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 temperature (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 temperatures. The results also show that for high stellar temperatures the He II Zanstra temperature underestimates the stellar temperature, even for high optical depths. The stellar temperature, as well as the optical depth, can be obtained from a Zanstra temperature ratio (ZR) plot ZR = TZ (He )/TZ (H) versus TZ(He II). The effects of departures from a blackbody spectrum, as well 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 TZ(He II) only provides lower limits for T*. In order to obtain better values for the optical depth and T*, we propose the use of the line intensity ratio He /He versus TZ(He II) diagram.