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InSlabiI1IieS in SN1987A and Other Supernovae
DAVID ARNETT,* BRUCE FRYXELL,* AND EWALD MULLERt
ABSTRACT
While giving a remarkably good description of many aspects of
SN1987A, calculations assuming spherical symmetry have a number of
flaws. Many of these problems naturally disappear in the development of
a twodimensional calculation, as nonspherical instabilities grow. This non
sphencal behavior, made evident in SN1987A, has implications for other
types of supernovae.
INTRODUCTION
For the first two weeks, onedimensional hydrodynamical models of
1987A provide an excellent description of luminosity, effective temperature,
and velocity (see the review of Arnett et al. 1989, denoted "ABKW,'). Lucy
(1987) has shown that the predicted spectra of SN1987A begin to deviate
after two weeks from those observed, suggesting an inadequacy in the
models. Figure 1, from Arnett (1988), indicates the two phenomena that
might show what is wrong. The ordinate is expansion velocity, which is
closely proportional to the radius. The top panel shows the abundance of
hydrogen, helium, 56Ni and 56Co; other nuclei are suppressed for clarifier.
The bottom panel shows temperature. Near the center (at low velocity)
*Department of Physics and Steward Obselvatory, University of Arizona' Tucson, AZ 85721,
USA
tMax Planck Institut fur Astrophysik, GarchingbeiMunchen, West Germany
1
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AMERICAN AND SOVIET PERSPECTIVES
1
.5
o

~6
a)
be:

so
so
a)
~4
Em
o
,,, I,,,, ,_1 %,,, l l l ' ' 1 ' ' ' ' ' ' ' ' '
_ I 'I _
_ Co I `, He H
_ ~ /
.\,~
_ _
I ~
_ / _
 Ni  I _
,,,,,, ~ I~ 1 ,1,, .,,,,, 1, ,,
8 9
7
Log u (cm/s)
1 1 1 1 ~T ~I ~
10
1 1 1 1
LMC 15 Ma
19.75 d ays
10
Log u (cm/s)
9
FIGURE 1 Temperature and composition versus velocity at 19.7 days (after Arnett 1988).
This shows the status of a one~imensional numerical model of SN1987A at a time when
its predictions for detailed spectral behavior debate Mom observations (Lupy 1987~. The
two dominant phenomena are the development of a hot "Ni bubble" near the center (low
reloaty) and the penetration of the photosphere to the edge of the helium core Both are
evident as "steps" in temperature. Other abundances are suppressed for cianty.
is the hot "Ni bubble," which is almost certainly Rayleigh~ylor unstable.
Further out, at the edge of the He core, is the photosphere, which appears
as a "step" in temperature.
DIE NICKEL BUBBLE
Consider the evolution of an opaque 56Ni sphere. The decay of 56Ni
(mean life of 8.8 days) releases about 2 Mew which corresponds to Q ~
4 x 10~6 erg/e. If this is converted to heat, and by expansion into buLk
motion, the mean velocit r is v ~ ,/~ = 2,800 Ems. Compare this to the
lower velocities of Ni and Co, shown in Figure 1.
Now suppose this opaque 56Ni sphere must "snowplow" into a sur
rounding shell of mass. For an initial velocity at the edge of the Ni of vO
1,000 knJs (see Figure 1) and homologous expansion (v proportional to r),
we have
.
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HIGHENERGY ASTROPHYSICS
and
so
3
m = mNi(V/vo)3
1/2mNiv2 + mNiQ = 1/2mv2
v = [2{1/2v2 ~ Q}v3] 1/5 ~ 1, 500km/s.
This is the case of perfect spherical symmetry.
InDared spectral lines show significant amounts of Ni and Co moving
at velocities in excess of 2,000 km/s (Witteborn et al. 1989~. 1b approach
the first case above rather than the second, we need (1) 56Ni energy to be
deposited in the Ni itself (this is true, because gamma attenuation lengths
are short for times t ~ TNi over which the 56Ni decay occurs), and (2)
holes in the mass distn~ution of overlying matter for t ~ rev`. Something
similar is needed for the early rise and flat shape of the gamma ray and
Xray luminosities.
The holes can be made by the expansion of the Ni and Co
eating, or a combination of the two.
THE LIGHT CURVE
, be pre
Figure 2 shows the UVOIR and gamma ray light curve for the first 700
days of SN1987N The good agreement of the theoretical model (solid line:
Arnett and Fu 1989) and the SAAO observations (the CllO and ESO data
are similar) has several important implications. First, the amount of 56Ni
needed is small, 0.0753:0.01 Me or onetenth that needed to reproduce a
SNIP light cube. This is due to a steep density gradient (so that little mass
has the correct conditions for 56Ni synthesis). This in turn is due to core
convergence, which is a direct consequence of veve cooling after helium
burning.
Second, the nearly exponential decay of the bolometnc luminosity,
from about 120 days on, limits the energy KlpUt from the neutron star
(pulsar?) to Lp~r < 1039 erg/e, which is 102 of that suggested prior to
1987.
Third, the good agreement of the analytic model and observations,
from 20 days to maximum, implies SN1987A is considerably less spherically
symntetnc than the onedimensional numerical models (Arnett 1988, and
especially see Figure 1 of Arnett and Fu 1989~. The numerical models
show dips and peaks not present in the observational data; disagreement
ranges up to a factor of two either way. The analytic model does better
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4
42
41
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AMERICAN AND SOVIET PERSPECTIVES
_ ,~ 1 1 1'

38
37
l

_~
SHIM L/IT FG
0/~

Ginga:
Inferred ~ 
1 ~_ 1 1 1 1 11 1 1 1 1 1
_c ~

200
400
Time (days)
600
FIGURE 2 Theoretical and observed WOIR and gamma ray luminosity from SN1987A
versus time. The solid squares are combined ultraviolet, optical, and infrared data
(UVOIR) from SAAO (Menzies et al. 1987; Catchpole et al. 1988; and Whitelock et al.
19883. The luminosity of escaping gamma rays as measured by Solar Maximum Mission
(SMM), Lc~cheedMarshall I, Caltech (CII), and FlondaGoddard (FG) are shown as
rhomboids: detailed references may be found in Amett et al. (1989~. Three cases are
shown (see Amett and Fu 1989) having pulsar luminosity (in gamma rays alone) of 2 x
1039 and 1038 erg/e, and zero. Ine higher solid lines correspond to the higher pulsar
luminosities; the opposite is true for gamma luminosities, which are shown as dashed lines.
The luminosity in X:ays is taken to be equal to that of gamma rays, in rough agreement
with numerical results (BUITOWS, private communication). At 700 days, the UVOIR curves
begin to deviate from the data.
because it uses an average opacity rather than We radially varying opacity
of the numerical models. Apparently me recombination wave does not
move Award through the abundance layers in complete synchronization.
NONSPHERICAL MODELS
Chevalier (1976) has adapted the stability analysis of Chandrasekhar
(1961) to show that the similarity solutions of Sedov develop a Rayleigh
~ylor instability. This work requires the presupernova structure to be
approximated by a power law in density (p oc rum. A realistic presupernova
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HIGHENERGY ASTROPHYSICS
s
model has a more complex structure, so that direct numerical computation
IS required ..
Nagasawa et at (1988, "NNM") have claimed to find violent Rayleigh
lkylor instabilities ~ n = 3 and 1.5 polytropes for all explosion energies.
Subsequent calculations with better resolution Sculler et al. 1989; Benz
and Thielema~ 1989) suggest that this claim is incorrect; the NNM result
may be due to asphericity in their initial "point" explosion due to poor
resolution.
The question of instabilities in polytropes is only of academic interest;
the results of a wellzoned calculation that began from a realistic pre
supernova structure is shown in Figure 3 (after Arnett et al. 1989~. This
twodimensional calculation used 380 x 380 zones and the Prometheus code
(Fryxell et al. 1989), which grew from the PPM of Colella and Woodward
(1984~. The density contours begin to show the characteristic "mushroom"
shape of the Rayleigh~ylor instability at 104 seconds after the explosion.
This is well before 56Ni decay (rNi = S.8 d). Hydrogen penetrates into
velocities as low as 1,000 km/s and heavy elements (Ni and Co) out to
velocities as high as 2,400 km/s. Density variations ("clumping") of order
~ip/p~ ~ 1 occur along many lmes of sight from the center. Because this
directly modifies the attenuation length, which itself appears in an exponen
tial function, the Xray and gamma ray light curves are strongly affected,
making them earlier and flatter, as needed.
The development of the instability is a dynamic process. Whether
a perturbation grows into the nonlinear regime depends upon its initial
amplitude and the number of efolding tunes available. For the shock
instability to dominate, it must go nonlinear in a time t < Phi. For this to
happen, the initial (nonspherical) perturbation must be of order of a few
percent or more, according to the numerical experiments. Figure 3 shows
the results of a 10% perturbation.
Such a value may be reasonable. Unlike most collapse calculations,
those of Arnett (1977) showed extensive shell flashing prior to core bounce.
These calculations bridged the gap between hydrostatic and hydrodynamic
evolution more consistently than has been usual. General relativistic dy
namics was used in both cases. Exactly the same reaction rates were used.
The equation of state was exactly the same, with no glitches in nuclear sta
tistical equilibrium, FermiDirac functions, or Coulomb corrections. This
gave a slower collapse initially, during which the silicon and eventually the
oxygen shell flashed (see Arnett 1977~. It is unlikely this will occur simul
taneously around the spherical surface, so that "seed" perturbations should
be formed. Crude estimates gave us the 1% and 10% values with which
we experimented, but better estimates will require numerical calculations
of precollapse through explosion in more than one dimension.
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6
3XlO
2XlO
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AMERICAN AND SOVIET PERSPECTIVES
1 1 1 1 AWNS, 'I ' ~'\~9
.
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FIGURE 3 Density contours (S percent spacing) at 9,814 seconds after explosion. There
is rotational symmetry about the vertical His; the equatorial plane lies along the horizontal
axis. A 15 M<3 B3 supergiant presupernova structure with a radius of 3 X 1012 cm and
a fairly realistic explosion (fitting the early light curve of 1987A) were used. At this time
the shock has already propagated off the grid. The "mushroom head" shapes, which are
characteristic of a RayleighTaylor instability (followed By KelvinHelmholtz) are evident.
These represent penetration of heavier elements into the He mass, and some H has already
been penetrated By the ascending "mushrooms" Subsequent decay of 56Ni is expected to
have further significant erects. Ihe clumping shown (a factor of two in density already)
will modify thermal opacities (removing "wiggles" in the one~imensional numerical light
curares, and favoring higher values of the ejected mass parameter). It will profoundly modify
the Xray and gamma ray escape (malting both "earlier,'). Even without the added effect
Of heating from 56Ni decay, it is sufficient to modify the expected spectral line shapes and
polarization.
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HIGHENERGY ASTROPHYSICS
7
The question is not whether nonspherical instabilities arise, but the
relative importance of at least two mechanisms.
SNIB AND Dlil?IISION MASSES
The width of a radioactive diffusion peak depends on the dimensionless
parameter
y2 = ~tM/(2pCV5CT2),
where ~ is the decay time, V5C the scale velocity, ~ the opacity, M the mass,
c the velocity of light, and ,0 is a constant (see Arnett 1982). We can get y
and V5C from observations. Then if ~ and ~ are ~own, M can be inferred:
we call this the "diffusion mass." The constant ,B comes from integration
over the structure of the ejecta. For an inhomogeneous medium such as we
have, the effective value of these parameters must change. With "holes"
such as in Figure 3, radiation diffuses out more readily, giving a "diffusion
mass" that is smaller than the actual mass.
Ensman and Woosley (1988) have attempted to use light curves of
Type IB supernovae to constrain the ejected mass to be between 4 and 7
Me,, originally 1525 Mat on the main sequence. This is a "diffusion mass"
and, based on the results shown in Figure 3, should be more lenient by
a factor of mot This would move their limit up to 35 or 40 M on the
main sequence. This effect is in addition to Mat of clumping on gamma
ray escape, which they speculate may improve agreement with observation.
Quantitative estimates require knowledge of the actual threedimensional
structure.
Finally, note that the instabilities discussed here occur inside the
hydrogen envelope, and therefore have a smaller effect on the diffusion
mass of supernovae which have a lot of hydrogen: SNII's and SN1987N
REFERENCES
Alcott, W.D. 1977. Ap. J. 218: 815.
AInett, W.D. 198Z Ap. J. 253: 785.
Arnett, W.D. 1988. In: Supernova 1987A in the Large Magellanic Aloud. University
Cambridge p. 301.
Arnett, ~D., J.N. Babcall, R.P. Kirshner, and S.E. Woosley. 1989. Ann. Rev. Astron.
Astrophys (in press, denoted ABKW)
Arnett, SUD., BE FIyxell, and E. Muller. 1989. Ap. J. (June 15).
Alcott, W.D., and A. Fu. 1989. Ap. J. (in press).
Bend, W., and F.K Thielemann. 1989. (in press).
Catchpole, R.M., et al. 1988. M.N.R^.S. 229: lSP.
Chandrasethar, S. 1961. Hydrodynamic and Hydromagnetic Stability. Clarendon Press,
Oxford.
Chevalier, R 1976. Ap. J. 207: 872.
Colella, P., and P.R. Woodward. 1984. J. Comp. Phys. 54: 174.
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AMERICAN AND SOYIET PERSPECTIVES
Ensman, LM., and S.E. Woosley. 1988. Ap. J. 333: 754.
Fryxell, B4, E. Muller, and W.D. Arnett. 1989. In: Woodward, P.R. (ed.~. Numerical
Methods in Astrophysics. Academic Press, New York.
Imay, LB. 1987. Astron. Astophys. 182: L31.
Mer~zies, J.W., e' al. 1988. M.N.RNS. 227: 39P.
Muller, E., W. Hillebrandt, M. Orio, P. Hoflich, R. Monchmeyer, B. F~xell. 1989. Astron.
and Astrophys (in press).
Nagasawa, M., 1: Nakamura, and S. Miyama. 1989. Publ. Astron. So~ Japan (in press).
Whitelock, P4, et al. 1988. M.NRNS. 234: SP.
Whitteborn, F.C et al. 1989. Ap. J. Lett. 338: L9.
Woosley, S.E. 1989. Ann. RRV. Astron. Astophys (in press, denoted ABKW).