 that is $T_{\rm e}[$O III$]=12\,300$K (ring) and $\lesssim20\,000$K (interior). Although the average temperature of $T_{\rm e}[$N II $]\simeq11\,700$K over the entire nebula is higher than that given in Table 4, the average temperature of $T_{\rm e}[$O III $]\simeq13,000$K is in decent agreement with that found by our plasma diagnostics.

It can be seen in Table 4 that the temperatures for the two main regions of the nebula are very different, although we assumed a homogeneous elemental abundance distribution for the entire nebula relative to hydrogen. The temperature variations in the model can also be seen in Fig. 11. The gas density structure and the location of the ionizing source play a major role in heating the central regions, while the outer regions remain cooler as expected. Overall, the average electron temperature of the entire nebula increases by increasing the helium abundance and decreasing the oxygen, carbon and nitrogen abundances, which are efficient coolants. We did not include any dust grains in our simulation, although we note that a large dust-to-gas ratio may play a role in the heating of the nebula via photoelectric emissions from the surface of grains.