Title: The RAINY3D code : The treatment of condensations in nova remnant during nebular. phase. Manoel Moraes 1, Marcos Diaz 2

Transcription

1 Anexo 2 Title: The RAINY3D code : The treatment of condensations in nova remnant during nebular phase Manoel Moraes 1, Marcos Diaz 2 1: Rua do Matão, 1226, Cidade Universitária, São Paulo, SP Brasil sala F 308 mcvmjr@astro.iag.usp.br 2: Rua do Matão, 1226, Cidade Universitária, São Paulo, SP Brasil sala F 303 marcos@astro.iag.usp.br Affiliation 1,2: IAG USP : Instituto de Astronomia, Geofísica e Ciências Atmosféricas Universidade de São Paulo

2 Abstract Classical novae remnants are important scenarios for improving the photoionization modeling. This work describes the pseudo 3D code RAINY3D which drives the photoionization code CLOUDY as subroutine. Photoionization simulations of old nova remnants are also presents and discussed. In these simulations we analyse the effect of condensations in the remnant spectra. The condensed mass fraction affects the Balmer lines by a factor greater than 4 when compared to homogeneous models and this directly impact the shell mass determination. The He II 4686/H ratio decrease by a factor of 10 in clumpy shells. These lines are also affected by the clump size and density distributions. The behavior of the strongest nebular line observed in novae remnants is also analyzed for heterogeneous shells. The gas diagnostic in novae ejecta are thought to be more accurate during the nebular phase, but we have obtained that at this phase the matter distribution can strongly affect the derived shell physical properties and chemical abundances.

3 1 Introduction Classical novae remnants are very important scenarios for improving photoionization modeling. There are still many difficulties to reproduce the observed novae remnant spectra. The emission line spectrum depends on the ionizing source, the chemical abundances of the shell and strongly on the mass distribution. There are some spatially resolved novae remnants (e.g. HR Del, Moraes & Diaz, 2009) that are far from having spherical and/or homogeneous mass distribution. Some of these remnants showed many structures and mass condensations in high resolution images (e.g. Slavin et al. 1995). The amount of condensed mass, their size and density distribution remain unknown. Spectral analysis of unresolved novae shells show the presence of emission lines of neutral and high ionization species simultaneously, during several stages of the shell expansion (e.g. Williams, 1992). The analysis of [O I] 6300,6364 doublet relative fluxes in several novae showed that the only process that can drive this line ratio towards unity is the line optical thickness (Williams, 1994). For typical oxygen novae abundances, such a large optical depth can not be achieved in homogeneous shells. Williams proposed that the [O I] lines come from high density condensations, where the gas is neutral. The condensations in the shell could explain the presence of strong lines from elements which are spread over a wide ionization range, specially in matter bound shells. The globules affect the transfer of stellar and diffuse continuum radiation in the cloud as well as the ionization structure, both locally and at large distances from the central source (Diaz, 2001). In fact, this represents a major discrepancy between current model spectra of nova shells and the observed line fluxes. The diagnostic lines used to obtain the shell physical and chemical parameters are affected by the mass distribution. Some emission line ratios observed in nebular phase novae are only possible if the shell has condensations. The shell physical and chemical diagnostics are thought to be more accurate during the nebular phase of the spectral evolution, since at the lower densities the spectra would be less affected by the condensations (Williams et al. 1991). The models often have a spherical or

4 cylindrical symmetry but the presence of condensations are treated with some extra free parameter in the density profile (e.g. equivalent radial power laws). Even with homogeneous shells, these 1D models can not always reproduce the shell geometry such as in the case of bipolar remnants. There are full 3D photoionization codes, for example those proposed by Gruenwald et al. (1997) and Ercolano et al. (2003). The last one is based on the Monte Carlo method and can calculate the diffuse component of the radiation field self consistently. However, because of the complexity of 3D calculations, the statistical treatment is in LTE, which is not correct for novae shells (Hauschildt et al. 1997). Full 3D codes demand a huge CPU time and enormous space parameters to consider, specially for highly structured environments. Diaz (2002) and Morisset et al.(2005) proposed pseudo 3D modeling based on several volume weighted 1D models. These codes have the advantage of being much faster than full 3D codes and the calculations could be done in NLTE. This Paper describes the pseudo 3D code named RAINY3D which build a 3D shell with condensations and drives the code CLOUDY (Ferland, 2005) to perform the matter radiation interaction calculations. The studies of the condensations distribution and how they affect the emission lines emitted by nova shells are presented. 2 The RAINY3D code and the condensations treatment. The problem of calculating 3D models of a nova envelope can be divided into two parts. The first consists of defining geometry and the mass density distribution in the shell and the second concern the calculus of photoionization, thermal equilibrium and matter radiation interactions and the emitted spectrum from the inhomogeneous shell for a given radiation source. The first task was performed by the code RAINY3D, the second by the code CLOUDY. The shell is built on the basis of a two component 3D mass distribution. The first is named background component and consists of a radial power density law r where is the background

5 power index. The second component is a random distribution of condensations. The shell is characterized by its condensed mass fraction parameter, fc, which is the mass ratio between the mass in the clumps and the total mass of the shell. The shell dimensions and mass are also input parameters of the models. The condensations are defined by its size and density contrast. Each clump has a Gaussian density profile. In fact, the shell may have many condensations, even for low fc values. The shell density as function of radius is defined by the expression: r = 0 r k 0 r 0 e where 0 is the density defined by the power law component and is normalized to 1 at r = r in, k is the clump contrast density, relative to the local background, defined in the input parameters, r o is the random position in the shell where the clump is located and c is related to the clump FWHM. The first term of this expression defines the background component. The clump size and density distribution are ruled by power laws probability functions: These probability power law indexes are input parameters for the shell assembling, as the maximum and minimum value of clump size and density. r r 0 2 2c 2 Y =[ H L H x L H ] H Where, L is the lower value, H the higher, x a random variable between 0 and 1 and the power law index. Y is the random value obtained biased by the distribution. Other shell has input parameters include the expansion velocity and the ejection age. Once they were configured, the 3D shell could be build. The code adopt a spherical coordinate system and the shell is divided in angular and radial grid. The shell grid resolution is defined by the dimensions of the remnant and clumps, a shell with smaller clumps will have a grid with higher angular resolution. The RAINY3D code put each condensation in a random spherical coordinate position in the

6 background density profile (see figure 1). The globule size and density contrast are defined by the power law probability. The code iterate until the mass in condensation achieve fc. The angular element resolution plus the inner and outer radius of the shell define the solid angle element that is sent to CLOUDY for calculation. Therefore all the 3D shell is divided in solid angle elements and the RAINY3D code accounts the emission lines contribution of each volume element. The total emission line flux is obtained by combining individual volume weighted contributions. The shell structure is solved locally, using a radial step calculated from the density and pressure gradients. The physical proprieties of the gas are considered to vary only in the radial direction in each solid angle element. So far, the transfer of ionizing diffuse field is restricted to the on the spot approximation or 1D radial transfer. This method allows the treatment of condensation shadows and partially ionized regions for a matter bounded shell. The NLTE calculation made by Cloudy (Ferland, 2005) is very important in the nova scenario, because LTE assumption leads to incorrect synthetic spectra (Hauschildt et al. 1997) and therefore incorrect model parameters. The RAINY3D code can run in spectral adjustment mode. The emission line data could be used to constrain the model parameters and the lines fluxes are an additional input parameter. The code read a input file that contains the line identification code for Cloudy, their flux and the relative error in the line flux determination. The data model comparison is made by a special 2, where the contribution of each line do not depend of their flux. The expression below are used to obtain the best model adjustment: 2 = 1 n [ f n i=1 oi f mi scf f oi e ]2 where, n is the number of lines, f oi and f mi are the line i observed and model fluxes, scf is a scaling factor (total flux observed/ total flux of model) and e is the observed line flux error in %.

7 3 Results The study of the effect of the inhomogeneities in novae spectra was made in two steps. The first analysis concerns the effect of the condensated mass in the line fluxes and line ratios. To perform this analysis we adopted a central source with luminosity log L = 36 erg.s 1 and effective temperature of 65000K. The ionizing field shape was defined by a high gravity and hot stellar atmosphere from the Rauch (2003) catalog. The shell abundances were CNO enhanced as observed in the novae ejecta. In all shell models with constant clump parameters, the increase of the condensed mass fraction, fc, increase the number of globules. For example, for fc = 0.05 the shell has typically between 50 and 150 clumps, while for fc = 0.9 there may be more than clumps. The increase of fc results in a less dense spherical component. The second analysis made explores the effect of the clump size and density distribution on the emission line ratios of an unresolved shell. All other shell and central source parameters were kept constant in the simulations. The only free parameters are the power law indexes that define the clump size and density probability distributions. 3.1 The effect of the condensed mass fraction on the emission line ratios The fc parameter affects the total line luminosity, because the presence globules affect the radiation matter interaction and the emission measure. The H luminosity increases with fc as shown in figure 2 (top). For instance, this line luminosity is more than 2 times larger than expected in homogeneous shell model. These results show that the shell mass obtained by using the Balmer line luminosities, could be strongly affected by the condensed mass fraction or the mass distribution of the ejecta.

8 The behavior of He II 4686/ H ratio can be seen in figure 2 (bottom). The increase of the condensed mass fraction yields a decrease in the line ratio. The presence of clumps reduce the He + ionization zones by shadowing the radiation field. The He II 4686/H ratio value may decrease by a factor 10 from an homogeneous shell to a shell with 90 % condensed mass. In the same fc range, the He I 5876/H presents only a 6% variation. Both helium and Balmer lines are used to obtain the helium abundance and therefore the condensed mass fraction affect this determination. The He/H abundance obtained by the He II and He I lines in the shell with fc = 0.9 is 45% lower. The [O III] 5007/H ratio is also affected by condensations. The figure 3 (top) shows the line ratio behavior with fc. The ratio found in a 90% clumpy shell is ten times larger than the ratio in a shell with fc = The [O III] total luminosity of the shell without condensations is 20 times smaller than the emission of the shell with 5% condensed mass. The collisionally excited line fluxex will increase with density. If the clump density increase to close to the critical density of 1 D 2 level, there will be a decrease in the [O III] 5007/H ratio. The [O II] 7325 line flux presented a factor 3 variation in the same fc range. However, the [O II]7325/H ratio increases by only 22 % from fc = 0 to fc = 0.9. The increase of mass fraction can rise the neutral oxygen population to a factor of 10 4 from fc = 0 to fc = 0.9 with a corresponding increase in [O I] 6300 emission. The nitrogen lines [N II] 6548, 6584 have the same behavior of [O III] For instance, the value of [N II] 6584 /H in the shell with fc = 0.9 is 20 times larger when compared to fc = Since the nebular [NII] excited levels have lower critical density than those of [OIII], the density increase inside clumps may affect the nitrogen line ratios by collisional de excitation. At low densities, the [O III] and [N II] auroral to nebular line ratios can be used to obtain the local electron temperature. The mass fraction ratio affects drastically the temperature of the shell. The figure 3 (bottom) shows how the fc increase render the shell cooler. Since the globules increase the emission of cooling lines and the density clump block the radiation field, the mean electron temperature is lower for clumpy shells.

9 3.2 The effect of the clump size and density distribution on the emission line ratios In this analysis, a constant condensed mass fraction of fc = 0.5 was adopted. All models simulate the nebular phase conditions, with low N H density ( cm 3 ) and a hot ionizing source. The maximum and minimum clump size and density are constant parameters, while the power law indices are the free parameters in the calculations. The table 1 shows the model parameters used. There are models with high density clumps, with globule density contrast between 10 and 50 larger times the background (k10 50 models) and models with clumps density between 3 and 10 times the background (model k3 10). The effect of the clumps size and density distribution depends on the average density of the shell and the emission line considered for comparison. The clump size distribution affects the total H luminosity in a constant shell mass models. Since this Balmer line is often used to estimate the shell mass, this is an useful simulation. In the lower density clump models, (hereafter, model k3 10), the H luminosity vary 30% between models. The higher line luminosity corresponds to a shell with a larger number of small clumps (and also larger total number of clumps). For the higher density clumps models (hereafter model k10 50), the effect is even more intense. The shells with a larger number of small clumps have twice the H luminosity than the shell with a flat size distribution, (see figure 4). Therefore, the clumps size distribution can affect significantly the shell mass determination in condensed shells. Now, considering only the density distribution slope ( k ) in k3 10 and k10 50 models, they have the same H flux variation is found; about 35% and 2 times the H flux. For both models, the shell with a higher number of denser clumps have higher H luminosity. The shell emission line ratios are affected by the power law indices of clump size ( fw ) and density ( k ) distribution. For k3 10 models, the He II 4686/ H ratio in the shells with smaller clumps is 2.2

10 times higher than the ratio in a shell with larger clumps. The calculations showed that there is a plateau, where an increase in the clump size does not affect the line ratio. The clump density distribution also affect the He II 4686/H ratio, by factor ~2 (see figure 6). The shell with a larger population of low density clump has a higher line ratio as expected, since the low density clumps block the ionizing radiation field less effectively and the He ++ population rise in number. The calculations indicate that there is a plateau for k > 2, because the volume shielded from the ionizing radiation field remains constant. The He I 5876/H ratio have no significant variation with clumps size and density distribution (only 2%). Both line ratios are used in helium abundance determination, but in this case the He ++ population is more affected by the clumps properties. The He II 4686 to H recombination coefficient ratio vary only 12 % from 5000 to K, but a factor 2 variation in this line ratio is seen in clumpy shell models, depending on the clump size and density characteristics. For example, the He abundance determinations using the k3 10 model indicate that the measured abundance may be underestimated by 30 40%. For the k10 50 models the correction is in the 40 50% range. The calculations showed that for the k3 10 models, the [O III]5007/H ratio vary by a factor of 3 from models with a less dense clump population to models with a denser clump population, until k = 2. For k > 2, where vast majority of the clumps are dense, the line ratio is constant. The k10 50 models show the same behavior, but the line ratio variation was smaller, (60%) and radiation block becomes important earlier, at k > 0, where there is a plateau (see figure 5). The clump size distribution also affects the line ratios derived from shell models. For the k3 10 models, the line ratio has a 70% increase from a population of smaller clumps to a population of large clumps. Large clumps shield the O ++ from the UV radiation more effectively in these models. For a denser shell, an increase of the clump size reduce the O ++ population and therefore the [O III] emissivity. This effect was observed in the k10 50 models. In the case of k3 10 models a 2000 K change was observed in the mean shell temperature by varying the clump density distribution. The models with a population

11 of less dense clumps have a higher temperature when compared to the models with a denser clump population. The size distribution of the clumps produces a 1200 K temperature variation (see figure 6). The hotter shell is the one with the majority of smaller clumps in its globule population. The colder shell models are those with a majority of large clumps. The [O II] 7319,7330 emission lines are strong in many old novae remnants and they are important lines for the oxygen abundance determination. The [O II]7325/H line ratio has a small variation with clump size and density distribution (around 10%). For k10 50 models the line ratio is somewhat larger, since the shell has a larger density and this increases the O + population. The maximum density in all models are lower than the critical density for this line. On the contrary, the [N II] 5755/H ratio is affected by clump size and density distribution, in a typical nova shell environment. The k3 10 models reveal a line ratio increase of 50% for models with denser clumps. In the k10 50 models, the clump density are below the critical density for collisional de excitation of the 1 S level and the calculations show the same behavior of the k3 10 models. The [N II] 6548,6584 lines present a larger sensitivity than the 5755 line. For the k3 10 models the [N II]6584/H ratio has a factor of 5 variation by changing the clump size distribution and a factor 7 with the clump density distribution. The k10 50 models have the same line ratio behavior, but the variations were smaller, a factor 2 and 3 for size and density distribution respectively. 3.3 Robust line ratios There are some line ratios that do not change significantly with the clump populations characteristics. The He + recombination lines show variations smaller than 5% in the models. The Balmer decrement remains roughly constant in the models. The H /H ratio varies only 10% for the whole range of parameters. Therefore the He + abundance determination is poorly affected and only

12 the collisional excitation corrections are needed to be done to obtain a fair He + abundance value. In the novae scenario, the H line is often blended with [N II] 6548,6584 lines because of the shell expansion velocity. In reddening corrected spectra, higher H /H values may be due to the [N II] contribution. 4 Discussion The presence of condensations in novae ejecta could be seen in spatial resolved shells, like HR Del, GK Per, RR Pic and T Pyx. Many nova without resolved shell show some emission lines ratios that are possible only if the ejecta have heterogeneous mass distribution (Williams 1994). There are many mechanisms that could form condensations in the nova expanding nebula. The Rayleigh Taylor instability (RTI), is one possibility. The RTI is excited in the first few seconds of the eruption and occurs because the ejecta has a velocity distribution and propagate in a stratified media (Chevalier et al. 1992). The ejecta expansion decreases the mean density and the shell becomes optically thin to emission lines. The corresponding cooling makes also the shell Kelvin Helmholtz unstable (Chevalier et al 1992). This thermal instability depends on the shell chemical composition, density and radiation field. The thermal instability was discussed in detail by Pistinner and Shaviv, 1995 who found that it has a much shorter scale (~10 4 factor) when compared to the shell radius. The density contrast for this instability is less than 100%. But it could give a rise of a large number of small clumps. The mass ejection itself is affected by the binary motion and white dwarf rotation. The orbital motion of the secondary in the common envelope phase was studied using 2.5D hydrodynamical simulations (Livio et al and Lloyd et al. 1997). The binary motion accumulate matter in the polar regions. If the WD is a fast rotator, the effective gravity gradient from polar to equatorial regions affects the local maximum temperature achieved at the outburst and therefore the strength of the thermonuclear reactions (Scott, 2000). This makes the ejection velocity

13 in the polar regions be greater than in the equatorial regions and the ejecta could be prolate or bipolar, (e.g. RS Oph and HR Del). The Lloyd et al. calculations that include the binary motion and the WD rotation, found that the density contrast is ~ 16 at a given shell radius and 64 from globules to the neighborhood diffuse regions. Recent observations of HR Del remnant show that it has a density contrast of 3 to 16 (Moraes & Diaz, 2009). The RTI are possibly related to the nova speed class, since the fast nova light curve are associated with higher expansion velocity speeds. Lloyd et al obtained a relation between the nova light curve decay time, t 3, and the maximum length of the RTI condensations. For very fast novae, or t 3 < 12, the maximum size of the condensations is 40% the shell radius (e.g. V1500 Cyg, Slavin et al. 1995). For slower novae, the maximum condensation length decreases. For very slow novae, the clumps size could be only 10 4 of the shell radius. All these effects could give a rise many nova remnant configurations, with a variety of condensed mass fraction and clump size and density distributions and justify the construction of a variety of shells models with parametrized clump population distributions. A full 3 D code demands huge CPU time and more parameters number to describe the novae remnant geometry. The pseudo 3D codes, like RAINY3D could treat the condensations by optimally assembling 1D models and allow to build many different shell geometries with clumps. The transfer of diffuse radiation is treated in 1D, although no significant differences were observed by comparing OTS to 1D transfer. The 2.5D simulations make possible to study the effect of the condensations population characteristics on the emitted spectrum and reveal the ionization structure both locally and at large scale (shadows). The calculations show that the globules affect the ionization structure inside and between inhomogeneities. The analysis of the novae spectral evolution by Williams et al. (1991) showed that any calculation of heavy element abundances with respect to hydrogen which involves forbidden lines at initial times of the nebular phase must consider the effect of condensations. The study of the condensations influence on the emitted spectra of the shell during the nebular phase (optical thin gas) made in this work shows that even

14 decades after the eruption, the condensations may affect the shell physical and chemicals diagnostics. The clumpy models show that the total H luminosity is affected by the condensed mass fraction and by the globules physical characteristics at the nebular phase. Even at lower density, the Balmer flux can vary by more than 4 by the presence of condensations in the shell. The presence of clumps can easily introduce a factor ~5 error in shell mass determinations. At the beginning of nebular phase the globules effect could be larger. Diaz et al. (accepted) models of recurrent nova U Sco show that the extreme nebular to auroral ratios observed require the presence of clumps. The calculations made in this work show that the He/H abundances determination are underestimated when the shell is clumpy. Since the He I lines do not show variations, the neutral He population may increase in the clumps. The He/H determination in different regions of the spatial resolved shell of HR Del show that in the clumps, the abundance obtained was ~20% lower than other regions (Moraes & Diaz, 2009). The oxygen abundance determinations also are affected by the condensations in the shell. The [O III] 5007/H ratio observed in the nova nebular phase is only obtained by the models when the shell is clumpy. Depending on the ionization structure of the shell, or O ++ ion abundance, this determination could affect the overall oxygen abundance determination. The nebular phase models calculated in this work show that the main oxygen population is at O + stage and also that the [O II] 7325 /H ratio has a small variation with the mass in clumps and their size and density distribution. The nitrogen lines, ([N II] 6548,6584), show the same behavior of oxygen lines. One important effect of the condensations is to decrease the electronic temperature of the shell, since some forbidden lines have their emission enhanced and cool the gas. One interesting result of the calculations is that the lower density contrast clumps models (k3 10) are much more affected by the condensations properties. These models show great temperature variations as function of the clump size and density distribution (see figure 6). If the emission of the line used to obtain the relative abundance is affected by local temperature, the abundance determination could

15 be wrong. There are some emission lines that are not strongly affected by presence of the condensations in the shell. The He I/H and Balmer decrement, show less than 5% variation with clumps characteristics. 5 Conclusion The code RAINY3D can built a 3 D shell with condensations and combine weighted Cloudy 1D calculations to obtain a pseudo 3D photoionization model of nova remnant. The condensations formation and survival mechanisms are not known and the RAINY3D simulations provided an insight on how the condensations parameters could affect the spectra from nova ejecta during the nebular phase. The condensation amount, defined as fc, and the globules size and density distribution can affect the emission line ratios and affect the accurate diagnostics of the nebula, including mass and chemical abundance determinations. Williams (1991) showed that the physical and chemical diagnostics are thought to be more accurate during the nebular phase, but the calculations made in this work show that even for low density shells, the condensations could affect significantly the results of quantitative spectral analysis. The characteristics of the gas condensations are very important to nebular photoionization models. The clump formation are related to the ejection mechanisms, binary interaction, white dwarf properties and interactions between ejected components of the shell. Therefore they could be a constrain for the modeling of several phenomena related to TNR's and shell expansion. The effort to observe resolved novae shells with large telescopes and spatially resolved spectroscopy can elucidate the clump size and density distribution in nova ejecta.

16 6 References 1992ApJ C Chevalier, Roger A.; Blondin, John M.; Emmering, Robert T. 2001ASPC D Diaz, Marcos 2002AIPC D Diaz, Marcos P. 2003MNRAS E Ercolano, B.; Barlow, M. J.; Storey, P. J.; Liu, X. W. 2005AAS F Ferland, G. J. 1997ApJ G Gruenwald, R.; Viegas, S. M.; Broguiere, D. 1997ApJ H Hauschildt, Peter H.; Shore, Steven N.; Schwarz, Greg J.; Baron, E.; Starrfield, S.; Allard, France 1990ApJ L Livio, Mario; Shankar, Anurag; Burkert, Andreas; Truran, James W. 1997MNRAS L Lloyd, H. M.; O'Brien, T. J.; Bode, M. F. 2009AJ M Moraes, Manoel; Diaz, Marcos 2005RMxAC M Morisset, C.; Stasińska, G.; Peña, M. 1995ApJ...448L..37P Pistinner, Shlomi; Shaviv, Giora 2003A&A R Rauch, T. 2000MNRAS S Scott, A. D. 1995MNRAS S Slavin, A. J.; O'Brien, T. J.; Dunlop, J. S. 1991ApJ W Williams, R. E.; Hamuy, M.; Phillips, M. M.; Heathcote, S. R.; Wells, Lisa; Navarrete, M. 1992AJ W Williams, Robert E. 1994ApJ W Williams, Robert E.

17 Tables: White dwarf temperature (K) Luminosity (log erg/s) 36 Shell mass (M ʘ ) 10 4 Condensed mass fraction 0,5 Local gravity (log g) 8 Continuum shape Rauch Atmosphere Abundance He/H (log He/H) 0,85 Abundance C/H (log C/H) 2,5 Abundance N/H (log N/H) 2 Abundance O/H (log O/H) 2,5 Other elements abundance Solar Shell size (cm) 4x10 16 Maximum clump size (log cm) 16 Minimum clump size (log cm) 14 Maximum clump density (bkg) 10 Minimum clump density (bkg) 3 Range fw 2 to 4 Range k 2 to 4 Table 1 Model parameter for clumps size and density distribution analysis.

18 Figures: Figure 1 Radial profile of the background (top) 3D shell component and the globules added to the background (bottom).

19 Figure 2 The luminosity of H as function of the condensed mass fraction (top). The increasing of clumps increase the mean density and therefore the recombination rate. The He II 4686/H ratio as function of condensed mass fraction (bottom). The increasing of condensed mass fraction decrease the He ++ population.

20 Figure 3 The [O III] 5007/ H ratio as function of condensed mass fraction (top). The shell temperature obtained from [O III] line diagnostics. The increasing of the mean density make the cooling of the shell more effective (bottom).

21 Figure 4 The H luminosity as function of the density ( k ) and size ( fw ) power law indexes. The line luminosity is more affected by the size distribution law of the clumps than the density distribution. A smaller clumps population have a large line luminosity. Figure 5 The [O III] 5007 / H ratio as function of the density power law index ( k ) for models with lower (k3 10) and higher (k10 50) density clumps.

22 Figure 6 The He II 4686/ H ratio (top) as function of the density ( k ) and size ( fw ) power law index of the clumps population distribution (models k3 10). The temperature obtained from [O III] diagnostics lines as function of the density and size power law index (bottom). The clumps size and density distribution affect more the lower density globules models (k3 10).

23