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Europhysics
News (2003) Vol. 34 No. 1
Hydrodynamics
of planetary nebulae
Vincent Icke
Sterrewacht Leiden, The Netherlands
Stellar evolution
Stars do not go gently into that
dark night. Those with masses above eight times the mass of the Sun
explode as supernovae, outshining for a few weeks their parent galaxy.
But even low- mass stars make a show of their departure. Because they
are so much more numerous than the heavies, the remains of defunct solar-type
stars litter the galactic landscape. Known as planetary nebulae,
they appear so frequently that many of them are found quite nearby;
with apparent sizes of many minutes of arc, these are excellent objects
for viewing with an amateur telescope. The moniker "planetary"
has nothing to do with planets; it means "planet-like", because
an emission line of doubly ionised oxygen gives these nebulae a greenish
glow, reminiscent of a planetary disk.
Stars shine due to the nuclear fusion of hydrogen, producing
helium, carbon, and heavier elements. The central temperature in low-mass
stars is too low to produce much beyond the CNO-group. When the hydrogen
in the core is exhausted–which takes about 10 billion years for the
Sun–the core contracts. The temperature is, as in Wien's Law, inversely
proportional to a length, in this case the core radius. The luminosity
is proportional to the fourth power of the temperature and the second
power of the radius, so that the core increases a hundredfold in brightness
for every factor 10 it contracts. Thus, when the core contracts, the
stellar atmosphere is heated from below and puffs up, ultimately reaching
a size comparable with the orbit of Venus. The luminosity of the cooling
core is radiated away through this enormous surface, so that the effective
temperature of the star is low, of the order of 2000 K or even less.
In its final ten million years, the star has become a red giant.
During the initial stages of this process, some fusion
still occurs in a shell around the core, but this state is overstable.
Ultimately, thermal pulses of increasing amplitude drive off
the outer shell, leaving behind a naked core with a surface temperature
up to several hundred thousand kelvin. Radiation pressure from this
brilliant pinpoint, soon to cool down and become a white dwarf,
drives a supersonic wind into the departing outer shell. It is the collision
between these two winds––a dense shell at tens of kilometres
per second, hit by a tenuous gas at a few thousand km/s––that
produces the beautiful planetary nebulae.
Colliding winds
Originally, it was thought that most PNs are spherical (Fig.1, left).
After all, that is what their parent stars are likely to be, and the
physics of the formation of such shells was well understood. The outer
'slow' wind expands with a velocity of about 20 km/s and a density of
1010 atoms per m3. The inner 'fast' wind races outward with speeds up
to 2000 km/s, and a much lower density, typically 107. The interacting
winds produce a typical shock/contact- discontinuity/shock configuration.
Seen from the central star, and following a radial flow line, the gas
is very supersonic, and moves outward with a Mach number of 40-100.
Then the gas passes through a strong shock, which converts most of the
kinetic into thermal energy. This creates a thick layer of very high
temperature and pressure, which pushes against the 'slow' gas at a contact
discontinuity where the gas density jumps upward but the pressure and
the velocity are continuous. Further out, the slow gas runs into whatever
external medium there is, and a second shock is formed.
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Fig 1 Left:
the spherical planetary nebula Abell 39; image courtesy of G.
Jacoby and Kitt Peak National Observatory. Right: the bipolar
NGC7009, observed by B. Balick with WFPC2 on the Hubble Space
Telescope.
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Using standard hydrodynamic conservation laws, these configurations
were readily analysed (e.g. Kwok 1982), and compared well with the observations
of spherical PNs. The fast wind is almost unobservable, and the region
beyond the inner shock, having a temperature in the millions of kelvin,
is properly observed only in X-rays, which has only recently become
possible with the advent of satellites that can observe X-rays with
large collecting area. The shocked slow wind, which forms the archetypical
'planetary nebula', stands out clearly in the light of various emission
lines, most notably Ha, [N+] and
[O2+]. The nebular spectrum is dominated by the reprocessed
ultraviolet radiation of the central white dwarf.
Kompaneyets revisited
But the spherical textbook cases turned out to be a minority (Fig. 1,
right). Many planetaries even have an extreme dumbbell shape (Fig.2;
for detailed reviews, see Schwarz 1993 or Sahai 2002). In 1987 Bruce
Balick (University of Washington, Seattle) published a classification
scheme in which, for the first time, the whole range of shapes was taken
seriously, all the way from purely spherical (Fig.1) to completely double-lobed
(Fig.2). In addition, Balick (1987) supposed that the shaping of bipolar
nebulae was due to a hypothetical aspherical shape of the outer slow
shell, even though a mechanism for producing this was––and
is––unknown.
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Fig 2 Bipolar
planetary nebula Hubble 5,observed by B. Balick, V. Icke and G.
Mellema with WFPC2 on the Hubble Space Telescope. Colour channels
indicate three emission lines: red, [O]; green, [N+];
blue, [O2+].
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Fig 3 Analytical
models for bipolarplanetary nebulae. From left to right, the pole/equator
density contrast increases; from top to bottom, time increases.
Images have been scaled to fit the boxes.
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When Balick visited Sterrewacht Leiden on a sabbatical leave,
he asked me to think about the hydrodynamics of interacting aspherical
winds. The main problem, as I saw it, was finding a suitable approximation
for the propagation of a very strong shock through an inhomogeneous
atmosphere. Of course I knew the elegant analytical solution that Kompaneyets
(1960) had found for the evolution of a strong shock in an exponential
atmosphere, and I formulated the PN problem in the same approximation.
The shape r(q, t) of the outer shock
can be found analytically from the equation
Here P is the pressure behind and r
the gas density ahead of the shock. In the PN case, the density r
in the outer atmosphere is far from exponential. Instead, it is cylindrically
symmetric, with a distribution dictated by the flow of the slow wind.
The velocity of this gas is almost constant, so that the density can
be approximated by K(q)/r2, where
K is an arbitrary function of the polar angle q
and r is the radial distance to the star. To my surprise and delight,
I discovered that this equation can be solved exactly by separation
of variables. The solution is
with separation constant E and arbitrary function g(E). This form
allows the easy generation of predicted shock shapes for a great variety
of confining density distributions (Fig.3).
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Fig 4 Snapshot
of adiabatic PN model.Symmetry axis horizontal, equatorial plane
vertical. Red is the density, green temperature, blue the absolute
value of the velocity.
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The sequence of bipolar shapes generated by
this analytical solution matched Balick's classification scheme almost
perfectly (Balick 1987; Icke 1991, and references therein).
Numerical hydrodynamics
But the analytic solution said nothing about what is perhaps the most
intriguing part of a PN, namely its interior. To solve that part of
the flow, numerical hydrodynamics was necessary. The difficulty here
is, that the gigantic jumps in pressure and density in PNs present severe
difficulties to ordinary, diffusive numerical methods (Soker & Livio
1988). Using my 'LCD' variant of Boris & Book's robust flux-corrected
transport code, I managed to keep the diffusive effects of numerical
hydrodynamics within acceptable bounds (Fig.4).
Subsequent work by Mellema and co-workers, using Roe's
approximate Riemann solver, greatly refined these results. Finally,
Mellema completed the picture by including radiative transfer and ionisation
effects in the equations of motion (Mellema 1995 and references therein).
In the early investigations of bipolar nebulae, the hydrodynamical
models concentrated on disk-wind interactions that are 'energy driven'.
A central star blows a spherical wind into a toroidal nebula; the wind
has such a low density and high speed that the surrounding (inner) shock
generates a very hot, pressure dominated layer that then drives the
outer shock. Because flow in the high pressure region is very subsonic,
inhomogeneities in this layer are mostly smoothed out, so that the outer
shock and the nebular shape are well rounded.
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Fig 5 the
Identification of features discovered in numerical models for
bipolar planetary nebulae. Most of these have now been identified
in observed images.
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These computations showed a wealth of structures in the
interior of the PN (Fig.5), most of which were subsequently identified
in observed images, especially those obtained at the European Southern
Observatory and with the Hubble Space Telescope. It was even discovered
that disk-wind interactions can produce sharply collimated supersonic
jets (Icke et al. 1992).
Cooling flows
If the flow is strongly compressible, a thick high-pressure smoothing
shell does not develop. This regime is often called 'momentum driven'.
In such flows the gas still has considerable momentum after passage
through the inner shock, and thereby shapes the outer nebula by direct
impact rather than by a mediating pressure. Strong cooling can provide
the requisite compressibility, and instabilities and other interesting
effects will certainly occur. In order to allow dimensionless scaling
of the simulations, the high compressibility can be mimicked by picking
a small value of the Poisson constant (usually g
= 1.1). A typical result is shown in Figure 6. When these computations
are used to generate images as they would be observed, the similarity
with actual nebulae is particularly striking (Fig.7).
It is expected that stars of higher mass (in the range
2-8 solar masses) produce nebulae that do not cool quickly, and are
therefore 'energy driven', whereas the stars that have smaller masses
produce the highly compressible, 'momentum driven' PNs. This prediction
remains to be verified.
Apotheosis: the
Red Rectangle
Astronomy is great, if only because there are always objects that look
so strange that they seem to defy explanation. One of these is the 'Red
Rectangle' (Icke 1981 and references therein), a nebula looking just
like what its name says on the Palomar plates on which it was discovered.
New observations (Van Winckel, priv.comm.) clearly show that the Red
Rectangle flow is biconical, but it also seems to pulsate : the
stream is intermittent, which at first sight would be very difficult
to produce in the interaction between a steady spherical outflow and
a stationary surrounding disk. However, the central star is a binary,
which could modulate the flow.
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Fig 6 Snapshot
of a highly compressible PN model (pseudo-cooling). All parameters
are the same as in Figure 4, except that the value of the adiabatic
index is 1.1 instead of 5/3. Symmetry axis horizontal, equatorial
plane vertical. Red is the density, green the temperature, blue
the absolute value of the velocity. Notice the collapsed shock-contact
configuration, and the instabilities of the outer shock front.
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Fig 7 Bipolar
planetary nebula MyCn18, as observed by R. Sahai & J. Trauger
with WFPC2 on the Hubble Space Telescope, compared with hydrodynamic
pseudo-cooling model.
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In the case of momentum-driven flow, the instabilities that
occur in the inner flow regime make the outflow far from stationary.
In some cases, the gas is focused into a double cone, when it passes
through the aspherical inner shock (cf. Fig.5). The density of the Red
Rectangle is so high, and the central star(s) so cool, that radiative
losses will be considerable, and highly compressible 'momentum driven'
outflow is almost guaranteed. Focusing of the primary gas through the
inner shock may create a biconical outflow pattern that can produce
the characteristic X-shape of the outer nebula.
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Fig 8 Left:
Observed image of the Red Rectangle, courtesy of Van Winckel.
Right: Pseudo- observation of a hydrodynamic simulation with g
= 1.1. Symmetry axis vertical, equatorial plane horizontal. The
image was produced by computing the scattering of white light
in the density field produced by the hydrocode. Note the 'Mach
stems' at the place where the recurving bow shocks intersect on
the symmetry axis.
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Observations such as those by Van Winckel
can thus be modelled in great detail. Computational experiments show
that a specific set of features is common to a wide range of compressible
biconical flows. Key features are (Fig.8): (1) cup-shaped density ridges
close to the central source, which become ragged rings at higher altitude;
(2) the recurving bow shocks ahead of the gaseous rings that are shot
upwards from the centre; (3) intersecting shocks on the symmetry axis,
forming Mach stems ('X-shocks'); (4) the gradual widening of the effective
cone opening angle at higher altitudes, due to the outward pressure
behind these shocks. One feature that I initially thought to be problematic
for my interpretation, namely the occurrence of X-shocks on the axis,
turned out to be present on closer inspection of Van Winckel's Ha image.
Conclusions
Bipolar nebulae are due to an interaction between two winds: a slow,
dense outflow deposited during the time immediately after the red giant
phase, and a very fast, tenuous wind driven by radiation from the dying
white dwarf. Analytical and numerical calculations allow the identification
of almost all flow features. Notable exceptions are the so-called 'FLIERs'
(fast, low-ionisation emission regions). These many forms of mass loss
fertilise the interstellar plasma with the materials of life, and show
the admirable ways in which low-mass stars rage against the dying of
their light.
References
Full colour-coded versions of some of the flow patterns shown here will
be kept as long as disk space permits on www.strw.LeidenUniv.nl/~176icke/,
from which they can be downloaded. QuickTime movies of some of my simulations
can be obtained on request from icke@strw.LeidenUniv.nl.
[1] Balick, B. 1987 Astron.J. 94, 671
[2] Icke, V. 1981 Astrophys.J. 247, 152
[3] Icke, V. 1991 Astron.Astrophys 251, 369
[4] Icke, V., Mellema, G., Balick, B., Eulderink, F., & Frank,
A.: 1992 Nature 355, 524
[5] Kompaneyets, A.S. (1960). Doklady Akad. Nauk SSSR
130, 1001
[6] Kwok, S. 1982 Astrophys.J. 258, 280
[7] Mellema, G., 1995 Mon.Not.Roy.astron.Soc. 277,
173
[8] Sahai, R., 2002 Rev. Mex. Astron. Astrophys. 13,
133
[9] Schwarz, H.E., p.223 in: Mass loss on the AGB and beyond,
H.E. Schwarz (Ed.), ESO Conf.Wrksh.Proc. 46 (1993)
[10] Soker, N., & Livio, M. 1989 Astrophys.J. 339,
268
Copyright EPS
and EDP Sciences,
2003
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