## 2. Derivation of the PNN masses## 2.1. Summary of the methodThe method is extensively described in GST97, and we give here only
the essentials. The basic idea - shared by all the methods for
deriving PNN masses - is that central stars of different masses
differ in their basic properties (effective
temperature and total luminosity
) at any time Our approach is to compare the properties of the observed objects
with those of simple models which receive exactly the same treatment
as the observations in the estimation of ,
and In the traditional H-R diagram, if the stellar temperature and
luminosity are obtained by the Zanstra method using the Shklovsky
distance, this amounts to comparing the positions of the objects to
"apparent" theoretical tracks for stars of different masses. These
apparent tracks differ significantly from the pure stellar tracks (see
Fig. 1a below), and are strongly dependent on
and (see GST97). This is why the conclusions on
PNN masses derived by authors using the H-R diagram (e.g. Kaler et al.
1990, Stanghellini et al. 1993, Cazetta & Maciel 1994) need to be
reexamined. A similar comment applies to the (,
Many PN have their expansion velocity measured. Such objects can be directly compared with a grid of apparent tracks built with the same as observed. The only unknown parameter in our approach is , and the estimated value of depends on the assumed . The advantage of our method is that it is self-consistent and that all the assumptions can be clearly stated. ## 2.2. The sample of PNWe have considered all the PN for which the relevant data (stellar apparent visual magnitude , total nebular flux F(H ), nebular angular radius and expansion velocity ) are available in the literature and of good quality, and for which the central stars are not known to be close binaries or to have H-poor atmospheres. The observational data, together with the references, are listed in Tylenda & Stasiska (1994), and we use the same extinction correction procedure as in that paper. This results in a sample of 125 objects. Fig. 1a shows all these objects in the (,
) plane, where is the
Zanstra temperature of the star derived from the H
line, and is its Zanstra
luminosity calculated with the Shklovsky distance. Superimposed are
the apparent tracks for central stars of masses 0.565, 0.605 and
0.645M , surrounded by a nebula of Fig. 1b is similar to Fig. 1a but in the (, ) plane, where is the absolute stellar visual magnitude, and the radius of the ionized part of the nebula, both calculated with the Shklovsky distance. Fig. 1c shows the same, but in the (, ) plane introduced by GST97, where is the nebular surface brightness in H , and is defined as , where is the stellar flux in the V band. Note that not only Fig. 1c is distance independant. Figs. 1a and 1b are also independent of the true distances, since they use the "Shklovsky distance", which is a mere combination of observed parameters.
## 2.3. The PNN masses of our sampleAs stated above, the only parameter on which we do not have any
observational constraint in our representation is the total nebular
mass. We derive the central star masses, ,
assuming a total nebular mass of 0.2M .
Table 1 (available also in electronic form) gives the values of
for all the PN of our sample that do not appear
in Table 1 of GST97. If more than one value of
is possible for a given object, the lowest value
is adopted, and the upper limit compatible with the data is indicated
in parenthesis. As an information, Table 1 also lists, in the
same format as Table 1 of GST97, the values of the following
parameters: the morphological type as defined in GST97
The uncertainty in PNN mass resulting from a change in by a factor two downwards or upwards is represented in Fig. 2a. We see that, for most objects, it becomes important only if is larger than about 0.65M . At high central masses, the uncertainty may become very large if the PN is in an evolutionary stage corresponding to a region of crowding of apparent tracks. The effects of the observational uncertainties in F(H ), and on the determination of are of 0.01M at most for 0.65M . They are much larger at higher central star masses. One may consider that for 0.65M , the problem of deriving central star masses with our method becomes degenerate, and that masses above 0.65M are very uncertain. Additional causes of uncertainty are of course deviations from our
adopted model. For instance, masses derived under the assumption of a
covering factor smaller than one are larger than if the covering
factor is assumed equal to one. Fig. 2b shows the effect of adopting a
covering factor of 0.3 for all the PN. While this leaves practically
unchanged the derived central star masses at
0.56M and some of
those at about 0.60M , it may increase
significantly the derived masses in other cases, especially those with
0.65M
. Actually, the true covering factor of many PN
is probably close to one, so that the errors are not as large as Fig.
2b might express. Covering factors drastically smaller than one are
expected for some bipolar PN (those of subclass
In Fig. 3, we plot the values of the evolutionary time , as a function of (assuming =0.2M ), for the low objects of our sample. We see a trend similar to the one found in GST97. PN with small (around 0.56M ) are seen at ages between from 5000 years to over 20000 years. PN with about 0.62M are seen between 2000 and 5000 years. Two curves are drawn in Fig. 3. The lower one represents the time when central stars have a temperature of 20000K. The other one, which is a sort of upper envelope of the PN ages in our sample, was obtained by dividing the sample into 5 bins of approximately 20 objects each, and by drawing a smooth curve below which are found 90% of the objects in each bin. One can see that both curves decrease with . The behaviour of the lower one simply reflects that low mass PNN evolve more slowly and become able to ionize the surrounding nebula only at larger ages. The behaviour of the upper curve is better understood when considering Fig. 4, which represents as a function of . This plot confirms the trend shown in GST97 that, for objects with 0.62M , tends to be larger for PN with nuclei of larger masses. Thus, at the same age, PN with central stars of higher masses tend to be more diluted, which discriminates against them in a sample where the H fluxes have to be measured.
© European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |