A&A 367, 973982 (2001)
DOI: 10.1051/00046361:20000431
R. Napiwotzki^{}
Dr. RemeisSternwarte, 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; GutiérrezMoreno 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.
/K  F_{5454}  /K  F_{5454} 
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:
(1) 

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 nonGaussian 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
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 postAGB 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.
Figure 2: Results of Monte Carlo simulations for several values of the maximum allowed "spectroscopic'' distance . The distributions are normalized to an integrated area of 1. For cosmetic reasons we used for this plot  
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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 preselection 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
(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
=  0.92  (4)  
=  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.
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
(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):
(7) 
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 postAGB 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, Sh268, and DeHt2
(open circles) and the non postAGB
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
(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 postAGB 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 stateoftheart 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.
Figure 5: The individual ratios of and as function of the nebula radius. The meaning of the symbols is the same as in Fig. 4. The line indicates our fit (Eq. (9))  
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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?
Figure 6: Sames as Fig. 5 but this time we compare the distances from trigonometric and spectroscopic parallaxes with . The Ciardullo et al. (1999) results are plotted as asterisks. For other symbols cf. Fig. 4. The line indicates the fit from Fig. 5  
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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
massradius relation
It is likely that during the PN evolution the complete material of the slow (10kms^{1}) AGB wind material is sweptup by the faster ( kms^{1}) expanding planetary shell (SchmidtVoigt & 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 massradius 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.
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 stateofthearttechniques. 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:  
=  10^{5}  
=  8  
Sample selection:  
=  17.5  mag  
=  1000  pc  
=  1.33  mas  
=  0.95  mas  
=  0.75  mag  
Stellar evolution:  
=  0.605  
=  3000  yrs  
=  50000  yrs  
Galactic model:  
R_{0}  =  4500  pc 
=  300  pc  
=  140  pc  
=  1.0  mag/kpc 
parameter  values  
std. model  
16  
32  
64  
16.5  mag  
18.5  mag  
750  pc  
1500  pc  
3000  pc  
1.5  mas  
2.0  mas  
3.0  mas  
4.0  mas  
5.0  mas  
7.0  mas  
10.0  mas  
0.1  mas  
0.2  mas  
0.3  mas  
0.4  mas  
0.5  mas  
0.7  mas  
1.2  mas  
0.5  mag  
1.0  mag  
1000  yrs  
5000  yrs  
25000  yrs  
100000  yrs  
150  pc  
600  pc  
60  pc  
210  pc  
0.50  mag/kpc  
1.50  mag/kpc 
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 R_{0} = 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 R^{3}. 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 postAGB age the absolute magnitudes of CSPNe can be computed from theoretical postAGB 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 postAGB track of Blöcker (1995) for our simulation. The postAGB age was varied within a given interval which approximately reproduces the M_{V} 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: