3. Results and discussion
We apply the above described 'general picture' of PNe to NGC 6543 and NGC 6826. For each object we first use method (a), then method (b) to derive the needed star and nebular parameters. We then analyze the nebular radius and expansion velocity in the frame of the ISW model (&2.2). We compare the resulting nebular age with that associated to the central star coming from the relevant evolutionary track. We also calculate the expected radiation from the hot nebular bubble in the two objects and compare with the observed X-ray radiation.
3.1. NGC 6543
Extended X-ray radiation has been definitely observed in this PN, according to Kreysing et al. (1992). This is important for our 'general picture' of a PN, which should make use of as many observables as possible. We then tentatively apply our model to this object, in spite of the difficulties with the complex morphology of its inner fine structures (Balick et al., 1997) and with the central star having WR characteristics. We would like to warn that there may be difficulties when applying the RDW theory to a CSPN with WR spectral characteristics. Also, the use of the standard AGB-post-AGB evolutionary tracks in such a case is not much justified considering the unclear evolutionary status of such CSPNs. On the other hand, this is at present the only possibility to carry out an analysis in the frame of the general PNe picture. Thus, we consider our analysis of NGC 6543 as a zeroth approach to the problem.
To derive the stellar and nebular parameters, we first apply method (a). A good set of parameters for the central star of NGC 6543 is likely : ; lg(L / ) = 3.75; / = 0.7; /d = 1.1 10-11 ; d = 1.44 kpc; yr-1 ; km s-1 (Perinotto et al., 1989). We recall that the effective temperature, , the ratio of the stellar radius to the distance to the object, /d, and are relatively well determined while the other quantities follow from the assumed typical evolutionary track of a post-AGB model of M = 0.6 (cf. Perinotto et al., 1989). The recent evolutionary track of a CSPN having stellar mass M = 0.605 (Blöcker, 1995) would give, combined with the above , the luminosity : lg(L / ) = 3.78 and then a set of parameters essentially identical to the above ones. The evolutionary stellar age corresponding to this recent track is of about 3500 years while the measured is consistent with a stellar age of about 4000 years if the RDW theory is applied to this 0.605 mass track.
We search now the nebular age. In NGC 6543 we accept for the expansion velocity and the nebular radius : km s-1 (Weinberger, 1989) and pc, where the latter value has been scaled from the one given by Weinberger (1989), to be consistent with the distance of 1.44 kpc. From Eq. (2) we then derive years, relatively close to the above years, from the stellar evolutionary track.
We apply now method (b). The mean distance from various determinations with statistical methods, excluding just the largest and the smallest one (see Acker et al., 1992), amounts to d = 1 kpc. Using and as above, we have the following values for the other stellar parameters : / = 0.49; lg(L / ) = 3.46 and yr-1 ( does not change since it does not depend on the distance). The above and the new lg(L / ) call for an evolutionary track with a mass smaller than 0.605 . The next available track from Blöcker (1995) is for 0.565 . This track implies years.
On the other hand the nebular radius corresponding to the distance of 1 kpc is pc. This provides years. Now the separation of the two ages is quite larger than before.
We consider this as an argument that method (b) works in this object less well than method (a). Probably method (a) applied to an evolutionary track a bit more luminous than that of 0.605/ , i.e. to a star a bit more massive would provide a still better solution. But in view of the uncertainties, we consider appropriate at present to accept the above discussed solution obtained with method (a).
Since we would like to have additional constraints on the PN physics, we have modelled the X-ray, EUV and IRCL emission of NGC 6543. In order to have an idea about the effect of the distance uncertainties on the hot-bubble emission we have done two sets of models related with two values for the distance to the object: 1.0 and 1.44 kpc. The corresponding models are numbered 1,2,3 (PN age of 3500, 4000 and 4500 years); 4,5,6 (PN age of 3500, 4000 and 4500 years) for the two distances, respectively. Additionally, we have considered the three different sets of the RDW model parameters (see Table 1). The results are presented in Fig. 1 and Table 3. Columns 3 and 4 in Table 3 show the internal consistency of the procedure ( and - signs mean whether the model value is bigger or smaller than that deduced from observations). We see that the nebular PN parameters, its radius and expansion velocity, are acceptably well reproduced in the frame of the ISW model along a 0.605 mass evolutionary track and applying the RDW theory with different force multiplier parameters. Also, column 2 in Table 3 gives the mean value of the slow wind quantity derived from the 'observed' and . If we assume a slow wind velocity of about 10 km s-1 then it is derived that yr-1, a value which can be considered typical for an AGB star. We mention that this value is at least one order of magnitude smaller than that expected for the 'superwind' of about yr-1.
Table 3. NGC 6543: results
Fig. 1 shows the theoretical X-ray spectra of NGC 6543. It can be seen that the shape of the X-ray spectrum is not sensitive to the model parameters while Models A and C give about 4-5 times higher fluxes and 'net' luminosities (see also columns 5 and 6 in Table 3). Given the values in Table 2, this result can be well understood in terms of the higher mass-loss rates of the CSPN wind as derived in Models A and C with respect to Model B. A higher mass loss rate suggests a higher mass of the hot bubble since the gas evaporation from the cold shell (optical PN) will be more efficient in this case. A comparison between the spectra in Fig. 1 and those in Fig. 3 of Kreysing et al. (1992) shows that the model predictions based on the ISM theory correspond fairly well to the observed X-ray characteristics of NGC 6543. Namely, the calculated and observed spectra are similar not only qualitatively, having similar shape and 'softness', but they show a quantitative correspondence as well (see the y-axes in the cited figures which give the flux density). On the other hand, X-ray observations with better spatial resolution are highly desirable because if the X-ray emission has a larger extension than the optical one (as claimed by Kreysing et al., 1992), then a more complex physics would be required to explain the X-ray properties.
Since the results presented here are based on the CSPN parameters (mass loss rate and wind velocity) predicted by the radiation-driven-wind theory (Pauldrach et al., 1988) another item is also of some interest. Namely, the results presented in Table 2 allow to estimate that for years the CSPN wind velocity is: km s-1 (Model A); km s-1 (Model B) km s-1 (Model C) and the corresponding values for the CSPN mass loss are: yr-1 (Model A); yr-1 (Model B); yr-1 (Model C). A comparison with the values of these quantities deduced from observations (see above) shows that the model predicted wind velocity value corresponds well to that observed especially for Models A and B (for Model C this is true only for a PN age of 4000-4500 years). On the other hand, the theoretical mass-loss rate, predicted by Model B, is an order of magnitude smaller than the 'observed' one while Models A and C (particularly Model C) show an acceptable correspondence with observations. This analysis definitely demonstrates that well defined force multipliers parameters are needed for the case of the CSPN winds. Further work in this direction is strongly desired.
We conclude that the X-rays data appear to confirm the general PN picture of NGC 6543 given above with a distance close to 1.4 kpc.
3.2. NGC 6826
NGC 6826 is another object with relatively well known parameters. Its rather regular shape suggests that applying a 1-D model would be quite suitable in this case.
We start applying method (a) to derive the basic parameters. A set of parameters for the CSPN in NGC 6826 is: ; lg(L / ) = 3.88; / = 1.45; /d = 1.09 10-11 ; d = 2.99 kpc; yr-1 ; km s-1. (Perinotto et al., 1989). They have been derived with the same procedure as the corresponding ones of NGC 6543. The recent evolutionary track of a CSPN having stellar mass M = 0.605 would indicate, combined with the above , an evolutionary stellar age of of about 3000 - 4000 years and lg(L / ) = 3.79 which will result in a corresponding change of the other CSPN parameters, i.e.: / = 1.31, d = 2.7 kpc and yr-1. The measured is consistent with a stellar age of about 3800 years if the RDW theory is applied to the case of a 0.605 mass star.
We adopt for NGC 6826 the nebular parameters: km s-1, pc at d = 1 kpc (Weinberger, 1989). By scaling from the distance d = 1 kpc, used by Weinberger, to d = 2.7 kpc, the radius becomes pc. Eq. (2) then gives years. We see that using method (a) we end up with quite different ages for the CSPN and the nebula in NGC 6826.
We then follow method (b) assuming as valid for NGC 6826 the distance based on various statistical determinations. Looking at Acker et al. (1992), we see that eigth different measurements concentrate around d = 1 kpc. Thus, the nebular NGC 6826 parameters: km s-1, pc at d = 1 kpc (Weinberger, 1989) suggest years. On the other hand, the measured and imply at this distance of 1 kpc, / = 0.54; lg(L / ) = 2.93 and yr-1. It is worth noting that these values of and L demand for this object an evolutionary track quite different from the standard one, i. e. with a quite smaller mass.
In the grid of models calculated by Blöcker (1995), the one for M = 0.546 would appear to have the proper luminosity, close to lg(L / ) = 2.93. The age corresponding to the observed would be however very much larger than any plausible value derived from the observed nebular and the nebular radius. The track relative to 0.565 is, from this point of view, quite more acceptable.
With this evolutionary track of 0.565 , the observed implies an age of about 8 000 years. The luminosity would be lg(L / ) = 3.58 which in turn suggests / = 1.03; and from the observed a value of 2.12 kpc for the distance to the object comes out. The mass loos rate of the fast wind would now be yr-1.
For the distance of 2.12 kpc, the PN radius becomes 0.129 pc. The PN age derived from the ISW model, if again a value of 27 km s-1 is assumed for the PN expansion velocity, is of 7 500 years, very close to the above value of 8 000 years.
In order to have more constraints on the physics of the PN system in NGC 6826 we will further proceed by considering some direct observables of the hot bubble (being a cornerstone for the ISW model). As in the case of NGC 6543, we have modelled the X-ray, EUV and IRCL emission of NGC 6826 for the three different sets of the RDW force multipliers parameters given in Table 1. The results are presented in Fig. 2 and Table 4. As seen from columns 3 and 4 of Table 4, the nebular PN parameters are acceptably well reproduced in the frame of the ISW model along the 0.565 mass evolutionary track and applying the RDW theory with different force multiplier parameters. Also, column 2 in the same table gives that the slow wind characteristic derived from the 'observed' and suggests that yr-1, if we again assume a slow wind velocity of about 10 km s-1. It is worth noting that as in the case of NGC 6543 this value is one-two orders of magnitude smaller than that usually quoted for the 'superwind'.
Table 4. NGC 6826: results
The theoretical X-ray spectrum can be seen in Fig. 2 and from there and Table 4 it is immediate that while the 'net' X-ray luminosity is about an order of magnitude smaller than that for NGC 6543 the X-ray flux is about 1.5-2 orders of magnitude less. Moreover the X-ray spectrum is expected to be softer than in the previous case. This results from the smaller interstellar extinction ( vs. ).
The above is consistent with the fact that an observed X ray flux from NGC 6826 has not been reported. NGC 6826 is in fact neither listed among the PNe detected nor among those observed but not detected by ROSAT (Conway & Chu, 1997). A further effort (Conway, priv. communications) shows that it does not belong presently to the Pointed Observations Catalogue, neither is listed in the ROSAT Bright Star Catalogue. The latter information, considering that it is extremely unlikely it falls in the few locations missed from the ROSAT all sky survey, allows to put for NGC 6826 an upper limit of 0.05 cts/s, a value a few times larger than the observed flux from NGC 6543 (Kreysing et al., 1992). This limit is however too high to offer some useful constraint with respecte to our predicted X-ray flux in NGC 6826. On the other hand the above information from the ROSAT satellite, taken together, is consistent with a flux from this source quite below that from NGC 6543, and then in agreement with our predictions.
In any case a comparison between Fig. 2a and Fig. 2b gives an idea about the differences in the theoretical X-ray characteristics of NGC 6826 between the 0.565 and 0.605 mass cases. A corresponding difference in the EUV and IRCL luminosities is evidently found.
Using the results from Table 2, we have the model predicted CSPN wind velocity, 1000-1400 km s-1 (the lower limit is for Model C) which is a bit lower than the value deduced from observations and the theoretical CSPN mass loss is yr-1 (the lower limit is for Model B). This confirms the previously drawn conclusion about the importance of the correct RDW modelling of the CSPN winds.
Thus, future observations of X-rays in NGC 6826 would be especially important to possibly confirm the general picture of this object we have offered.
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998