Journal of Physical Studies 24(1), Article 1905 [8 pages] (2020) DOI: https://doi.org/10.30970/jps.24.1905 PHOTOIONIZATION MODELING OF PLANETARY NEBULAE WITH THE DUST GRAINS PRESENCE. II. DETERMINATION OF THE MASSES OF A NEBULAR ENVELOPE AND ITS PROGENITOR STAR USING ELECTRON TEMPERATURE IN THE CASE OF HOMOGENEOUS DISTRIBUTION OF THE NEBULAR MATTER A. R. Kuzmak1 , B. Ya. Melekh2  1Department for Theoretical Physics, Ivan Franko National University of Lviv, 12, Drahomanov St., Lviv, UA-79005, Ukraine, e-mail: andrijkuzmak@gmail.com, 2Department for Astrophysics, Ivan Franko National University of Lviv, 8, Kyrylo and Mefodiy St., Lviv, UA-79005, Ukraine, e-mail: bmelekh@gmail.com Received 26 December 2019; in final form 13 February 2020; accepted 17 February 2020; published online 02 April 2020

The study has calculated the grid of photoionization models for the homogeneous planetary nebulae in the Milky Way galaxy with the dust grain along the evolutionary tracks of their central stars. Based on these calculations, the age dependencies of the electron temperature of doubly-ionized oxygen atoms (O III) were obtained.

For nebulae with low-mass progenitor stars (up to $\approx 1.5\,M_☉$), these age dependencies contain domains with an almost constant temperature. However, this temperature is sensitive to the masses of both the progenitor star and the nebular envelope. Thus, the behavior of the electron temperature obtained for the doubly-ionized oxygen ionization zone can be used to determine the masses mentioned above. Here we test our assumptions in the case of the Milky Way homogeneous planetary nebulae. As a result, a method for the determination of the masses of a planetary nebula envelope as well as its progenitor star using the value of the electron temperature within the doubly-ionized oxygen ionization zone is proposed. This method consists of two steps. At the first step, for a given planetary nebula we define the ratio of the emission line intensities of ionized sulfur atoms $I_{\rm [SII]λ 6716}/I_{\rm [SII]λ 6731}$. This ratio is sensitive to the age of the nebula, which allows us to obtain the limit on its age. At the second step, based on the temperature diagnostic ratio $λ 4363$\AA$/(λ 4959$\AA$ + λ 5007$\AA) between the emission line intensities of the doubly-ionized oxygen in the planetary nebula envelope, the value of the electron temperature is determined.

Taking into account the limit on the age of the planetary nebula mentioned above, the obtained result is compared with the age dependencies of the averaged electron temperature within the nebular gas for different masses of the progenitor star and the planetary nebula envelope.

Matching this value with a specific curve within the respective age range allows us to determine the masses of both the progenitor star and the planetary nebula envelope. This procedure is verified for two cases of the electron temperature determination: using the method of crossing of $n_{\rm e}-T_{\rm e}$ dependences obtained for various diagnostic ratios between emission lines as well as the popular two-zones Pagel's method (the so-called $T_{\rm e}$-method), that is based on diagnostic ratios between auroral and nebular emission lines of the doubly-ionized oxygen. As a result, using this method the mass of a progenitor star is determined with some accuracy. However, the mass of a planetary nebula envelope is determined within the accuracy of $\pm 0.1\,M_☉$.

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