D. Schönberner - R. Jacob - M. Steffen
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 22 March 2005 / Accepted 11 July 2005
Abstract
A detailed theoretical study of the basic internal kinematics of
planetary nebulae is presented, based on 1D radiation-hydrodynamics
simulations of circumstellar envelopes around central stars of
0.595 and 0.696 .
By means of observable quantities like radial
surface-brightness distributions and emission-line profiles computed from
the models, a comparison with real objects was performed and revealed
a reasonable agreement. This allowed to draw important conclusions
by investigating the kinematics of these models in detail.
Firstly, it is shown that the determination of kinematical
ages, normally considered to be simple if size and expansion
rate of an object are given, can seriously be flawed. Secondly, the
expansion law of a planetary nebula is different from what is
assumed for deriving spatio-kinematical models.
Thirdly and most importantly, our
hydrodynamical models help to correctly use existing angular expansion
measurements for distance determinations. The mere combination of the
angular expansion rates with the spectroscopic expansion velocities
leads always to a serious underestimate of the distance,
the degree of which depends on the evolutionary state of the object.
The necessary correction factor varies between 3 and 1.3.
Individual correction factors can be estimated with an accuracy
of about 10% by matching our hydrodynamical models to real objects.
As a result, revised distances for a few objects with
reliable angular expansion rates are presented.
But even these corrected distances are not always satisfying: they still
appear to be inconsistent with other distance determinations
and, even more disturbing, with the accepted theory of post-asymptotic
giant branch evolution.
As a byproduct of the angular expansion measurements, the
transition times from the vicinity of the asymptotic giant branch
to the planetary-nebula regime could be estimated. They appear
to be shorter than assumed in the present evolutionary calculations.
Key words: hydrodynamics - radiative transfer - planetary nebulae: general - stars: AGB and post-AGB
To date the success of trigonometric parallaxes is rather limited
because only few objects are close enough. More promising are the
spectroscopic and expansion parallaxes. The first method
needs very sophisticated model atmospheres to analyze the photospheric
spectrum of the central stars and has successfully been applied to a
number of objects (cf. Méndez et al. 1992).
The second method uses the observed nebular expansion rates and is much simpler:
it relates the radial angular expansion on the sky,
,
to a
spectroscopically measured expansion velocity along the line-of-sight, V.
From the geometry of expansion, the distance to the nebula follows as
The implicit assumption is that spectroscopy and imaging are
sampling the same physical regime within the object, so that it appears
justified to use the (small angle) relation
.
This assumption, however, cannot be taken for granted because it is well
known that measured Doppler splits do often depend on the ion
considered, indicative of velocity gradients within the expanding
shell. The question is which velocity to take, a question
difficult to answer without detailed knowledge of the velocity field
and the ionization stratification of the PN under consideration.
The determination of expansion parallaxes is, of course, only meaningful for objects with a spherical or ellipsoidal shape, allowing to relate the expansions in the plane of the sky and along the line-of-sight at the position of the central star (cf. Palen et al. 2002). But even in this case assumptions about the internal expansion law have to be made that are not verified.
The most severe drawback of this method, not considered at all in previous applications, is the fact that the measured edges of the expanding shells are either ionization or shock fronts, the expansion speeds of which are not measurable by spectroscopy. In both cases one compares pattern with matter velocity, where the latter is always less than the former, leading to a systematic underestimate of the distance. This fact has been mentioned at several occasions in the past but did not find proper attention (cf. Marten et al. 1993; Steffen et al. 1997).
Very recently Mellema (2004) investigated the expansion properties of the shell edges of PNe by applying the jump conditions across shock and ionization fronts. Assuming that the spectroscopically measured expansion speeds refer to the material immediately behind the shock front, he estimated that correction factors between 1.3 and 1.5 need to be applied to the distances.
In the present work we utilize detailed 1D-radiation-hydrodynamics simulations of planetary nebulae to give a more realistic estimate of the correction factors to be expected. Selecting models that have a very close similarity to real objects, we demonstrate that, due to the density and velocity structure typical of PNe, material velocities do not refer usually to regions immediately behind the shock fronts. This fact increases the disparity between pattern and material velocity, and distance corrections up to a factor of two and more are easily possible (cf. Perinotto et al. 2004, hereafter Paper I, Fig. 22 therein).
The present paper is organized as follows: we first introduce in Sect. 2 our method of modelling the PN evolution and discuss to some extent the properties of our hydrodynamical models, in particular their kinematical ages and expansion laws. We verify in Sect. 3 the usefulness of our models by comparing observable quantities like surface brightnesses and emission-line profiles computed from the models with those observed. Section 4 deals then with the important issue of the expansion parallaxes. The paper closes in Sect. 5 with a short summary and discussion.
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Figure 1:
Top: evolutionary path for the
0.595
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In the present paper we employed again the 0.595
post-AGB model
introduced in Schönberner et al. (2005, hereafter Paper II)
as the typical representative of central stars. This model was generated
by interpolation between the 0.605
track of Blöcker
(1995b) and the 0.565
track of Schönberner
(1983). Its general properties are repeated in
Fig. 1 showing the evolution in the Hertzsprung-Russell
diagram and the corresponding temporal evolution of the photon and wind
luminosities, and the mass-loss rate and the wind speed. We followed the
recommendations of Pauldrach et al. (1988) for the parameters of
the central-star wind, while we assumed a Reimers' wind (Reimers 1975)
during the transition from the vicinity of the AGB to higher effective
temperatures where the wind becomes line driven. For more details,
the reader is again referred to Paper I.
We computed with this central-star model
a new sequence with an initial AGB envelope model which had been used
earlier in conjunction with a 0.605
central-star model
(Paper I, T YPE C initial model of sequence No. 6 from Table 1).
This model which is shown in Fig. 2 has an initial density and
velocity structure based on consistent radiation-hydrodynamics simulations
along the upper AGB using the mass-loss prescription of Blöcker
(1995a). The important
property of this model is that the radial density distribution
falls off much steeper than obtained under the assumption of stationary
winds: at larger distances from the star we have
.
This is the consequence of the luminosity increase after the
the last thermal pulse on the AGB whose signature in the wind envelope is
seen at a distance of 1018 cm, corresponding to
yr.
Further details are given in Steffen et al.
(1998) and in Paper I.
We followed the hydrodynamical evolution of the nebular envelope along the
central-star's track across the HR diagram and down along the
white-dwarf cooling path until the model nebula reached the outer boundary of
the computational domain after 22 000 yr. The morphology of the model PN
turned out to be very similar to that of sequence No. 6
discussed in Paper I. The only noticeable difference is the consequence of
the lower evolutionary speed of the central-star:
the nebular model has more time to expand and becomes optically thin
at a lower effective temperature, viz. at 48 000 K
instead of at
60 000 K in the case of the faster
0.605
central star, which is more consistent with the
observations.
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Figure 2: Initial model used in this paper. Plotted are the density of the heavy particles (thick), of the electrons (dotted), and the flow velocity (thin) against the distance from the central star. |
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In order to facilitate the comparison with observed quantities, several supplementary codes were used for the computation of surface brightnesses and especially the emission-line profiles along preselected lines-of-sight with different spatial resolutions. For the line profiles we employed an updated version of the code used by Gesicki et al. (1996). We always assumed infinite spectral resolution, although it is possible to broaden the theoretical profiles by prescribed functions in order to simulate finite spectral resolutions. Additionally, several IDL routines were developed for the automatic determination of shock front positions from the model structures and of the flow variables immediately before and behind these shocks. Furthermore, we fitted the computed line profiles by multi-component Gaussians (usually four) in order to determine the bulk motions within the individual nebular shells. In general, the determination of bulk velocities within the respective shells by decomposition into Gaussians worked very well.
By decomposition of the line profile into individual components we are able to "measure'' expansion velocities of models in the same manner as one would do for a real objects, with the important difference that we know the true expansion rate. The latter is given by the speed of the shocks which can not be measured spectroscopically. By comparing both velocities we can estimate the systematic error inherent in the expansion parallax method. We preferred in our analysis of the model properties the strong lines [N II] 6583 Å and [O III] 5007 Å, computed for the central line-of-sight through the model nebulae. These lines are usually easy to observe with sufficiently high resolution and signal-to-noise ratio.
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Figure 3:
Heavy-particle density (thick), electron density
(dotted), and velocity field (thin) together with the
[O III] and [N II] surface brightnesses and emission-line
profiles of hydrodynamical models at selected positions along the
horizontal part of the 0.595 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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These snapshots illustrate the important phases
in the life of a PN. The top panel shows the early, optically thick stage
when the ionization front is still trapped and the high pressure of the
ionized gas is driving a shock into the neutral ambient AGB wind (D-type
ionization front). The ionized shell is highly stratified as can be seen
by the shapes of the surface brightnesses in the lines of [N II] and
[O III]. The peak at
cm, only seen in
[O III], delineates the inner edge of the model, i.e. the position of
the contact surface or discontinuity.
Consequently, the [N II] and [O III] lines monitor completely
different regions of the H II region with their different gas
velocities. Gas close to the outer edge is expanding with large speed,
(v=26.5 km s-1), as indicated by the outer components of the [N II]
line, whereas it is streaming only very slowly near the the contact surface
(v=5.0 km s-1 from [O III]). At
cm the gas is
weakly shocked because the slowly moving shell gas is being abruptly swept up
by faster gas accelerated by the contact discontinuity.
One recognizes also from the top panel of Fig. 3 that only the
[O III] line probes the kinematics of the developing rim. The strong
inner components of the [N II] line monitor the region behind the
ionization front where the gas velocities are still quite large. In the rim,
nitrogen is mainly in the second ionization stage.
The next panel gives the moment shortly after the beginning of the optically
thin stage: the ionization front left the shock behind, and the latter
constitutes now the outer edge of the PN proper. The velocity profile behind
the shock front becomes nearly linear with a positive gradient.
The ionization stratification
is still persistent, as the different profiles of [N II] and
[O III] show. Because of the now more vigorous stellar wind, the
contact surface moves faster and compresses the gas of the inner regions into
the "rim'' bounded at
cm by a weak shock.
The PN has now entered its main evolutionary phase where it consists of two
independently expanding shells. The (outer) shell, driven by the thermal
pressure maintained by heating from the ionizing photons emitted from the
central star, expands supersonically into the ionized AGB flow, and its shock
(at
cm) accelerates the gas from
15 to 27 km s-1. From the strong outer [N II] components we deduce
25.6 km s-1. The rim is a "bubble'' blown into the shell by the pressure
of the thermalized stellar wind gas and is steadily accelerated. The
[O III] line indicates a velocity of 7.8 km s-1, intermediate between
the postshock value of 5.5 km s-1and that at the contact surface of 9.5 km s-1.
The last two panels illustrate more advanced phases of the double-shell
stage. The rim is now always the most dense and prominent part of the PN,
the shell is faint and sometimes called "attached''
(Chu et al. 1987). Rim and shell continue to
expand independently of each other: the expansion speed of the shell is mainly
determined by the radial slope of the AGB density distribution,
whereas the rim is accelerated by the steadily increasing wind power of the
central star (Schönberner et al. 2004; Paper I).
The surface brightness of the shell turns into one with
a nearly (negative) linear slope which is typical for AGB envelopes with
radial density profiles significantly steeper than
.
In these advanced stages with very hot central stars the ionization degrees depend smoothly on distance from the star. The [N II] and [O III] line profiles are similar, but the [N II] line intensities are very low since all nitrogen atoms are mainly in their second and third ionization stage.
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Figure 4:
Detailed illustration of how the kinematical properties of our model PN
from Fig. 3 develop with time while the central star
evolves across the HR diagram as shown in Fig. 1.
Material velocities are indicated by V, motions of structures by
![]() Top, left: for the shell, the velocities (in the stellar rest frame) of the shock front, ![]() ![]() Top, right: corresponding velocities for the rim, except that instead of the post-shock velocity we plotted its pre-shock velocity, ![]() ![]() Middle: mach numbers of the resp. shocks, ![]() ![]() ![]() Bottom: kinematic ages, ![]() |
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During the optically-thick phase which lasts till t = 4000 yr, the
leading shock of the shell is steadily accelerated. Its speed is highly
supersonic (middle panel) and reaches finally 38 km s-1 when the transition
to the optically-thin stage occurs. During the following "champagne'' phase
of expansion, the shocks resumes its acceleration until the star fades at
yr, causing the electron temperature in the shell to drop.
The main cause of the shocks acceleration during this "champagne'' phase of
expansion is the steepening of the density slope of the AGB wind
with distance from the star (cf. Fig. 2), but
also the slowly increasing electron temperature as the star becomes
hotter contributes to a lesser degree (for a comprehensive discussion see
Paper II). The shock's Mach number with respect to the upstream flow,
,
with
being the
isothermal sound speed, and
the pre-shock flow speed,
is always a little bit below 2.
Concerning the spectroscopic expansion speed of the shell, both ions give
similar results, except
during the optically-thick stage (more precisely for yr) where
the shell is highly ionization stratified. The [O III] lines
probe only the inner more highly ionized region where the gas velocity is
quite small (Fig. 3, top panel). Using [N II] only,
the real expansion speed of the PN, as given by the motion of the shell's
shock front, will be underestimated by a factor F that depends on the
evolutionary stage and may amount up to a factor of 1.4 (middle panel).
Using proper motion measurements of the shell,
,
the distance would then be given by
.
The left bottom panel of Fig. 4 contains the important
information how kinematic ages of PNe, as determined by the radial position
of the shock front, i.e. the outer edge of a PN,
divided by an expansion velocity, compare with the true post-AGB ages.
It turns out that, using lower ionized species like [N II], the
agreement between both ages is not too bad during the whole course of
evolution shown in the figure. Using higher ionized elements like
[O III] gives, as expected, poor agreement only during the early,
optically thick period. We have here obviously two effects that compensate
each other to a certain degree: the lower expansion speed as determined from
the emission lines makes up for the fact that due to the shock's acceleration
the kinematical age of the shell falls well behind the true age, as is evident
from the time behaviour of
.
Once formed, the speed of the rim's leading shock,
,
is steadily increasing from a rather low value,
10 km s-1, to over
35 km s-1, driven by the expansion of the bubble,
.
However, the shock's relative speed to the ambient shell gas stays close
to the (isothermal) sound speed since
increases as well
with time. Thus the shock remains always very
weak, with Mach numbers close to unity, with one exception
(middle panel). Shortly after t=6000 yr the He II ionization
front is approaching the shock, leading to an additional acceleration
because of the larger electron temperature behind the ionization front.
After the passage of the He II ionization front the rim's acceleration
and speed return to the normal values.
The top panel contains also information about the Doppler speeds, measured
from the split strong line components (cf. Fig. 3).
As we have found for the shell, the expansion velocities based on line-profile
analyses are significantly below
.
Actually they are
close to the speed of the contact surface because the highest rim density
is there.
The discrepancy between the rim's motion and the matter flow within the rim
is quite large and invalidates all attempts to combine both velocities
for a distance determination. We will come back to this point in
Sect. 4.
We mention here only that the mismatch between shock and matter speed, as
expressed by
,
is
3 during the early phase,
declining later to more modest values around 1.4 (right middle panel of
Fig. 4). This behaviour is consistent with Mellema's
analysis of weak isothermal shocks
(Mellema 2004, Fig. 3 therein)
.
For weak isothermal shocks,
and
,
one gets from
the jump conditions in the stellar frame
(cf. Eq. (4) in Mellema 2004),
The bottom panel of Fig. 4 contains again information about the relations between derived kinematical ages and the true evolutionary age. As it turns out, the rim of a PN is not at all suited to determine the object's age. Using size and Doppler expansion, the kinematical age is not indicative for the real age. Instead, the kinematical age may even decrease with time if one uses the splitting of the [O III] line. Using the rim's proper motion instead is not very useful either: because of the roughly constant acceleration of the rim's front the kinematical age falls well behind the real age (see also the discussion in Sect. 3.2).
The difficulties of determining realistic ages from sizes and spectroscopic measurements have already be noted earlier (cf. Marten et al. 1993; Mellema 1994; Schönberner et al. 1997). Especially Mellema discussed this problem in some detail, and we confirm his conclusions. We defer a more extensive discussion on kinematical ages to Sect. 2.3.4.
Real planetary nebulae are, however, ruled by hydrodynamics instead of
ballistics, and their expansion properties can be checked by means of the
hydrodynamical models displayed in Fig. 3. We see clearly that,
although the shell as a thermally-driven shock wave develops a velocity field
with a nearly constant and positive slope,
does not
really hold because the velocity is not vanishing at the
origin, i.e. at the position of the central star. The rim's velocity field
is not similar to
at all because the rim matter is being
compressed by the expanding bubble of wind-heated gas and the confining shell.
Based on the wind model used here, the flow is fastest close to the
contact discontinuity (cf. Fig. 3).
The expansion of our hydrodynamical model is illustrated in Fig. 5. Plotted are the radial positions of the outer edges, i.e. the shock fronts, of the shell and the rim, and also the position of the contact discontinuity. It is evident from this figure that none of the plotted structures expand ballistically! Instead, the expansion is accelerated: that of the shell by the increase of the circumstellar density gradient with distance from the star, and that of the rim mainly by the increase of the stellar wind power with time.
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Figure 5:
Radial displacements of shell, rim and contact surface of a model PN
as a function of time. Plotted are
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For the rim, the expansion law is of the form
The sizes of the exponents, b=1.40 and 1.95, reflect the fact that the
rim is driven by increasing stellar wind power (cf. Fig. 1)
into an expanding confining medium. The shell disturbs the original
circumstellar density profile to such an extent that the density decrease
ahead of the rim is different from the original one.
In fact, we derived from the model that the gas density in front of the rim
decreases roughly with
,
although the original radial
density decline of the wind envelope was close to
.
We note that similarity solutions for adiabatic wind-driven bubbles predict
b=1.0 if
and b=1.2 if
for
the rim shock (Koo & McKee 1992), well below the result of our
hydrodynamic simulation, b=1.4. Note that these similarity solutions
assume expansion into a static medium under the influence of constant wind
power. We conclude that analytical solutions based on simplified conditions
are not suited to describe the expansion of planetary nebulae.
We investigated also the rim expansion laws for the simpler models with
initial power-law density profiles,
,
used
in Paper II. For fixed
,
we found b to vary slightly between
1.2 and 1.4, but virtually independently of central-star mass and initial
envelope density. The dependence on
is, however, significant:
we found
for
.
For elliptical objects it is generally assumed that the equatorial (e) and
polar (p) radii and (spectroscopic) expansion velocities behave like
.
This implies
and uniform acceleration starting in the origin. Both assumptions
are not supported by our hydrodynamical models. However, considering the
weak dependence of b on the environment conditions as follows from our
spherical models, a relation
is very likely to hold provided R is substantially
larger than
.
More problematic is the fact that we found, as long as the shock-heated
wind gas continues to expand,
(cf. Fig. 4, right upper panel). This offset in velocity
results in
,
but the
error introduced thereby cannot be estimated without
realistic 2D hydrodynamics simulations.
The expansion velocities of PNe listed in existing catalogues are usually
measured from the bright nebular parts, i.e. they belong to the rim in the case
of double shell objects, while the radii are exclusively those of the shell.
Using our model sequence, we thus combined shell radii,
,
with the corresponding spectroscopic expansion velocities,
,
and the result is plotted in Fig. 6. The models appear to be very old
right from the beginning: the kinematical ages vary
between 10 000 and 15 000 years, depending on the line used, and are
virtually independent of the real evolutionary age! Only for a very
advanced stage of evolution when the central star is approaching
its maximum temperature (
yr) we find a reasonable
agreement between the real post-AGB age and
.
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Figure 6:
Comparison of different kinematical ages derived from the shell radius.
They are computed from our hydrodynamical models and plotted as a
function of their real evolutionary age. The curves for
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Judging from this very disappointing result it appears to be not
straightforward at all to determine empirically ages for planetaries even
if the distances and hence the dimensions are known. According to the model
sequence presented here, reasonable results for the whole range of
evolutionary times considered are in principle only to be expected if the radius
of the shell is combined with the shell's spectroscopic expansion speed
(see Fig. 6,
).
For this purpose, the [O III] lines should not be used for
young, optically thick objects because of their ionization stratification.
We thus caution here that extreme care should be taken when interpreting
kinematical ages of double shell PNe since the spectroscopic velocities of
shells are usually not known.
In this context we would like to mention the work of McCarthy et al. (1990) where kinematical ages of PNe, most of which are double shell
objects, were compared to evolutionary ages, the latter being determined
from positions of the central stars in a
diagram of
post-AGB evolutionary tracks. These authors found
systematic discrepancies in the sense that in general the empirical ages are
larger than the evolutionary ones. This has been interpreted in the sense
that the residual envelope masses of the AGB remnants are in most cases
larger than used in the evolutionary calculations.
Not considering the fact that the determination of evolutionary ages demands
a very accurate evaluation of the surface parameters of the central stars,
it is clear from the above results that this method is not able to
provide results useful for such a study. For instance, although McCarthy
et al. (1990) do not provide the radii and velocities used for
their age determination, the only possible combinations for the double shell
objects they could have used are either
,
since the shell velocities were not known, or
,
both of which are unsuited for estimating kinematical ages
(cf. Figs. 4 and 6).
Another illuminating example is the case study performed for M 1-46
by Guerrero et al. (1996). This PN appears to be a young
double shell object with a rather cool central star of
K.
The [N II] spectroscopic velocities of rim and shell are 11.5 and
26.0 km s-1, respectively (Guerrero et al. 1996; Paper II).
Adopting a distance of 4.4 kpc, Guerrero et al. found ages of 12 650
and 5700 years, depending on whether the rim or the shell velocity is
combined with the outer radius, i.e. with the shell radius.
Our models offer a simple explanation for this discrepancy: M 1-46
is still close to the optically thick/thin transition phase, and based on
our hydrodynamical models we have shown that a large discrepancy between
and
exists, which in the case depicted in Fig. 6 are about 4000 and
12 000 yr, i.e. a difference of about 8000 yr, in astonishing agreement
with what has been found for M 1-46.
Considering that only
provides a
reasonable age estimate, it follows that the real age of M 1-46 is
close to 5700 years (for the assumed distance), and that there is no reason
to assume two distinct mass-loss episodes separated by several thousands
of years.
It may also happen that the kinematical age, based on
,
becomes smaller during later stages than
that based on
if
increases sufficiently faster than
.
The corresponding
curves in Fig. 6 would then intersect at larger ages.
Candidates are NGC 7662 (Guerrero et al. 2004) and other
evolved PNe with very hot central stars and
similar expansion rates for shell and rim.
For this purpose, we selected three planetaries with rather well-defined double shell structures not too much deviating from sphericity: NGC 6826, IC 2448, and NGC 3242, which cover a certain range of evolutionary stages as is evident from the different effective temperatures of their central stars (Table 1). We also discuss in some detail NGC 7027 which is a young but evolved PN around a massive central star.
Table 1: Objects selected for comparison with our hydrodynamical models. The effective temperatures of the central stars are from Méndez et al. (1992), and the spectroscopic expansion velocities from the decomposition of [O III] line profiles (Paper II, Table 4 therein).
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Figure 7:
Top:
normalized surface brightness distributions in [O III] of three
selected planetary nebulae from HST monochromatic images (F502N),
with effective temperatures of their central stars increasing
from left to right (cf. Table 1). For NGC 3242
the He II image (F469N) is added.
The cuts are taken along or close to the minor axes and
scaled to the model sizes.
Bottom:
normalized surface-brightness distributions in [O III]
5007 Å (and additionally in He II 4686 Å in the case of
NGC 3242) for selected hydrodynamical models that match
the observations as closely as possible.
The positions of the models along the track shown in
Fig. 1 are ( from left to right) at
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The models along the evolutionary path shown in Fig. 1 are selected such as to provide a reasonable match to the observations (bottom panels of Fig. 7). As it turned out, the models follow also a sequence with increasing stellar temperatures, although these temperatures are generally somewhat larger than those of the observed objects. The agreement of our models with these objects is astonishing, especially if one considers that we have not aimed at fitting particular objects from the outset: relative sizes and brightnesses of the shells and rims are remarkably well matched.
There are, however, differences to be noted: the "cavities'' enclosed by the rim appear deeper than the models predict, NGC 6826 excepted. Also the observed brightness profiles of the shells are somewhat flatter (e.g. in IC 2448). We have shown in Paper II that the shape of the shell's surface brightness depend on the radial density profile of the initial configuration set up during the final AGB mass-loss phase. From the small mismatch between the models and the observation that is evident from Fig. 7 we conclude that the original density gradient should be a little bit steeper than the one in our initial model (see Fig. 2).
The density gradient in the wind envelope rules also the expansion speed (see Paper II), hence we expect also a small mismatch between modelled and observed expansion speeds. This is shown in Fig. 8 where the spectroscopic velocities listed in Table 1 are compared with the model predictions. It is evident from the figure that the spectroscopic velocities of the shells and the rims of the models are falling behind the observations for the two more evolved objects. A larger density gradient of the initial model would be needed to accelerate the expansion of the models.
In any case, the relative speeds between shell and rim are rather well matched by our models. We conclude therefore that also the shock properties, especially that of the rim, are adequately described by our models.
We found already in Paper II that most likely all double
shell PNe have very similar properties concerning morphology and expansion
properties. Obviously they form an evolutionary sequence, and
their evolutionary status can be inferred from their morphology,
expansion property, and temperature of the central star. The combination of a
post-AGB model of about 0.6
with an appropriate model for the
AGB-wind envelope enables obviously a successful hydrodynamical modelling
of the planetary-nebula evolution.
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Figure 8: Spectroscopic expansion velocities of the objects from Table 1 compared with predictions from the models used in Fig. 7. The dashed line is the 1:1 relation. |
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Morphologically, NGC 7027 consist of a bright rim-like structure with an attached faint shell, surrounded by a molecular shell seen in emission from H2 (Latter et al. 2000, Fig. 7 therein). We have shown in Paper I that envelopes around massive, very fast evolving central stars remain optically thick and that most likely the interface between the neutral and the ionized domains, i.e. the photo-dissociation region, is a place where all sorts of molecules, including H2, are being formed and destroyed (cf. Hasegawa et al. 2000).
For the case of NGC 7027 we employed sequence No. 10 from Paper I
which combines a simple AGB-wind envelope with a rapidly evolving post-AGB
object of 0.696 .
A detailed description of the evolutionary properties
of this hydrodynamical model sequence is presented in the Appendix.
Here we show in Fig. 9 (top panel) a number of normalized
H
surface-brightness profiles computed from this hydrodynamical
sequence and selected at positions along the high-temperature part of the
evolutionary track, as shown in Fig. A.1, which embrace
the estimated position of NGC 7027 at
K and
(Latter et al. 2000).
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Figure 9:
Top:
radial H![]() ![]() ![]() ![]() ![]() |
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The H
brightness profiles displayed in the top panel of
Fig. 9 are reflecting the underlying nebular structures as
illustrated in Fig. A.2: development of a very bright and
(geometrically) thick rim due to the action of the powerful stellar wind
while the central star crosses the HR diagram within about 700 years,
surrounded by a much fainter shell.
The morphology resembles indeed that of the double-shell objects
discussed earlier, with the difference that the outer edge of the shell
appears to be sharp due to the D-type ionization front typically for
optical thick conditions (t=304 and 450 yr).
As the stellar temperature exceeds 100 000 K, the number of hydrogen-ionizing photons decreases to such an extent that recombination begins to dominate, leading to a smoothing of the ionization front (t=584 yr). Recombination increases further in the shell as the central star approaches and passes its maximum effective temperature, and the emission from the shell disappears quickly (t>584 yr).
We selected the model with t=584 yr and compared in the bottom panel of
Fig. 9 its (scaled) H
surface-brightness profile with
a cut along the semi-major axis of the HST P
image of
NGC 2027 from the observations of Latter et al. (2000).
The model matches astonishingly well the relative sizes and brightnesses of
rim and shell. The rather smooth outer edge of NGC 2027's
brightness profile indicates that recombination has already started there,
just as in the model. We note further that the model's electron densities
and temperatures are
105 cm-3 and 16 000 K in the rim, and
104 cm-3 and 13 000 K in the shell, values that are consistent
with those reported in the literature, viz.
cm-3 (Basart & Daub 1987) and 14 000 K (Masson 1989).
The total mass of the ionized region is 0.015
.
With 155 000 K, the central star of our selected model is cooler than usually
inferred for NGC 7027 from recent observational analyses which suggest
effective temperatures of 200 000 K or more (see Latter et al. 2000; Volk & Kwok 1997). However, the exact
position along an evolutionary track where recombination starts depends on
the local density and stellar luminosity, and may thus differ considerably
from object to object.
There appears also some hydrogen-line emission beyond the edge of the shell
(at
)
not accounted for by our models. Obviously the real
object is not as homogeneous as our (spherical) model, and we suggest some
leakage of ionizing photons to be responsible for this weak emission.
In any case, our hydrodynamical model provides an overall better fit to the observed Paschen brightness profile than the rather sophisticated kinematic model of Latter et al. (2000, Fig. 4 therein) which does not account for the shell emission (ibid. Fig. 7). We can not provide a comparison with the H2 molecular emission which occurs just at the outskirts of the recombining shell where the excitation conditions appear to be most favorable (Hasegawa et al. 2000). The reason is that our radiation-hydrodynamics code does not include formation and destruction of molecules.
Table 2:
Expansion time scales from proper-motion measurements.
The data are either from Palen et al. (2002), optical
measurements, or from the compilation of Terzian (1997),
radio measurements. The angular radii,
,
and their
expansion rate,
,
refer to the semi-minor axes.
The effective temperatures are spectroscopic values (Méndez
1992) or He II Zanstra temperatures (Gorny, priv. comm.)
Although the studied sample is too small as to allow definitive conclusions, it appears from Fig. 10 that, within the errors, and again with the assumption that all objects from the plotted sample (with the exception of NGC 7027) have very similar evolutionary histories, the time scale of the rim's expansion does not depend on the effective temperature, i.e. the time scale remains virtually constant with evolution, at least in the temperature range between 60 000 and 100 000 K. With other words, size and proper motion of the rim does not appear to be a good age indicator either, a fact stated already in Sect. 2.3.2. Based on the rim expansion properties found from our model simulation (Fig. 4, right bottom), the true ages are substantially larger than the observed expansion times listed in Table 2.
The observations are, qualitatively, supported by our model simulations. From Fig. 10 it is evident that indeed the kinematical time scale of the rim increases only slowly with effective temperature and falls significantly behind the true age for all sequences shown in the figure. The reason is, of course, the steady acceleration of the rim with time (see Fig. 4, right top).
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Figure 10:
Kinematic expansion time scales,
![]() ![]() ![]() ![]() ![]() ![]() |
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However, assuming that all the objects plotted in Fig. 10,
(except NGC 7027), have typical masses of about 0.6 ,
it is also evident that the two low-mass sequences have kinematical ages which
are definitively too large. It is easy to remedy this discrepancy:
reducing the transition time needed for the models to evolve from the
vicinity of the AGB to, say, 30 000 K, by about 1000 years,
leaving the further evolution unaltered, would shift the sequences downwards
to such an extent that a reasonable agreement with the observations could
be achieved. Thus it appears that, as a byproduct
of the proper-motion measurements, one can pose limits on the
duration of the transition from the AGB to the PN regime.
Two of the objects from Table 2 deserve additional comments:
NGC 3242 and NGC 7027, both of which have very small
kinematical ages.
The angular expansion of NGC 3242 is the largest of all objects
listed in Table 2, and its expansion time of
700 yr implies a relatively large central-star mass of >0.65 ,
similar
to what is assumed for NGC 7027. This is in sharp contrast to
our finding in Sect. 3.1 where NGC 3242 appears as
an ordinary double shell PN of intermediate age, representing simply a
more evolved stage of, e.g., IC 2448 or NGC 6826.
All these 3 objects have electron
densities of about
cm-3 which correspond perfectly with the
densities of the hydrodynamical models illustrated in Fig. 3.
A real young object like NGC 7027 has much larger electron densities
exceeding 104 cm-3, and is most likely still optically thick.
In view of these evidences for a substantially larger age of NGC 3242
we consider the proper motion measurements of Hajian et et al. (1995)
as problematic, and new measurements, preferably with the HST, are
urgently needed.
NGC 7027 is the object with the hottest central star for which
angular expansion measurements exist. Its low age,
combined with its very hot central star, is consistent with the theoretical
expectation for more massive, rapidly evolving post-AGB stars.
Since the expansion time is the upper limit of the transition time needed
for post-AGB stars to reach the PN regime, this transition must have been very
short for NGC 7027, and most likely also for other PNe with massive
central stars as well. For the 0.696
sequence shown in
Fig. 10 this transition time amounts to about 100 yr (cf.
Fig. A.1).
Although a direct quantitative agreement of the predictions of our new hydrodynamics simulations with the observations could not be achieved, the models are close enough to the observations as to allow a detailed assessment of the expansion parallax method and the determination of correction factors. We begin with a detailed discussion for the objects shown in Fig. 7 and continue then more generally with other objects. Please note that in any case the correction factors for the distances deduced from our models are always larger than unity! This is a consequence of the jump conditions for shocks, as emphasized by Mellema (2004). Additional corrections may be necessary if departures from sphericity exist (cf. Palen et al. 2002), which, however, are not addressed here.
![]() |
Figure 11:
Detailed illustration of the different possible correction
factors
![]() |
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In Fig. 11 the various combinations of shock and matter speeds
based on our 0.595
model sequence are collected and their variations
during the course of evolution plotted.
The effective temperature of the central star
is used again as an indicator for the evolutionary stage in order to
facilitate the comparison with real objects. We repeat here that
the ratio
is the factor by which the distance to an object in
question has to be enlarged to correct for the assumption that shock
and matter speed are equal.
Figure 11 illustrates in more detail what is already seen in
the middle panels of Fig. 4. We have already noted in
Sect. 2.3 that shell and rim have different expansion
properties and hence different correction factors F. That of the rim
is generally larger and begins with about 3 when the rim is forming
during the optically-thick stage. It decreases later until it approaches the
value of F for the shell. After the passage of the He II
ionization, both factors vary only slowly between 1.4 and 1.3. The correction
factor for the shell remains always rather modest: It remains close to unity
during the optically-thick stage, and does not exceed 1.4 later on.
Mellema (2004) proposed a smaller correction factor,
,
since he used the post-shock velocity (
in
Fig. 4) which is always larger than the spectroscopic
velocity used here.
From this discussion we conclude that proper motion measurements of the shell are more promising since the necessary correction is small and virtually independent of the evolutionary state of the object in question for a rather long period of evolution. Further advantages are that the shell's asphericity is usually small and that it expands faster than the rim. A disadvantage is that the shells are much fainter than the rims. Hence most existing proper motion measurements relied on the bright rims, making a thorough check of the object's evolutionary state necessary.
As a first guess, the evolutionary state of an object can be determined from the effective temperature of its central star, and the corresponding value of F can then be read of from Fig. 11. More relevant, however, is the dynamical state of the nebular envelope, which depends mainly on the initial envelope configuration, viz. AGB-wind velocity and radial density distribution. We therefore preferred to use the models shown in Fig. 7 with their larger central-star temperatures to estimate the correction factors F for the PNe shown there.
The errors of the F determinations based on our still simple spherical
models are difficult to quantify. We implicitly assume in the following,
if not otherwise stated, that our 0.595
central-star model sequence
describes real objects sufficiently well. Considering then the rather smooth
variation of F with temperature during the optically thin stage, we estimate
uncertainties between 10 and 20%. Because they are smaller than the
present errors of the angular expansion determinations, we will neglect
them in the following considerations.
We can also not quantify the error caused by assuming ballistic expansion
to estimate
for elliptical rims. The accelerated expansion
begins at a certain distance from the star and will lead to a
not consistent with the assumption of
ballistic expansion (Sect. 2.3.3). Although 2D simulations are
necessary to quantify the error, we estimate from the expansion properties of
our models that
will most likely be somewhat
underestimated.
In the following detailed discussion of individual objects ballistic expansion is always assumed for deriving equatorial expansion velocities.
The central star is hotter than those of the two objects discussed above. Helium
is doubly ionized in the rim, but not in the shell (Balick et al.
1987), and possible values for F are between 1.6 and 1.9,
larger than Mellema's estimate,
.
Our selected model with its He II profile shown in Fig. 7, in
conjunction with Fig. 11, suggests F=1.7. In any case, the
distance to NGC 3242 would then be between 0.5 and 0.6 kpc, a still
rather small distance, considering that the luminosity of the central star
would be just marginally about 103
.
The spectroscopic distance to NGC 3242 has recently been revised downwards to 1.1 kpc by Pauldrach, Hoffmann, & Méndez (2004). But even this new spectroscopic distance is well above our corrected expansion distance. A significantly smaller angular proper motion would bring the expansion distance into better agreement with other distance determinations and would also increase the expansion age to values found for similar objects.
We analyzed the expansion behaviour of the shock fronts of the
nebular envelope around the 0.696
post-AGB model (see Appendix)
in the same way as we did
for the 0.595
sequence, and the corresponding correction
factors are plotted in Fig. 12 for both the rim and the shell.
Because the model remains always highly ionization stratified, we can use
for the shell only [N II] and for the rim only [O III]. Note that the overall
behaviour of F is very similar to that of the models with the less massive
central-star shown in Fig. 11. During the
high-temperature phase of evolution the rim's correction factor varies
only slowly from 1.5 to 1.3.
Mellema (2004) proposes a larger correction factor of
,
which, according to our models, is not consistent
with the high temperature of the central star (cf. Fig. 12).
![]() |
Figure 12:
Same as in Fig. 11, but now for hydrodynamical models
with a central star of 0.696 ![]() |
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An equatorial expansion velocity of 13 km s-1, based on [O III] 4959 Å, has
been measured recently by Bains et al. (2003). Together with
the angular expansion listed in Table 2 and a correction
factor of 1.4 as read of from Fig. 12 a new distance to
NGC 7027 of
pc follows, with an error
of
pc.
Because of the different spectral type of its central star, it does not
appear to be wise to apply the results of our hydrodynamical simulation
directly to BD +303639
.
The evolutionary properties and also the wind power of its central star are
certainly too different. The nebular shell has, however, a single-shell
morphology and is still optically thick with a corresponding ionization
stratification, i.e. low-ionization stages are expected to be located close
to the ionization front, higher ones closer to the star (Harrington et al. 1997; Bryce & Mellema 1999).
In this respect, the measurements of Bryce & Mellema (1999) are interesting. They found expansion velocities along the central line of sight of 28 km s-1 from [N II] profiles and 35.5 km s-1 from [O III] profiles. From our optically-thick models (top panels of Fig. 3) we would expect the larger velocity from [N II]. Thus the measurements of Bryce & Mellema (1999) hint to a very powerful wind from the [WC] central star which has already compressed and considerably accelerated the innermost parts of the ionized shell.
For a distance estimate we proceed as follows. Because of the ionization stratification, the outer regions of the shell are better probed by ions of lower ionization. We used the Doppler split of the [N II] lines, 28 km s-1, and combined this spectroscopic radial expansion with the angular expansion of 4.25 mas yr-1 measured by Li et al. (2002), yielding a distance of 1.4 kpc.
The factor necessary to correct for the improper use of a spectroscopic
velocity can be read of from Fig. 11, viz.
for
the shell at
K, consistent with Mellema
(2004) who gives
.
We have thereby implicitly
assumed that the D-type ionization front of BD +30
3639
behaves in a similar fashion as our models predict
.
The distance would
then be 1.8 kpc. If we correct for the axial ratio of 1.5 as Li et al. (2002) recommend, the final distance to BD +30
3639
is then back to the Li et al. value of 1.2 kpc. The error is difficult
to quantify, but we estimate it close to 20 %, i.e.
kpc.
We found that an evolving central star model of 0.6 ,
coupled to a
AGB wind envelope which accounts fully for the final mass-loss history on
the AGB and which consequently has a radial density profile much steeper
than the usually assumed r-2 law, appears to be an acceptable choice
for interpreting the properties of typical double shell PNe if the full
radiation-hydrodynamics is considered.
Mellema (1994) has also pointed out that his model sequence
with a r-3 initial density distribution gives a better agreement with
the observations than the sequences based on
.
A more massive central-star model of, say 0.7
,
which evolves very
quickly across the HR diagram, is necessary to explain the
structure of objects like NGC 7027.
Further conclusions drawn from this work and other detailed radiation-hydrodynamics parameter studies (e.g. Mellema 1994; Paper I; Paper II) concerning the internal kinematics of PNe are as follows:
From the theoretical side, the situation appears to be better for the shell where the correction factor does not exceed 1.4. The shell's edge is, however, difficult to observe properly because of its often rather low contrast to the halo emission.
We note that the correction factors estimated by Mellema (2004) using the jump conditions of shocks and reasonable assumptions for the flow velocities are, with exceptions, in good agreement with our more elaborate determinations based on radiation-hydrodynamics simulations.
We believe that further proper motion studies with a longer time base are needed before any definite conclusion on this matter can be drawn. Equally important is the fact that different epoch observations must be taken with exactly the same instrument and filter setting in order to avoid possible systematic errors.
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Figure A.1:
Same as in Fig. 1 but for a 0.696 ![]() |
In this appendix we present the details of a radiation-hydrodynamics
simulation of a planetary nebula with a star of
0.696
used in this article to interprete the angular expansion of
NGC 7027. The gross properties of this model have already been
explained in Paper I (cf. sequence no. 10 from Table 1 therein).
The initial nebular model is of T YPE A, which means constant
mass-loss rate and wind speed during the end of the AGB evolution.
Specifically, we have selected for this sequence
yr-1 and
km s-1.
![]() |
Figure A.2:
Same as in Fig. 3 but for a 0.696 ![]() ![]() ![]() ![]() ![]() |
During the early ionization phase (top panel) the model develops a structure similar to the one seen for less massive stars, except that its size is ten times smaller and its density about hundred times larger than for a less massive and more slowly evolving post-AGB star (cf. Fig. 3). However, because of the very rapid evolution of the central star, the circumstellar matter remains so dense during the high-luminosity part of the star's evolution that the ionization front is always bounded by neutral (or molecular) material (D-type ionization front). The shock is nearly isothermal and compresses the AGB matter into a very thin, dense shell.
Nevertheless, a double-shell structure develops, with a rather dense and thick rim consisting of wind-compressed and accelerated shell material (middle panels). Contrary to the optically-thin models of Fig. 3, an important property of the models with a massive central star is their extreme ionization stratification, with species of low ionization close to the ionization front and those with high ionization in the rim close to the hot star. This stratification is evident from the surface-brightness profiles shown in Fig. A.2: in the light of [O III] both rim and shell are visible, but not in [N II]. The N II zone behind the ionization front is so narrow that the model appears as a ring structure with an extreme center-to-limb brightness variation (top panels). This stratification is responsible for the different expansion velocities measured by the Doppler splits of the [O III] and [N II] profiles.
Later on the ionization front cannot advance further (in mass), and recombination weakens its steepness (panels next to bottom). Consequently, the shell fades in the light of [O III] (and hydrogen as well), while the N II zone extends somewhat inwards, leading to a less extreme ring structure.
With the continuing luminosity drop of the central star, recombination
dominates until only the rim remains fully ionized and the shell
is not visible anymore (bottom panels). The outer leading shock,
at
cm, which is decelerating because of the decreasing
post-shock electron temperature, is nevertheless still travelling with a
relatively high speed of about 25 km s-1 (stellar frame). It is isothermal
and compresses and accelerates the neutral/molecular
AGB matter considerably: the density inside the geometrically
thin shell is with a few 105 cm-3 substantially larger
than the density of the ionized region, and the gas velocity
is increased from 15 to about 25 km s-1 across the shock. The rim travels
much faster and is approaching the outer shock.
During the approach of the central star to its maximum effective
temperature, the whole circumstellar structure is changing from the
double-shell morphology into one which is very similar to the empirical model
constructed by Volk & Kwok (1997) and Hasegawa et al. (2000) for NGC 7027:
a very compact H II region of high density and temperature,
cm-3 and
K, is surrounded by a high-density, geometrically
very thin shell of neutral material compressed by the outer shock, and in
which the (electron) temperature falls off rapidly over a distance of less
than 1016 cm to the low values of the still undisturbed AGB wind.
The total size of the ionized/neutral shells is a few hundredth of parsecs
in diameter, well within the range estimated for NGC 7027.
The partially ionized and neutral shells embracing the PN proper are, as a photon-dominated region (PDR), a site of active circumstellar chemistry (cf. Latter et al. 2000; Hasegawa & Kwok 2001). The detailed study of the CO molecular lines by Jaminet et al. (1991) revealed that there exists, in addition to the main molecular flow of the AGB wind with 15 km s-1, a region where the gas expands faster, i.e. with 23 km s-1. Jaminet et al. argue that this faster expanding gas comes from a thin high-density shell which, as an interface between the AGB wind and the H II region, is being compressed and accelerated by the thermal pressure of the ionized gas.
This interpretation is in accordance with our model, although it is,
for the particular evolutionary stage of NGC 7027 considered here,
the outer shock set up earlier during the ionization phase
which compresses and accelerates the neutral/molecular AGB gas.
The post-shock speed decreases with the shock speed:
at the beginning of the recombination phase we found (in the stellar frame)
km s-1 which then later decreases to 25 km s-1
(bottom panel of Fig. A.2).
![]() |
Figure A.3:
Same as in Fig. 4 but now for the sequence
with the 0.696 ![]() |
We should mention that the hydrodynamical models used here were not intended to describe properly the PDRs which obviously form in the circumstellar envelopes around fast evolving central stars since our code does not contains all the physical processes necessary to describe formation and destruction of molecules, including H2. However, since the basic structure forming processes are photo-ionization and wind interaction we are confident about the usefulness of our models also for the dense envelopes around massive central stars.
There are, however, also differences, mainly caused by the fact that the nebular
shell becomes never optically thin, i.e. the shell is always bounded by an
D-type ionization front whose associated shock moves highly supersonically
through the neutral AGB wind (left middle). This shock is accelerated during
the early ionization phase ( yr), proceeds with roughly constant
speed for another 150 years until it decelerates when the
ionization front weakens and detaches slowly from the shock because of the
increasing recombination rate. Because of the slower shock speed the now
neutral matter behind the shock becomes even more compressed into a
(geometrically) thin shell (bottom panel of Fig. A.2).
Because of recombination the shell disappears completely as an
ionized structure at about t=630 yr and is not plotted any further
(cf. also Fig. A.2).
Like in the 0.595
case of Fig. 4 the kinematic
age can be estimated quite reliably from size and spectroscopic velocity of
the shell (left bottom).
The evolution of the rim is illustrated in the right panels of
Fig. A.3. Driven by the expanding contact surface, the
weak rim shock is steadily accelerated (top) and "snowploughes'' the shell
matter. Once the shell's temperature and pressure is reduced because of
recombination, the shock speed and Mach number increases somewhat
yr). After 750 years the outer edges of the rim start to recombine
as well, and the shock position becomes difficult to determine.
Like in the case with the low-mass central star, the correction factor between
shock and matter speed can be large (middle), and also the determination of
kinematic ages from the rim's size and expansion speed is problematic (bottom).
![]() |
Figure A.4:
Same as in Fig. 5 but now for the sequence
with the 0.696 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
As Fig. A.4 shows, the expansion laws are qualitatively the same as those for the low-mass sequence from Fig. 5. A fit to the rim has only be done for the first 400 years since the physical environment changes later substantially due to recombination.