Download The structure of steady detonation waves in Type Ia supernovae

Mon. Not. R. Astron. Soc. 310, 1039±1052 (1999)
The structure of steady detonation waves in Type Ia supernovae:
pathological detonations in C±O cores
Gary J. Sharpew
Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT
Accepted 1999 July 23. Received 1999 July 5; in original form 1999 April 15
A B S T R AC T
The structure of steady, one-dimensional detonation waves in C±O is investigated for initial
densities in the range 2 107 to 1 109 g cm23 . At these and greater densities, the selfsupporting detonation wave is of the pathological type. For such waves the detonation speed
is an eigenvalue of the steady equations, and the reaction zone contains an internal frozen
sonic point where the thermicity vanishes. The self-supporting flow downstream of this
singular point is supersonic, and is very different from that in supported (overdriven)
detonations. A method for determining the structure of pathological detonation waves is
described. These waves are examined, and the self-sustaining wave is compared with and
contrasted to the supported detonations considered previously by Khokhlov. We show that
the thickness of the self-sustaining detonation is a few times the thickness of supported
detonations, and that the self-sustaining detonation produces more of the iron-peak and less
of the intermediate mass elements than do supported detonations. Implications for the
cellular detonation instability are also discussed.
Key words: instabilities ± nuclear reactions, nucleosynthesis, abundances ± shock waves ±
supernovae: general ± white dwarfs.
1
INTRODUCTION
It is generally accepted that Type Ia supernovae are the result of
the thermonuclear explosion of C±O white dwarf stars. Whether
the explosion occurs as a deflagration (subsonic burning), a
detonation (supersonic burning), or a combination of both
(delayed detonation) is still a matter of some debate. In the
simplest models the white dwarf accretes matter from a
companion, which triggers an explosion at the centre when the
star is close to the Chandrasekhar mass. However, pure detonation
models fail to produce the observed abundances of intermediate
mass elements because of the high densities involved (Arnett,
Truran & Woosley 1971), while for pure deflagration models the
flame speed has to be very nicely tuned in order to reproduce the
observed characteristics of the supernova (Nomoto, Sugimoto &
Neo 1976; Nomoto, Thielemann & Yokoi 1984; Thielemann,
Nomoto & Yokoi 1986). All is not lost for detonations, however,
since at lower densities (& 107 g cm23 according to Khokhlov
1989) the thickness of the detonation wave becomes comparable
to the size of the star, which can lead to the production of
intermediate mass elements as a result of incomplete burning. This
is exploited in delayed detonation models, which seem to give the
best overall agreement with observations (Khokhlov 1991; Arnett
w
E-mail: [email protected]
q 1999 RAS
& Livne 1994; Bychkov & Liberman 1995; Niemeyer & Woosley
1997; Khokhlov, Oran & Wheeler 1997). In these models the
explosion is initially a deflagration which burns part of the star
and causes it to expand to lower densities, whereupon the rest of
the burning proceeds as a detonation. An alternative scenario, in
which detonations produce intermediate mass elements because of
low densities, concerns helium- or double-detonation models
(Livne 1990; Livne & Glasner 1990, 1991; Woosley & Weaver
1994; Livne & Arnett 1995; Wiggins & Falle 1997a; Wiggins,
Sharpe & Falle 1998). In these models a detonation in a helium
envelope induces a detonation in a low-density, sub-Chandrasekhar C±O core, either at the core±envelope boundary or at the
centre of the core. It therefore seems very likely that detonations
play a role in Type Ia supernovae.
Khokhlov (1989) investigated the steady one-dimensional
structures of nuclear detonations in C±O mixtures and in pure
He. In C±O the detonation wave consists of three burning layers: a
carbon-burning stage, followed by an oxygen-burning or nuclear
statistical quasi-equilibrium (NSQE) relaxation layer, and finally a
silicon burning, or nuclear statistical equilibrium (NSE) relaxation, stage. Khokhlov (1989) found that the length-scale for silicon
burning is very much greater than that for oxygen burning, which
is in turn very much greater than that for carbon burning. He also
found that, for initial densities greater than about 107 g cm23, the
self-sustaining detonation is of the pathological type as a result of
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G. J. Sharpe
endothermic photo-disintegration reactions. Such detonations
travel at the minimum possible speed, which is greater than the
Chapman±Jouguet (CJ) speed, and have an internal frozen sonic
point (the so-called pathological point) where the thermicity
vanishes. Downstream of the sonic point the flow is supersonic.
Unlike CJ detonations, for which the detonation speed does not
depend on the form of the reaction rates, for pathological
detonations the detonation speed is an eigenvalue of the steady
equations. This results in the structure of the self-sustaining
detonation (the object of interest in the case of Type Ia
supernovae) downstream of the sonic point being quite different
from that of supported, or overdriven, detonations, which are
subsonic throughout. However, because of the singular nature of
the sonic pathological point, Khokhlov (1989) could not find the
self-sustaining structure by direct integration and considered only
supported detonations. No conclusions about the downstream selfsustaining structure can be drawn from such calculations,
however. For example, the thickness of the self-sustaining
detonation may be much greater because of lower temperatures.
Although the governing equations for detonations admit such
steady, one-dimensional waves, terrestrial experiments show that
detonations usually have a time-dependent, multidimensional
structure. In fact the steady detonation waves are prone to both
pulsating, or one-dimensional, and cellular, or multidimensional,
instabilities. Pulsating detonations are rare in nature, but they are
seen for instance when blunt bodies are fired into detonatable
gases at supersonic velocities (e.g. Lehr 1972). Usually detonations propagate in the cellular mode (Fickett & Davis 1979). In
these cases the leading shock is wrinkled, consisting of alternate
weak incident shocks and stronger Mach stems, joined at triple
points by transverse waves which travel back and forth
perpendicular to the front and extend back into the reaction
zone. The tracks of these triple points map out the boundaries of
the `cells', which can be remarkably regular. Reactions are
essentially complete within one cell length, which is one to two
orders of magnitude greater than the corresponding steady, onedimensional reaction zone length (Lee 1984). Linear stability
analyses of steady detonations show that they are unstable to onedimensional disturbances provided that the reaction rates are
sufficiently temperature sensitive, whereas, unless the heat of
reaction is unreasonably small, they are always unstable to
multidimensional disturbances, even when the reaction rates are
independent of temperature (Erpenbeck 1965; Sharpe 1997; Short
& Stewart 1998, 1999).
Kriminski, Bychkov & Liberman (1998) investigated the onedimensional linear stability (i.e. the pulsational instability) of
detonations in carbon white dwarfs. However, they considered
only the C+C reaction. They found that the linear stability
spectrum contains a purely real (non-oscillatory) mode for initial
densities greater than about 2 107 g cm23 , and hence suggest
that detonations cannot propagate for such densities. They also
suggest that this may be a reason why burning starts as a
deflagration at high densities, with a deflagration-to-detonation
transition occurring at densities of about 2 107 g cm23 . They
base this conclusion on the one-dimensional numerical simulations of He & Lee (1995), in which the detonation dies to a shock
followed by a deflagration when the linear spectrum has nonoscillatory modes. However, the results of He & Lee (1995) have
been shown to be incorrect as a result of poor numerical resolution
(Williams, Bauwens & Oran 1996, Sharpe & Falle 1999). In fact
the detonation always re-ignites after an induction time and
propagates as a series of failures followed by re-ignitions. Also,
Short & Quirk (1997) have shown that the real eigenvalue of linear
stability analyses does not correspond to the detonation failure/reignition regime in the non-linear simulations. Most importantly, in
multidimensions the detonation propagates in the cellular mode,
and failure does not occur (Williams et al. 1996).
Numerical simulations were used to investigate the onedimensional instability by Khokhlov (1993) and Koldoba,
Tarasova & Chechetkin (1994). Both considered only the
instability arising from the carbon burning layer for supported
detonations. Khokhlov (1993) found the detonation was unstable
to one-dimensional perturbations for initial densities
.2 107 g cm23 . Boisseau et al. (1996) carried out two-dimensional numerical simulations of detonations in C±O. They
resolved only the carbon-burning stage, which was overdriven
because of the assumed energy release of the subsequent burning
stages. Not surprisingly, they found that nuclear detonations in
C±O are unstable to the cellular instability. Their simulations
showed that pockets of unburnt carbon were produced, and the
effective burning length was increased by about three times the
steady reaction length. However, they considered detonations
propagating down narrow tubes. When the widths of such tubes
are much smaller than the natural cell size, the cell width is
constrained to be the tube width. Hence one cannot make any
conclusions about the average natural cell size of unconfined
detonations from such calculations (the natural cell widths of
terrestrial experiments are of the order of one to two magnitudes
greater than the corresponding induction zone length of the steady
one-dimensional detonation: see Lee 1984). In order to determine
the natural cell sizes, simulations with tube widths much larger
than the natural cell size are required.
It is very likely that the oxygen- and silicon-burning stages also
have their associated instabilities. However, it is not clear that one
can separate each burning stage into supported detonations and
investigate the instabilities separately. The instability of one
burning stage may affect the others. For instance, experiments
show that the transverse waves can themselves be detonations and
have their own transverse waves moving back and forth across
them (see Fickett & Davis 1979 and references therein). This
suggests that when there are reactions with disparate length-scales
there may be a `cells within cells' structure, such as that seen in
detonations in nitromethane mixtures (Presles et al. 1996).
Furthermore, Sharpe (1999) carried out a linear stability analysis
of pathological detonations, which showed that, unlike the case in
which the unsupported detonation is of the CJ type, predictions of
the cell size are very sensitive to the detonation speed near the
self-sustaining detonation speed. Hence one should be very careful
about drawing any conclusions regarding cell sizes, etc., of selfsustaining, pathological detonations by studying even only slightly
overdriven detonations. Unfortunately, however, because of the
large differences in reaction lengths of the burning stages in C±O,
it is currently impossible to resolve more than one stage at a time
in numerical simulations.
Although such structures are almost always hydrodynamically
unstable, the correct underlying steady, one-dimensional structure
is an essential starting point for any stability analysis or timedependent numerical simulation. Since the structure of the steady,
self-sustaining detonation is quite different from that of steady
supported detonations, one would also expect the time-dependent,
multidimensional structure of self-sustaining detonations to be
quite different from their supported counterparts, especially in
light of the linear stability analysis of Sharpe (1999). Also, the
structure of the one-dimensional pathological detonation can be
q 1999 RAS, MNRAS 310, 1039±1052
Detonations in Type Ia supernovae
exploited to estimate the nucleosynthesis caused by large-scale
multidimensional effects. The burning up to the sonic pathological
point occurs on a very short length-scale (compared to the size of
the white dwarf and to the total detonation length) and Khokhlov
(1989) found that very little heat is absorbed in the subsequent
burning, which occurs on a relatively large length-scale. Hence the
burning up to the sonic point can be modelled as an instantaneous
`detonation' and, to a first approximation, the subsequent burning
can be decoupled from the flow. For instance, Wiggins et al.
(1998) used this to estimate the effects of the cusp in the
detonation front found by Wiggins & Falle (1997a), and the
associated rarefactions at the core±envelope boundary, which
occur when the detonation is ignited at a point in the helium
envelope.
In this paper we investigate the steady, one-dimensional
detonation waves in C±O when the self-sustaining detonation is
of the pathological type. In Section 2 we give the governing
equations for such waves. In Section 3 we determine the
pathological detonation speeds, and then in Section 4 we describe
a method for determining the solutions of the equations near
pathological points. This method is then used to calculate the
complete steady, one-dimensional detonation waves, which are
examined in Section 5. Conclusions and ideas for future work are
given in Section 6.
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find our results for supported detonations are in very good
agreement with those of Khokhlov (1989), who used an extended
network. Note that not all the Yi are independent, since
13
X
X i ˆ 1;
…5†
iˆ1
where
X i ˆ Ai Y i
is the mass fraction and Ai is the atomic mass of the ith species.
Equation (5) can be used to eliminate one of the Yi in terms of the
other nuclei abundances, and hence equation (4) reduces to 12 (for
the a -network) independent equations. It appears that neither
Khokhlov (1989) nor Wiggins et al. (1998) took this into account,
so that their partial derivatives with respect to one of the Yi refers
to changing that Yi while keeping all of the other nuclei
abundances constant, which from (5) we cannot do. In this
paper we choose to eliminate the 56Ni nuclei abundance, so that
from now on Y refers to (Y1,¼,Y12).
Equation (1)±(4) are closed by specifying an equation of state.
We use an equation of state which includes contributions from
radiation, non-degenerate ions and arbitrarily degenerate and
relativistic electrons and electron-positron pairs. We then have
p ; p…r; T; Y†
2
G O V E R N I N G E Q UAT I O N S
and
A detonation is a supersonic regime of burning in which a strong
shock heats the fuel to high temperatures, causing reactions to
proceed at a rapid rate behind the shock. The heat released thereby
helps to support the detonation. The governing equations for
gaseous detonations are the reactive Euler equations. For steady,
one-dimensional flow these are
dr
du
‡r
ˆ 0;
dx
dx
…1†
du dp
ru ‡
ˆ 0;
dx dx
…2†
dE
p dr
2
ˆ0
dx r2 dx
…3†
u
and
dY R
ˆ ;
dx
u
…4†
where x is the distance behind the shock, u the fluid velocity in the
shock rest frame, r the density, p the pressure, E the internal
energy per unit mass,
Yi ˆ
ni
rN A
is the nuclei abundance of the ith species, with ni the number
density of the ith species, NA is Avogadro's number, and Ri ;
Ri …r; T; Y † the corresponding reaction rates. We define here the
mean molecular weight of ions, m , by
X
m ˆ 1= Y i :
i
We use an a -network consisting of the 13 species 4He, 12C,
O, 20Ne, 24Mg, 28Si, 32S, 36Ar, 40Ca, 44Ti, 48Cr, 52Fe and 56Ni,
linked by 27 reactions. The reaction rates are taken from Fryxell,
MuÈller & Arnett (1989). Although this is a simple network, we
16
q 1999 RAS, MNRAS 310, 1039±1052
E ; e…r; T; Y† 2 q…Y†;
where e is the internal energy of the ions, photons, electrons and
positrons and
q ˆ NA
13
X
Qi Y i
iˆ1
the total binding energy of the mixture, Qi the binding energy per
nucleus …. 0† of the ith species and Y13 (the 56Ni nuclei
abundance) is given by (5).
Equations (1)±(3) can be integrated to give the usual
conservation relations for steady, one-dimensional flow
ru ˆ r0 D;
…6†
p…r; T; Y† ‡ ru2 ˆ p…r0 ; T 0 ; Y 0 † ‡ r0 D2
…7†
and
E…r; T; Y† ‡
p…r; T; Y† u2
ˆ
‡
r
2
E…r0 ; T 0 ; Y 0 † ‡
p…r0 ; T 0 ; Y 0 † D2
;
‡
r0
2
…8†
where D is the detonation speed and the 0 subscript denotes
quantities in the upstream (initial) state. For a given initial state
and detonation speed, eliminating u using (6), equations (7) and
(8) give two relations between the 14 quantities r , T and Y. For
instance, to find the post-shock density and temperature, we solve
equations (7) and (8) iteratively using a Newton±Raphson
method, given that the composition across the shock is unchanged
(Khokhlov 1989).
In this paper we will consider the initial material to be 0:512 C‡
0:516 O by mass and the initial temperature to be 1 108 K. Note
that in the upstream state the electrons are highly degenerate so
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G. J. Sharpe
that the equation of state is not sensitive to the temperature. From
equations (7) and (8), this means that the post-shock temperature
and hence the detonation wave structure are themselves insensitive
to the initial temperature, and so the only parameter left to vary is
the initial density.
Alternatively, using the differential relations
12 X
­p
­p
­p
dp ˆ
dr ‡
dT ‡
dY i
…9†
­r T; Y
­T r; Y
­Y i r; T; Y i
iˆ1
and
de ˆ
­e
­r
T; Y
dr ‡
­e
­T
r;Y
dT ‡
12 X
­e
iˆ1
­Y i
r; T; Y i
dY i ;
…10†
where the subscripts refer to the quantities being kept constant and
Y i ˆ …Y 1 ; ¼; Y i21 ; Y i‡1 ; ¼; Y 12 †;
equations (1)±(3) give
dr
ra2 s ´ R
ˆ2 f
dx
u h
and
dT
ˆ
dx
­p
­T
21 ("
r;Y
…11†
­p
u 2
­r
2
#
T;Y
)
12 dr X
­p
dY i
2
:
dx
­Y i r;T;Y i dx
iˆ1
…12†
Here
h ˆ a2f 2 u2
is the sonic parameter, where
"
# 21
­p
p
­e
­p
­e
2
af ˆ
‡
2
­r T; Y r2
­r T; Y ­T r; Y ­T r; Y
is the frozen sound speed (i.e. the sound speed evaluated at
constant composition), and s ´ R is the thermicity, which is
roughly proportional to the rate at which nuclear binding energy is
transformed into heat and kinetic energy (Fickett & Davis 1979).
The thermicity coefficients are given by
( 1
­p
si ˆ
­Y i r; T; Y i
ra2f
" # 21 )
­e
­q
­p
­e
:
2
2
­Y i r;T;Y i
­Y i Y i ­T r;Y ­T r;Y
Given the post-shock state as initial conditions, equations (4),
(11) and (12) can then be integrated into the reaction zone …x . 0†
to determine the steady, one-dimensional solutions, with u given
by (6).
Note that it is also possible to define an equilibrium sound
speed ae, which is the sound speed evaluated with the composition
shifting in order to maintain equilibrium (i.e. R ˆ 0), with ae # af
(Fickett & Davis 1979). The Chapman±Jouguet condition is then
that the flow is equilibrium sonic at the end of the reaction zone,
i.e. ae ˆ u when R ˆ 0. However, for pathological detonations
this state is unattainable and the reaction zone has an interior
frozen sonic point where the thermicity vanishes.
For a detailed discussion of the properties of pathological
detonations based on the p 2 1=r plane see, for example, Fickett
& Davis (1979), Khokhlov (1989) or Sharpe (1999). For the sake
of clarity we briefly summarize. Immediately behind the shock the
flow is frozen subsonic. If the detonation speed is too low, then a
frozen sonic point is reached inside the wave at which s ´ R ± 0,
so that from (11) the derivatives of the thermodynamic quantities
diverge there (but note that the derivatives of the nuclei
abundances do not), and no physically valid solution exists (Wood
& Salsburg 1960). The minimum, or self-sustaining, detonation
speed is then given by the value of D for which s ´ R ˆ 0 at
the frozen sonic point. Note that the derivatives of the
thermodynamic variables are indeterminate at such a pathological
point. The thermicity can become zero with R ± 0 due to
endothermic or dissapative effects. In the case of detonations in
C±O the thermicity passes through zero as a result of an
endothermic stage of the reactions. Fig. 1 shows a schematic
diagram of the possible pathological wave structures. The flow
downstream of the pathological point can either be frozen
supersonic (dashed line in Fig. 1), corresponding to the selfsustaining detonation, or frozen subsonic with a discontinuity in
the thermodynamic derivatives at the pathological point for a
supported pathological detonation (dotted line in Fig. 1), which is
the limit of supported detonations as the detonation speed tends to
the self-sustaining speed from above. The structure of the
detonation downstream of the pathological point is then quite
different for the self-sustaining and supported pathological
detonations. Detonations with speeds greater than this minimum,
pathological speed, corresponding to piston-supported detonations, are frozen subsonic throughout and the thermodynamic
quantities have an interior turning point where the thermicity
passes through zero.
3
D E T O N AT I O N S P E E D
The Chapman±Jouguet detonation speed can be found by solving
equations (6)±(8) together with the conditions
u ˆ ae …r; T; Y†
and
R…r; T; Y† ˆ 0:
The pathological detonation speed, however cannot be found by
solving a set of algebraic equations. In order to determine the selfsustaining speed we follow Khokhlov (1989). Given a value of the
initial density and an initial guess for the corresponding
pathological detonation speed we integrate the equations into
the reaction zone. If the detonation speed is too low then the
solution will terminate at a frozen sonic point, whereas if it is too
high the thermicity will vanish, corresponding to a supported
detonation wave. We can thus iterate using bisection to obtain the
Figure 1. Diagram of pathological detonation wave structures.
q 1999 RAS, MNRAS 310, 1039±1052
Detonations in Type Ia supernovae
1043
pathological detonation speed to within any desired degree of
accuracy. Fig. 2 shows the detonation speed obtained by this
method as a function of the initial density. We find that the selfsustaining detonation becomes Chapman±Jouguet below initial
densities of about 2 107 g cm23 .
4 S O L U T I O N S N E A R PAT H O L O G I C A L
POINTS
Since we are interested in the self-sustaining detonation, we
require the solution at, and downstream of, the frozen sonic
pathological point. We cannot obtain the downstream selfsustaining structure by direct integration from the shock because
of the singular nature of (11) and (12) at such points. It is therefore
necessary to analyse the structure of the solutions in the
neighbourhood of the pathological points.
We note first that equations (6)±(8) can be viewed as giving the
thermodynamic variables as functions of the composition, Y, for a
given initial state and detonation speed. Where they are defined,
these functions are double valued for a given composition, with a
frozen subsonic and a frozen supersonic value, and can only
change continuously from one solution branch to the other at a
frozen sonic point (Wood & Salsburg 1960).
For N independent reactions there is then an …N 2 2† surface or
locus of `pathological points' (note for N ˆ 2 this locus reduces to a
single point, cf. Sharpe 1999), which can be determined by solving
hˆ0
Figure 2. Detonation speed versus the initial density.
…13†
and
s ´ R ˆ 0;
…14†
together with equations (6)±(8). For the pathological detonation
speed, the solution which starts at the post-shock state reaches a
point on the corresponding pathological locus, which we denote
by Yp, with corresponding density and temperature r p and Tp and
detonation speed Dp. Note that for N . 2, knowledge of the
pathological detonation speed does not allow one to determine the
state at the pathological point algebraically. Note also that the
pathological point is a frozen sonic point, but from equations (4),
unlike the derivatives of the thermodynamic variables, the reaction
rates are well defined there. Hence the nuclear composition can be
determined in the neighbourhood of such a point by a linear
expansion. Since the composition is thus determined and the
thermodynamic variables are double-valued functions of the composition, there are two solutions entering or leaving the pathological point,
a frozen subsonic one and a frozen supersonic one.
Fig. 3 shows the solutions obtained by integrating from the
shock for detonation speeds very near the pathological speed when
the initial density is 1 109 g cm23 , from which it can be seen that
the pathological point has a `saddle-like' nature. Note that any
thermodynamic variable plotted against any nuclei abundance has
a similar behaviour. Away from the pathological point, the
solutions lie very close to one another, i.e. they form a
`separatrix'. For D , Dp, the solution eventually leaves the
separatrix from below and the gradients of the thermodynamic
variables become infinite as a non-pathological frozen sonic point
is reached, while for D . Dp the solution leaves the separatrix
from above as the thermodynamic variables pass through a turning
point where the thermicity is zero (note that downstream of this
turning point, these supported detonation solutions again converge
on another separatrix). The closer the detonation speed is to the
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Figure 3. Solutions integrated from the shock for detonation speeds very
close to the pathological speed, solid lines D ˆ …1 ^ 10215 †Dp , dotted
lines D ˆ …1 ^ 10210 †Dp , and dashed lines D ˆ …1 ^ 1025 †Dp . (a) Density
versus distance behind shock, and (b) density versus 4He nuclei abundance.
r0 ˆ 1 109 g cm23 .
pathological speed, the later the solution leaves the separatrix.
One can see from Fig. 3 that although we can obtain the
pathological detonation speed to very high accuracy by simply
shooting at the pathological point from the shock, we cannot
actually reach the singular pathological point because of its
saddle-like nature, although we can get very close to it.
One idea for overcoming this problem is to start at a
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G. J. Sharpe
pathological point, shoot towards the shock and try to satisfy the
shock conditions. Unfortunately, equations (4), (11) and (12) are
unstable to numerical integration in the upstream direction.
Besides which, the pathological point has N 2 2 (i.e. 10 for the
a -network) degrees of freedom so it would be very difficult to
determine the correct values in this way.
We thus need to find a very good guess for the state at the
pathological point given the initial conditions and the pathological
detonation speed. Fig. 4 shows values of the state obtained at the
frozen sonic point (for D , Dp ) or the zero thermicity point (for
D . Dp ) against the detonation speed when the initial density is
1 109 g cm23 . Note that the state at such points becomes
increasingly sensitive to the detonations speed as D approaches
Dp, which is a result of the saddle-like nature of the pathological
point. However, from Fig. 4, it can be seen that the sonic value of
the density (and also the pressure) is not very sensitive to the
detonation speed as D tends to Dp from below (the reason for this
is clear if one considers Rayleigh lines and Hugoniot curves in a
p 2 1=r plane). Hence we can determine the value of the density
at the sonic pathological point to very good accuracy. We can now
determine a good guess for the composition at the pathological
point, and for the distance between the shock and the pathological
point. The procedure is to integrate the solutions from the shock
for detonation speeds very close to the pathological speed, so that
we approach as near to the pathological point as possible (cf. Fig.
3). We can then interpolate the separatrix to the known
pathological density, in the space of the density and each of the
nuclei abundances or the distance from the shock, in order to
determine the values at the pathological point.
We make a brief aside here. From Fig. 4 it can be seen that the
state at the turning point in the thermodynamic variables where
the thermicity vanishes is extremely sensitive to the detonation
speed for supported detonations with detonation speeds near the
pathological speed, and becomes increasingly sensitive as D ! Dp
from above. In particular we find that the minimum in
h ˆ a2f 2 u2 , which occurs at the zero thermicity point, is itself
extremely sensitive to the detonation speed. We believe it is this
sensitivity of the sonic nature of the flow which is, in turn, the
cause of the sensitivity in the linear stability spectrum of such
detonations on the detonation speed, as found by Sharpe (1999).
We therefore expect the stability of overdriven waves in C±O to
be extremely sensitive to the detonation speed near the pathological speed, and we reiterate that one must be very careful in
drawing any conclusions about the stability of the self-sustaining
detonation by using even slightly overdriven detonation.
Now, we note that (for an a -network) the pressure and the
frozen sound speed depend on the composition only through the
mean molecular weight of ions, m . Hence, given the initial
conditions, pathological detonation speed and sonic density, we
can solve the momentum equation, (7), together with condition
(13), for the frozen sonic values of the temperature and the mean
molecular weight (which gives one of the nuclei abundances, Yj
say, as a function of the other abundances). We can then determine
a point on the pathological locus by solving the energy equation,
(8) together with condition (14) for two of the nuclei abundances,
Yk and Yl say, with the remaining abundances given by the
interpolated pathological values, found as described above. One
can see from Fig. 4 that the sonic values of the temperature and
the composition are quite sensitive to the detonation speed near
the pathological speed, and so also on the sonic density. Hence by
fine tuning the sonic density (by which we mean changes in the
density by the order of only about 103 g cm23) we find we can
Figure 4. State at sonic point (dashed line) or zero thermicity point (solid
line) against detonation speed: (a) density, (b) temperature, (c) distance
behind shock and (d) mean molecular weight. r0 ˆ 1 109 g cm23 .
q 1999 RAS, MNRAS 310, 1039±1052
Detonations in Type Ia supernovae
1045
make the values of Yj, Yk and Yl simultaneously very close to their
interpolated values, which lends credence to the interpolation
procedure. We can thus find a point on the pathological locus which
is very close to the desired pathological point. We use the nuclei
abundances with the highest interpolated values for Yj, Yk and Yl,
usually 4He, 28Si and 32S, since these are the most important.
Figs 5 and 6 show the dependence of the state at the
pathological point, as determined by this method, on the initial
density. The temperature and the log of the density at the
pathological point depend almost linearly on the log of the initial
density. The distance between the pathological point and the shock
increases quite dramatically as the initial density decreases. Also,
as the initial density decreases, the state at the pathological point
becomes more completely burnt; the abundances of carbon and
oxygen decrease, while the abundances of the products of silicon
burning and the iron-peak elements increase.
Given the state at the pathological point, it remains to determine
the downstream solutions. Since equations (11) and (12) are
indeterminate at such points it is necessary to linearize about the
pathological values in order to determine the behaviour of the
solutions in the neighbourhood of the pathological point. We
follow partly the analysis of Wood & Salsburg (1960), who
considered linear expansions about such points for a very general
system with an arbitrary number of reactions.
Near the pathological point we write
r ˆ rp …1 ‡ dr†;
T ˆ T p …1 ‡ dT†
and
Y ˆ Y p …1 ‡ dY†;
where the p superscript refers to quantities evaluated at the pathological point and dr , d T and d Yi …i ˆ 1; ¼; 12† are small. We note
first that, since the reaction rates are well defined at the pathological
point, from equations (4) we have, to first order in d Ym,
dY i p
Rp
Y pi dY i ˆ Y pm dY m
ˆ Y pm dY m pi ; i ˆ 1; ¼; 12
dY m
Rm
for any m, provided Rpm ± 0 (since the pathological point is a not
an equilibrium point, certainly not all the reaction rates can be
zero there). Then from equations (6), (7) and (9) we have, to first
order in the small quantities,
" #21 ("
p #
­p p
r0 Dp 2
­p
p
T dT ˆ
2
rp dr
p
­T r;Y
r
­r T; Y
)
p
12
X
Y pm
p ­p
:
…15†
Ri
2 p dY m
Rm
­Y i r;T;Y i
iˆ1
Hence we have two unknowns, dr and d Ym. We next define a
pseudo-time variable, t , by
dx
dt ˆ :
h
Then, from (11) and (4) we obtain
dr
ra2
ˆ 2 f s´R
dt
u
…16†
and
dY m
Rm
:
ˆh
dt
u
q 1999 RAS, MNRAS 310, 1039±1052
…17†
Figure 5. Values of (a) the density, (b) the temperature and (c) the distance
behind shock, at the pathological point versus the initial density.
Clearly, the pathological point is a critical point of these equations
(in fact the pathological locus is a locus of critical points).
Linearizing in the small quantities about the pathological point we
get
­s ´ R p p
­s ´ R p p
r dr ‡
T dT
­r
­T
p #
12
X
Y pm
p ­s ´ R
‡ p dY m
Ri
Rm
­Y i
iˆ1
ddr
…ap †2 rp
ˆ f p
dt
r0 D
1046
G. J. Sharpe
where the matrix Ap depends only on the state at the pathological
point. As expected we find that Ap has real eigenvalues, one
positive and one negative, so that the pathological point is a saddle
point of equations (16) and (17). There are thus two solutions, the
separatrices of the saddle, which enter or leave the pathological
point, corresponding to the frozen subsonic and supersonic
branches. Note that, for subsonic …h . 0† solutions, t -stability
implies x-stability, whereas, for supersonic solutions …h , 0†
t -stability implies x-instability and vice versa. Hence two
solutions leave the pathological point as x increases (i.e. in the
downstream direction), corresponding to the self-sustaining
(supersonic) and supported pathological (subsonic) solutions. Of
the two solutions which enter the pathological point from
upstream, the subsonic one is the branch of the solution between
the shock and the pathological point, while the supersonic one
corresponds to a non-physical unshocked wave.
We can thus use the eigenvectors of Ap very near the
pathological point to determine initial conditions for these
solutions. We then integrate towards the strong or weak
equilibrium points to determine the downstream solutions. We
find that these solutions are not very sensitive to the exact location
of the pathological point, or the nuclei abundances used for Yj, Yk
and Yl. Also, as expected, the supported pathological detonation
quickly tends to, and eventually becomes indistinguishable from,
the (downstream) separatrix formed by the solutions for overdriven detonations with D very slightly greater than Dp. One
should also integrate back upstream to the shock along the
subsonic solution to ensure that the original initial conditions are
met. However, as pointed out earlier, the equations are
numerically unstable to integration in this direction, but since
the subsonic solution downstream quickly tends to the subsonic
separatrix, we expect the upstream solution to also quickly tend to
the upstream separatrix, i.e. to pick up the solutions found by
integrating from the shock with D . Dp . Indeed, the subsonic
eigenvector of Ap is directed back towards this upstream
separatrix. However, this is not too important since we are
interested in the difference between the self-sustaining and
supported downstream solutions. Hence we can find the complete
structure by integrating from the shock with D ˆ Dp , as far as
possible along the separatrix towards the pathological point,
continuing this solution by interpolation to the pathological point,
then linearizing about the pathological point to integrate to
equilibrium. Fig. 7 shows the structure of the different solution
branches determined by this method near the pathological point
for an initial density of 1 109 g cm23 (compare with Fig. 3).
Figure 6. Composition at the pathological point versus the initial density,
(a) nuclei abundances of 4He to 20Ne, (b) nuclei abundances of 24Mg to
36
Ar and (c) nuclei abundances of 40Ca to 56Ni.
and
ddY m
ˆ
dt
rp Rpm
r0 Dp Y pm
"
­h
­r
p
p #
p
12
X
­h
Yp
­h
;
rp dr ‡
T p dT‡ pm dY m
Rpi
­T
Rm
­
Yi
iˆ1
with d T given by (15), or
!
!
dr
dr
d
p
ˆA
;
dt dY m
dY m
5 PAT H O L O G I C A L D E T O N AT I O N WAV E
STRUCTURES
In this section we examine both the self-sustaining and the
supported pathological detonation wave structures for three
representative initial densities, log…r0 ; g cm23 † ˆ 9:0, 8.0 and
7.6. Figs 8±10 show the density, temperature, nuclear energy
release and mean molecular weight throughout the waves for these
initial densities. Fig. 11 shows the abundances of He, C, O, Si, Ca,
Fe and Ni for both the unsupported and supported waves, while
Fig. 12 shows the detailed nuclear composition for the
unsupported detonation, for the case log …r0 † ˆ 8:0. Table 1
gives the detonation lengths (defined as the distance behind the
shock at which the differences between all the nuclei abundances
and their equilibrium values become less than 1 per cent) and the
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Detonations in Type Ia supernovae
1047
Figure 7. Pathological detonation wave structure near the pathological
point. The solid line is the branch of the solution between the shock and
the pathological point, the dashed line is the self-sustaining downstream
solution and the dotted line is the supported downstream solution. (a)
Density versus distance behind shock, and (b) density versus 4He nuclei
abundance. r0 ˆ 1 109 g cm23 .
equilibrium density, temperature and composition. Note that
downstream of the pathological point there can be additional
points where the thermicity again passes through zero. Indeed,
from Fig. 9 it is clear that, for the self-sustaining solution, the
density, temperature and heat release pass through a minimum and
then increase slightly as equilibrium is approached. As the initial
density decreases this effect seems to become more pronounced.
For the log…r0 † ˆ 7:6 case it is so great that the solution again
encounters, and terminates at, a frozen sonic point before
equilibrium is reached. Hence for lower initial densities there
seems to be no completely steady self-sustaining wave. However,
since the flow is frozen sonic at the pathological point, it is also
possible to have a time-dependent continuation of the solution
downstream of the pathological point (Wood & Salsburg 1960).
Indeed, for any initial density, the steady continuation downstream
of the pathological point of the self-sustaining solution is not
unique; time-dependent continuations are also possible, depending
on the downstream conditions.
For a given initial density, the temperature and the density
downstream of the pathological point are consistently lower for
the self-sustaining wave than for the supported wave. Even though
the difference in the temperature is not large, because the reaction
rates are extremely temperature sensitive, this results in the
structures of the self-sustaining and supported waves being quite
q 1999 RAS, MNRAS 310, 1039±1052
Figure 8. Pathological detonation wave structure for r0 ˆ 1 109 g cm23.
The solid line is the branch of the solution between the shock and the
pathological point, the dashed line is the self-sustaining downstream
solution and the dotted line is the supported downstream solution. (a)
Density, (b) temperature, (c) nuclear energy release and (d) mean
molecular weight versus distance behind shock.
1048
G. J. Sharpe
Figure 9. Pathological detonation wave structure for r0 ˆ 1 108 g cm23.
The solid line is the branch of the solution between the shock and the
pathological point, the dashed line is the self-sustaining downstream
solution and the dotted line is the supported downstream solution. (a)
Density, (b) temperature, (c) nuclear energy release and (d) mean
molecular weight versus distance behind shock.
Figure 10. Pathological detonation wave structure for log…r0 † ˆ 7:6. The
solid line is the branch of the solution between the shock and the
pathological point, the dashed line is the self-sustaining downstream
solution and the dotted line is the supported downstream solution. (a)
Density, (b) temperature, (c) nuclear energy release and (d) mean
molecular weight versus distance behind shock.
q 1999 RAS, MNRAS 310, 1039±1052
Detonations in Type Ia supernovae
1049
Figure 11. Pathological detonation wave structure for r0 ˆ 1 108 g cm23. The solid line is the branch of the solution between the shock and the
pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) 4He, (b) 12C, (c)
16
O and (d) 28Si, (e) 40Ca, (f) 52Fe and (g) 56Ni nuclei abundances versus distance behind shock.
different. First, the lower temperature of the self-sustaining
detonation means that the fuel takes much longer to burn to
equilibrium than for the supported detonation. From Table 1, the
self-sustaining detonation lengths are about 2.4 and 3.0 times that
q 1999 RAS, MNRAS 310, 1039±1052
of the corresponding supported detonation wave lengths for
log…r0 † ˆ 9:0 and 8.0, respectively. Hence one would expect the
detonation length to become comparable with the length-scale of
the star at slightly higher initial densities than the supported
1050
G. J. Sharpe
Table 1. Detonation length and equilibrium density, temperature and
composition by mass for the self-sustaining and supported
pathological detonation waves.
log(r 0, g cm23)
Figure 12. Nuclear composition in the unsupported detonation wave for
r0 ˆ 1 108 .
detonation wave structures would predict. Secondly, since the
more tightly bound nuclei are favoured in equilibrium for lower
temperatures (Wiggins et al. 1998), the self-sustaining detonation
produces more 56Ni and slightly less of the intermediate mass
elements. This also results in much less energy being absorbed
downstream of the pathological point in the self-sustaining
detonation than in the supported wave. From Figs 8±10 it can
be seen that very little heat indeed is absorbed for the selfsustaining case. Although the difference in equilibrium compositions is not large, one can see from Fig. 11 that the composition in
the wave at a given distance behind the pathological point is quite
different. Hence if the detonation was quenched for some reason,
e.g. as a result of the expansion of the star, the resulting
composition would be quite different for the self-sustaining
detonation than for the supported detonation.
Finally, from Fig. 11 it is clear that the pathological point
occurs near the start of silicon burning; both carbon and oxygen
are virtually completely burnt by the time the pathological point is
reached. Note the very different length-scales of carbon, oxygen
and silicon burning, which can also be seen from Fig. 12.
6
CONCLUSIONS
In this paper we have investigated steady, one-dimensional
detonation waves in C±O for initial densities greater than
2 107 g cm23 . For such densities the self-sustaining detonation
is of the pathological type. We have developed a method for
determining the self-sustaining and supported pathological
detonations. The structure for the self-sustaining detonation
downstream of the pathological point is quite different from that
of the corresponding supported detonation. The lower temperatures of the self-sustaining detonation waves mean that the
detonation length is significantly larger than that in the supported
wave. More of the iron-peak and less of the intermediate elements
are produced, and hence there is a greater total nuclear energy
release by the self-sustaining wave than by the supported
detonation.
In this paper we have used a simple a -network for the burning.
This should give the correct qualitative results, including
(importantly) the energy release. Large differences in the energy
release, compared to a complete network, would only be
important near the end of the burning, because of the neglect of
xdet, cm
r , 109 g cm23
T, 109 K
XHe, %
XC, %
XO, %
XNe, %
XMg, %
XSi, %
XS, %
XAr, %
XCa, %
XTi, %
XCr, %
XFe, %
XNi, %
xdet, cm
r , 109 g cm23
T, 109 K
XHe, %
XC, %
XO, %
XNe, %
XMg, %
XSi, %
XS, %
XAr, %
XCa, %
XTi, %
XCr, %
XFe, %
XNi, %
xdet, cm
r , 109 g cm23
T, 109 K
XHe, %
XC, %
XO, %
XNe, %
XMg, %
XSi, %
XS, %
XAr, %
XCa, %
XTi, %
XCr, %
XFe, %
XNi, %
9.0
8.0
7.6
Self-sustaining
Supported
1:9 103
1.270
8.145
26.6
0.006
0.017
0.001
0.054
5.5
8.5
7.0
8.7
1.0
3.6
12.0
27.0
8:0 102
1.543
8.344
28.5
0.007
0.020
0.002
0.061
5.6
8.6
7.1
8.7
1.1
3.6
11.6
25.1
2:2 105
0.1415
6.658
19.7
0.002
0.006
0.0002
0.018
4.9
7.6
5.8
7.4
0.49
2.4
11.6
40.1
7:3 104
0.1783
6.862
23.0
0.003
0.008
0.0003
0.023
5.3
8.1
6.1
7.6
0.55
2.5
11.3
35.4
2:9 105 *
0.06178*
6.139*
12.9*
0.003*
0.009*
0.0002*
0.026*
10.1*
14.0*
9.0*
10.4*
0.47*
2.0*
8.9*
32.2*
9:3 105
0.07155
6.274
17.4
0.001
0.003
0.00009
0.011
4.2
6.7
5.0
6.6
0.37
2.0
11.3
46.4
* State at frozen sonic termination point.
such processes as electron capture in the a -network. Since the
pathological point occurs near the start of silicon burning, this
should have little qualitative effect on the structure near such
points, but may produce rather different predictions for the
equilibrium abundances. However, it is difficult to assess the
differences that would occur if a complete network was employed
instead of an a -network, without determining the complete
detonation wave. While the method described in this paper could
be used to determine the complete pathological detonation wave
structures, the large number of species (a few hundred, with a few
q 1999 RAS, MNRAS 310, 1039±1052
Detonations in Type Ia supernovae
thousand reactions) would make it rather difficult to handle the
determination of the structure around the pathological points.
Secondly, in any time-dependent numerical simulation, one really
is restricted to a simple network like the a -network.
The detonation wave structures presented here are for
detonations propagating into material with a constant pre-shock
density. Of course, in a real supernova the density drops with the
radius of the white dwarf. However, for the density range
investigated here the detonation thickness, or more importantly
the distance between the shock and the pathological point (since
the flow downstream of this point cannot affect the front), is very
much smaller than the size of the white dwarf and also much
smaller than the scale of density variations in the star (,107 cm,
Khokhlov 1989). Hence such density gradients will have little
effect on the propagation of the detonation wave, and the
structures presented here can be viewed as snapshots of the
detonation wave when it has reached a given density. For lower
densities, when the self-sustaining detonation is Chapman±
Jouguet, the detonation wave thickness can become comparable
with the scale of density variations (Khokhlov 1989). To
investigate the effect of the gradient at such densities, one could
perform one-dimensional (i.e. spherically symmetric), timedependent simulations in a portion of the star corresponding to
the low-density outer layers. Fortunately, for these densities the
detonation is hydrodynamically stable to one-dimensional perturbations (Khokhlov 1993). An adaptive gridding technique would
be required to resolve the different length-scales. However, at such
densities, other effects, such as the curvature of the front, are also
likely to become important. A small amount of curvature can
cause the detonation to become pathological, even when the
planar detonation is CJ, because of the divergence of the flow
(Fickett & Davis 1979). In this case, it is the distance between the
shock and the sonic pathological point which is the relevant
length-scale. In the slightly curved detonation, this distance may
be much shorter than the detonation length as a whole (as in the
cases studied in this paper), and also much shorter than the scale
of the density variations, in which case such variations in the
density would be unimportant to the propagation of the
detonation.
We intend to use the information presented here in several
ways. First, using the method of Wiggins & Falle (1997b), we
intend to perform two-dimensional simulations of double detonations in Type Ia supernovae, where a detonation is ignited at a
point in the He envelope and triggers a detonation in the C±O core
at the core±envelope boundary. We have shown in this paper that
the burning up to the pathological point occurs on a very short
length-scale, and only a small amount of heat is absorbed
downstream of the frozen sonic pathological point. Hence we can
model the burning up to the frozen sonic point as a thin
detonation, with the downstream state given by the onedimensional pathological point values. We can then evaluate the
burning beyond the pathological point from the subsequent flow
calculated from the hydrodynamics. This can be used to
investigate the effects of two-dimensional phenomena, such as
asymmetrical expansion of the star and the cusps in the wave front
found by Wiggins & Falle (1997a), on the nucleosynthesis and the
range of expansion velocities of the intermediate mass elements.
Secondly, we intend to perform a linear stability analysis of
both CJ and pathological C±O detonations. Sharpe (1997, 1999)
has developed methods for dealing with both cases. Preliminary
two-dimensional numerical simulations show that such analyses
give very good predictions of the natural cell size, at least when
q 1999 RAS, MNRAS 310, 1039±1052
1051
the steady detonation is stable to one-dimensional perturbations
(Sharpe & Falle, in preparation). It should also reveal the
multiscale instabilities of each of the burning layers. We then
intend to perform two-dimensional numerical simulations to see
how these instabilities on different length-scales affect each other.
However, because of the vastly disparate length-scales it is not
possible to resolve more than one burning layer at a time. The only
way of overcoming this difficulty that occurs to us is to perform
successive simulations with the length-scale of the carbon-burning
layer artificially increased, and see if we can scale these to the
correct value.
It would, of course, be easier to first perform one-dimensional
simulations. However, the one-dimensional, or pulsating, instability is a very different instability and propagation mechanism than
the cellular instability in multidimensions. As mentioned in the
introduction, such pulsating detonations are seen only very rarely
in terrestrial experiments (when blunt bodies are fired into
detonatable gases). Almost always the wave propagates as a
cellular detonation, even when the detonation is stable to onedimensional perturbations. Hence the pulsating instability has no
relevance in the multidimensional progenitors of Type Ia supernovae. Indeed the one-dimensional instability leads to erroneous
predictions about how the detonation would propagate in multidimensions, such as the detonation failure seen in one dimension,
which does not occur in multidimensions because of the constant
re-ignition by the transverse waves.
AC K N OW L E D G M E N T S
I acknowledge funding from the Particle Physics and Astronomy
Research Council (PPARC) during the course of this work. I
would also like to thank S. A. E. G. Falle for many useful
discussions.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
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