436
Progress of Theoretical Physics, Vol. 84, No.3, September 1990
Three Dimensional Simulations of Supernova Explosion. I
Yoshiyuki YAMADA, Takashi NAKAMURA*'*) and Ken-ichi OOHARA*
Department of Physics, Kyoto University, Kyoto 606
Laboratory for High Energy Physics, Tsukuba 305
*National
(Received May 26, 1990)
§ 1.
Introduction and summary
Various observations of SN1987a indicate the existence of strong mixing in the
exploding shell of the supernova. One of possible causes of such a mixing is the
occurrence of hydrodynamic instability such as Rayleigh-Taylor instability in the
expanding shell. Chevalier suggested that the pressure gradient plays a role of
gravity in an exploding shell and that the configuration where the pressure gradient
and density gradient have opposite sign is Rayleigh-Taylor unstable. l ) From his
results, a self-similar solution of a point explosion in power law envelope pcx r- w with
adiabatic index y=4/3 and 2.53<w<3.0 has a Rayleigh-Taylor unstable pressure
gradient. Richtmyer showed that when a shock propagates across the density gap,
Rayleigh-Taylor like instability occurs. 2) In this paper, we discuss hydrodynamic
instability in a shocked shell moving in matter with spatially varying density. A
supernova progenitor is a star in the last satge of its evolution and its density
distribution is very steep. It also has density gaps corresponding to discontinuous
change of chemical compositions. Assuming the existence of small perturbations in
velocity and pressure at the time of explosion, we can anticipate the growth of this
perturbation. We do not treat other possibilities of mixing in this paper. But the
hot bubble scenario, proposed by Colgate,3) may have something to do with initial
sources of fluctuations.
Nagasawa et al. showed that an exploding shell of N=3 polytropic star is
Rayleigh-Taylor unstable using a SPH code with a small number of partic1es. 4 ) This
result has not been confirmed and must be checked by other codes. So we compute
instability in N =3 case first. A density profile of an N =3 star is nearly flat in its
inner part. The density gradient becomes steeper and steeper in large r, and the
value of - alogp/alogr becomes infinite at the surface of the star. So if enough
perturbations exist when the shock reaches the steep part, the initial perturbation
*)
Present address: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606.
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We have performed three dimensional numerical simulations of supernova explosions using (200)3
rectangular grids. As initial density distribution, we consider N =3 polytrope and the realistic
presupernova star model (MIl) computed by Nomoto. For each calculation we put small random
perturbations in pressure and velocity. In the case N =3 polytropic star, although the fluctuation
grows once, it stops growing. As a result no significant growth of fluctuations occurs except for
extremely large initial perturbations. This is completely different from the preliminary simulations
by Nagasawa et a!. using a smoothed particle method with a small number of particles. In realistic
star cases, perturbations put at the density gap corresponding to the hydrogen burning shell grow very
large, which is consistent with linear analysis.
Three Dimensional Simulations of Supernova Explosion. I
437
§ 2.
Computational method
In supernova explosions, the gravitational energy is negligible coinpared with the
explosion energy. So we can treat this problem by solving following hydrodynamic
equations,
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seems to grow up to a nonlinear stage. But in fact, the exploding shell of N =3
polytropic star has almost Rayleigh-Taylor stable pressure gradient asa whole. In
this paper, we computed explosions of N =3 polytropic star by a mesh code and no
significant growth of instability is detected. Muller et al. also computed this process
and reached consistent results with ours. 5 ) Though a shell has Rayleigh-Taylor
unstable pressure gradient at some intervals, such a situation does not continue
enough time for the instability to grow up to a nonlinear stage. While the pressure
has Rayleigh-Taylor stable distribution, the perturbation does not grow. This is the
reason why an N =3 polytropic star does not show the significant growth of instability.
_
We also computed explosions of realistic star models with steep density gradients
and density gaps corresponding to shell burning. Mixing due to hydrodynamic
instability is related to details of density profile such as gradients and gaps. So it is
necessary to treat a realistic star model. As a model, we use the stellar evolution
model MIl computed by N omoto. This model can be considered as one of the
standard models of 1987a progenitor. Nomoto's model gives good agreement with
many observational data such as hard X-ray, r-ray line emission,6) optical light
curve. 7 ) This model is consistent with Woosley's model, which is also widely used for
various analysis of Supernova SN1987a.
As for three dimensional simulations, Muller et al. 5 ) have already computed
explosion of N =3 polytropic star by three dimensional PPM code with (74)3 equidistance orthogonal grids. They also computed explosion of realistic star model using
PPM code with 200 x 20 x 20 r8¢ grids. B) We calculated supernovae explosion in our
three dimensional code which is second order in space and time with various mesh
numbers as test calculations. In the case of small number of mesh less than 50, we
cannot remove effects of orthogonal mesh. While the shock propagates in the region
of inner 20 or 30 cubic meshes, physical quantities fluctuate non-spherically due to the
orthogonal grids. Small number of mesh calculations have hazard to picking up such
a perturbation in addition to given initial perturbations, and we will not able to
distinguish whether resulting instability is due to initial perturbation or the effect of
meshes. The purpose of this paper is to see this problem by three dimensional code
with fine grids. Den et al.,9) Arnett et al.lO) and Hachisu et al. l l ) have also performed
simulations of supernova explosion using two dimensional mesh code with fine grids.
But the simulations by three dimensional code are strongly anticipated because the
restriction of axial symmetry may affect the stabilities we are interested in. .
In § 2, we discuss our method of computation. In §§ 3 and 4, we give numerical
results.
Y. Yamada, T. Nakamura and K. Oohara
438
a
Ttup+f7- u up+f7P=O,
a
TtE+f7u(E+ P)=O,
(2-1)
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where p is density, u is velocity, E is total energy density including thermal and
kinetic one and r is adiabaticindex which is assumed to be 4/3. We use the following
numerical method, details of which are shown in Oohara and N akamura. 12) To keep
the second order accuracy in time and in space, all numerical fluxes are fnterpolated
linearly within each cell (piecewise linear) with modification for monotonicity condition. 13 ) To increase the computational speed and saving memory, we apply donor cell
method to these modified fluxes. Integration with respect to time direction is explicit.
A computational grid is orthogonal and equi-distance in XYZ-coordinates. Our
computational domain is cubic which encompasses an octant of initial star applying
symmetric condition on the respective boundary in N=3 cases. Available mesh
number is 8 X 106 , i.e., 200 meshes in each direction using 460 MB core memory and it
takes about 2 hours for 3000 time steps by HITAC S820 / 80. In the explosion of
polytropic star, 200 meshes in one direction seems to be sufficient since half mass
radius of N =3 star is 0.28 times the radius of the star. On the other hand, a realistic
star is highly centrally concentrated. The density of the inner most density gap
(carbon burning shell) is of the order of 105 g/cm3 • On the other hand, the surface
density of the hydrogen rich envelope is about 10-9g/cm3 • Thus the density contrast
is 1014 •. The radius of the inner most gap corresponding to the carbon shell burning
is 3.0 X 108 cm, while the hydrogen shell burning radius is 4.6 X 10 10 cm. Half mass
radius of such a star is less than 1/15 times the radius of the star. We cannot resolve
such a density distribution with "200 meshes in one direction. The central part of
progenitor star has very steep density gradient. If we take initial density and
pressure distribution on three dimensional grid with 200 meshes, large numerical
errors at the center will occur. Especially, when we put the explosion energy at the
central sphere with 20 grid length radius, deviation from the sphere due to orthogonal
grids gives uncontrollable large initial perturbation. So we start three dimensional
computations from a few seconds after the explosion using the corresponding solution
of the spherically symmetric calculation. The initial length of the cube's edges is
twice as large as the shock radius at the starting time of three dimensional computations. When the shock reaches computational boundary, we define a new grid with
its mesh space twice as large as old one, an computational data are copied on the new
grid. Then we contine our computation with 5 to 8 times successive remapping. Our
scheme is a kind of finite volume method and such remapping may not introduce extra
viscosity.
We put random noises on kinetic and thermal energy as initial perturbations.
Three Dimensional Simulations of Supernova Explosion. I
439
We generate uniform random number Rndi and put it on four physical quantities,
pressure and three velocity components on all grid points through the following
formulae,
P(x, y, z)=(1 +cRnd4i)H(r),
~ cRnd4i+l)vI(r) ~,
Vx(x, y, z)=( 1 +
Vz(x, y, z)=( 1 +
~,
~ cRnd4i+3)vl(r) ~ .
(2·2)
In the above formulae, P, Vx, Vyand Vz denote physical quantities of 3D calculations,
H and VI denote the results of spherical symmetric computations, and i denotes ith
grid point. Amplitude c is chosen to be 0., 0.1 and 0.5. This value of c should not be
considered as the amplitude of perturbation. In the mesh calculations, we can see the
growth of perturbation with its wave length n( ~ 10) times larger than mesh spaces
Llx. The perturbations with wave length shorter than ~ 10Llx decay due to the effect
of numerical viscosity included in the scheme. In three dimensional computations,
the amplitude of initial perturbations with such long wave length is 1/ H ~ 1/30
times smaller than the amplitude of uniform random perturbations. It means that
initial perturbation with c=0.5 corresponds to the effective £=0.016 in interested
modes.
As a test of our method, we computed Rieman's shock tube problem and point
explosion. We confirmed our code reproduces known analytic solutions for these
problems. For example, point explosion in the medium of constant density reproduces Sedov's self-similar solution within 1% error in density. In this case, the shock
is captured within two mesh length. We also computed self-similar adiabatic expanding shell. 14 ) When shell expands twice as large as initial state, relative error in
density is less than 1%.
§ 3.
Table I. A list of models. The time when we
starts 3D calculation is denoted as to. Ro is the
radius of the sphere where we deposited explo·
sion energy.
Model
Initial model
toCsec)
Ro(cm)
A
N=3
0
1 x lOll
B
N=3
800
4x10·
C
N=3
1500
4X10·
D
Mll
7.5
6.4 x 107
E
Mll
75
6.4 x lOB
Numerical results
We show the results of N=3 polytropic star first. We performed three
different kinds of computations.. In all
cases, total mass, radius and explosion
energy are M=10Me , r=10 I2 em, and
Eexp=105I ergs. In the first case, we
deposi ted the energy as thermal energy
in the central sphere of radius 20 grids
length (~1Oll em), with random pe~tur­
bation (model A, see Table 0. In other
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Tfy(x, y, z)=( 1 + ~ cRnd4i+2)vI(r)
Y. Yamada, T. Nakamura and K. Oohara
440
MODEL:
N=3CB):1%
RSTAR = 1. 00E+1Z CM
ENERGY= 1. 00E+S1 ERG
VCMAX)= 6. 41E+0B CM~S
RCUNIT) 1. 00E+11 CM
(f]
TIME
4601
STEP
1.S3E+03 SEC
(f]
....
·X
([
}-
X-AXIS
Z-AXIS
Fig. 1. Density contour and velocity field of three dimensional calculation of N =3 star model B on the
x- y, y - z and z- x planes at 1550 sec after the explosion. Radius of initial star is 10'2 cm and
total mass is 10M o • Explosion energy is 1051 ergs and is deposited as thermal energy. The value
of E (the amplitude of initial perturbation, see text in detail) is 0.016 and length of a side is 6.43
X lO" cm.
two cases, we deposited energy in the sphere with radius 4 X 109 cm and calculate with
spherically symmetric code at first. In these two cases, we switch from 1D calculation to 3D one at 800 seconds (model B) and 1500 seconds (model C) after the explosion
respectively. The former one is appropriate for the comparison with the results by
Muller et al.
We computed above all cases with the value of 6"=0.003 and 0.016. But the
results are essentially the same. For example, we show the model in which the value
of 6"=0.016 and 3D calculation starts at 800 sec after the explosion in Fig. 1. Even
the case with 6"=0.016 shows no significant growth of perturbation till t=1500 sec.
In the case of N=3 star, there is not enough time for growth of instability as
predicted by the linear analysis. The final amplitude of the density fluctuation in the
case with 6"=0.016 is less than 10% in models Band C. The perturbation grows at
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IT
Three Dimensional Simulations of Supernova Explosion. I
441
Downloaded from http://ptp.oxfordjournals.org/ at Pennsylvania State University on September 11, 2016
most a factor 5 to 10. No significant density fluctuation can exist except for extremely large initial perturbations. These results are completely different from the preliminary simulations using a smoothed particle method. 4l The reason for this difference
is due to the small number of particles.
In the case of the realistic supernova model, a region with large density gradient
exists near the center. Thus the details of the energy deposition affect the results of
simulations with small number of meshes. For avoiding the affect, we use the results
of 1D hydrodynamic computation as initial conditions of 3D computation. Model
Mll has three density gap corresponding to carbon, helium and hydrogen shell
burning, respectively. Our 3D calculation starts when the shock propagates just
behind outer two gaps. The radius of the gap between Carbon-Oxygen layer and
Helium rich layer is 1.5 X 109 cm, and it is the surface of C-O layer. The shock reaches
the gap at 7.5 sec after the explosion. At 75 sec after the explosion, the shock reaches
the interface between hydrogen rich envelope and He rich core. As the starting time
of 3D computation, we choose 75 sec after the explosion (model E). The initial
perturbation are put at pressure and velocity. This means that initial perturbation is
put only within the shock sphere at the time when 3D computation starts. We also
computed the model in which we started 3D computation at 7.5 sec after the explosion
(model D). The physical difference between these two models is the location of the
initial perturbation. As a source of initial fluctuations, we can consider, for example,
convective velocity in the progenitor star. But convective velocity is 10-2 or 10-3
times as small as shock velocity. If we assume the spherical shock is propagating in
the convective zone of the progenitor star, convective velocity cannot be the initial
perturber unless the case that initial fluctuations can be amplified 102 or 103 times in
magnitude. Another possibility of initial fluctuations is explosion mechanism. Initial shock is formed by the core collapse and collapse is very unstable phenomena.
Therefore it does not seem so strange that the shape of the initial shock is far from
spherically symmetric. Though very large initial fluctuation is needed, it is important to consider the relation between initial fluctuation and resulted mixing in the case
that fluctuation is put only in the inner region.
Numerical results show that significant growth of perturbation can be detected in
both cases. After the fluctuation becomes nonlinear, change of density profile is
small and the explosion seems to enter the homologous expansion phase. When the
shock passed the region where density dropssteepIy, the post shock profile has also
steep density gradient. On the other hand, the pressure has positive gradient in the
post shock region. So after the shock passes the region with the density gap, the
pressure and the density profile satisfy the Rayleigh-Taylor unstable condition.
Because the density gradient in the post shock region is very large, growing time scale
of the instability is so short that the instability grows up to the nonlinear stage.
Figures 2 shows the result of model E at 3680 sec after the explosion. At the
same time the shell becomes clumpy, and the shape of contact density surface of
H/He also becomes non-spherical. We can also see finger-like structures. Thus we
can expect the occurrence of the global mixing in the ejecta. Model D in which 3D
calculation starts at 7.5 sec after the explosion also shows the clumpy shell. But
contact surface of H/He does not have such a finger like structure in this case.
Y. Yamada, T. Nakamura and K. Oohara
442
MODEL:
MllC:
1%
RSTAR = 2. S0E+12
ENERGY= 1. 00E+Sl
VCMAX)= 4. 06E+0B
RCUNIT) 1. 00E+ll
CM
ERG
CM ..... S
CM
IT
STEP
X
([
N ..
3. 69E+03 SEC
..~
~~~~~~~~~07~~~"~.,
,.j
m
[flU!
X
([
)-
..
N
z
B
z
10
4
6
B
10
Z-AXIS
Fig. 2. Density contour and velocity field of three dimensional calculation of realistic star model E on
the x-y, y-z and z-x planes at 3690 sec after
explosion. Model of realistic star is MIl
computed by N omoto. Explosion energy is 1051 ergs and is deposited as thermal energy. Three
dimensional calculation starts at t=75 sec after the explosion and the value of £"=0.016 and length
of a side is 1 X 10 12 cm.
the
We summarize our conclusions.
1. In the case of N =3 polytropic star, there is no significant growth of instability
because of lack of growth time. The preliminary results by N agasawa et al.
may be considered due to the small number of particles.
2. Existence of a density gap is essential for occurring global mixing. The
existence of a gap may shorten the growth time scale of the instability. Two
realistic models (MIl in this paper and Arnett's Model in Ref. 10» of supernovae progenitor show the significant growth of instabilities opposite to the
case N =3. Both have density gaps.
3. Sigificant growth of initial perturbation is detected in the case of the explosion
of realistic progenitor model MIl of SN1987 A. This is consitent with preliminary two dimensional results by Den et al}) Arnett et al. 10) and Hachisu et al. l1 )
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6701
TIME
Three Dimensional Simulations of Supernova Explosion. I
443
Our results only show the non-spherically shape of the contact surface and do
not show mushroom heads like two dimensional simulations. The reason of
this difference may be due to lack of number of meshes in our simulations.
Acknowledgements
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This work was supported in part by the Grant-in-Aid for Scientific Research on
Priority Areas (Observation of Supernova Remnants) of the Japanese Ministry of
Education, Science and Culture (01652505, 01652509). The numerical computation of
this work is processed using F ACOM M780 and F ACOM VP400E of the data processingcenter of the Kyoto University and HITAC S820/80 of Data Handling Center of
National Laboratory for High Energy Physics (KEK). We would like to thank
Professor K. N omoto for showing us a computer output of stellar evolution models,
and T. Yoshida for showing us a spherical symmetric hydrodynamic code by a flux
split method.