Temperature of the central stars of planetary nebulae and the effect of the nebular optical depth

The effect of the nebula optical depth on the determination of the temperat ure (T-*) of the central stars in planetary nebulae is discussed. Based on photoionization models for planetary nebulae with different optical depths, we show, quantitatively, that the details of the distribution of the H and He II Zanstra temperatures are mainly explained by an optical depth effect ; in particular, that the discrepancy is larger for low stellar temperature s. The results also show that for high stellar temperatures the He II Zanst ra temperature underestimates the stellar temperature, even for high optica l depths. The stellar temperature, as well as the optical depth, can be obt ained from a Zanstra temperature ratio (ZR) plot ZR = T-Z(He II)/T-Z(H) ver sus T-Z(He II). The effects of departures from a blackbody spectrum, as wel l as of the He abundance in the nebulae, are also discussed. For nebulae of very low optical depth and/or high stellar temperature the distribution ZR versus T-Z(He II) only provides lower limits for T-*. In order to obtain b etter values for the optical depth and T-*, we propose the use of the line intensity ratio He II/He I versus T-Z(He II) diagram.