Continuing our series of papers on the three-dimensional ( 3-D ) structures of and accurate distances to Planetary Nebulae ( PNe ) , we present our study of the planetary nebula NGC 6781 .
For this object we construct a 3-D photoionization model and , using the constraints provided by observational data from the literature we determine the detailed 3-D structure of the nebula , the physical parameters of the ionizing source and the first precise distance .
The procedure consists in simultaneously fitting all the observed emission line morphologies , integrated intensities and the 2-D density map from the [ SII ] line ratios to the parameters generated by the model , and in an iterative way obtain the best fit for the central star parameters and the distance to NGC 6781 , obtaining values of 950 \pm 143 pc and 385 L _ { \odot } for the distance and luminosity of the central star respectively .
Using theoretical evolutionary tracks of intermediate and low mass stars , we derive the mass of the central star of NGC 6781 and its progenitor to be 0.60 \pm 0.03 M _ { \odot } and 1.5 \pm 0.5 M _ { \odot } respectively .