The evolution of planetary nebulae IV. On the physics of the luminosity function

Abstract

The nebular evolution is followed from the vicinity of the asymptotic-giant branch across the Hertzsprung-Russell diagram until the white-dwarf domain is reached, using various central-star models coupled to different initial envelope configurations. Along each sequence the relevant line emissions of the nebulae are computed and analysed. Maximum line luminosities in Hbeta and [OIII] 5007A are achieved at stellar effective temperatures of about 65000K and 95000-100000K, respectively, provided the nebula remains optically thick for ionising photons. In the optically thin case, the maximum line emission occurs at or shortly after the thick/thin transition. Our models suggest that most planetary nebulae with hotter (>~ 45000K) central stars are optically thin in the Lyman continuum, and that their [OIII] 5007A emission fails to explain the bright end of the observed planetary nebulae luminosity function. However, sequences with central stars of >~ 0.6 Msun and rather dense initial envelopes remain virtually optically thick and are able to populate the bright end of the luminosity function. Individual luminosity functions depend strongly on the central-star mass and on the variation of the nebular optical depth with time. Hydrodynamical simulations of planetary nebulae are essential for any understanding of the basic physics behind their observed luminosity function. In particular, our models do not support the claim of Marigo et.al (2004) according to which the maximum 5007A luminosity occurs during the recombination phase well beyond 100 000K when the stellar luminosity declines and the nebular models become, at least partially, optically thick. Consequently, there is no need to invoke relatively massive central stars of, say > 0.7 Msun, to account for the bright end of the luminosity function.

Full text

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arXiv:0708.4292v1 [astro-ph] 31 Aug 2007
Astronomy & Astrophysics manuscript no. 7437
February 1, 2008
c ? ESO 2008
The evolution of planetary nebulae
IV. On the physics of the luminosity function⋆
D. Sch¨ onberner, R. Jacob, M. Steffen, and C. Sandin
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
e-mail: deschoenberner@aip.de, msteffen@aip.de
Received .....; accepted .....
ABSTRACT
Context. The luminosity function of planetary nebulae, in use for about two decades in extragalactic distance determinations, is still
subject to controversial interpretations.
Aims. The physical basis of the luminosity function is investigated by means of several evolutionary sequences of model planetary
nebulae computed with a 1D radiation-hydrodynamics code.
Methods. The nebular evolution is followed from the vicinity of the asymptotic-giant branch across the Hertzsprung-Russell diagram
until the white-dwarf domain is reached, using various central-star models coupled to different initial envelope configurations. Along
each sequence the relevant line emissions of the nebulae are computed and analysed.
Results. Maximum line luminosities in Hβ and [O] 5007 Å are achieved at stellar effective temperatures of about 65000 K and
95000...100000 K, respectively, provided the nebula remains optically thick for ionising photons. In the optically thin case, the
maximum line emission occurs at or shortly after the thick/thin transition. Our models suggest that most planetary nebulae with hotter
(>∼45000 K) central stars are optically thin in the Lyman continuum, and that their [O] 5007 Å emission fails to explain the bright
end of the observed planetary nebulae luminosity function. However, sequences with central stars of>∼0.6 M⊙and rather dense initial
envelopes remain virtually optically thick and are able to populate the bright end of the luminosity function. Individual luminosity
functions depend strongly on the central-star mass and on the variation of the nebular optical depth with time.
Conclusions. Hydrodynamical simulations of planetary nebulae are essential for any understanding of the basic physics behind their
observed luminosity function. In particular, our models do not support the claim of Marigo et al. (2004) according to which the
maximum 5007 Å luminosity occurs during the recombination phase well beyond 100000 K when the stellar luminosity declines and
the nebular models become, at least partially, optically thick. Consequently, there is no need to invoke relatively massive central stars
of, say >0.7 M⊙, to account for the bright end of the luminosity function.
Key words. hydrodynamics – radiative transfer – planetary nebulae: general – planetary nebulae: individual – stars: AGB and post-
AGB
1. Introduction
Since the pioneering paper by Jacoby (1989) in which the the-
oretical fundament of the planetary nebulae luminosity function
(PNLF) has been laid out, the use of this tool for establishing
cosmic distances has been proven to be extremely successful.
Its main success stems from the fact that it works not only for
spirals, but also for ellipticals, despite the fact that both sys-
tems consists of completely different stellar populations. A re-
cent summary of the use of the PNLF can be found in Ciardullo
(2003, and references therein).
Jacoby (1989) defined the line magnitudes as
m = −2.5 logF − 13.74,
(1)
with F (in ergcm−2s−1) being the line flux from the object.
Based on 13 galaxies with well-determined Cepheid distances,
an absolute cut-off brightness of M⋆(5007) = −4.45±0.05 mag
Send offprint requests to: D. Sch¨ onberner
⋆Based in parts on observations made with the NASA/ESA Hubble
Space Telescope, obtained at the Space Science Institute, which is oper-
ated by the Association of the Universities for Research in Astronomy,
Inc., under NASA contract NAS 5-26555. The data are retrieved from
the ESO/ST-ECF Science Archive Facility.
has been derived (Ciardullo 2003). This brightness corresponds
to 620 L⊙emitted in the [O] 5007Å line!
Despite its use for about two decades, the physical basis of
the PNLF is still mysteriousand subject to controversalinterpre-
tations. A major uncertainty is related to the question whether
the brightest planetary nebulae (PNe) are optically thick or thin
for Lyman continuumphotons because the efficiency of convert-
ing stellar UV radiation into optical line emission is heavily de-
pendent on the optical depth.
Jacoby (1989) considered only optically thick nebular mod-
els for the constructionof a theoreticalPNLF. Based on the post-
asymptotic giant branch (post-AGB) tracks available, Jacoby
computed the line luminosities from expanding filled spheres
along tracks with hydrogen-burningand helium-burning central
stars, andfound,by comparingthe brightestPNe in Local Group
galaxies with these models, upper mass limits for central stars of
≈0.65 M⊙.
Also in the work of Dopita et al. (1992) for Magellanic
Cloud PNe only optically thick nebular models were consid-
ered. These authors found that the maximum stellar luminos-
ity belonging to the [O] cutoff is ≃ 104L⊙, corresponding
to ≃ 0.7 M⊙if the standard core-mass luminosity relation of
hydrogen-burningcentral stars is used.

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2 D. Sch¨ onberner et al.: The evolution of planetary nebulae IV.
Stanghellini (1995) modelled only the Hβ luminosity func-
tion, assuming also optically thick nebulae, and investigated the
dependenceonstar formationrate,post-AGBtransitiontimeand
initial-finalmassrelation.TheobservedHβcut-offat≃−2.5mag
was achievedby nebularmodels aroundnuclei with 0.65M⊙for
a fairly broad range of assumptions.
Considering only models that are optically thick to ion-
ising radiation oversimplifies the problem certainly. Thus
M´ endez et al. (1993) and M´ endez & Soffner (1997) used a dif-
ferent method. They modelled luminosity functions based on
hydrogen-burning post-AGB evolutionary tracks and empirical
properties of PNe without resorting to nebular models. Their
main conclusion is that the bright end of the luminosity function
is predominantly populated by optically thin PNe (not thin in all
directions around the central star, but thin in at least some direc-
tions to allow for some leaking of Lyman continuum photons)
with maximum central-star masses between 0.63 and 0.66 M⊙.
A completelynovel approach
Marigo et al. (2001, 2004). These authors constructed sim-
ple nebular models, taking into account their outer boundary
conditions set by the AGB and the central-star winds, using
analytical expressions for interacting winds in a similar way
as described by Volk & Kwok (1985). The pressure increase
within the nebular shell by photoionisation is approximately
considered. This new “synthetic” approach allows to compute
the evolution of model PNe together with their observable
quantities in a very fast and efficient way.
Using their new tool, Marigo et al. (2004) computed nebu-
lar sequences along the post-AGB tracks of Vassiliadis & Wood
(1994) and constructed PNLFs for stellar populations with var-
ious metallicities and star formation histories. It turned out that
the bright end of the PNLF is populated by objects with central-
star masses between 0.70 and 0.75 M⊙. Consequently, the value
of the bright cut-off of the PNLF must depend critically on
the properties of the parent stellar population: large differences
between the maximum PNe luminosities of elliptical and spi-
ral galaxies are to be expected (see Marigo et al. 2004, Fig. 25
therein), which, however, are not observed.
This completely different interpretation of the luminosity
function prompted us to employ our hydrodynamical models
presented in Perinotto et al. (2004, Paper I hereafter) to in-
vestigate the physical basis of the PNLF with a more realis-
tic approach. We believe that hydrodynamical simulations are
ideally suited to tackle this task since it has been demon-
strated on several occasions that a reasonably good match to
observed properties of PNe is achieved with such models, pro-
vided appropriate initial and boundary conditions are chosen
(Sch¨ onberner et al. 1997; Sch¨ onberner & Steffen 1999, 2003a,b;
Steffen & Sch¨ onberner 2006). However, we do not aim at con-
structing a theoretical luminosity function because the number
of available combinations of nebular models and central-star
masses is too low. Instead, we will concentrate on a description
of the processes responsible for the strength of the relevant line
emissions.
We begin in Sect. 2 with a short description of our hydro-
dynamical modelling including a brief description of the evolu-
tionary properties of PNe as they follow from these models. We
outline in Sect. 3 the basic differences to the Marigo et al. ap-
proach and investigate in Sect. 4 in detail how the luminosities
of important lines depend on the model properties and how they
evolve with time. Section 5 is devoted to individual luminosity
functions as predicted by our simulations. Section 6 introduces
the corresponding Hβ luminosity functions. The paper is closed
by Sect. 7 with an extensive discussion. A short presentation of
was introducedby
the basic results of this investigation has already been given by
(Sch¨ onberner et al. 2006a).
2. Modelling the planetary nebulae evolution
A detaileddescriptionhowwe simulatethe formationandevolu-
tion of planetary nebulae has already been given in previous pa-
pers and shall not be repeated here (Perinotto et al. 1998, 2004).
Instead, we emphasise here only the basic ingredientsof our cal-
culations which are importantfor the general appreciationof our
models and for understanding the differences to the “synthetic”
approach of Marigo et al. (2001).
In this context it is important to remember that a planetary
nebula is a very complex dynamical system, even if we approx-
imate real objects by spherical configurations. The whole object
consists of a rapidly evolving post-AGB star and an expanding
circumstellarenvelopeoriginallyset upbyamassivestellarwind
when the object was still on or close to the AGB. The radiation
field and the wind fromthe central star initiate a shock wave pat-
tern at the inner edge of the slowly expanding AGB wind enve-
lope.Itsstructureandexpansionpropertiesdependmainlyonthe
radial density distribution of the AGB material, on the electron
temperature inside the ionised matter, and on the pressure sup-
port from the central-star wind. The PN proper is confined be-
tween an inner contact surface which separates the nebula mat-
ter from the shock-heated wind matter, and an outer shock front
which propagates through the ambient AGB material, thereby
increasing the PN mass with time.
Any approach with the aim to understand at least the basic
physics of the formation and evolution of PNe must therefore
rely on radiation-hydrodynamics simulations with the proper
initial and boundary conditions, with all the relevant physical
processes treated fully time-dependently.
2.1. The hydrodynamical models
In short, the basic philosophy of our hydrodynamical PN mod-
els is to couple a spherical circumstellar envelope, assumed
to be the relic of a strong AGB wind, to a post-AGB model
and to follow the evolution of the whole system across the
Hertzsprung-Russell diagram towards the white-dwarf cool-
ing path, employing an 1D radiation-hydrodynamics code (see
Perinotto et al. 1998). We emphasize that our code is designed
to compute ionisation, recombination,heating, and cooling fully
time-dependently. For each volume element, the cooling func-
tion is composed of the contributions of all the ions considered
(nexttoHandHe,alsoC,N,O,Ne,S,Cl,andAr).Foreachindi-
vidual element up to 12 ionisation stages are taken into account.
More details on the radiation part of our code can be found in
Marten & Szczerba (1997).
We used in all of our computations a chemical composi-
tion typical for Galactic disk PNe (Table1). Although the line
emission of a PN and the luminosity and radiation field of its
central star depend on the metal content, the effect is relatively
small and will not influence the basic properties of the PNLF
(cf. Dopita et al. 1992). Any abundance variations, notably that
of oxygen, are thus not considered in the present work.
The hydrodynamical model sequences selected for the
present work from Paper I are listed in Table2. The table pro-
vides the sequence numbers from Paper I (col. 1), the central-
star masses (col. 2), the central-star luminosities at 30000 K
(col. 3), the AGB mass-loss rates (col. 4) and the AGB wind
velocities (col. 5) of the initial models, and the envelope types

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D. Sch¨ onberner et al.: The evolution of planetary nebulae IV.3
Table 1. Elemental abundances, ǫi, used in the computations of
our hydrodynamical models, in (logarithmic) number fractions
relative to hydrogen,logǫi= logni/lognH+ 12.
H HeCNO NeS ClAr
12.00 11.04 8.898.398.65 8.017.045.326.46
Table 2. Hydrodynamicalsequences of model planetary nebulae
used in this work. The sequencenumbers(col. 1) refer to Table1
in Paper I. Central star parameters at Teff= 30000 K are given
in cols. (2) and (3), and cols. (4) and (5) indicate the AGB mass-
loss parameters. The peak mass-loss rate of the hydrodynamical
simulation is about 1 × 10−4M⊙yr−1.
No.
ML
˙ Magb
[M⊙yr−1]
(4)
Vagb
T
[M⊙]
(2)
[L⊙]
(3)
[kms−1]
(5) (1) (6)
22 0.56538833 × 10−5
10A
4
6
0.605
0.605
0.595
6280
6280
5593
1 × 10−4
10A
C
C
Hydro. Simulation
Hydro. Simulation
≃12
≃12 6a
8 0.62579001 × 10−4
15A
10 0.696 116151 × 10−4
15A
(col. 6). The initial envelopemodels have either an ad hoc radial
power-law density distribution, ρ ∝ r−2(T A), or are the re-
sultofdetailedhydrodynamicssimulationsalongtheupperAGB
(T C), as described in Sch¨ onberner et al. (1997) or in Paper
I. The structure of the T C envelope reflects the mass-loss
history of the preceding 50000 years of AGB evolution while
T A correspondsto the simple case of constant mass loss and
outflow velocity.
Anadditionalstellar
Sch¨ onberner et al. (2005a, Paper II hereafter) by replacing
the 0.605 M⊙track of sequence No. 6 by a 0.595 M⊙track
interpolatedfrom the 0.605M⊙and 0.565M⊙tracks of Bl¨ ocker
(1995) and Sch¨ onberner (1983). The PNe models of this new
sequence (No. 6a in Table 2) match the observations even better
than the models of sequence No. 6 with a 0.605 M⊙central star
(Sch¨ onberner et al. 2005b, Paper III hereafter).
For the purpose of this work we recalculated the sequences
listed inTable 2bymeansofan updatedversionofourradiation-
hydrodynamicscode.In particular,the radiationtransportis now
treated in the ‘outward only’ approximation, and heat conduc-
tion as described in Sch¨ onberner et al. (2006b) is also included.
These improvements of the physics did not lead to any signifi-
cant changesofthe dynamicalstructuresof the PN models.Thus
all the conclusion obtained in earlier publications and which are
based on the older simulations remain valid.
All the central-star models used in our PNe simulations are
burning hydrogen,and they are assumed to radiate as black bod-
ies. This choice is justified since Gabler et al. (1991) showed
by means of so-called Unified NLTE model atmospheres that
a black-body energy distribution with the effective temperature
of the photosphere provides a good empirical description of the
stellar UV flux. This holds at least for effective temperatures be-
tween approximately 40000 and 100000 K.
The success of these detailed radiation-hydrodynamics sim-
ulations in describing planetary nebulae is illustrated in Fig.1
trackwas introducedin
wherewe comparetheHα brightnessdistributionsof3 PNe with
well-developed double-shell structures with the model predic-
tions. The models are selected from sequence No.6 of Table 2
which started with an initial envelope computed by means of
two-component radiation-hydrodynamics simulations along the
upper AGB (T C, see Steffen et al. 1998, for more details).
Although the models are spherically symmetric, they rep-
resent the observed general structures as indicated by the Hα
brightness distribution astoundingly well: the brightness ratio
between both shells, i.e. ‘rim’ and ‘shell’, is well matched, and
also the linear brightness slope of faint ‘shells’, typical for many
nebulae, is reproduced. Such a linear radial brightness profile
of the ‘shell’ develops if the radial density gradient of the cir-
cumstellar envelopesteepens with distance from the star, indica-
tive of increasing mass loss towards the end of the AGB evolu-
tion (Steffen et al. 1998; Sch¨ onberner et al. 2005a). More com-
parisons between observed structures of planetary nebulae and
the predictions of our radiation-hydrodynamics simulations can
be found in Steffen & Sch¨ onberner (2006).
2.2. General behaviour of our nebular models
For a better appreciation of the differences between our hydro-
dynamical models and those developed by Marigo et al. (2001)
it appears to be useful to give a brief description of the basic
principles of the formationand evolutionof planetarynebulae as
they follow from recent realistic hydrodynamics simulations. In
PaperIweintroducedseveralphasesofthePN evolutionaccord-
ing to the main physical processes acting on the whole system.
We briefly repeat them here, using the models of sequence No. 6
for illustration.
Neglecting the proto-planetary-nebula phase which is of no
interest here, the first important phase, the ionisation phase, be-
gins when the intense flux of ionising photons from the central
star starts to ionise and to heat the inner parts of the AGB wind
envelope. The large thermal pressure of the ionised gas deter-
mines shape and expansionof the newly created PN. The ionisa-
tion front is of type D and trapped by a strong, nearly isothermal
shock (see Fig. 2, top panel).
The bottom panel of the figure depicts the ionisation struc-
ture of oxygen, which is of particular interest for the present
study. The degree of ionisation is still rather moderate: neutral
in the undisturbed AGB wind, singly ionised preferently in the
outer and doubly ionised in the inner parts of the ionised shell.
Theionisationphasefinishesiftheionisationfrontpassesthe
shock and propagatesquickly through the still undisturbedAGB
matter (R-type ionisation front). The PN is then optically thin
for ionising radiation and said to be density bounded. Figure 3
depicts a moment well after the thick/thin transition. The outer
boundary of the PN is now the shock front enclosing the ‘shell’,
and not the ionisation front which has already left the compu-
tational domain. The main ionisation stage of oxygen is O+2
throughout the shell and the halo, except in the inner part of the
rim where we have already O+3. Helium is also doubly ionised
in this inner region, and the extra heat deposited by the ionisa-
tion of He+drives a weak shock (at r = 1.3 · 1017cm in Fig. 3,
top panel).
ThemodelPN isnowinthecompressionphase inwhichthe
high pressure of the shock-heated central-star wind accelerates
and compresses the inner parts of the shell into the so-called
‘rim’. The rim becomes the dominant structure of a PN, and the
objects displayed in Fig.1 are in this stage. The compression
phase is terminated when the wind power declines because the

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4 D. Sch¨ onberner et al.: The evolution of planetary nebulae IV.
Fig.1. Top: normalized 3D rep-
resentations of the Hα images
of NGC 6826 (left), NGC 3242
(middle), and NGC 1535 (right).
The images are from an unpub-
lishedcatalogue
B¨ assgen (priv.
Bottom: corresponding Hα surface
brightness profiles of optically
thin models selected from the
hydrodynamic sequence
listed in Table 2. Post-AGB ages
increase from left to right. The
normalizations of the models are
adjusted as to match those of the
observed objects.
compiled
communication).
by
No. 6
Fig.2. Top: radial profiles of heavy-particledensity (thick),elec-
tron density (dotted), and velocity (thin) of an optically thick
model selected from sequence No. 6. The stellar parameters of
the 0.605M⊙central star are Teff= 40245K and L = 6240 L⊙,
with a post-AGB age t = 2478yr. Bottom: ionisation frac-
tions of neutral (dotted), singly ionised (dashed), and doubly
ionisedoxygen(dash-dotted).Theionisationfractionswithinthe
shocked central-star wind domain (r ≤ 0.5 · 1017cm), though
computed, are not shown for clarity.
central star is approaching its maximum effective temperature
and fades towards a white-dwarf configuration.
Figure 4 shows a model very close to maximum stellar tem-
perature at the end of the compression phase. At the large effec-
tive temperature well above 100000 K a significant fraction of
oxygen is now triply ionised. There exists, however, still some
singly ionised oxygen close to the outer edge of the shell.
Fig.3. Same as in Fig. 2, but for an optically thin model with
well developed rim and shell structures. The stellar parameters
are Teff = 79708K and L = 5797 L⊙, with a post-AGB age
t = 4241yr. The dash-triply dotted line indicates triply ionised
oxygen.
During the whole compression phase the leading edge
of the shell continues to propagate supersonically into the
AGB wind with a (relative) speed ruled only by the ther-
mal properties of the gas and the radial density profile of the
AGB wind (Franco et al. 1990; Chevalier 1997; Shu et al. 2002;
Sch¨ onberner et al. 2005a). The expansion of the shell is not at
all influenced by the wind interaction responsible for the forma-
tion of the rim. Usually,the shell expandsfaster than the rim (see
Paper II, Fig.12 therein).
The final fading of the central star causes recombination
in the outer parts of the PN provided the nebular densities are

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D. Sch¨ onberner et al.: The evolution of planetary nebulae IV.5
Fig.4. Same as in Fig. 2, but for an optically thin model at
the end of the compression phase close to maximum stellar
temperature. The stellar parameters are Teff = 156662K and
L = 2075 L⊙, with a post-AGB age t = 7124yr.
sufficiently large, and/or the luminosity declines very rapidly.
Recombination turns eventually into re-ionisation after the fad-
ing of the central star has slowed down and the nebular density
becomes sufficiently low due to the continued expansion.
The model shown in Fig.5 illustrates the situation af-
ter the end of the recombination and at the beginning of
the re-ionisation stage. The shell is mainly neutral, although
not completely, and forms for a while a recombination halo
(Corradi et al. 2000). The rim which remained fully ionised to a
large extent continues to expandand will eventuallyswallow the
shell material because the shell’s shock is slowed down during
recombination. The ionisation is highly stratified (bottom) and
reflects the physical situation given by a central star of low lu-
minosity but still very high effective temperature: mainly O0in
the shell, but the degree of ionisation increases inwards over O+
until O+3at the inner rim.
These main evolutionary stages as described here may not
always occur, or they may even occur at the same time. For in-
stance,aPN arounda low-mass,slowly evolvingcentralstarwill
not recombine at all. On the other hand, a PN around a massive,
veryquicklyevolvingcentralstarmaynotbecomeopticallythin.
In this case the ionisation front remains always of type D, i.e.
trappedby the outershock,and we haveionisation andcompres-
sion at the same time. However,the doubleshell configurationis
very robust and develops also in these cases.
This discussion shows thatthe formationof typicalPN struc-
tures is quite complex even in the spherical approximation. Two
processes are relevant, viz. heating by ionisation and compres-
sion by wind interaction. Their relative importance will depend
on the metallicity, i.e. on the content of coolants. For instance,
at lower metallicities than we have used here we expect (i) less
Fig.5. Same as in Fig. 2, but for a model after the end of recom-
binationand at the beginningof re-ionisation.The stellar param-
eters areTeff= 122269K and L = 249L⊙, with a post-AGBage
t = 9557yr.
dense rims simply because the central-star wind power is most
likely lower, and (ii) higher expansion rates of the shell because
the temperature, and sound speed, is larger.
A completely different situation is encountered in systems
with a Wolf-Rayet central star. The virtuallyhydrogen-freewind
from such a star is up to two orders-of-magnitude more in-
tense compared with objects of normal surface abundances (cf.
Leuenhagen et al. 1996). Consequently,wind interaction will al-
ways be dominant for shaping a PN around a Wolf-Rayet cen-
tral star, independently of the general metallicity. Since origin
and evolution of hydrogen-poorcentral stars is not known, neb-
ular models around them can not be computed to date. The se-
quences listed in Table 2 und used here have exclusivelycentral-
star models with normal surface composition burning hydrogen
to provide their luminosity.
3. Comparison with the Marigo et al. models
As already mentioned in the Introduction, the approach of
Marigo et al. (2001) to compute the evolution PNe analyti-
cally goes back to methods developed by Volk & Kwok (1985).
Marigo et al. introduced several improvements, the most impor-
tant one being the approximate consideration of radiative pro-
cesses, viz. heating by ionisation and cooling by line emissions.
However, an analytical treatment of the dynamics of photoioni-
sation oversimplifies the problem and has severe consequences.
For instance, the basic nebular structures of the Marigo et
al. models are shells of constant density which are partially or
fully ionised. They are in full contrast to real objects which have
complicated density and velocity profiles and which can only be
approximated by hydrodynamical models (cf. Figs. 2 till 5). In