A&A 367, 973-982 (2001)
DOI: 10.1051/0004-6361:20000431
Dr. Remeis-Sternwarte, Sternwartstr. 7, 96049 Bamberg, Germany
Received 12 March 1999 / Accepted 4 December 2000
Abstract
We use the results of our recent NLTE model atmosphere analysis of
central stars of old planetary nebulae (PN) to calculate distances.
We perform a comparison with three other methods (trigonometric parallaxes,
interstellar NaD lines, and Shklovsky distances) and discuss the
problem of the PNe distance scale.
The result of the comparison of our spectroscopic distances
with the trigonometric distances is that the spectroscopic distances are
55% larger. Since using trigonometric parallaxes
with large relative measurement errors
can introduce systematic errors, we carried out a Monte Carlo simulation
of the biases
introduced by selection effects and measurement errors.
It turns out
that a difference between both distance scales of
the observed
size is expected for the present day data if the underlying distance
scales are identical. Thus our finding is essentially a confirmation of the
spectroscopic distance scale! Good agreement is found between the
spectroscopic distances and distances derived from the interstellar
NaD lines.
All three independent methods of distance measurement
indicate that the widely used ``statistical'' distance scales of the
Shklovsky type are too short for old PNe. A correlation with nebular radii
exists. The most likely explanation
is an underestimate of the nebula masses for large PN. Implications for the
nebula masses are discussed. Estimates of the PNe space
density and birthrate, which are based on Shklovsky type distances,
therefore give too large values.
Key words: stars: distances - planetary nebulae: general - white dwarfs - stars: fundamental parameters
The question of PNe space densities and birth rates is closely related
to the distance scale problem.
Ishida & Weinberger (1987)
compiled a list of nearby PNe, which contains mostly old,
evolved nebulae (actually many of our PNe were selected from this list).
Ishida & Weinberger collected distance determinations from literature and
computed the space density and birth rates of this local sample of PNe.
The derived birth rate of
is
too high to be in accordance with estimates of
white dwarf birth rates (
;
Weidemann 1991).
Since every central star of a PN (CSPN) should become a
white dwarf this yielded a real dilemma. Taken at face value this
would indicate that current white dwarf samples are very incomplete
and the white dwarf birth rates are seriously underestimated. A
certain fraction of white dwarfs may be hidden in binaries, indeed.
Weidemann (1991) used a very local sample of white dwarfs (d<10pc)
for his estimate and applied corrections for incompleteness and binarity.
Pottasch (1996) reevaluated the PN space density and
derived a value of
,
lower
than Ishida & Weinberger's result, but still higher than the
estimated white dwarf birthrate. The difference in the PNe birthrate
can be traced back to the use of different distance scales.
While Ishida & Weinberger's collection is mainly based on statistical distance
estimates (mainly Shklovsky distances and derivates of this method),
Pottasch excluded statistical distance determinations.
In Paper IV of this series (Napiwotzki 1999) we presented the results of an NLTE model atmosphere analysis of 27 central stars of old PNe. This analysis of a reasonably sized sample of central stars of old PNe enables us to address the question of the distance scale of these objects. In Paper III (Napiwotzki & Schönberner 1995) we proposed the use of the interstellar NaD lines for distance determinations. These distances are free of assumptions about the nebula or the central star and are on average larger than the Shklovsky distances by a factor of 2.5 for our sample of old PNe. During the last years a number of trigonometric parallax measurements became available for a sample of central stars of old PNe (Harris et al. 1997; Pottasch & Acker 1998; Gutiérrez-Moreno et al. 1999). We will show that the CSPNe distances derived form our model atmosphere analysis, the NaD distances, and the trigonometric parallaxes are in agreement, but all three distances scales are much larger than those based on the Shklovsky method.
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F5454 |
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F5454 |
40000 | 2.18 | 120000 | 6.37 |
50000 | 2.76 | 140000 | 7.21 |
60000 | 3.31 | 170000 | 8.50 |
70000 | 3.87 | 200000 | 9.67 |
80000 | 4.51 | 250000 | 11.93 |
90000 | 5.05 | 300000 | 13.98 |
100000 | 5.52 |
Spectroscopic distances can be derived from the model atmosphere analyses described in Paper IV and offer an approach to the PN distance scale independent from the properties of the nebulae. After the stellar mass is estimated from a comparison with evolutionary models (cf. Paper IV) the distance can be calculated in a straightforward way from effective temperature, gravity, and the dereddened apparent magnitude of the stars:
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(1) |
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Recently Harris et al. (1997) published trigonometric
parallaxes of 11 CSPNe. Eight of these stars belong to our sample (the
value given for A7 is actually only an upper limit).
Pottasch & Acker (1998)
discussed HIPPARCOS parallaxes of three CSPNe including
PHL932 which belongs to our sample.
The resulting
distance values are listed as
in Table 2 and
are compared with the model atmosphere results in Fig. 1. The
error limits were computed from the measurement errors of the
parallaxes.
As it is evident from Fig. 1 and the following figures
the error distribution of
distance determinations is highly non-Gaussian with many outliers,
which can skew the common weighting techniques to erroneous
values. The reason is that the usually (implicitly) adopted Gaussian
probability distribution gives high weight to deviant points.
Therefore we decided
to minimize the
absolute deviations, which corresponds to a double sided exponential
probability distribution and provides a more robust estimate
(see discussion in Press et al. 1992).
Since the distance errors are highly
asymmetric, we did the comparison with the parallaxes, which have
roughly symmetric error limits. Since we in all cases compare two measurements
suffering from large uncertainties,
we performed pro forma a linear regression with
allowance for errors in both directions and with the intersection fixed
at zero.
The error ranges given below correspond to the error of the mean.
The measured trigonometric distances are always
smaller than the
NLTE distances (cf. Fig. 1).
The weighted mean of the distance
ratios
,
computed as described above, amounts to
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Figure 1: Distances computed from the results of the NLTE analysis compared to trigonometric distances. The error bars correspond to the measurement errors given in Table 2. The solid line indicates the average ratio of the NLTE distances and the trigonometric distances, the dotted line equality |
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We took a different approach and performed a Monte Carlo simulation to derive biases caused by the selection of CSPNe for parallax measurements and the accompanying measurement errors. Our working hypothesis is that the spectroscopic distance scale is essentially correct, i.e. no systematic errors are present. The results of the Monte Carlo simulations are used to test if this hypothesis is compatible with observations.
In our simulation CSPNe were randomly created according to a simple model of
the Galactic stellar density distribution and a theoretical post-AGB track.
A spectroscopic analysis is simulated, i.e. random measurement errors are
added. The star is selected for parallax measurement if the "spectroscopic
distance'' is below a threshold value
.
A measurement error
of the parallax
is added to the true parallax and the resulting
distance used for a comparison with the spectroscopic parallax, if the
"measured parallax'' is larger than a threshold value
.
stars are collected and the mean ratio
is computed analogous to the procedure
applied for the observed data. This process is repeated
times
and a probability distribution is evaluated. Sample probability
distributions are given in Fig. 2 (these are discussed below).
A detailed description of the Monte Carlo simulations, the input parameters,
and their standard values is given in the Appendix.
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Figure 2:
Results of Monte Carlo simulations for several values of the
maximum allowed "spectroscopic'' distance
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An investigation of the influence of the different input parameters on the
results can be found in Table A.2. The derived bias between
spectroscopic and trigonometric distances is most sensitive to the chosen
threshold value
for the pre-selection of stars, the
parallax measurement error
and the threshold parallax
.
Our standard model assumes
pc.
Note that one star in the Harris et al. (1997)
sample has a much larger
value of the spectroscopic distance (A74;
pc).
For
the mean of the error limits in Table 2,
,
was adopted. The threshold parallax was set to
(corresponds to A74).
Our standard sample size,
,
corresponds to the observed sample and the
simulation is carried out for
samples.
The first qualitative conclusion, which can be drawn from
Table A.2 and Fig. 2 is that the "measured'' ratio
is always considerably larger than
unity, although the underlying distance scales used in the Monte Carlo
simulations are identical. Our standard model yields
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(3) |
A paradoxical effect results if the threshold parallax
is
increased: the bias increases for increasing
,
if the spectroscopic threshold distance is left unchanged! E.g. for
mas we derive
.
The reason is that we
introduce a strong selection effect for stars with much too large measured
parallaxes. This effect is only overcome, if
is increased
to even larger values (
mas).
In principle, the expected bias can be reduced by lowering
and
.
However, if we e.g. adopt
pc
and
mas our observed sample is reduced to 3 stars!
We conclude that the mean value of
derived
from our observed sample (Table 2) is well within the range
predicted by Monte Carlo simulations for perfect agreement of both distance
scales. Thus the trigonometric parallax measurements provide no evidence
that the spectroscopic distances are in error, but confirm the spectroscopic
distance scale, instead!
However, due to the statistical uncertainties we cannot
provide a definitive proof of an agreement of better than about 20%.
Another method independent of assumptions about central star or nebula
properties is based on interstellar lines. In Paper III we used
the interstellar NaD lines at 5890/96Å for this purpose. The
CSPNe are too hot to show any photospheric NaD lines and the PNe
material is extremely dispersed. In their survey Dinerstein et al. (1995) detected circumstellar NaD lines in nine PNe. However, all
investigated PNe were young objects, and if there is any neutral sodium at all
in the circumstellar matter of the central star of old PNe, the column density
would be very low and the resulting nebula contribution negligible.
Naturally this method is restricted to stars at low galactic
latitude. The distances derived in Paper III are given as
in Table 2.
Our NaD distances in Paper III were determined from the map of the interstellar NaD line strength of Binnendijk (1952). One might ask, whether the distance scale adopted in that work is still valid. To our knowledge no more recent collection of interstellar NaD equivalent widths is available. The reason is that using the equivalent width is out of fashion, because resolved interstellar lines in high resolution spectra provide more information. However, for our faint CSPNe equivalent widths are still a useful tool.
Binnendijk (1952) adopted spectroscopic distances of B stars,
which were
determined by the Yerkes group (Ramsey 1950;
Duke 1951, and unpublished
distances provided by W. W. Morgan). It is save to assume that systematic
differences between these authors are small. We performed a check of the
Ramsey (1950) and Duke (1951) distances.
For this purpose we selected a
representative subsample of 20 stars from each collection
with Strömgren
measurements in the Hauck & Mermilliod
(1998) catalogue.
Temperatures and gravities were calculated with an updated version
(Napiwotzki & Lemke, in prep.) of the photometric calibration of Napiwotzki
et al. (1993). The new gravity calibration is based on accurate
trigonometric parallaxes measurements of B stars with HIPPARCOS. Since the
calibration doesn't cover O stars and supergiants, these stars were excluded
from the comparison. Masses were derived by interpolation in the
evolutionary tracks of Schaller et al. (1992) and distances
computed as described in Sect. 2.1. Finally these distances were compared
with the spectroscopic distances provided by Ramsey (1950) and Duke
(1951)
and mean ratios were computed as described in Sect. 2.2. The results are
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= | 0.92 | (4) |
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= | 1.17 | (5) |
Although the scatter of the individual determinations is large we can
conclude that altogether systematic differences between the old Yerkes
system and modern photometric distances, which are tied to accurate HIPPARCOS
parallaxes, are small.
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Figure 3: Distances computed from the results of the NLTE analysis compared to NaD distances. The solid line indicates the average distance ratio |
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Figure 3 shows
that the agreement of the NaD distances with the model atmosphere
analysis is good. The scatter is easily explained by the spatially highly
variable extinction in the galactic plane. The average ratio of both
distance scales amounts to
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(6) |
The Shklovsky (1956) method allows the
calculation of PN distances from the measurement of the recombination
line H
and the angular diameter. The distance can be computed from
(see e.g. Pottasch 1984, p. 115):
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(7) |
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Figure 4: Distances computed from the results of the NLTE analysis compared to Shklovsky distances. Filled diamonds: ordinary white dwarf CSPNe of spectral type DA and DAO, open circles: hybrid/high luminosity objects, open squares: non post-AGB objects. The dotted line indicates equality, the solid the average distance ratio for the ordinary white dwarf CSPNe |
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A comparison of the Shklovsky and the model atmosphere distances is
shown in Fig. 4. We distinguish between "ordinary''
white dwarf CSPNe of spectral type DA and DAO (filled symbols)
and the hybrid/high luminosity objects
A43, NGC7094, Sh2-68, and DeHt2
(open circles) and the non post-AGB
objects (open squares). The evolutionary history of both latter
classes are likely very different from standard evolution and may
cause very different PN properties. The comparison shows that almost
all Shklovsky distances of the ordinary white dwarf CSPNe are smaller
than the model atmosphere distances. The average (weighted by the
error limits of our analysis) amounts to
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(8) |
Since
is directly related to
the ionized mass of the PN by the Shklovsky formula
,
we interpret this as evidence that the ionized masses of the old PNe
are much higher than the adopted typical mass of
.
The
distance scales would be in agreement if we increase the adopted PNe
mass to
.
However, the large scatter is a first
indication that this is an
oversimplification of the real situation, as we will show below.
Note in passing that the non post-AGB objects (open squares in
Fig. 4) have considerably lower values of
(mostly <1) indicating
masses lower than the adopted PN mass.
We have now shown that the distances
derived from
our NLTE model atmosphere analyses are (within today's error limits)
in good agreement with the trigonometric parallaxes measured by
Harris et al. (1997) and Pottasch & Acker (1998)
after biases are taken into account,
and with distances derived from the strength of
the interstellar NaD lines. Both distance scales are model
independent and thus demonstrate that the NLTE analysis are not
subject to large systematic errors. On the other hand
Napiwotzki et al. (1999)
have shown that state-of-the-art analyses of hot
white dwarfs performed independently by different groups can yield
surface gravities, which differ systematically by up to 0.1dex
which translates into a distance error of 12%. Such errors of the
model atmosphere distance scale would be compatible with the
trigonometric and NaD distances of our stars. Much larger
systematic errors can be excluded. The
Shklovsky distance scale of old PNe is shown to be too short,
most likely caused
by an underestimate of the PNe masses.
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Figure 5:
The individual ratios of
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Further insight can be gained when plotting (Fig. 5)
the individual ratios
as function of the PN radii
(given in Table 2). A strong correlation is present. A simple
(double logarithmic) linear fit results in
How does our result compare with other tests of the Shklovsky distance scale?
Ciardullo et al. (1999) determined PN distances from main sequence
companions of the CSPN and concluded that statistical distance determinations
overestimate the PN distances. Stasinska et al. (1991) and
Pottasch & Zijlstra (1992)
applied the Shklovsky method to PN in the bulge
of our Galaxy. Stasinska et al. concluded that, besides considerable
scatter, the mean Shklovsky distance reproduces the distance of the galactic
bulge well, while Pottasch & Zijlstra claimed that the Shklovsky distances
are systematically too high. Do these results contradict our findings
in Fig. 5?
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Figure 6:
Sames as Fig. 5 but this time we compare
the distances from trigonometric and spectroscopic parallaxes
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Ciardullo et al. (1999) derived spectroscopic
parallaxes of main sequence
companions of 14 CSPN and compared their results (and 7 trigonometric
parallaxes) with four different statistical distance scales based on the
prescriptions of Cahn et al. (1992),
Maciel & Pottasch (1980), van der Steene & Zijlstra (1995) and Zhang (1995).
These are variants of the Shklovsky method (Eq. (2)) in which the
ionized mass
grows as a function of radius (Cahn et al. 1992; Maciel & Pottasch 1980; Zhang 1995) or of
the radio brightness
temperature of the PN. Cahn et al. (1992)
assumed that the ionized mass of small, ionization bounded PNe grows with
radius until an upper limit is reached for larger density bounded PN.
This has the effect that these distance determinations for PNe in the sample
discussed by Ciardullo et al. (1999)
are essentially classical Shklovsky
distances with
.
We performed a small
correction to the Cahn et al. (1992) distances to transform them to
the
common
scale and added the Ciardullo et al. (1999) data points to Fig. 6.
Two conclusions can be drawn:
Since one wants to exclude foreground objects, PNe with an angular diameter
larger than 20'' are excluded from bulge samples. This translates into a
radius R = 0.41pc. Thus the old and large PNe of our sample
(Table 2) are explicitly excluded from investigations of bulge PNe.
Another strong selection effect against old PNe might already be at work,
because
of their low surface brightness combined with the large extinction. A lower
limit of the angular diameter of 1'' is set by the need to resolve
the PNe. This constrains the radii of bulge PNe, which could be used to test
the Shklovsky method approximately to the range
pc. From Fig. 5 we would
predict that the Shklovsky distances of bulge PNe are moderately too large,
in qualitative agreement with the Pottasch & Zijlstra (1992) result.
However, one should keep in mind that the stellar population of the galactic
bulge is quite different from the local population. Therefore one should be
aware that the properties of bulge PNe might be different from local samples.
The correlation given in Eq. (9) translates into a
mass-radius relation
It is likely that during the PN evolution the complete material of the
slow (10kms-1) AGB wind material is swept-up by the faster
(
kms-1) expanding planetary
shell (Schmidt-Voigt & Köppen 1987;
Marten & Schönberner 1991)
and could finally be incorporated into the PN. However,
is much
larger than the mass a typical CSPN precursor (
)
can loose during it's evolution. There are some points which should be
considered when interpreting the mass-radius relation (Eq. (10)):
We used the results of our model atmosphere analysis to compute distances of the central stars and showed that our model atmosphere distance scale is in good agreement with measured trigonometric parallaxes (after the effect introduced by biases has been taken into account via a Monte Carlo simulation) and distances derived from the interstellar NaD lines. All these three independent methods of distance measurement indicate that the widely used "statistical'' distance scales of the Shklovsky type are too short for old PNe. The most likely explanation is an underestimate of the nebula masses.
Estimates of the PNe space
density and birthrate, which are based on Shklovsky type distances,
therefore give too large values. If a more realistic distance scale is
applied, discrepancies between white dwarf and PNe birthrates are resolved.
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Figure 7: Probability distribution for our standard model and for the possible future improvements discussed in the text |
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Due to statistical uncertainties and biases of the trigonometric distances we
could test the spectroscopic distance scale
only on the 20% level. Do we have to
wait for space missions like GAIA or SIM to improve this situation? A test
on the 10% level might be achieved with state-of-the-art-techniques. Harris
et al. (1997) announced that parallax measurement
accuracies of 0.5mas are within
reach, but this alone is not enough (cf. Table A.2). However,
if a sample with
pc is produced and the sample
size is doubled to
the bias
is
reduced to 4% and the scatter to 10% (the probability distribution is
compared with the present situation in Fig. 7).
Thus, if we work on both fields,
measuring of more accurate parallaxes and analyzing and selecting more
candidates from spectroscopic investigations, this goal is within reach.
Acknowledgements
The author thanks Joachim Köppen for inspiring discussions and Detlef Schönberner, Uli Heber, and Klaus Werner for useful comments on previous drafts of this paper.
Our Monte Carlo simulation of the local PN distribution proceeds in three steps:
Sample sizes: | |||
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= | 105 | |
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= | 8 | |
Sample selection: | |||
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= | 17.5 | mag |
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= | 1000 | pc |
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= | 1.33 | mas |
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= | 0.95 | mas |
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= | 0.75 | mag |
Stellar evolution: | |||
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= | 0.605 | ![]() |
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= | 3000 | yrs |
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= | 50000 | yrs |
Galactic model: | |||
R0 | = | 4500 | pc |
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= | 300 | pc |
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= | 140 | pc |
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= | 1.0 | mag/kpc |
parameter | values |
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|
std. model |
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||
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16 |
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|
32 |
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||
64 |
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||
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16.5 | mag |
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18.5 | mag |
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|
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750 | pc |
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1500 | pc |
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|
3000 | pc |
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|
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1.5 | mas |
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2.0 | mas |
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|
3.0 | mas |
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|
4.0 | mas |
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|
5.0 | mas |
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|
7.0 | mas |
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|
10.0 | mas |
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|
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0.1 | mas |
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0.2 | mas |
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|
0.3 | mas |
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|
0.4 | mas |
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|
0.5 | mas |
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|
0.7 | mas |
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|
1.2 | mas |
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|
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0.5 | mag |
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1.0 | mag |
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|
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1000 | yrs |
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5000 | yrs |
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25000 | yrs |
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100000 | yrs |
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|
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150 | pc |
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600 | pc |
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|
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60 | pc |
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210 | pc |
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|
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0.50 | mag/kpc |
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1.50 | mag/kpc |
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Our simple description of the galactic disk is based on the
Galaxy model of Bienaymé et al. (1987). An exponential density law
with a scale height
of 300pc is adopted for the CSPN.
This corresponds to
a stellar population with an age of
yrs. We included
extinction by a dust component with a scale height
pc, which is the value appropriate for the interstellar matter. A
dust opacity at the position of the sun
mag/kpc
was chosen.
The
stellar and dust density decreases exponentially with the distance from the
galactic center and a scale length R0 = 4.5kpc. For the distance of the
sun from the galactic center we adopted the standard value of 8.5kpc.
Let us note that Köppen & Vergely (1998) could successfully
reproduce the extinction properties of galactic bulge PNe with our
parameter values.
Our parameter study in Table A.2 shows that the influence of particular values of the parameters of our "galactic model'' are very small. Due to the exponential decrease of stellar density with height above the galactic plane the number of stars within a sphere with a given radius Rincreases less then R3. Since this lowers the number of far away stars, a lower bias is expected for lower values of the scale height. Extinction introduces another selection against far away stars (through the limiting magnitude). However, as Table A.2 proofs the effect of varying these parameters within reasonable limits is quite small.
For a given post-AGB age the absolute magnitudes of CSPNe can be computed from
theoretical post-AGB tracks. In principle a mass distribution for the
CSPNe should be used. However, since the mass distributions of CSPN and white
dwarfs are quite narrow, we considered it sufficient to use only one
theoretical track for simplicity. Spectroscopic studies of CSPNe (Paper IV),
white dwarfs (Bergeron et al. 1992; Napiwotzki et al. 1999)
and an
investigation of PNe based on a distant independent method by Stasinska
et al. (1997)
derived peak masses in the range
.
Thus we selected the
post-AGB track of Blöcker (1995)
for our simulation. The post-AGB age was varied within a given interval
which approximately reproduces the
MV distribution of our CSPNe. However the simulation results are quite
insensitive to their particular values (cf. Table A.2).
After "producing'' a CSPN we had to simulate the selection for parallax measurement and the distance determination with their measurement errors. The following scheme was adopted: