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Astron. Astrophys. 337, 517-538 (1998)

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2. The model

2.1. Shell morphology and evolution

The central star is assumed to have had a prior red giant epoch of mass loss [FORMULA] [FORMULA] yr-1 and wind speed [FORMULA] km s-1 (Loup et al. 1993). In our model where a constant velocity shell overtakes and shocks the red giant wind, these wind parameters only affect the intensities of the lines produced in the shock, and we can therefore separately calculate the FUV/Xray-induced emission from the PDR in the shell and the shock emission. It is the calculation of the thermal, chemical and dynamical evolution of the PDR shell which constitutes the bulk of this paper. We briefly discuss how the shock intensities vary with [FORMULA] and [FORMULA] in Sect. 4.4 and Sect. 5.9.

The red giant wind epoch is followed by the onset of a rapid phase of high mass loss rate, or the ejection of a shell of mass [FORMULA] [FORMULA]. The shell formation and acceleration to speeds in excess of [FORMULA] may be associated with the rise in the effective temperature of the central star and the consequent increase in the radiation pressure on the outflowing gas and dust (see, e.g., Kwok 1993). Kwok et al. (1978) and Kwok (1982) have postulated that a high-velocity ([FORMULA] km s-1), radiation-driven wind is produced once the central star radiates primarily in the UV. The action of this wind on the slow red giant wind drives an overtaking dense shell into the coasting red giant wind. We have assumed that the end of the red giant epoch is marked by the "ejection" of a shell or by a period of extremely high mass loss, which begins as the effective temperature of the central star rises above [FORMULA] K ([FORMULA]). The shell moves outward initially at 8 km s-1 (the speed of the red giant wind), but accelerates smoothly to 25 km s-1 as [FORMULA] rises from 20,000 K to 30,000 K (Kwok 1993). Once [FORMULA]30,000 K, the shell moves outward at constant velocity [FORMULA] km s-1 (Pottasch 1984). This occurs at a radius [FORMULA]= 8[FORMULA], [FORMULA], [FORMULA], and [FORMULA] cm and time [FORMULA]= 2200, 760, 120, and 120 yr for the four cases of central stellar mass [FORMULA]=0.6, 0.64, 0.696, and 0.836 [FORMULA] we consider. For [FORMULA], R is given simply:


As a function of time, as the shell expands, the shell density n decreases. We model two cases which should represent the extrema in the density evolution of the gas. We assume that


with [FORMULA] or 3 representing, respectively, the case of a significant driving pressure (radiation or wind) from the central star or the case of a freely expanding, coasting shell. For constant mass shells with thickness dominated by the neutral gas, [FORMULA] is constant for [FORMULA] and [FORMULA] is constant for [FORMULA]. For convenience, we choose the fiducial radius [FORMULA] cm, as a typical size of a young PNe. Because of the different evolution in [FORMULA] for stars of different mass, and therefore the different times [FORMULA] when the shell accelerates to 25 km s-1, the time [FORMULA] corresponding to [FORMULA] varies somewhat with [FORMULA]: [FORMULA] yr for [FORMULA] [FORMULA], 2450 yr for [FORMULA] [FORMULA], and 2050 yr for both [FORMULA] and 0.836 [FORMULA].

The neutral gas density [FORMULA] at [FORMULA] is related to the mass ejected in the shell, the solid angle of the shell, and the thickness of the shell. The neutral shell subtends solid angle [FORMULA] as seen from the central star; a similar equation with [FORMULA] applies for the solid angle subtended by the ionized shell. The parameter [FORMULA] can therefore be utilized to model the ejection of toroidal or clumpy shells ([FORMULA]) as well as spherical shells ([FORMULA]=1). We will use [FORMULA] as our major free parameter. However, there is a minimum value [FORMULA] required to fit the ejected mass in the specified volume at [FORMULA]. Assuming the shell is entirely neutral and that the density throughout the shell is [FORMULA], it is


However, [FORMULA] refers to the H2-emitting ([FORMULA] K) PDR density, and the outer (shielded) parts of the shell may be much cooler and denser. A weaker criterion is that the inner, hot PDR column of [FORMULA] cm-2 (Tielens & Hollenbach 1985) must fit inside [FORMULA]. This criterion is simply [FORMULA] cm-3.

We assume that the total mass [FORMULA] of the shell, including ionized and neutral components, is fixed. At any given time, the total thickness of the shell is [FORMULA], with an inner ionized portion of thickness [FORMULA] (the PN) and the possible existence of an outer neutral region of thickness [FORMULA]. The density of the ionized region and the PDR portion of the neutral region are fixed at any instant of time: the electron density [FORMULA] in the inner HII region is related to the assumed constant hydrogen nucleus density n in the FUV-illuminated inner neutral region by the relation


The parameter [FORMULA] is generally taken to be 0.5, which reflects the expectation that either the temperature in the neutral photodissociation region (PDR) will be [FORMULA] K, or that the microturbulent velocities, which provide the neutral pressure, will be of order [FORMULA] km s-1 (Tielens & Hollenbach 1985). The implicit assumption is that pressure equilibrium applies between the HII plasma and the neutral PDR gas, and within the neutral shell. These assumptions are approximate, but necessary to speed the computation. When model results are compared to data, n should be interpreted as the average density of the H2 emitting gas in the shell. It should be noted again that we have allowed for the ionized gas to expand away from the neutral torus or clumps and effectively fill a larger solid angle than the neutral gas, but we assume the electron density is constant throughout the HII volume that dominates the emission measure.

In summary, the shell evolution is defined by the following parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. In this paper, we assume [FORMULA] [FORMULA], [FORMULA] km s-1, [FORMULA], [FORMULA], and vary the PDR density [FORMULA] at [FORMULA] cm ([FORMULA] cm-3) and the filling factor [FORMULA] ([FORMULA]). Fig. 1 shows a schematic diagram of the shell model.

[FIGURE] Fig. 1. Sketch of the model geometry. See text for the symbols.

2.2. The evolution of the central star

We have used the evolutionary tracks of Schönberner (1983) and Blöcker (1995) to determine the evolution of the FUV photon luminosity [FORMULA] and the Lyman continuum photon luminosity [FORMULA], as well as the effective temperature of the central star [FORMULA]. In these models, the mass of the progenitor star affects the resultant evolution and we have chosen four tracks, having core masses [FORMULA]=0.6, 0.644, 0.696 and 0.836 [FORMULA], with precursor masses of 3, 3, 4 and 5 [FORMULA], respectively. These models span the expected range of spectral evolution. Fig. 2 shows the evolution of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] with time for the four models. The last two quantities have been computed using the model atmospheres of Clegg & Middlemass (1987) for [FORMULA][FORMULA]180,000 K, and blackbody curves for higher [FORMULA]. The basic evolution of the four stars is similar, with a sharp increase of [FORMULA] and [FORMULA] as the stellar photospheric temperature increases, followed by a quick decline because, although [FORMULA] is continuing to rise, the photosphere is shrinking to maintain roughly constant bolometric luminosity. The rapid decline of [FORMULA] and [FORMULA] ceases when the star reaches the white dwarf cooling sequence.

[FIGURE] Fig. 2. The four panels describe the evolution with time of the stellar radiation field for the four core masses we have considered: [FORMULA]=0.6 [FORMULA] (dot-short-dashed curve), [FORMULA]=0.64 [FORMULA] (dot-long-dashed curve), [FORMULA]=0.696 [FORMULA] (dashed curve), and [FORMULA]=0.836 [FORMULA] (solid curve). The top left panel plots the effective temperature [FORMULA], the top right panel the core luminosity [FORMULA], the bottom-left panel the number of ionizing photons [FORMULA], the bottom-right panel the number of FUV photons [FORMULA].

There is considerable uncertainty in the soft X-ray spectrum emitted from the central star of a planetary nebula (cf., Husfeld et al. 1984, Henry & Shipman 1986, Clegg & Middlemass 1987). As a standard case, we assume that the central star has a soft X-ray spectrum given by a blackbody of temperature [FORMULA].

Although the pattern followed by the four stars is similar, the evolutionary time scales are very different. Less massive stars peak in [FORMULA], [FORMULA] and [FORMULA] at later times, reach lower maximum [FORMULA] (300,000 K for [FORMULA]=0.836 [FORMULA] vs. 150,000 K for [FORMULA]=0.6 [FORMULA]), and then decline more slowly in all three parameters than more massive stars. The difference in maximum [FORMULA] has a very large effect on the soft X-ray heating of the neutral gas.

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© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998