White dwarf masses derived from planetary nebulae modelling
Astronomy & Astrophysics manuscript no. gesicki c© ESO 2021
December 4, 2021
Letter to the Editor
White dwarf masses derived from planetary nebulae modelling
K. Gesicki1 and A.A. Zijlstra2
1 Centrum Astronomii UMK, ul. Gagarina 11, PL-87-100 Torun, Poland
e-mail: Krzysztof.Gesicki@astri.uni.torun.pl
2 School of Physics and Astronomy, University of Manchester, P.O. Box 88, Manchester M60 1QD, UK
e-mail: a.zijlstra@manchester.ac.uk
Received ; accepted
ABSTRACT
Aims. We compare the mass distribution of central stars of planetary nebulae (CSPN) with those of their progeny, white dwarfs (WD).
Methods. We use a dynamical method to measure masses with an uncertainty of 0.02 M�.
Results. The CSPN mass distribution is sharply peaked at 0.61 M�. The WD distribution peaks at lower masses (0.58 M�) and shows
a much broader range of masses. Some of the difference can be explained if the early post-AGB evolution is faster than predicted by
the Blöcker tracks. Between 30 and 50 per cent of WD may avoid the PN phase because of too low mass. However, the discrepancy
cannot be fully resolved and WD mass distributions may have been broadened by observational or model uncertainties.
Key words. Planetary nebulae: general – Stars: evolution – Stars: white dwarfs
1. Introduction
White Dwarf (WD) mass distributions have been determined us-
ing a variety of different methods. Discrepancies exist between
the different determinations in particular between the photo-
metric and spectroscopic WD masses. Boudreault & Bergeron
(2005) compared the masses derived by fitting the observed
Balmer lines with masses derived from trigonometric parallaxes
and photometry. They found differences of ∼ 50 per cent for cool
(6 500–14 000 K) DA white dwarfs. Spectroscopic masses are
believed to be more accurate, especially for WDs in the temper-
ature range between 15 000 and 40 000 K (Liebert et al. 2005).
Atmospheric models are less well established for stars outside
this range. For hotter WDs the atmospheric structure is modi-
fied by an (often unknown) amount of metals and by non-LTE
effects. For cooler WDs the convection has to be considered and
the models are sensitive to the mixing length and the amount of
helium convected to the surface (Boudreault & Bergeron 2005).
Central stars of planetary nebulae (CSPN) provide a way to
test the mass distributions. CSPNe evolve directly into WDs,
with only very minor mass changes, allowing one to measure
masses of currently forming white dwarfs. However, CSPN mass
distributions have also been uncertain. For example, Napiwotzki
(2006) shows that the very high CSPN masses (close to the
Chandrasekhar limit) derived spectroscopically with state-of-
the-art model atmospheres by Pauldrach et al. (2004) are physi-
cally implausible and masses close to the peak of the CSPN/WD
mass distribution are more likely.
CSPN masses are normally obtained from the luminosi-
ties. But more accurate masses can be derived using the age–
temperature diagram, obtainable from the surrounding planetary
nebula (PN). Gesicki et al. (2006) applied this to a sample of
101 PNe. In this Letter we discuss the resulting mass distribu-
tions for hydrogen-rich and hydrogen-poor CSPNe and compare
with published WD masses.
2. Methods and results
2.1. Models
The method requires the age of the nebula and the temperature
of the central star to be determined. Together these provide the
heating time scale for the star.
We derive the age of the PNe using a combination of line
ratios, diameters (taken from the literature), and new high res-
olution spectra (Gesicki et al. 2006). The diameters and line
ratios are used to fit a spherically symmetric photo-ionization
model. The model assumes a density distribution and finds a
stellar black-body temperature. For each ion, the model finds a
radial emissivity distribution. The observed line profiles for each
ion represent the convolution of the thermal broadening and the
expansion velocity at each radius. Thus, the line profiles for dif-
ferent ions are used to fit a velocity field. An iterative procedure
is used to improve the ionization model. The emissivity distribu-
tions of different ions overlap, and this gives a strong constraint
on the shape of the wings of the line profiles. A genetic algo-
rithm, PIKAIA, is used to arrive at the optimum solution for
ionization model and velocity field. A turbulent component is
added if needed: turbulence is indicated by a Gaussian shape of
the line profiles. The expansion velocities are found to increase
with radius, due to the overpressure of the ionized region.
¿From the velocity field v(r), we derive the mass-weighted
average over the nebula, vav. This parameter has been shown
to be robust against the simplifications. Different models which
provide comparable quality fits give the same vav to within
2 km s−1 (Gesicki et al 2006). Applying this to a radius of 0.8
times the outer radius (equivalent to the mass-averaged radius)
allows us to define a kinematic age t to the nebula. A linear ac-
celeration is assumed to have occurred from the AGB expansion
velocity (10–15 km s−1) to the PN velocity vav (20–25 km s−1).
The derived nebular age and stellar temperature are com-
pared to the the H-burning tracks of Blöcker (1995), which pro-
vide the largest and most uniform collection available. We inter-
2 K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae
0.565
0.605
0.625
0.696
0.836
2 4 8
Fig. 1. Comparison of the 101 modelled PNe with the evolu-
tionary tracks in the HR diagram. The model black-body tem-
peratures are plotted against the luminosities interpolated from
tracks. Filled circles indicate [WR] stars, open circles are wels
and pluses indicate non-emission-line stars. The dotted lines
show H-burning evolutionary models of Blöcker (1995), labeled
by mass in units of M�. The solid lines are isochrones, labeled
by the time after the nebula ejection, in units of 103 yr.
polate between different tracks to find for each (t,Teff), the CSPN
luminosity and mass.
2.2. Different CSPN types
The CSPNe fall into two broad categories: the hydrogen-rich
O-type stars and the emission-line central stars which are gen-
erally hydrogen-deficient. The second group consists of [WR]-
type stars with strong emission lines and wels (weak emission
line stars). The [WR] are subdivided into hot [WO] and cool
[WC]. [WR] stars are in most cases hydrogen-free (three possi-
ble exceptions are mentioned by Werner & Herwig 2006). The
wels may contain some hydrogen. Gesicki et al. (2006) show
that one group of wels is located in the temperature gap between
[WC] and [WO] stars. The other wels stars form a non-uniform
group, including higher-mass objects where the high luminosity
drives a wind but the star is not necessarily hydrogen-poor. The
hydrogen-rich stars are believed to be related to the DA white
dwarfs, while the [WR] may evolved into DB’s.
2.3. The HR diagram
The full analyzed sample contains 101 PNe, of which about 60
are in the direction of the Galactic Bulge and the remainder are in
the Galactic disk. Foreground confusion among the Bulge PNe
is estimated at 20%. The sample contains 23 [WR]-type, 21 wels
and 57 non-emission-line central stars1. The CSPN classification
was adopted from literature. The last group contains also objects
without any information about their spectrum.
In Fig.1 we show the photoionization temperatures and inter-
polated luminosities, plotted on the HR diagram. The H-burning
tracks of Blöcker (1995) are also shown: the luminosities and
masses of CSPNe fall into a rather restricted range of values.
Isochrones of 1,2,4, and 8 × 103 yr are also shown.
A previous HR diagram of CSPNe presented by Stanghellini
et al. (2002) shows a much broader range of luminosities and,
in consequence, masses. They use Zanstra temperatures and
luminosities. The Zanstra method of locating a CSPN in the
HR diagram was criticized by Schönberner & Tylenda (1990).
1 The data file is available from web page
www.astri.uni.torun.pl/∼gesicki/modelled pne.dat
Table 1. Comparison between our dynamical masses and spec-
troscopic masses from Kudritzki et al. (2006). Observed mass-
loss rates from the same paper are also listed and compared to
values from the model tracks of Blöcker (1995). He 2-108 is
classified as wels, the other three are non-emission-line stars.
Object M [M�] Teff [103 K] log Ṁ [M� yr−1]
dyn. spec. dyn. spec. spec. evol. tracks
Tc 1 0.59 0.81 32 34 −7.46 −7.91
He 2-108 0.57 0.63 32 34 −6.85 −8.16
IC 418 0.61 0.92 37 36 −7.43 −7.82
NGC 3242 0.61 0.63 79 75 −8.08 −7.86
Observationally, the accuracy of the luminosity determinations
is about a factor of 2. On the Schönberner tracks, a CSPN mass
change from, e.g., 0.57 to 0.7 M� corresponds to a factor of 3
in luminosity. The masses determined directly from luminosity
are thus accurate to only 0.1 M�. This is less than the typical
dispersion of masses. In contrast, for the same mass range, the
dynamical time scales differ by a factor of 60. Even for a factor
of 2 uncertainty in the nebular age, the mass changes by only
0.02 M�. Therefore, the dynamical method improves the accu-
racy.
Schönberner & Tylenda (1990) also developed a method to
improve the CSPN mass determination. This method (Tylenda et
al. 1991) results in masses similar to ours.
Table 1 compares, for four objects in common, our dynami-
cal masses with the spectroscopic masses derived by Kudritzki et
al. (2006). The spectroscopic masses are larger, in two cases very
much larger. The lower masses are supported by the kinematical
properties of Tc 1 and He 2-108 (see Fig. 5 of Napiwotzki 2006),
which favour an old thin disk population. Kudritzki et al. also
derive Teff : our photo-ionization values are in good agreement.
Pauldrach et al. (2004) find from a spectroscopic analysis,
five CSPNe with masses close to the Chandrasekhar limit. This
result is implausible, as argued by Napiwotzki (2006). Three of
their objects are also in our sample, and all are found to have
regular masses.
2.4. The mass distributions
In Fig.2 the upper panel presents the mass distribution of our
whole sample of 101 PNe. All CSPNe masses fall into a narrow
range, 0.55 − 0.66 M�, with a mean mass of 0.61 M�. The range
of masses is almost identical to that of Tylenda et al. (1991) but
they obtained a smaller mean mass of 0.593 M� and their distri-
bution peaks at 0.58 M�.
The lower panel of Fig.2 presents masses for the same types
of CSPNe as shown in Fig. 1. The non-emission-line stars show
a Gaussian mass distribution. The hydrogen-deficient emission-
line stars seem to consist of two populations: one sharply peaked,
containing [WR] stars, and the other showing a wider spread,
composed of [WR] and wels. The sharp peak consists, with a
single exception, of hot [WO] stars only.
The presented histograms seem to suggest that hot [WO]
stars form a different group from the combined cooler [WC] and
wels CSPNe.
K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae 3
Fig. 2. The CSPN mass histograms. Upper panel: the histogram
of all modelled PNe. Lower panel: the histogram of different
subgroups of the 101 PNe. The dashed line indicates [WR] stars,
the dotted line wels and the solid line non-emission-line stars.
3. Comparing CSPNe and WDs
3.1. The histograms
The comparable birth rates of PNe and WDs suggests that most
white dwarfs go through the PN phase (e.g. Liebert et al. 2005).
The mass distribution in both samples should therefore be simi-
Fig.3 presents the histograms of our interpolated O-type
CSPN masses and the masses of DA white dwarfs from recent
surveys. The WD data of Madej et al. (2004), kindly provided
by the authors, contain 1175 new DA WDs extracted from the
Sloan Digital Sky Survey. The data of Liebert et al. (2005) taken
from the electronic version of their article, contain 347 DA WDs
from the Palomar Green Survey. For Fig.3 we selected the ob-
jects with temperatures between 15 000 K and 40 000 K. The two
WDs histograms are not identical, but both peak at similar val-
ues and show extended low- and high-mass tails. We plot the
histograms using narrower bins than usually done for WDs, op-
timized to the mass resolution of our CSPN data. The difference
between the WD and CSPN distributions is striking.
First, the obtained CSPN masses are restricted to a much nar-
rower range of values than WDs, and are also much more sharply
peaked. At face value, this implies that only some of the WDs
have gone through the PN phase, in contrast to the conclusion
from their similar birth rates (Liebert et al. 2005). Second, the
two distributions peak at different masses. Here a systematic er-
ror cannot be excluded, as discussed below.
3.2. Hydrogen-rich vs. hydrogen-deficient
Hansen & Liebert (2003) point to a variety of WD mass distri-
butions with clear differences between hydrogen- and helium-
rich cool stars. Beauchamp et al. (1996) found for hot helium-
atmosphere DB stars a sharp peak lacking almost entirely of low-
and high-mass components. They also found that the DBA stars,
which exhibit traces of atmospheric hydrogen, show a distinctly
different, broad and flat distribution.
The CSPN show an apparent difference between hydrogen-
rich and hydrogen-deficient mass distributions. The hydrogen-
deficient stars show a very narrow mass distribution; it is tempt-
ing to relate this to the helium-rich DB and DBA populations.
We use hydrogen-burning tracks to derive these masses. The
Fig. 3. The mass distribution of non-emission-line O-type
CSPNe (shaded area) is compared to two DA white dwarf distri-
butions of intermediate temperatures: thin line: data from Liebert
et al. (2005); dotted line: data from Madej et al. (2004) which are
more numerous, and are rescaled.
evolution after the thermal pulse leading to helium burners is
very complicated and not well understood (Werner & Herwig
2006). This may not affect the derived masses too much: the ef-
fect of a thermal pulse is to change the temperature of the star,
but as shown in Fig. 1, the isochrones have only a weak depen-
dence on temperature. The resulting offset in time (still very un-
certain) when accounted for can shift those CSPN masses to-
wards higher values.
4. Discussion
4.1. Uncertainties in mass determinations
When comparing the CSPNe and WDs we have to remem-
ber that we compare different spatial distributions. Because of
their faintness the WD observations are restricted to our near-
est neighbourhood while PNe are observed across the whole
Galaxy. Nevertheless we didn’t obtain significantly different dis-
tributions for PNe at different distances.
Our mass determination relies on a single set of evolutionary
tracks. There are two possible sources of errors in the Blöcker
tracks. The first is the early post-AGB evolution where the time
scales depend on how and when the AGB wind terminates. The
Blöcker tracks end this at Teff ∼ 6000 K, (pulsation period of
50 days) to agree with the observations of detached shells around
hotter stars but not around cooler stars. A later termination would
lead to an earlier start of the ionization: in this case we would
systematically overestimate the masses. For a reduction of the
post-AGB transition time by 103 yr, the typical mass would re-
duce by 0.01 M�.
The second uncertainty is the mass-loss rate during the post-
AGB phase. For M ∼ 0.6 M�, the post-AGB mass-loss rate in
the Blöcker models is 0.1 times the nuclear burning rate, but
for high-mass models the mass loss accelerates the evolution
by 50% (Blöcker 1995). A higher post-AGB mass loss than as-
sumed would reduce our masses, but for the typical masses we
find a very large increase would be required. Table 1 compares
the Blöcker mass-loss rates with observed values, where we used
the dynamical mass to calculate the Blöcker rate. For the three
non-emission-line stars, observed rates are higher by up to a fac-
tor of 3. This appears to be in part related to the high luminos-
ity derived by Kudritzki et al: if we compare their rates with
Blöcker tracks at similar luminosity, then the Blöcker rates tend
to be higher. The nuclear burning rate of ṀH ∼ −6.8 exceeds
the observed wind by a factor of four (more for NGC 3242). For
4 K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae
Table 2. Blöcker track time scales: PN visibility is defined as
between log Teff = 4.4 and either a nebular age t = 104 yr or a
stellar luminosity log L = 3.0, whichever occurs earlier
Mass [M�] tstart [yr] tend [yr] tvisibility [yr]
0.546 90 103 - -
0.565 4 103 10 103 6 103
0.605 1.5 103 7.4 103 5.9 103
0.625 660 3.6 103 2.9 103
0.696 100 880 780
0.836 100 840 740
0.940 12 90 78
this factor, the Blöcker tracks would underestimate the speed of
evolution by only 10 per cent. We conclude that the post-AGB
mass-loss rates have little effect on the derived masses. The ex-
ception is the wels star in the sample, where the wind mass loss
rate is comparable to the nuclear burning rate.
There is also an uncertainty in the dynamical age estimate.
A later acceleration would increase the ages by up to 50 per cent
and shift the mass peak from 0.61 to 0.60 M�.
The WD mass determinations also suffer from simplifica-
tions and model assumptions, in addition to the uncertainties
concerning cool and hot WDs as described in the Introduction.
One uncertainty is in contemporary plasma physics, concerning
the pressure broadening in a very high density plasma (Madej et
al. 2004). The mass-radius relations used depend on the assumed
mass of the hydrogen layer. Napiwotzki et al. (1999) compared
estimates from different studies and concluded that the gravities
obtained from spectroscopic method suffer from systematic er-
rors of up to 0.1 dex in log g. This corresponds to an offset in
masses of about 0.02 M� and could, in principle, explain the dif-
ference in peak masses between WDs and CSPNe. The width of
the peak may also be narrower than derived from the models.
Nevertheless, the wide tails of the mass distribution are not in
doubt.
4.2. Time scales, birth rates and binarity
The derived CSPN mass distribution combines the effects of the
birth rate as function of mass, and the observable life time of
the PN. The latter depends on mass as indicated in Table 2. The
period of visibility is defined here as beginning when the star
reaches Teff = 25 103 K, and ending either when the star enters
the cooling track (defines as log L = 3.00) or when the age of the
nebula is 104 yr, whichever comes earlier. Our histogram should
be corrected for the difference in visibility time. This increases
the number at high CSPN mass only by a factor of up to 10,
and brings the high mass tail in somewhat better agreement. We
may also have a sample bias against high masses, as these are
not expected in the Bulge objects. The de-selection of bipolar
objects may have removed a few higher-mass nebulae in the disk.
CSPNe with M < 0.56 M� would not produce a visible PN,
as the post-AGB transition time becomes too long (’lazy PNe’).
In the sample of Liebert et al. (2005), 30 per cent of white dwarfs
have masses in this range, and 50 per cent in the sample of Madej
et al. (2004). However, the sharp drop in the CSPN mass distri-
bution below 0.60 M� occurs at too high mass to be affected.
Hansen & Liebert (2003) argue that both the high- and low-
mass tails in WDs distribution can be a result of binary evolu-
tion. Merging leads to high-mass WDs while a close compan-
ion stripping the envelope can cause an early termination of the
evolution and produce a low-mass helium WD. Both channels
together may account for some 10 per cent of all WDs (Moe &
de Marco 2006). Therefore the histogram for single WDs could
be narrower. Close binary evolution can affect the PN phase as
well, leading to strongly non-spherical nebulae. Our model anal-
ysis assumes spherical symmetry, and we did not analyze bipo-
lar nebulae. Our selection therefore favours single CSPNe and
rejects low-mass CSPNe in interacting binaries. Thus, the CSPN
histogram (Fig. 3) is biased toward single-star evolution, while
the WD histogram includes binary broadening. This may affect
the tails of the WD histogram but is not expected to affect the
main peak.
Moe & de Marco (2006) predict a number of PNe in the
Galaxy of around 46000. Based on local column densities,
Zijlstra & Pottasch (1991) derive an actual number of 23000,
suggesting that only about half the stars which could produce
a PN, do so. This comparison is limited by our knowledge on
the time a PN remains observable. Moe & de Marco (2006) pre-
dict a birth rate of PNe of 1.1× 10−12 PNe yr−1 pc−3, comparable
to the current, local WD birth rate of 1.0 × 10−12 PNe yr−1 pc−3.
Again assuming only half their predicted number of PNe is actu-
ally observed, the expectation is that half of all WDs have passed
through the PN phase.
5. Conclusions
We show that the mass distribution of CSPNe is sharply peaked
at M = 0.61 M�. The published WD mass distributions show
a much broader distribution peaking at a lower mass of M =
0.59 M�. Part of the difference in the peak may indicate faster
evolution during the early post-AGB phase than assumed in the
Blöcker tracks. CSPN mass-loss rates cannot explain the dif-
ference. However considering the uncertainty of 0.02 M� in the
WD mass estimations both peaks are in reasonable agreement.
About 30 per cent of WDs have too low masses to have passed
through the PN phase.
Acknowledgements. We thank our referee Ralf Napiwotzki for important com-
ments. This project was financially supported by the “Polish State Committee
for Scientific Research” through the grant No. 2.P03D.002.025 and by a NATO
collaborative program grant No. PST.CLG.979726. AAZ and KG gratefully ac-
knowledge hospitality from the SAAO.
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	Introduction
	Methods and results
	Models
	Different CSPN types
	The HR diagram
	The mass distributions
	Comparing CSPNe and WDs
	The histograms
	Hydrogen-rich vs. hydrogen-deficient
	Discussion
	Uncertainties in mass determinations
	Time scales, birth rates and binarity
	Conclusions
