Astron. Astrophys. 334, 239-246 (1998)

2. PNe: general picture

In order to have a 'general' picture of PNe we have to consider their different parts in a self-consistent manner. This means that the physical parameters (effective temperature, luminosity etc.) of a CSPN must be consistent with its wind characteristics which in turn have to be consistent with the PN parameters (such as nebular radius, expanding velocity etc.). So, in the frame of the general PN picture we make the following basic assumptions:

• the evolution of the CSPN follows according to the standard evolutionary tracks for an AGB-post-AGB star
• the radiation-driven-winds (RDW) theory is applicable to the CSPN wind
• the ISW model is valid

Using these assumptions, the observables of an individual PN have to be analyzed in a self-consistent manner.

2.1. Analyses of the CSPN characteristics

The basic CSPN physical parameters are: effective temperature, , luminosity, L and radius . From analyses of the optical and ultraviolet spectra, the least model-dependent parameters: the effective temperature, , and the ratio /d (where d is the distance to the object) can be derived. We remind that the value is based on the assumed validity of the black-body emission approximation and /d is derived by making use of some evolutionary track for the CSPN but it does depend only on and the measured flux (in some spectral range). So, these two quantities are evolutionary track independent. Then to obtain the other parameters we have two choices : a) to adopt an evolutionary track, i.e. to assume a mass for the central star; b) to use the distance usually available only from statistical methods. In the first case we read out L in the adopted track from the known and thus derive and finally d from the above mentioned ratio. In the second case from the distance d, we first derive from the same ratio and next L. These and L will specify an evolutionary track.

Since the mass distribution of observed CSPNs is believed to be quite concentrated around 0.6   , the assumption of the corresponding evolutionary track (case a) is considered less uncertain than to accept the distance given by the statistical methods (case b).

By assuming as 'typical', in the mentioned sense, the evolutionary track of a 0.6  star, we have immediately a value for the age of the central star, . On the other hand, using the CSPN parameters along the chosen evolutionary track, the CSPN wind characteristics (mass-loss rate, wind velocity) can be derived by making use of the RDW theory. We recall that the mechanical luminosity of the CSPN (fast) wind can be expressed with a power-law function in time within the first few thousands years of the CSPN evolution and this allows a similarity solution for the hot-bubble structure of PNe to be derived (see ZhP for details). This will provide another estimate of age : . In a self-consistent picture the two ages must be similar and the comparison of them will help to find the best 'general solution' for our object.

We start then considering the most recent evolutionary tracks of a CSPN. They have been calculated for masses in the range between 0.53 to 0.94  (Blöcker, 1995). Our first step has been to derive the 'fast' wind parameters along the tracks for 0.565, 0.605, 0.625  which are nearest to the standard mean mass of CSPNs generally accepted to be very close to 0.6  . This was done using the RDT theory with the force multiplier parameters and (cf. Kudritzki et al., 1992), presented in Table. 1 designated as A, B ,C. Case A is representative of an O3 star; case B of an O3 star but includes the UV emission from shocks formed in its wind; case C is as B but uses new iron opacities (for details see Kudritzki et al., 1992 and Pauldrach et al., 1994). Clearly there are uncertainties on how the above satisfactory represents real winds particularly in the hottest CSPNs. We have anyhow a representation of fast winds along some significant evolutionary tracks which allows us to apply the ISW theory as formulated in ZhP. The results are shown in Table 2. Note the small differences between the values of and given by ZhP for the 0.605  case and those listed in Table 2. They are due to the use here of newer evolutionary tracks.

Table 1. RDW: adopted force multiplier parameters

Table 2. CSPN wind parameters

2.2. Analyses of the PN characteristics

As with the nebula, we consider its radius and expansion velocity. It is well known that the gas velocity may not be uniform through the nebula and this is why the expansion velocity of a given PN might have a bit different values if optical lines of various ions are used. Unfortunately, our 1D model does not allow to treat this item in detail and the value of the expansion velocity is attributed to the gas near the outer radius of the optical shell. The latter radius is assumed to be the radius of the PN. Thus, from the ZhP model we have that the radius of the PN can be expressed through the fast and the slow wind parameters:

where , , t 1000 yr , and (see Table 2 for the values of corresponding parameters) . And from here one gets that

where is the observed PN expansion velocity and is the corresponding nebular age according to the ISW model.

Thus having from observations the expansion velocity and the radius of a PN, we can check whether the nebula age, , is consistent with the CSPN age, , as derived from an evolutionary track. We note that in the frame of the ISW model the 'traditional' expansion age, , is not equal to the true nebular age since the CSPN wind parameters are function of time.

Finally, from the consistency of the mentioned stellar and nebular ages, one can estimate the slow wind parameters which then allows quantitive predictions for the hot-bubble characteristics to be made. In fact, in the frame of the ISW model, once having the age of the PN, formulae (1) and (2) can be used to have two independent estimates of the slow wind parameter if the PN expansion velocity and the radius are known. Then, having calculated further observables of a given PN (X-ray, EUV and IRCL characteristics), they can be confronted with those observed and thus further constraints can be placed on the ISW model. Of course, the applicability of the general PNe picture is limitted by the uncertainties of the object parameters due both to observations and theories applied. As a first attempt we apply the proposed scheme to two PNe which have relatively well known parameters.

© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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