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
.
For a given subset of (
)
a line ratio may describe the locus of the subsets of (
),
which then defines (constrains) a contour of possible parameters. CP00
present 3-dimensional plots of the line ratio vs. (
), for
given (
). While it is clear that the
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
), 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.
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).
Source |
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r/pc | R/![]() |
W | Reference |
NGC 6741 | 6.3 | 12.5 | 140 | 0.0063 | 0.063 |
![]() |
Hyung & Aller 1997a; a |
NGC 6886 | 5 - 10 | 13 | 150 | 0.001(0.0345) | 0.046 |
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Hyung et al. 1995; b |
NGC 6884 | 10 | 10 | 110 | 0.002(0.020) | 0.13 |
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Hyung et al. 1997; c |
IC 351 | 2.5 - 20 | 13 - 16 | 58.1 | 0.05 | 0.72 |
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Feibelman et al. 1996; d |
NGC 2440 | 5 | 14.2 | 180 | 0.015(0.0425) | 0.038 |
![]() |
Hyung & Aller 1998; e |
NGC 7662 | 3 - 17 | 13 | 105 | 0.025(0.035) | 0.15 |
![]() |
Hyung & Aller 1997b; f |
Hyung & Aller (1997a);
Hyung et al. (1995);
Hyung et al. (1997);
Feibelman et al. (1996);
Hyung & Aller (1998);
Hyung & Aller (1997b).
A comprehensive study of most of the possible line ratios was carried out as
functions of
,
,
and with and without FLE, at various radiation temperatures
and dilution factors W(
.
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
5146
(as in
Nussbaumer & Storey 1978).
The partial Table 6 given in the text
contains only those line ratios for which observed values are available.
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||||||||
Transition |
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104 | 103 | 104 |
![]() |
![]() |
104 |
![]() |
![]() |
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: 1.048a; 1.094d; 8.33-1e; 1.652f | |||||||
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: 7.39-1a, 6.55-1d,1.478f | |||||||
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: 3.20-1a, 5.56-1d | |||||||
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: 7.65-1a;1.083b;9.52-1c; 5.67-1d; 1.0f | |||||||
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: 4.85-1a; 4.33-1d; 7.61-1f | |||||||
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: 4.34-1a; 3.98-1d | |||||||
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: 4.60-1a; 6.08-1d; 7.83-1f | |||||||
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: 4.85-1a; 7.42-1e; 4.78-1f | |||||||
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: 4.49-1a; 6.67-1d; 3.91-1f |
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
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.
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
at 5177, 5278, 5335, 5425, 5427, 5485, 5631 and 5677
and from
the
at 4973 and 5146
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
= 12 500 K,
= 6300 cm-3, and a
stellar
=
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
zone, and
within the inner radius, i.e. r(Fe VI)
.
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
line), where observational
values are available (Hyung & Aller 1997a).
![]() |
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
(an unphysical
result), whereas with FLE we obtain
a consistent
cm-3,
suitable for the high ionization [Fe VI] zone. The derived
is somewhat lower than the
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
(with respect to the 5146
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
and
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
([Fe VI]) and
substantiate the spectral model with FLE.
While the electron density of the [Fe VI] in NGC 6741 is determined to be
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
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.
Level Index | Wavelengths | A-ratios | NS | CAL | OBS |
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1.748 | 1.793 | 1.797 | 1.663a, 0.933d,1.277f |
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0.454 | 0.490 | 0.467 | 0.660a, 1.284d |
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0.917 | 0.947 | 0.952 | 1.0a, 1.592f |
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0.283 | 0.294 | ||
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0.446 | 0.474 | 0.467 | 0.967a, 0.587d |
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0.932 | 0.967 | 0.964 | 1.048a, 1.094d,0.833e,1.652f |
a Hyung & Aller (1997a); c Hyung et al. (1997); d Feibelman et al. (1996); e Hyung & Aller (1998); f Hyung & Aller (1997b).
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
)
for 5278
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
should be reduced by about 70% or more, and the value 0.122 for the
5677
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
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
could have a much
higher uncertainty (intensity larger or lower).
After determining
and W(r), the same method as used in NGC 6741
is applied to determine the electron density
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 (
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
should be reduced by 60%; line 5336
increased by 20%;
line 5278
reduced by
175%. The intensity of the line 5425
is accurate. From these
4 line ratios,
is determined to be
1000 cm-3.
Using this
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
should be reduced by 80%; line 5677
reduced by 40%; and line 5177
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).
Finally we apply our spectral diagnostics, and the same procedures used
above, to NGC 7662. In this PNe, the effective temperature
and
emission region distance
to the central star seem to have been determined within low uncertainties.
Hence, we
adopt here
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
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
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
are also very accurate (within 10%) according
to the procedures described above.
Thus only the 5335
line intensity needs to be
increased by 5%, and the line
5177
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
.
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
cm-3. Using revised ratios, and Table 7, one
can then deduce possible observational errors in other
lines.
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