4 [Fe VI] line ratios: Temperature and density diagnostics

Some of the salient features of the spectral diagnostics with FLE are discussed in CP00. It is shown that the line ratios can be parametrized as a function of $T_{\rm e},N_{\rm e},T_{\rm eff},W(r)$ . For a given subset of ( $T_{\rm e},N_{\rm e}$ ) a line ratio may describe the locus of the subsets of ( $T_{\rm eff},W(r)$ ), which then defines (constrains) a contour of possible parameters. CP00 present 3-dimensional plots of the line ratio vs. ( $T_{\rm eff},W(r)$ ), for given ( $T_{\rm e},N_{\rm e}$ ). While it is clear that the $T_{\rm eff}$ or the W(r) can not be determined uniquely and independently, it was found that the observed value of the line ratio cuts across the surface (double-valued function in $T_{\rm eff},W(r)$ ), along the contour of likely values that lie within. The variation of the intensity ratio with effective temperature and the distance of the emitting region may constrain these two macrospopic quantities, in addition to the determination of the local electron temperature and density.

The spectral diagnostics so developed is applied to the analysis of [Fe VI] lines from planetary nebulae as described below.

4.1 Planetary nebulae

The central stars of planetary nebulae correspond to high stellar radiation temperatures (e.g. Harman & Seaton 1966), of the order of 10⁵ K. Resonant absorption was first discussed by Seaton (1968), who pointed out the efficacy of this mechanism in line formation of [O III], in addition to electron scattering and recombination, estimated the oxygen abundance taking this into account. We might expect, a priori, that if the atomic structure of the emitting ion is subject to FLE then the PNe might be good candidates for radiative fluorescence studies in general, as shown in CP00.

In recent years, Hyung and Aller in particular have made a number of extensive spectral studies of PNe, and in nearly all of those [Fe VI] optical emission lines have been detected (Table 5).

Table 5: Physical conditions of gaseous nebulae
Source $N_{\rm e}/10^3~{\rm cm}^{-3}$ $T_{\rm e}/10^3~{\rm K}$ $T_{\rm eff}/{\rm K}$ r/pc R/ $R_{\odot}$ W Reference

NGC 6741 6.3 12.5 140 0.0063 0.063 $1.3\,\, 10^{-14}$ Hyung & Aller 1997a; a

NGC 6886 5 - 10 13 150 0.001(0.0345) 0.046 $2.7\,\, 10^{-13}$ Hyung et al. 1995; b

NGC 6884 10 10 110 0.002(0.020) 0.13 $5.4\,\, 10^{-13}$ Hyung et al. 1997; c

IC 351 2.5 - 20 13 - 16 58.1 0.05 0.72 $2.6\,\, 10^{-14}$ Feibelman et al. 1996; d

NGC 2440 5 14.2 180 0.015(0.0425) 0.038 $8.2\,\, 10^{-16}$ Hyung & Aller 1998; e

NGC 7662 3 - 17 13 105 0.025(0.035) 0.15 $4.6\,\, 10^{-15}$ Hyung & Aller 1997b; f

**Table 5:** Physical conditions of gaseous nebulae
Source	$N_{\rm e}/10^3~{\rm cm}^{-3}$	$T_{\rm e}/10^3~{\rm K}$	$T_{\rm eff}/{\rm K}$	r/pc	R/ $R_{\odot}$	W	Reference
NGC 6741	6.3	12.5	140	0.0063	0.063	$1.3\,\, 10^{-14}$	Hyung & Aller 1997a; a
NGC 6886	5 - 10	13	150	0.001(0.0345)	0.046	$2.7\,\, 10^{-13}$	Hyung et al. 1995; b
NGC 6884	10	10	110	0.002(0.020)	0.13	$5.4\,\, 10^{-13}$	Hyung et al. 1997; c
IC 351	2.5 - 20	13 - 16	58.1	0.05	0.72	$2.6\,\, 10^{-14}$	Feibelman et al. 1996; d
NGC 2440	5	14.2	180	0.015(0.0425)	0.038	$8.2\,\, 10^{-16}$	Hyung & Aller 1998; e
NGC 7662	3 - 17	13	105	0.025(0.035)	0.15	$4.6\,\, 10^{-15}$	Hyung & Aller 1997b; f

$^{\rm a}$ Hyung & Aller (1997a); $^{\rm b}$ Hyung et al. (1995); $^{\rm c}$ Hyung et al. (1997); $^{\rm d}$ Feibelman et al. (1996); $^{\rm e}$ Hyung & Aller (1998);
$^{\rm f}$ Hyung & Aller (1997b).

Physical conditions in some of the PNe's are listed in Table 5. Observed line ratios are used to develop the temperature-density diagnostics for [Fe VI]. An earlier study of [Fe VI] line ratios was carried out by Nussbaumer & Storey (1978) who calculated a number of line emissivities relative to the $\lambda$ 5146 ${\rm\AA}$ line. Owing to new atomic collisional and radiative data our line ratios differ significantly from the earlier work for many lines. Also, Nussbaumer & Storey (1978) did not take the FLE mechanism into account. On examination of the observed [Fe VI] lines in several PNe, we noted that the $\lambda$ 5146 ${\rm\AA}$ line was mis-identified and assigned to [O I] in the PNe labeled a, c, d in Table 5 (in a private communication we confirmed the new identification with Prof. Lawrence Aller, who noted that [Fe VI] was the more likely source, particularly in NGC 6741 which is a high excitation object).

A comprehensive study of most of the possible line ratios was carried out as functions of $T_{\rm e}$ , $N_{\rm e}$ , and with and without FLE, at various radiation temperatures $T_{\rm eff}$ and dilution factors W( $R_{\ast}/r)$ . In Table 6 we present line ratios for lines which frequently appear in various kinds of PNe's, with different physical conditions, with respect to the line $\lambda$ 5146 ${\rm\AA}$ (as in Nussbaumer & Storey 1978). The partial Table 6 given in the text contains only those line ratios for which observed values are available.

Table 6: Partial Table 6 (complete table available electronically from CDS). Line intensity ratios for transitions relative to $\lambda$ 5146 AA (8-3: $^4{\rm F}_{7/2}$ - $^2{\rm G}_{7/2}$ ), with $T_{\rm eff}=150\ 000$ K, and with observed values from planetary nebulae in the fifth column (the full Table 6 contains a number of other line ratios). The first four entries are for no FLE, and FLE with $W=5\,\, 10^{-16}, 10^{-13}, 10^{-10}$ , respectively. Entries in the fifth column are values calculated by Nussbaumer & Storey (1978)

$T_{\rm e}=10000$ $T_{\rm e}=20000$ $T_{\rm e} = 12\ 000$ $T_{\rm e}=16000$

Transition $\lambda({\rm\AA})$ $n_{\rm e}=10^3$ 10⁴ 10³ 10⁴ $2\,\, 10^3$ $6\,\, 10^3$ 10⁴ $2\,\, 10^3$ $6\,\, 10^3$ 10⁴

8-2 4973 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1

9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1

9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1

9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1 9.64-1

9.67-1 9.67-1 9.67-1 9.67-1 Obs: 1.048^a; 1.094^d; 8.33-1^e; 1.652^f

9-4 5177 7.06-1 8.19-1 9.27-1 1.02-0 7.85-1 8.34-1 8.80-1 8.77-1 9.23-1 9.66-1

7.04-1 8.18-1 9.26-1 1.02-0 7.84-1 8.34-1 8.80-1 8.76-1 9.23-1 9.66-1

5.59-1 8.05-1 8.20-1 1.01-0 7.05-1 8.09-1 8.67-1 8.11-1 9.02-1 9.54-1

1.54-0 1.57-0 1.54-0 1.57-0 1.55-0 1.56-0 1.57-0 1.55-0 1.56-0 1.57-0

6.12-1 6.33-1 7.77-1 7.99-1 Obs: 7.39-1^a, 6.55-1^d,1.478^f

7-2 5234 6.55-2 7.23-2 5.65-2 6.10-2 6.37-2 6.65-2 6.91-2 5.98-2 6.22-2 6.44-2

6.60-2 7.24-2 5.67-2 6.10-2 6.39-2 6.66-2 6.92-2 5.99-2 6.22-2 6.44-2

1.25-1 8.51-2 8.39-2 6.53-2 9.30-2 7.94-2 7.80-2 7.97-2 7.05-2 7.00-2

1.81-1 1.81-1 1.81-1 1.79-1 1.81-1 1.81-1 1.80-1 1.81-1 1.80-1 1.80-1

Obs:

6-1 5278 1.87-1 1.94-1 1.56-1 1.61-1 1.79-1 1.82-1 1.85-1 1.66-1 1.69-1 1.71-1

1.92-1 1.95-1 1.58-1 1.61-1 1.81-1 1.83-1 1.85-1 1.67-1 1.69-1 1.71-1

7.27-1 2.94-1 3.98-1 1.93-1 4.39-1 2.88-1 2.54-1 3.40-1 2.36-1 2.14-1

5.05-1 4.98-1 5.05-1 4.95-1 5.04-1 5.01-1 4.97-1 5.04-1 5.00-1 4.96-1

2.19-1 2.21-1 1.83-1 1.84-1 Obs: 3.20-1^a, 5.56-1^d

5-1 5335 5.10-1 5.09-1 4.89-1 4.86-1 5.05-1 5.04-1 5.03-1 4.97-1 4.96-1 4.95-1

5.66-1 5.14-1 5.06-1 4.88-1 5.23-1 5.10-1 5.07-1 5.08-1 5.00-1 4.97-1

6.64-0 1.42-0 3.19-0 7.90-1 3.35-0 1.56-0 1.13-0 2.39-0 1.17-0 8.92-1

1.72-0 1.68-0 1.72-0 1.67-0 1.72-0 1.70-0 1.68-0 1.72-0 1.69-0 1.67-0

7.17-1 7.09-1 6.25-1 6.20-1 Obs: 7.65-1^a;1.083^b;9.52-1^c; 5.67-1^d; 1.0^f

6-2 5425 4.01-1 4.16-1 3.35-1 3.44-1 3.84-1 3.90-1 3.96-1 3.57-1 3.62-1 3.66-1

4.11-1 4.17-1 3.38-1 3.44-1 3.88-1 3.92-1 3.97-1 3.59-1 3.62-1 3.67-1

1.56-0 6.31-1 8.53-1 4.14-1 9.40-1 6.18-1 5.44-1 7.28-1 5.06-1 4.59-1

1.08-0 1.07-0 1.08-0 1.06-0 1.08-0 1.07-0 1.07-0 1.08-0 1.07-0 1.06-0

4.47-1 4.50-1 3.73-1 3.75-1 Obs: 4.85-1^a; 4.33-1^d; 7.61-1^f

7-3 5427 2.23-1 2.46-1 1.92-1 2.08-1 2.17-1 2.27-1 2.35-1 2.04-1 2.12-1 2.19-1

2.25-1 2.46-1 1.93-1 2.08-1 2.18-1 2.27-1 2.36-1 2.04-1 2.12-1 2.19-1

4.24-1 2.90-1 2.86-1 2.22-1 3.17-1 2.71-1 2.66-1 2.71-1 2.40-1 2.39-1

6.16-1 6.15-1 6.16-1 6.11-1 6.16-1 6.15-1 6.14-1 6.15-1 6.14-1 6.13-1

1.57-1 1.62-1 1.39-1 1.43-1 Obs: 4.34-1^a; 3.98-1^d

5-2 5485 2.84-1 2.83-1 2.72-1 2.71-1 2.81-1 2.81-1 2.80-1 2.77-1 2.76-1 2.76-1

3.15-1 2.86-1 2.82-1 2.72-1 2.91-1 2.84-1 2.82-1 2.83-1 2.78-1 2.77-1

3.69-0 7.92-1 1.77-0 4.40-1 1.87-0 8.68-1 6.31-1 1.33-0 6.49-1 4.97-1

9.58-1 9.34-1 9.58-1 9.31-1 9.55-1 9.44-1 9.33-1 9.55-1 9.43-1 9.32-1

4.00-1 3.93-1 3.47-1 3.44-1 Obs: 4.60-1^a; 6.08-1^d; 7.83-1^f

6-3 5631 4.21-1 4.37-1 3.51-1 3.61-1 4.04-1 4.10-1 4.16-1 3.75-1 3.80-1 3.85-1

4.32-1 4.38-1 3.55-1 3.62-1 4.07-1 4.11-1 4.17-1 3.77-1 3.81-1 3.85-1

1.64-0 6.62-1 8.96-1 4.35-1 9.87-1 6.49-1 5.71-1 7.64-1 5.31-1 4.82-1

1.14-0 1.12-0 1.14-0 1.12-0 1.14-0 1.13-0 1.12-0 1.13-0 1.13-0 1.12-0

4.72-1 4.76-1 3.94-1 3.97-1 Obs: 4.85-1^a; 7.42-1^e; 4.78-1^f

7-4 5677 4.77-1 5.27-1 4.12-1 4.45-1 4.64-1 4.85-1 5.04-1 4.36-1 4.54-1 4.70-1

4.81-1 5.28-1 4.13-1 4.45-1 4.66-1 4.86-1 5.04-1 4.37-1 4.54-1 4.70-1

9.08-1 6.21-1 6.12-1 4.76-1 6.78-1 5.79-1 5.69-1 5.81-1 5.14-1 5.11-1

1.32-0 1.32-0 1.32-0 1.31-0 1.32-0 1.32-0 1.32-0 1.32-0 1.32-0 1.31-0

3.30-1 3.42-1 2.93-1 3.02-1 Obs: 4.49-1^a; 6.67-1^d; 3.91-1^f

**Table 6:** Partial Table 6 (complete table available electronically from CDS). Line intensity ratios for transitions relative to $\lambda$ 5146 AA (8-3: $^4{\rm F}_{7/2}$ - $^2{\rm G}_{7/2}$ ), with $T_{\rm eff}=150\ 000$ K, and with observed values from planetary nebulae in the fifth column (the full Table 6 contains a number of other line ratios). The first four entries are for no FLE, and FLE with $W=5\,\, 10^{-16}, 10^{-13}, 10^{-10}$ , respectively. Entries in the fifth column are values calculated by Nussbaumer & Storey (1978)

		$T_{\rm e}=10000$	$T_{\rm e}=20000$	$T_{\rm e} = 12\ 000$	$T_{\rm e}=16000$
Transition	$\lambda({\rm\AA})$	$n_{\rm e}=10^3$	10⁴	10³	10⁴	$2\,\, 10^3$	$6\,\, 10^3$	10⁴	$2\,\, 10^3$	$6\,\, 10^3$	10⁴

8-2	4973	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1
		9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1
		9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1
		9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1	9.64-1
		9.67-1	9.67-1	9.67-1	9.67-1	Obs: 1.048^a; 1.094^d; 8.33-1^e; 1.652^f
9-4	5177	7.06-1	8.19-1	9.27-1	1.02-0	7.85-1	8.34-1	8.80-1	8.77-1	9.23-1	9.66-1
		7.04-1	8.18-1	9.26-1	1.02-0	7.84-1	8.34-1	8.80-1	8.76-1	9.23-1	9.66-1
		5.59-1	8.05-1	8.20-1	1.01-0	7.05-1	8.09-1	8.67-1	8.11-1	9.02-1	9.54-1
		1.54-0	1.57-0	1.54-0	1.57-0	1.55-0	1.56-0	1.57-0	1.55-0	1.56-0	1.57-0
		6.12-1	6.33-1	7.77-1	7.99-1	Obs: 7.39-1^a, 6.55-1^d,1.478^f
7-2	5234	6.55-2	7.23-2	5.65-2	6.10-2	6.37-2	6.65-2	6.91-2	5.98-2	6.22-2	6.44-2
		6.60-2	7.24-2	5.67-2	6.10-2	6.39-2	6.66-2	6.92-2	5.99-2	6.22-2	6.44-2
		1.25-1	8.51-2	8.39-2	6.53-2	9.30-2	7.94-2	7.80-2	7.97-2	7.05-2	7.00-2
		1.81-1	1.81-1	1.81-1	1.79-1	1.81-1	1.81-1	1.80-1	1.81-1	1.80-1	1.80-1
						Obs:
6-1	5278	1.87-1	1.94-1	1.56-1	1.61-1	1.79-1	1.82-1	1.85-1	1.66-1	1.69-1	1.71-1
		1.92-1	1.95-1	1.58-1	1.61-1	1.81-1	1.83-1	1.85-1	1.67-1	1.69-1	1.71-1
		7.27-1	2.94-1	3.98-1	1.93-1	4.39-1	2.88-1	2.54-1	3.40-1	2.36-1	2.14-1
		5.05-1	4.98-1	5.05-1	4.95-1	5.04-1	5.01-1	4.97-1	5.04-1	5.00-1	4.96-1
		2.19-1	2.21-1	1.83-1	1.84-1	Obs: 3.20-1^a, 5.56-1^d
5-1	5335	5.10-1	5.09-1	4.89-1	4.86-1	5.05-1	5.04-1	5.03-1	4.97-1	4.96-1	4.95-1
		5.66-1	5.14-1	5.06-1	4.88-1	5.23-1	5.10-1	5.07-1	5.08-1	5.00-1	4.97-1
		6.64-0	1.42-0	3.19-0	7.90-1	3.35-0	1.56-0	1.13-0	2.39-0	1.17-0	8.92-1
		1.72-0	1.68-0	1.72-0	1.67-0	1.72-0	1.70-0	1.68-0	1.72-0	1.69-0	1.67-0
		7.17-1	7.09-1	6.25-1	6.20-1	Obs: 7.65-1^a;1.083^b;9.52-1^c; 5.67-1^d; 1.0^f
6-2	5425	4.01-1	4.16-1	3.35-1	3.44-1	3.84-1	3.90-1	3.96-1	3.57-1	3.62-1	3.66-1
		4.11-1	4.17-1	3.38-1	3.44-1	3.88-1	3.92-1	3.97-1	3.59-1	3.62-1	3.67-1
		1.56-0	6.31-1	8.53-1	4.14-1	9.40-1	6.18-1	5.44-1	7.28-1	5.06-1	4.59-1
		1.08-0	1.07-0	1.08-0	1.06-0	1.08-0	1.07-0	1.07-0	1.08-0	1.07-0	1.06-0
		4.47-1	4.50-1	3.73-1	3.75-1	Obs: 4.85-1^a; 4.33-1^d; 7.61-1^f
7-3	5427	2.23-1	2.46-1	1.92-1	2.08-1	2.17-1	2.27-1	2.35-1	2.04-1	2.12-1	2.19-1
		2.25-1	2.46-1	1.93-1	2.08-1	2.18-1	2.27-1	2.36-1	2.04-1	2.12-1	2.19-1
		4.24-1	2.90-1	2.86-1	2.22-1	3.17-1	2.71-1	2.66-1	2.71-1	2.40-1	2.39-1
		6.16-1	6.15-1	6.16-1	6.11-1	6.16-1	6.15-1	6.14-1	6.15-1	6.14-1	6.13-1
		1.57-1	1.62-1	1.39-1	1.43-1	Obs: 4.34-1^a; 3.98-1^d
5-2	5485	2.84-1	2.83-1	2.72-1	2.71-1	2.81-1	2.81-1	2.80-1	2.77-1	2.76-1	2.76-1
		3.15-1	2.86-1	2.82-1	2.72-1	2.91-1	2.84-1	2.82-1	2.83-1	2.78-1	2.77-1
		3.69-0	7.92-1	1.77-0	4.40-1	1.87-0	8.68-1	6.31-1	1.33-0	6.49-1	4.97-1
		9.58-1	9.34-1	9.58-1	9.31-1	9.55-1	9.44-1	9.33-1	9.55-1	9.43-1	9.32-1
		4.00-1	3.93-1	3.47-1	3.44-1	Obs: 4.60-1^a; 6.08-1^d; 7.83-1^f
6-3	5631	4.21-1	4.37-1	3.51-1	3.61-1	4.04-1	4.10-1	4.16-1	3.75-1	3.80-1	3.85-1
		4.32-1	4.38-1	3.55-1	3.62-1	4.07-1	4.11-1	4.17-1	3.77-1	3.81-1	3.85-1
		1.64-0	6.62-1	8.96-1	4.35-1	9.87-1	6.49-1	5.71-1	7.64-1	5.31-1	4.82-1
		1.14-0	1.12-0	1.14-0	1.12-0	1.14-0	1.13-0	1.12-0	1.13-0	1.13-0	1.12-0
		4.72-1	4.76-1	3.94-1	3.97-1	Obs: 4.85-1^a; 7.42-1^e; 4.78-1^f
7-4	5677	4.77-1	5.27-1	4.12-1	4.45-1	4.64-1	4.85-1	5.04-1	4.36-1	4.54-1	4.70-1
		4.81-1	5.28-1	4.13-1	4.45-1	4.66-1	4.86-1	5.04-1	4.37-1	4.54-1	4.70-1
		9.08-1	6.21-1	6.12-1	4.76-1	6.78-1	5.79-1	5.69-1	5.81-1	5.14-1	5.11-1
		1.32-0	1.32-0	1.32-0	1.31-0	1.32-0	1.32-0	1.32-0	1.32-0	1.32-0	1.31-0
		3.30-1	3.42-1	2.93-1	3.02-1	Obs: 4.49-1^a; 6.67-1^d; 3.91-1^f

The complete Table 6 (available electronically from the CDS library), gives a number of other line ratios computes in a similar manner relative to the $\lambda$ 5146 ${\rm\AA}$ .

Line ratios are calculated with different dilution factors within the CR model in order to evaluate the influence of FLE under different conditions. The first four entries are: no FLE, FLE with $W=5\,\, 10^{-16}, 10^{-13}, 10^{-10}$ respectively. The Nussbaumer & Storey (1978) values are given as the fifth set of entries for comparison. Also, observational values are give in these entries (under "Obs") for various planetary nebulae, wherever available.

4.2 NGC 6741

Observations of this high excitation nebula by Aller et al. 1985 and 1997 show several optical [Fe VI] lines in the spectrum from the multiplet ${\rm 3d^3 \ (^4F - ^{4}P)}$ at 5177, 5278, 5335, 5425, 5427, 5485, 5631 and 5677 ${\rm\AA}$ and from the ${\rm (^4F - ^{2}G)}$ at 4973 and 5146 ${\rm\AA}$ for NGC 6741. The basic observational parameters, in particular the inner and the outer radii needed to estimate the distance from the central star and the dilution factor, are described in these works, and their diagnostic diagrams based on the spectra of a number of ions give $T_{\rm e}$ = 12 500 K, $N_{\rm e}$ = 6300 cm^-3, and a stellar $T_{\rm eff}$ = 140 000 K. As the ionization potentials of Fe V and Fe VI are 75.5 eV and 100 eV respectively, compared to that of He II at 54.4 eV, Fe VI emission should stem from the fully ionized ${\rm He^{2+}}$ zone, and within the inner radius, i.e. r(Fe VI) $\leq r_{\rm in}$ . With these parameters we obtain the dilution factor to be W = 10^-14; the dominant [Fe VI] emission region could be up to a factor of 3 closer to the star, with W up to 10^-13, without large variations in the results obtained.

Figures 1 and 2 show all the line ratios for NGC 6741 (with respect to the 5146 ${\rm\AA}$ line), where observational values are available (Hyung & Aller 1997a).

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig1.ps} \end{figure}$

Figure 1: NGC 6741: Line ratios with fluorescent excitation (FLE), with $T_{\rm eff}$ = 140 000 K, W = 10^-14, at $T_{\rm e}$ = 12 000 and 14 000 K - solid and dotted lines respectively; without FLE ( $T_{\rm e}$ = 12 000 K), W = 0 - dashed line; observed values from sources in the text (Table 5) - OBS

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig2.ps} \end{figure}$

Figure 2: NGC 6741: line ratios with and without FLE, as in Fig. 1

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig3.ps} \end{figure}$

Figure 3: IC 351: line ratios with fluorescent excitation (FLE), with $T_{\rm eff} = 105\ 000$ K, W = 10^-14, at $T_{\rm e} = 13\ 000$ and 16 000 K - solid and dotted lines respectively; without FLE ( $T_{\rm e} = 13\ 000$ K), W = 0 - dashed line; observed values from sources in the text (Table 5) - OBS

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig4.ps} \end{figure}$

Figure 4: IC 351: line ratios with and without FLE, as in Fig. 3

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig5.ps} \end{figure}$

Figure 5: NGC 7662: line ratios with fluorescent excitation (FLE), with $T_{\rm eff} = 80\ 000$ K, W = 10^-13, at $T_{\rm e} = 12\ 000$ and 14 000 K - solid and dotted lines respectively; without FLE ( $T = 12\ 000$ K), W = 0 - dashed line; observed values from sources in the text (Table 5) - OBS

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig6.ps} \end{figure}$

Figure 6: NGC 7662: line ratio 5177/5335 with FLE, as in Fig. 5

$\begin{figure} \par\includegraphics[width=14cm,clip]{fig7.ps} \end{figure}$

Figure 7: Comparison of Length vs. Velocity gf-values for 867 dipole allowed and intercombination E1 transitions in Fe VI

[Fe VI] line ratios are presented as a function of several parameters, in particular with and without FLE. In all cases the FLE = 0 curve fails to correlate with the observed line ratios, and shows no dependence on $N_{\rm e}$ (an unphysical result), whereas with FLE we obtain a consistent $N_{\rm e} \approx 1000-2000$ cm^-3, suitable for the high ionization [Fe VI] zone. The derived $N_{\rm e}$ is somewhat lower than the $N_{\rm e}$ range 2000 - 6300 cm^-3 obtained from several ionic spectra (including [O II] and [S II]) by 1997. The total observational uncertainties cited by Hyung & Aller (1997a) are 17.6%, 19.5%, 38.9%, 15.6%, 23.2%, 25.5%, 10.2%, 14.5%, and 36.5% for 4973, 5177, 5278, 5335, 5425, 5427, 5485, 5631 and 5677 ${\rm\AA}$ (with respect to the 5146 ${\rm\AA}$ line), respectively (Hyung & Aller 1997a). However, an indication of the overall uncertainties may be obtained from the first line ratio, 4973/5146, which is independent of both $T_{\rm e}$ and $N_{\rm e}$ since both lines have the same upper level, and which therefore depends only on the ratio of the A-values and energy separations. The observed value of 1.048 agrees closely with the theoretical value of 0.964. Whereas the combined observational and theoretical uncertainties for any one line ratio can be significant, most measured line ratios (except three line ratios 5278/5146, 5427/5146 and 5677/5146 which will be analyzed in the next paragraph) yield a remarkably consistent $N_{\rm e}$ ([Fe VI]) and substantiate the spectral model with FLE. While the electron density of the [Fe VI] in NGC 6741 is determined to be $\approx 1000-2000$ cm^-3 from most of the observational line ratios as demonstrated above, we note from Fig. 2 that three line ratios 5278/5146, 5427/5146 and 5677/5146 deviate from this $N_{\rm e}$ considerably. It is interesting to estimate the possible errors in these three observational line ratios from our theoretical method and model. Several pairs of line ratios are shown in Table 7 that have a common upper level.

Table 7: Line intensity ratios for transitions with common upper level. A-ratios - ratios of transition probabilities from the present calculation; NS - line ratios from Nussbaumer & Storey (1978); Present - line ratios from the present results; Obs - observational line ratios for various planetary nebulae
Level Index Wavelengths A-ratios NS CAL OBS

$\displaystyle\frac{5-1}{5-2}$ $\displaystyle\frac{I(5335)}{I(5485)}$ 1.748 1.793 1.797 1.663^a, 0.933^d,1.277^f

$\displaystyle\frac{6-1}{6-2}$ $\displaystyle\frac{I(5278)}{I(5425)}$ 0.454 0.490 0.467 0.660^a, 1.284^d

$\displaystyle\frac{6-2}{6-3}$ $\displaystyle\frac{I(5425)}{I(5631)}$ 0.917 0.947 0.952 1.0^a, 1.592^f

$\displaystyle\frac{7-2}{7-3}$ $\displaystyle\frac{I(5234)}{I(5427)}$ 0.283 0.294

$\displaystyle\frac{7-3}{7-4}$ $\displaystyle\frac{I(5427)}{I(5677)}$ 0.446 0.474 0.467 0.967^a, 0.587^d

$\displaystyle\frac{8-2}{8-3}$ $\displaystyle\frac{I(4973)}{I(5146)}$ 0.932 0.967 0.964 1.048^a, 1.094^d,0.833^e,1.652^f

**Table 7:** Line intensity ratios for transitions with common upper level. A-ratios - ratios of transition probabilities from the present calculation; NS - line ratios from Nussbaumer & Storey (1978); Present - line ratios from the present results; Obs - observational line ratios for various planetary nebulae
Level Index	Wavelengths	A-ratios	NS	CAL	OBS

$\displaystyle\frac{5-1}{5-2}$	$\displaystyle\frac{I(5335)}{I(5485)}$	1.748	1.793	1.797	1.663^a, 0.933^d,1.277^f

$\displaystyle\frac{6-1}{6-2}$	$\displaystyle\frac{I(5278)}{I(5425)}$	0.454	0.490	0.467	0.660^a, 1.284^d

$\displaystyle\frac{6-2}{6-3}$	$\displaystyle\frac{I(5425)}{I(5631)}$	0.917	0.947	0.952	1.0^a, 1.592^f

$\displaystyle\frac{7-2}{7-3}$	$\displaystyle\frac{I(5234)}{I(5427)}$	0.283		0.294

$\displaystyle\frac{7-3}{7-4}$	$\displaystyle\frac{I(5427)}{I(5677)}$	0.446	0.474	0.467	0.967^a, 0.587^d

$\displaystyle\frac{8-2}{8-3}$	$\displaystyle\frac{I(4973)}{I(5146)}$	0.932	0.967	0.964	1.048^a, 1.094^d,0.833^e,1.652^f

^a Hyung & Aller (1997a); ^c Hyung et al. (1997); ^d Feibelman et al. (1996); ^e Hyung & Aller (1998); ^f Hyung & Aller (1997b).

These line ratios depend only on the ratio of the A-values and energy differences and are independent of the detailed physical conditions in PNe. They can be used to determine possible, systematic errors in observed line intensties. From the line ratio 4973/5146 and the other five line ratios in Table 7, we conclude that the intensity of the reference line 5146 ${\rm\AA}$ should be very accurate. It is very unlikely that the error for each pair of these line ratios are the same and show the same tendency. An error estimate of 6.8% for this line given by Hyung & Aller (1997a) is consistent with our justification. As such, the observed intensities of lines $\lambda\lambda$ 4973, 5177, 5335, 5425, 5485, and 5631 ${\rm\AA}$ should be of high accuracy (within 20%). This conclusion is also supported by the good agreement between the two other theoretical and observational line ratios 5335/5485 and 5425/5632 shown in Table 7.

Based on these arguments, we infer that the observed ("Hamilton") line intensity of 0.087 (Hyung & Aller 1997a) (relative to the uniform flux of I(H $\beta)=100$ ) for 5278 ${\rm\AA}$ should be reduced by about 40% from a comparison of the line ratio 5278/5425 in Table 7. Similarly, the reported line intensity of 0.118 for the line 5427 ${\rm\AA}$ should be reduced by about 70% or more, and the value 0.122 for the 5677 ${\rm\AA}$ should be increased by 20% or so from the comparison of the line ratio 5427/5677 in Table 7. If our justifications for the errors in the intensities of these three lines are correct, the corresponding three line ratios shown in Fig. 2 will also yield the same and consistent $N_{\rm e}$ as do the other line ratios, particularly those in Fig. 1. It is interesting to note that the uncertainties given by Hyung & Aller (1997a) are also large (as inferred above), although the line 5427 ${\rm\AA}$ could have a much higher uncertainty (intensity larger or lower).

4.3 IC 351

The physical conditions of IC 351, especially the effective temperature $T_{\rm eff}$ and the distance of PNe emission region to the central white dwarf (WD), are highly uncertain. We first apply our method, as developed in CP00, to determine the appropriate $T_{\rm eff}$ and the dilution factor W(r). $T_{\rm eff}$ is thereby determined to be 80 000 $\pm$ 10 000 K. This is considerably different from $T_{\rm eff}$ = 58 100 K cited by 1996. It is interesting to note that the $T_{\rm eff}$ determined by our spectral method with FLE agrees with the He II Zanstra temperature, which is 85 000 K according to Preite-Martinez & Pottasch (1983). As pointed out by Preite-Martinez & Pottasch (1983), different methods (Zanstra method, color temperature and energy balance method, etc.) used to determine the effective temperature of PNe remain discrepant; but the He II Zanstra method is applicable to optically thick PNe's. With the $T_{\rm eff}$ as above, we obtain the dilution factor W(r)to be 10^-13 - 10^-14.

After determining $T_{\rm eff}$ and W(r), the same method as used in NGC 6741 is applied to determine the electron density $N_{\rm e}$ of the [Fe VI] emission nebula in IC 351, and possible errors in the observed line intensities. There are 7 observational line ratios for IC 351 as shown in Figs. 3 and 4; but we calculated the same 8 line ratios theoretically as for NGC 6741.

The only reported uncertainty for a line ratio given by 1996 is 36.6% for the pair 5335/5146 (Fig. 3). Comparing observed and calculated pairs of line ratios with common upper levels (using Table 7), we find good agreement for 4973/5146 ( $\approx$ 10%), implying accurate intensities for both lines. However, the diffierence is 26% for 5427/5677 (Fig. 4), 92% for 5335/5485 (Fig. 3), and 175% for 5278/5425 (Fig. 4). From these differences, and Figs. 3 and 4, one can estimate possible errors in some observed lines: line 5485 ${\rm\AA}$ should be reduced by 60%; line 5336 ${\rm\AA}$ increased by 20%; line 5278 ${\rm\AA}$ reduced by 175%. The intensity of the line 5425 ${\rm\AA}$ is accurate. From these 4 line ratios, $N_{\rm e}$ is determined to be $\approx$ 1000 cm^-3. Using this $N_{\rm e}$ and Fig. 4, and the comparison of the line ratio 5427/5677 in Table 7, the possible errors in the intensities of the other 3 observed lines can be deduced as follows: line 5427 ${\rm\AA}$ should be reduced by 80%; line 5677 ${\rm\AA}$ reduced by 40%; and line 5177 ${\rm\AA}$ increased by 20%. In summary, the above error analysis is based in Table 7 and the theoretically computed line ratios reported in this work (Figs. 3 and 4).

4.4 NGC 7662

Finally we apply our spectral diagnostics, and the same procedures used above, to NGC 7662. In this PNe, the effective temperature $T_{\rm eff}$ and emission region distance to the central star seem to have been determined within low uncertainties. Hence, we adopt here $T_{\rm eff}=$ 105 000 K and W=10^-14 (Hyung & Aller 1997b) in our present calculation; some results are shown in Fig. 5.

However, the observational uncertainties in line intensities are even larger than those in Feibelman et al. (1996) as shown from both the rms uncertainties given in Hyung & Aller (1997b) and our detailed spectral analysis by using the method developed by Chen & Pradhan (2000). However, the reference line 5146 ${\rm\AA}$ intensity highly uncertain in NGC 7662, in contrast to NGC 6741 or IC 351, as deduced from Table 7. On the other hand we find the observed intensity of line 5425 ${\rm\AA}$ to be very accurate; at most too high by 5 - 10% (consistent with the rms uncertainty given by 1997b, a remarkbly low 3.9%). The lines 5335 and 5177 ${\rm\AA}$ are also very accurate (within 10%) according to the procedures described above. Thus only the 5335 ${\rm\AA}$ line intensity needs to be increased by 5%, and the line 5177 ${\rm\AA}$ intensity decreased by 5%. To further confirm our justification, we plot a new line ratio 5177/5335 in Fig. 6, which the leads to a reasonable determination of $N_{\rm e}$ .

In all the above arguments, we have a consistent spectral analysis from the line ratios of 5335/5146, 5177/5146 and 5425/5146 as shown in Fig. 5, as well as the line ratio 5177/5335 in Fig. 6. As such, we conclude that: (1) the observed intensity of line 5146 could be higher by 80% (the rms given in 1997b is 21.7%); (2) the electron density of the [Fe VI] emission region is $N_{\rm e}\approx 3 000$ cm^-3. Using revised ratios, and Table 7, one can then deduce possible observational errors in other lines.

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