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Astron. Astrophys. 342, 823-830 (1999)

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4. Comparison of the H2 spectrum with models

The H2 emission in the 1-0S(1) line of the central torus of NGC 2346 is compared in this section to the predictions of the NH98 theoretical models.

The NH98 models compute the emission expected from tori (or shells or clumps) exposed to the radiation field of the hot central star and expanding inside the remnant of the wind ejected by the central star in its previous red giant phase. As the central star evolves initially at constant luminosity toward higher effective temperatures and then along the white dwarf cooling track, the torus expands radially with roughly constant velocity and decreasing density. The models consider the effects of UV and soft X-ray radiation on the neutral gas and follow the time dependent chemistry for H2, solving for the chemical and temperature structure and the emergent spectrum (in the following, the PDR spectrum) of the evolving torus.

The torus expands inside the material ejected by the star in its previous phase as red giant. Since its velocity is larger than that of the wind itself, it gives origin to a shock which heats and compresses the gas, which then emits intense lines of vibrationally excited H2. NH98 consider the emission of J shocks, since the importance of magnetic field in PNe is not clear and, in this range of shock velocities, the H2 line intensity is maximum in J shocks.

4.1. PDR models

The NH98 PDR models require the specification of a number of parameters. We fix the mass of the central core, which determines the time scale of the evolution of the stellar radiation field, to be about 0.7M[FORMULA] (Calvet and Peimbert 1983; Bachiller et al. 1989). For a core mass M[FORMULA]=0.7 M[FORMULA], NH98 show the results of only one model, which has a neutral gas density at the time t=2500 yr (roughly the age of NGC 2346) [FORMULA] cm-3 (note that in NH98 the density varies with time as [FORMULA]). We used the NH98 code to compute a number of additional models, varying the [FORMULA] yr density [FORMULA] from [FORMULA] cm-3 (Model 1) to [FORMULA] cm-3 (Model 2), [FORMULA] cm-3 (Model 3) and [FORMULA] cm-3 (Model 4). The model parameters are summarized in Table 2. In all cases, He/H=0.13, C/H=[FORMULA], O/H=[FORMULA]. The results are shown in Fig. 4, which plots the intensity of the H2 1-0S(1) line as a function of time.


[TABLE]

Table 2. Model parameters
Notes:
a: Mod 5 assumes a luminosity of the central core equal to L[FORMULA](M[FORMULA]=0.7)/5 at all times.


[FIGURE] Fig. 4. Model predictions for the H2 1-0S(1) line intensity as function of the age of the nebula. Model parameters are given in Table 2. All the models have a neutral gas density decreasing as [FORMULA] as the torus expands with constant velocity [FORMULA]km s-1. The square shows the value of the observed H2 1-0S(1) intensity averaged over the central torus.

The square in Fig. 4 shows the observed intensity of the 1-0S(1) line averaged over the torus. We have estimated the age of NGC 2346 by taking the separation of the two 1-0S(1) peaks (about [FORMULA]) at [FORMULA] pc and an expansion velocity of [FORMULA]25 km s-1 (as in the NH98 models), consistent with the observed range (15-35 km s-1) of CO expansion velocity (Bachiller et al. 1989). The result is a dynamical age of [FORMULA]2500 yr. The uncertainty on the age is certainly large, but difficult to estimate. We plot in Fig. 4 an error bar corresponding to an uncertainty of [FORMULA]500 yr. The observed intensity of the 1-0S(1) line is reproduced quite well by the low-density models, especially if we take into consideration the large uncertainties that affect the models as well as the age estimate of the nebula itself.

One problem that arises immediately with these models has to do with the estimated luminosity of the central star. At [FORMULA] yr, a core of 0.7 M[FORMULA] has already reached the white dwarf cooling track; it has a luminosity of about 250 L[FORMULA], an effective temperature T[FORMULA][FORMULA] K and a number of ionizing photons [FORMULA][FORMULA] photons s-1 (Blöcker 1995). While the effective temperature is roughly in agreement with the Zanstra HeII temperature (Méndez 1978), the luminosity is significantly larger than the values 17-90 L[FORMULA] quoted in the literature. However, we are somewhat suspicious of those very low values. The luminosity derived from the HeII [FORMULA]4685 Å intensity by Méndez (1978) is a lower limit L[FORMULA][FORMULA]43 L[FORMULA] (for D=800 pc) and is very sensitive to the extinction. The number of ionizing photons we derive from the observed radio flux at 6 cm (86 mJy; Milne and Aller 1975) and from the total H[FORMULA] flux (Walsh 1983) is at least [FORMULA] photons s-1 (assuming no escape of ionizing photons and an average optical depth in H[FORMULA]), consistent with L[FORMULA][FORMULA]250 L[FORMULA], but not with lower values of L[FORMULA] (see Fig. 3 of NH98). If L[FORMULA]=250 L[FORMULA], the non-detection of the white dwarf star in the visual is not surprising; assuming, for simplicity, that the A star and the white dwarf spectrum can be represented by black-bodies at 8500 K and [FORMULA] K, having luminosities of 15 and 250 L[FORMULA], respectively, we find that the white dwarf is a factor 54 weaker than the A-type star at 5500 Å, a factor of 10 at 3000 Å and that the two stars become comparable only at [FORMULA]2000 Å. This last is consistent with the UV excess (with respect to the flux expected for the A star) measured by the ultraviolet satellite ANS and reported by Méndez (1978). All together, we suspect that the white dwarf luminosity is roughly of the order of 250 L[FORMULA]. In any case, we have also computed a model where we have artificially reduced the stellar luminosity by a factor 5 at all times; the density of this model (Model 5), shown as a dashed curve in Fig. 4, is [FORMULA] cm-3. The predicted line luminosity scales approximately with the luminosity of the central core. A value L[FORMULA]=50 L[FORMULA] (although not predicted by any evolutionary track) is still roughly consistent with the observed line intensity, especially if we consider the uncertainty on the PN age estimate.

The best fit to the H2 1-0S(1) observations is provided by models with low density ([FORMULA] cm-3), in good agreement with the low electron density ([FORMULA] cm-3) derived for the ionized part of the nebula (Liu et al. 1995; McKenna & Keenan 1996). Assuming pressure equilibrium between the ionized and the neutral gas, and a PDR temperature [FORMULA]500 K, we expect a neutral density about 40 times the electron density. The low density we require is in rough agreement with the Bachiller et al. (1989) estimate that the neutral density is [FORMULA] cm-3.

In these low-density models, at [FORMULA] yr, the 1-0S(1) emission is mostly due to collisionally excited H2, kept warm by the heating of the gas by the soft X-rays emitted by the central core. As discussed in NH98, X-rays determine the chemical and physical evolution of the neutral gas around high-mass PN cores, after a short initial phase (about 1000 yr for a 0.7 M[FORMULA] core) where UV photons dominate. If the X-rays effects are neglected, PDR models predict a much lower intensity of the H2 molecular lines.

Time-dependent effects in the H2 chemistry are important. For M[FORMULA]=0.7 M[FORMULA] and [FORMULA], [FORMULA] yr, the mass of ionized gas increases with time. In these conditions, at each time step a new layer of molecular gas is exposed to the X-ray heating radiation, as H2 molecules are advected from deep in the PDR slab toward the irradiated surface. Therefore, compared to the predictions of equilibrium calculations, a larger amount of hot molecular gas is formed, which emits stronger H2 vibrationally excited lines. This effect, discussed in detail in NH98, is larger in models with lower density, so that the 1-0S(1) intensity is higher in models with lower n. The opposite is true in models where the H2 chemistry is treated under the assumption of stationary equilibrium. These models predict at [FORMULA] yr a 1-0S(1) line about 7-10 times weaker (for [FORMULA] cm-3), mostly due to fluorescence in H2 pumped by UV photons, and lower in models with lower n.

The predicted intensities of all the H2 observed lines have been computed using a code which calculates the level population for the N=299 bound states with rotational quantum number [FORMULA]29 of the H2 molecule. The code includes the effects of UV pumping by an external radiation field as well as collisions with H, H2, He, electrons and protons (Draine & Bertoldi 1996). We have used as input the physical conditions (namely, the radiation field at the inner edge of the PDR and the run with the depth in the PDR of temperature and fractional abundances of H, H2, He, electrons and protons) computed with the NH98 code for [FORMULA] yr.

The models agree rather well with the observations for all the lines. Fig. 5 shows a Boltzmann plot for the H2 transitions, where the column density of the upper level of the transition (divided by its statistical weight) is plotted as function of the energy of the level. The filled dots are the observed values, the open symbols show the prediction of the best-fitting model (Mod 1, [FORMULA] cm-3). The agreement is somewhat worse for models with higher density, which tend to predict weaker lines from the higher excitation levels.

[FIGURE] Fig. 5. Boltzmann plot. The values of the column density of the upper level of the transition (divided by its statistical weight) are plotted as function of the energy of the level. The filled dots are the values derived from the observed lines. Arrows are 3[FORMULA] upper limits. The prediction of the best-fitting model ([FORMULA] cm-3) are shown by the open triangles for lines of the v=0-0 band, open squares for the 1-0, open circles for the 2-1 and pentagons for the 3-2 band.

4.2. Shock models

In the NH98 models of the H2 shock emission, the relevant parameters that determine the intensity of the lines are the shock velocity [FORMULA] (i.e., the difference between the torus expansion velocity and the red-giant wind velocity) and the pre-shock density, which, at any given distance R from the star, is determined by the red-giant wind properties ([FORMULA], where [FORMULA] is the rate of mass-loss, [FORMULA] the wind velocity, [FORMULA] the fraction of solid angle over which the wind was ejected and µ the mean molecular weight). The H2 line intensity does not depend on the torus properties, but only on its expansion velocity, as long as the matter in the torus does not become completely ionized or photodissociated.

In NGC 2346, the expansion velocity of the neutral gas derived from CO data ranges from [FORMULA] 15 to [FORMULA] km s-1 (Bachiller et al. 1989). We show in Fig. 6 the predicted 1-0S(1) intensity at [FORMULA] yr as a function of the wind parameter [FORMULA] and two shock velocities, [FORMULA]=10 km s-1 (solid line), and [FORMULA]=15 km s-1 (dashed line). Higher values of [FORMULA] will appreciably dissociate H2 in J shocks. The intensity is computed according to Eq. 17 of NH98. For [FORMULA]=10 km s-1, one needs [FORMULA] M[FORMULA] yr-1 in order to reproduce the observed 1-0S(1) intensity. Unless [FORMULA], this implies an unusually high rate of mass-loss in the red-giant wind (Loup et al. 1993). The density of the pre-shock gas ([FORMULA]) is about [FORMULA] cm-3; this value is also unconfortably high, given the fact that the observed density of the CO torus (postshock gas), which these models predict to be [FORMULA], is [FORMULA] cm-3 (Bachiller et al. 1989). A lower mass-loss rate is required if the shock velocity is higher. For [FORMULA]=15 km s-1, the observed intensity is reproduced by [FORMULA] M[FORMULA] yr-1 and a pre-shock density of about [FORMULA] cm-3. The shock models of NH98 predict a ratio of the 2-1S(1)/1-0S(1) intensity of 0.10 for [FORMULA] km s-1, and 0.19 for [FORMULA] km s-1, somewhat larger than the observed value ([FORMULA]0.07).

[FIGURE] Fig. 6. Intensity of the 1-0S(1) line emitted in the shock between the expanding torus and the precursor red-giant wind at [FORMULA] yr as function of the red-giant wind parameter [FORMULA] (see text). The top scale gives the corresponding value of the pre-shock density [FORMULA] for [FORMULA] km s-1. The two curves correspond to two values of the shock velocity, as labelled. The observed 1-0 S(1) intensity is shown by the horizontal arrow.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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