3. Comparison with spectroscopic parallax
The analysis of the stellar spectra gives values for the effective temperature () and the surface gravity g. To derive a distance from these quantities a model of the atmosphere is necessary, as well as the stellar mass and the magnitude corrected for reddening. Since all three stars have been studied by Mendez et al. (given in Table 3), the model atmospheres calculated by these authors will be used. This leads to the following relation between surface gravity and distance (Mendez et al., 1988b):
where M is the stellar mass, g the surface gravity (cm s-2), the visual magnitude of the star corrected for extinction, and d is the distance in Kpc. The extinction correction is small, and is based on the measured , assuming = -0.38 (see Pottasch, 1996). F is the stellar flux of a model atmosphere in the visual in units of 108 erg cm-2 s-1 . To a first approximation F depends only on the effective temperature, and its value is given in Table 2.
Table 2. Values of Eddington flux at 5480 Å
We have chosen to make the comparison by first converting the Hipparcos distance into a surface gravity. The results are given in Table 3, where the second column gives the Hipparcos distance, the third column the effective temperature of the star. The fourth column gives the gravity determined from Eq. (1), assuming M = 0.6 M . The values of T given are taken from the spectroscopic determinations. The values could have an error of 30%, which is the difference between what Hoare et al. (1995) and Mendez et al. (1992) give for the central star of NGC 1360. Incidentally, Hoare et al. (1995) notice the difference, and explain it by noting that the model profiles for the H line from the = 80 000K and = 110 000K models are almost identical, and state that it was pure chance that Mendez et al. picked the lower temperature. Hoare et al. were able to distinguish between the two values by fitting more than 6 Balmer line profiles. In the case of the other two central stars only a single value of temperature is available. Since in the analysis the temperature and the gravity are coupled, if one is wrong, so is the other. If the gravity should be increased, the temperature should be increased as well.
Table 3. Comparison of Hipparcos and spectral data
The values of surface gravity reported in the literature are given in the last column of Table 3. Those from Mendez et al. (1988a and 1992) and Herrero et al. (1990) are found by fitting the observed H line profile to an NLTE model. The range of values given by Hoare et al. (1995) are from comparing ROSAT X-ray measurements with NLTE models, none of which was completely satisfactory. The value given by Hoare et al. (1996) is from a comparison of EUV (70 to 760 ) measurements with NLTE models.
The conclusions which can be drawn from Table 3 are subtle. Each of the PN will be discussed separately. Consider first NGC 1360. Here the parallax error is large and the gravity has sufficient error to encompass any of the literature values listed in Table 3 for this object. However, we feel that the evidence points to a more likely value of log g between 6.0 and 6.3; the value of log g = 5.4 is seen to be 0.5 too low.
The parallax of A 36 is better determined although the error is still large: the distance being between 150 pc and 600 pc. Even at the largest distance the derived gravity (log g = 5.6) is still somewhat high compared to the value of 5.3 0.2 given by Herrero et al. (1990). The parallax of PHL 932 has a much smaller error and the distance is between 84 and 160 pc, which leads to a log g of between 6.5 and 7.0 assuming a mass M = 0.6M . This is much higher than the value of 5.5 0.2 given by Mendez et al. (1988a). These authors have suggested that the mass is much lower. Using the Hipparcos distance, the gravity derived from Eq. (1) can be reduced to log g = 5.5 by lowering the mass by a factor 20. This would reduce the stellar mass to M = 0.03 M . Such low mass is unlikely, since such a star would evolve very slowly and would still be on the main sequence. In that case it should not be so hot. Mendez et al. (1988a) suggest that it is unlikely that the mass of the star could be less than 0.2M . With this mass, a value of log g = 6.3 would be derived from the Hipparcos distance, still considerably higher than found from the spectroscopic analysis of Mendez et al.
© European Southern Observatory (ESO) 1998
Online publication: November 24, 1997