3.4 Dust modeling

PB 8 is known to be very dusty (e.g. Lenzuni et al., 1989; Stasinska & Szczerba, 1999), which must influence the radiative processes in the nebula. Lenzuni et al. (1989) studied the IRAS measurements (25, 60 and 100 $\mu$m fluxes), and derived a dust temperature of $T_{\rm d} = 85 \pm 0.4$K, an optical depth of $\tau($Ly-c$)=0.63$ and a dust-to-gas mass ratio of $\rho_{\rm d}/\rho_{\rm g} =0.0123$ from a blackbody function fitted to the IRAS data. Similarly, Stasinska & Szczerba (1999) determined $T_{\rm d} = 85$K, but $\rho_{\rm d}/\rho_{\rm g} =0.0096$ from the broad band IRAS data. From the comparison of the mid-IR emission with a blackbody model of $150$K, Todt et al. (2010) suggested that it possibly contains a warm dust with different dust compositions. We notice that the models MC1 and MC2 cannot provide thermal effects to account for the Spitzer IR continuum, so a dust component is necessary to reproduce the spectral energy distribution (SED) of the nebula observed in the IR range. The third model (MC3) presented here treats dust properties of PB 8 using the dust radiative transfer features included in the MOCASSIN (Ercolano et al., 2005). Discrete grain sizes have been chosen based on the size range given by Mathis et al. (1977). The absence of the 9.7 $\mu$m amorphous silicate feature in the IR spectrum of PB 8 is commonly observed in O-rich circumstellar envelopes, which could imply this PN has a carbon-based dust. However, the strong features at 23.5, 27.5, and 33.8 $\mu$m are mostly attributed to crystalline silicates (Molster et al., 2002). The features seen at 6.2, 7.7, 8.6, and 11.3 $\mu$m are related to polycyclic aromatic hydrocarbons (PAHs) (García-Lario et al., 1999), together with broad features at 21 and 30 $\mu$m corresponding to a mixed chemistry having both O-rich and C-rich dust grains. The far-IR emission fluxes at 65 and 90 $\mu$m (Yamamura et al., 2010) could be related to relatively warm forsterite grains, which emit at a longer wavelength. The 65 $\mu$m emission may be related to a crystalline water-ice structure, although its presence cannot be confirmed at the moment.

Figure: Observed Spitzer spectrum (black line) of PB8 are compared with the continuum predicted by the model MC2 (blue line) and MC3 (red line). It also shows the photometric measurements for 12, 25, 60, and 100 $\mu$m (denoted by green diamonds) from IRAS (Helou & Walker, 1988), 8.3, 12.1, 14.7, and 21.34 $\mu$m (orange downward triangle) from MSX (Egan et al., 2003), and the far-IR measurements (blue squares) $F$(65$\mu$m $)=5.60\pm 0.19$, $F$(90$\mu$m $)=5.83\pm 0.16$, and $F$(140$\mu$m $)=1.74\pm 3.33$ Jy from AKARI/FIS (Yamamura et al., 2010). Note that the predicted nebular SED does not contain any nebular emission line fluxes.

The thermal IR emission of PB 8 was modeled by adding a mixed dust chemistry to the pure-gas photoionization model described in the previous sections. We explored a number of grain sizes and species, which could provide a best-fitting curve to the Spitzer IR continuum (see Figure5). As seen, the model MC2 cannot produce the IR continuum, whereas the model MC3 fairly produces it. We tried to match the far-IR emission flux at 65 $\mu$m, while the 140 $\mu$m flux is extremely uncertain, $F$(140$\mu$m $)=1.74\pm 3.33$ Jy. The dust-to-gas mass ratio was varied until the best IR continuum flux was produced. Table 5 lists the dust parameters used for the final model of PB8, the dust-to-gas ratio is given in Table 3. The geometry of the dust distribution is the same as the gas density distribution. The value of $\rho_{\rm d}/\rho_{\rm g} = 0.01$ found here is in agreement with Lenzuni et al. (1989). The final dust model incorporates two different grains, amorphous carbon and crystalline silicate with optical constants taken from Hanner (1988) and Jaeger et al. (1994), respectively. We also note that the nebular SED (shown in Figure5), which is computed by MOCASSIN, does not contain any contributions from the nebular emission line fluxes.

For PB8, Lenzuni et al. (1989) estimated a grain radius of $0.017$ $\mu$m from the thermal balance equation under the assumption of the UV absorption efficiency $Q_{\rm UV}=1$. Stasinska & Szczerba (1999) argued that the method of Lenzuni et al. underestimates the grain radius, and one cannot derive the grain size in such a way. Our photoionization modeling implies that dust grains with a radius of $0.017$ $\mu$m produce a very warm emission higher than $T_{\rm d} = 85$K. The final dust model uses two discrete grain sizes, namely grain radii of 0.16 $\mu$m (warm) and 0.40 $\mu$m (cool), which can fairly reproduce the observed thermal infrared SED with wavelengths less than 80 $\mu$m. Smaller grain sizes can produce hot emission that increases the continuum at shorter wavelengths ($<10$ $\mu$m), whereas larger grain sizes add cooler emission that may depict as a rise in the continuum at longer wavelengths ($>80$ $\mu$m). Although the current two grain sizes can well reproduce the Spitzer SED of PB 8, the solutions may not be unique. A dust model with more than two grain sizes may also be possible, but it needs more computational simulations to find the best-fit model. Moreover, inhomogeneous dust distribution and viewing angles (see Figure 2 in Ercolano et al., 2005) can also change the predicted SED. As there is no information on the inclination of dust grains and their geometry, so we assumed that they follow the gas density distribution.

There is a discrepancy in fluxes with wavelengths higher than 80 $\mu$m, which could be attributed to a possible inhomogeneous dust distribution. We note that the band measurements with wavelengths higher 100 $\mu$m could have high uncertainties, e.g., $F$(140$\mu$m $)=1.74\pm 3.33$Jy. We also note a small discrepancy for fluxes with wavelengths less than 15 $\mu$m, which could be related to the difference between the SL aperture ( $3.7 \times 57$ arcsec$^2$) and the LL aperture ( $10.7 \times 168$ arcsec$^2$) used for the SL spectrum (5.2-14.5$\mu$m) and the LL spectrum (14.0-38.0$\mu$m), and uncertainties in scaling the LL spectrum (see Sec. 2). As both the LL and SL apertures covering some areas larger than the optical angular diameter of PB8 (7 arcsec), they could also be contaminated by the ISM surrounding the nebula.

Table 6: Comparison of predictions from the models and the observations. The observed, dereddened intensities are in units such that $I$(H$\beta$)$=100$. Columns (6)-(12) give the ratios of predicted over observed values in each case.

Line	$\lambda_{0}$(Å)	Mult	$I_{\rm obs}$	Err(%)	MC1	MC2			MC3
						Normal	M-rich	Total	Normal	M-rich	Total
H, He recombination lines
H$\beta$	4861.33	H4	100.000	5.0	1.000	0.772	0.228	1.000	0.771	0.229	1.000
H$\alpha$	6562.82	H3	282.564	6.0	1.031	0.801	0.247	1.047	0.799	0.247	1.047
H$\gamma$	4340.47	H5	45.666	5.0	1.019	0.784	0.228	1.013	0.784	0.229	1.013
H$\delta$	4101.74	H6	24.285	5.0	1.057	0.813	0.235	1.048	0.813	0.236	1.049
HI	3970.07	H7	14.466	6.0	1.089	0.838	0.242	1.080	0.837	0.243	1.080
HI	3835.39	H9	6.784	6.0	1.069	0.822	0.238	1.060	0.822	0.239	1.061
He I	3888.65	2	19.892	6.0	0.702	0.531	0.235	0.766	0.532	0.236	0.768
He I	7065.28	10	4.265	7.0	0.752	0.552	0.202	0.754	0.555	0.203	0.758
He I	5875.64	11	17.127	6.0	1.089	0.852	0.473	1.325	0.850	0.473	1.323
He I	4471.47	14	6.476	5.0	1.014	0.787	0.411	1.199	0.786	0.412	1.198
He I	4026.21	18	3.116	6.0	0.976	0.757	0.366	1.123	0.756	0.367	1.123
He I	7281.35	45	0.815	8.0	1.126	0.842	0.339	1.181	0.844	0.340	1.185
He I	6678.15	46	5.233	6.0	1.020	0.802	0.432	1.233	0.799	0.432	1.231
He I	4921.93	48	1.737	5.0	1.014	0.790	0.402	1.191	0.788	0.403	1.191
Heavy-element recombination lines
C II	6578.05	2	0.545	9.0	0.575	0.438	0.126	0.563	0.437	0.126	0.563
C II	7231.34	3	0.234	17.0	1.088	0.840	0.257	1.096	0.837	0.257	1.094
C II	7236.42	3	0.464	10.0	0.988	0.762	0.233	0.995	0.760	0.233	0.993
C II	4267.15	6	0.781	7.0	0.857	0.670	0.218	0.888	0.667	0.218	0.885
N II	5666.64	3	0.192	25.0	0.114	0.048	0.683	0.731	0.048	0.685	0.732
N II	5676.02	3	0.084	:	0.116	0.049	0.693	0.741	0.048	0.694	0.743
N II	5679.56	3	0.260	18.0	0.157	0.066	0.940	1.006	0.066	0.942	1.007
N II	4601.48	5	0.099	21.0	0.073	0.031	0.420	0.451	0.030	0.421	0.452
N II	4607.16	5	0.083	25.0	0.070	0.029	0.400	0.429	0.029	0.401	0.430
N II	4613.87	5	0.063	30.0	0.069	0.029	0.395	0.424	0.029	0.396	0.424
N II	4621.39	5	0.085	24.0	0.068	0.028	0.390	0.418	0.028	0.391	0.419
N II	4630.54	5	0.289	10.0	0.075	0.031	0.429	0.460	0.031	0.430	0.461
N II	4643.06	5	0.122	18.0	0.059	0.025	0.338	0.362	0.025	0.339	0.363
N II	4994.37	24	0.099	21.0	0.066	0.028	0.420	0.448	0.028	0.421	0.449
N II	5931.78	28	0.151	30.0	0.045	0.019	0.284	0.303	0.019	0.285	0.304
N II	5941.65	28	0.115	:	0.110	0.047	0.696	0.743	0.046	0.697	0.743
O II	4638.86	1	0.206	12.0	0.181	0.197	0.527	0.724	0.196	0.527	0.723
O II	4641.81	1	0.380	8.0	0.248	0.270	0.720	0.990	0.268	0.721	0.989
O II	4649.13	1	0.458	8.0	0.391	0.426	1.136	1.562	0.423	1.137	1.561
O II	4650.84	1	0.221	12.0	0.169	0.184	0.491	0.675	0.183	0.491	0.674
O II	4661.63	1	0.222	12.0	0.215	0.234	0.624	0.858	0.232	0.625	0.857
O II	4676.24	1	0.184	13.0	0.218	0.237	0.633	0.870	0.236	0.633	0.869
O II	4319.63	2	0.081	26.0	0.364	0.397	1.067	1.465	0.395	1.068	1.463
O II	4336.83	2	0.054	36.0	0.161	0.176	0.472	0.648	0.175	0.472	0.647
O II	4349.43	2	0.197	13.0	0.346	0.378	1.016	1.395	0.376	1.017	1.393
O II	3749.48	3	0.281	11.0	0.132	0.143	0.375	0.518	0.142	0.376	0.518
O II	4414.90	5	0.036	:	0.489	0.524	1.275	1.799	0.521	1.278	1.799
O II	4416.97	5	0.090	24.0	0.109	0.116	0.283	0.399	0.116	0.284	0.399
O II	4072.15	10	0.265	11.0	0.331	0.363	0.987	1.350	0.360	0.988	1.348
O II	4075.86	10	0.275	11.0	0.460	0.505	1.374	1.879	0.501	1.375	1.876
O II	4085.11	10	0.086	26.0	0.190	0.209	0.568	0.776	0.207	0.568	0.775
O II	4121.46	19	0.163	16.0	0.063	0.069	0.191	0.260	0.069	0.191	0.260
O II	4132.80	19	0.202	13.0	0.099	0.109	0.301	0.410	0.108	0.301	0.410
O II	4153.30	19	0.250	12.0	0.115	0.126	0.348	0.474	0.125	0.348	0.473
O II	4110.79	20	0.147	17.0	0.060	0.066	0.182	0.248	0.066	0.182	0.248
O II	4119.22	20	0.087	25.0	0.374	0.411	1.133	1.544	0.408	1.133	1.541
O II	4699.22	25	0.026	:	0.093	0.103	0.283	0.386	0.102	0.283	0.385
O II	4906.81	28	0.096	21.0	0.096	0.105	0.289	0.394	0.104	0.289	0.394
O II	4924.53	28	0.154	15.0	0.101	0.111	0.307	0.418	0.111	0.307	0.417
Collisionally excited lines
$^{{d}}$	5754.64	3F	0.346	14.0	0.495	0.186	0.506	0.692	0.199	0.509	0.708
	6548.03	1F	7.667	6.0	0.926	0.393	0.613	1.006	0.418	0.631	1.048
	6583.41	1F	22.318	6.0	0.972	0.412	0.643	1.055	0.438	0.662	1.100
	3726.03	1F	17.103	6.0	0.897	0.947	0.061	1.008	1.035	0.065	1.100
	3728.82	1F	9.450	6.0	0.782	0.813	0.044	0.857	0.890	0.047	0.936
$^{{d}}$	7318.92	2F	0.227	18.0	0.530	0.491	0.538	1.029	0.546	0.539	1.085
$^{{d}}$	7319.99	2F	0.811	8.0	0.451	0.418	0.536	0.954	0.465	0.538	1.003
$^{{d}}$	7329.66	2F	0.387	12.0	0.518	0.480	0.537	1.017	0.535	0.539	1.074
$^{{d}}$	7330.73	2F	0.471	10.0	0.419	0.388	0.536	0.924	0.432	0.537	0.969
	4363.21	2F	0.528	7.0	1.733	0.927	0.007	0.935	0.995	0.008	1.003
	4958.91	1F	116.957	5.0	1.130	0.858	0.137	0.995	0.886	0.143	1.028
	5006.84	1F	348.532	5.0	1.132	0.859	0.137	0.996	0.887	0.143	1.030
[Ne II]	12.82$\mu$m		24.370	...	0.516	0.696	0.209	0.905	0.716	0.207	0.923
	3868.75	1F	19.164	6.0	1.131	0.784	0.006	0.790	0.816	0.006	0.822
	3967.46	1F	5.689	6.0	1.147	0.795	0.006	0.801	0.828	0.006	0.835
	15.55$\mu$m		110.660	...	1.006	1.054	0.203	1.257	1.050	0.206	1.255
	36.02$\mu$m		7.360	...	1.275	1.331	0.236	1.566	1.326	0.239	1.565
[S II]	4068.60	1F	0.223	:	1.004	0.699	0.011	0.710	0.772	0.012	0.784
	6716.47	2F	0.957	7.0	0.978	0.742	0.026	0.769	0.809	0.028	0.837
	6730.85	2F	1.441	7.0	1.010	0.773	0.032	0.805	0.843	0.034	0.877
	6312.10	3F	0.639	9.0	3.617	1.984	0.015	2.000	2.109	0.016	2.126
	18.68$\mu$m		54.820	...	2.495	1.974	0.393	2.367	2.008	0.395	2.403
	33.65$\mu$m		30.360	...	2.076	1.624	0.225	1.849	1.652	0.226	1.878
[Cl III]	5517.71	1F	0.366	14.0	0.940	0.711	0.014	0.725	0.736	0.014	0.750
	5537.88	1F	0.366	14.0	1.054	0.806	0.020	0.826	0.834	0.021	0.855
[Ar III]	7135.78	1F	15.477	7.0	1.048	0.775	0.035	0.809	0.796	0.035	0.831
	7751.10	2F	3.493	7.0	1.113	0.822	0.037	0.859	0.845	0.038	0.883
	8.99$\mu$m		14.970	...	1.657	1.483	0.351	1.834	1.490	0.351	1.841
H$\beta$ $^{\mathrm{a}}$/10$^{-12}$ ergcm$^{-2}$s$^{-1}$			19.7	13.0	0.884	0.863	0.255	1.118	0.856	0.254	1.110
$\tau$(HI) $^{\mathrm{b}}$					0.556			0.581			0.651
$\tau$(He I) $^{\mathrm{c}}$					0.336			0.343			0.422

$^{\mathrm{a}}$

The intrinsic H$\beta$ line flux of the entire nebula.

$^{\mathrm{b}}$

Optical depth at the HI photoionization threshold (13.6eV).

$^{\mathrm{c}}$

Optical depth at the He I photoionization threshold (24.6eV).

$^{\mathrm{d}}$

Recombination contribution estimated by equations (1) and (2) included in the predicted lines.