524
Progress of Theoretical Physics, Vol. 71, No.3, March 1984
Effects of the Temperature and Mass of Iron Cores
on the Mass Ejection of Neutrino-Trapping Supernovae
Mariko T AKAHARA and Katsuhiko SATO*
Department of Earth Science and Astronomy
College of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153
*Department of Physics, University of Tokyo, Tokyo 113
(Received October 6, 1983)
Effects of the neutrino trapping and the temperature, T c , and/or mass of iron cores, M, on the mass
ejection of neutrino· trapping supernovae are investigated with the aid of an idealized equation of state on
the assumption of adiabatic collapse. It is found that 1) for a fixed Tc and/or M, 'the final stages of the
collapse of Fe·cores are classified into the following three cases according to the trapped lepton fraction,
YL ; in the increasing order of YL , i) no ejection, ii) mass ejection due to the thermal expansion and iii) mass
ejection by shock waves. 2) The critical values of YL distinguishing these cases increase with Tc and/or
M. Therefore, if we adopt the value of the trapped lepton fraction YL = 0.39, only cores less massive than.
1.4 M0 can make violent explosions. The physical interpretation of the above results and the detailed
dependence of mass ejection on Tc and/or M are discussed.
§ 1.
Introduction
Many calculations of the collapse of iron cores have ever been performed incorperating the detailed physical processes1)-lO) such as complicated equation of state (hereafter
we abbreviate it as EOS), processes of electron capture and photodisintegration of iron
nuclei and transfer of neutrinos. However, almost all calculations showed no explosions
and no neutron stars formed. Why iron cores do not explode? It is, however, difficult to
analyze the results due to the complicated computations and detailed physical processes.
Therefore recently there have been made many attempts of the interpretation of these
calculations and the analytical researches of the collapse. 11l- 15)
However, the conditions for explosion, namely physical conditions for the formation
and propagation of the strong shock waves, have not known precisely. Especially the
effects of neutrino trapping and the resultant suppression of electron capture reaction on
the dynamics of the supernovae and mass ejection have not been clarified yet in spite of
their importance recognized.
The collapse of the stellar core using the idealized EOS has been already simulated by
Van Riper 16 ) and Lichtenstadt et al.l7)on the assumption of adiabatic collapse. HO'Yever,
they have been interested only in the problem how dynamics changes with the adiabatic
index and the thermal stiffness; the effects of the neutrino trapping on dynamics of the core
have not been examined at all.
Therefore in a previous papee S) (hereafter we refer to it as Paper I) we investigated
the effects of neutrino trapping on supernova explosions by the use of an idealized EOSon
the assumption of adiabatic collapse. It was found that mass ejection becomes violent
only if the trapped lepton fraction YL (the number of leptons per baryon after neutrinos
are trapped) lies in an appropriate range and that mass ejection is difficult if E lies out
of this range. Furthermore this range becomes wider with the thermal stiffness of the
shocked matter.
Effects of the Temperature and Mass of Iron Cores
525
In Paper I we investigated only the collapse of the iron core of 1.4 Me supported by
the pressure of degenerate electrons. However, in the realistic case the core is of finite
temperature and electrons are not degenerate near the surface of the core; the surface
layers of the core are supported by the thermal pressure which increases the core mass
more than the Chandrasekhar mass. Then as the next step we investigate the effects of
the temperature, namely the effects of the core mass. Furthermore it has been recently
shown that the Fe-core of small mass does make an explosion7 )-10) indicating the importance of the effects of the core mass. Therefore it is the purpose of this paper to
investigate the effects of the temperature and/or core mass of the presupemova models on
the dynamics of the core in addition to the effects of neutrino trapping. We also investigate the effects of the electron fraction of the initial core, Y e ,;, because it changes the
Chandrasekhar mass.
We describe our idealized EOS and initial models in §2 and the numerical results in
§3. The summary of the results and discussion are given in §4.
§ 2.
Idealized equation of state and initial models
As discussed in Paper I it is adequate to use an idealized EOS rather than realistic
ones21 )-23) for our purpose. In the present work we adopt the same EOS as used in Paper
1. The total pressure P is assumed to be the sum of the cold pressure Pc which is
originated from the degenerate leptons and nuclear force, and the thermal pressure P T ,
(1)
The cold pressure Pc is determined only by the density and assumed to be described
by the following equation (see also Fig. 1 in Paper I),
Pc(p)=
K1p7"
K 2p 7.,
{ K 3P 73,
K4p 7.,
(P<P1
; region
0*)
(P1~P<P2; region II)
(P2 ~ P < P3; region III)
(P3~P
; region IV)
(2)
where K; and r;<i=l, 2,3,4) are constants and adiabatic indices, respectively. In region
I the Fe-core is supported by the pressure of degenerate electrons. In region II the
pressure deficit increases gradually as electron capture reaction proceeds. In regions III
and IV neutrinos are trapped and degenerate in the core. In region III the pressure of
degenerate leptons dominates that of nuclear force, while in region IV the latter dominates
the former. The degree of the progress of electron capture process in region II is
described by the amotint of pressure deficit in region III, hence the trapped lepton fraction,
YL.
The parameters adopted are 1'1=1'3=4/3, 1'4=2.5, P1=4.0X10 9 gcm- 3, P2=1.0X10 12
g~m-3 and P3=2.7x1014 gcm- 3. The value of the parameter 1'2 depends on the trapped
lepton fraction Y L which we treat as the parameter.
On the other hand, the thermal pressure P T is described as a function of density P and
the specific thermal energy eTas
*) Strictly speaking, in the case p4;.p, our EOS deviates from this representation because we used the EOS of
completely degenerate electrons of T=OK in region I (see the Appendix).
526
M. Takahara and K. Sato
(3)
10
where /'T is the parameter denoting the thermal stiffness. As· the estimation of Sack et
al. 24 ) gave 1.2 ~ 1.3 for plausible value of /'T,
we adopted the middle value 1.25 in this
investigation. The specific thermal energy.
fbO
lOT is described by the density and the
.2
temperature, lOT=lOT{P, T); lOT is increased
by the compression during the collapse and
also produced by the shock dissipation.
9
It is to be noticed that the models of /'T
=
1.25
were not investigated in Paper I and P3
6
7
.8
9
is 2.7 times as large as that in Paper 1. As
log p (g om- 3)
the effects of nuclear force on the EOS
Fig. 1. Structures of presupernova models. Those becomes significant above the nuclear dencalculated by Weaver et al. are illustrated by
sity, this choice of P3 is more reasonable than
the dotted (15 M 0 ) and dot-dashed curves (25
the previous value.
M 0 ) and those calculated by N omoto et al!O)
To construct the initial models of iron
are illustrated by the dashed (3 M0 He-star) and
solid curves (10 M0 He-star), respectively. The cores we must know the distribution of Y e ,;,
solid line represents the relation assumed in this P and temperature, T_ To know the distriinvestigation; Trxp4/30.
bution of T, we plotted in Fig. 1 the struc-·
tures of the presupemova models of Weaver et al. l9 ) (models of 15 Me and 25 Me) and
those of N omoto et al. 20 ) (He-stars of 3 Me and 10 Me). They are approximated as Tcx
a
p , where a is the constant of 4/30. Then by the procedure described in the Appendix we
construct many initial models with various temperature at the center.
The parameters contained in our models are the trapped lepton fraction, Y L , and the
temperature at the center of the initial model, Te. The former is described by the
pressure deficit parameter d at the neutrino trapping density, p = P2 = 10 l2 gcm-a, i.e., the
ratio of the cold pressure at the neutrino trapping density, P2=Pc(P2), to the cold pressure
necessary for the hydrostatic equilibrium, P O =Kl P2 71 ; d=P2/PO which is approximately
described at P = P2 by the following equation,
1
.)
d ~ ( Yd 0.4643 )4/3
,
(4)
where the numerical value in denominator represents the lepton fraction in the initial Fecore. If we specify the value of d, /'2 is determined by connecting the point (Pl = Pe(Pl),
Pl) and (P2, P2) in the log P-Iog P diagram.
The latter parameter T e determines the initial model; the structure and therefore the
core mass. The adopted values of Te and
Table I. Initial models of iron cores.
the relevant core masses are summarized
in Table I and corresponding density disTc(K)
Ye
M/M
Model
tributions are illustrated iIi Fig. 2. The
0.0
1.24
A
0.4643
models of Te=OK(Models A and E) are
6.0X 10·
1.39
B
0.4643
8.3X10·
1.57
0.4643
C
also calculated for comparison. . The ini1.2 X 1010
2.00
D
0.4643
tial central density is fixed at 4.0 X 109gcm- 3
1.05
0.0
0.4280
E
and Y e .; is assumed to be 0.4643 which
0
Effects of the Temperature and Mass of Iron Cores
527
10.0...---,,-----,---,
to r-----r---....,
9.0
P.,/P
/""-.
0.5
7
E
o
bO
'-' 8.0
t;
bO
.2
1.0
2.0
7.0
Fig. 2. The ratio of the thermal pressure P T to the
total pressure P versus mass coordinate Mr of
Model C of d=O.85. The curves denoted by ti,
t. and tf represent the distribution of the ratio at .
the initial time, at the bounce time and at the
final stage of the calculation, respectively. At
the initial time the thermal pressure contributes
only near the surface of the core. However,
after the bounce its contribution becomes
significant in the shocked matter. The peak of
the curve denoted by t. represents the beginning
of the shock dissipation.
6 ..0 ~-=-7.'"=S---::-'-::-...1.L..L...1.--="
log r
(em)
Fig. 3. Structures of the initial iron cores. The
distribution of density is shown by the solid
turves with the symbols corresponding to the
models summarized in Table 1. The dashed
curves connect the same mass coordinates of
these models. The higher central temperature
causes the larger radius and therefore result in
more massive core.
corresponds to the electron fraction of Fe-nuclei. The effects of temperature is significant
near the surface of the core because electrons are nondegenerate there. To show the
contribution of the thermal pressure we have plotted the distribution of the ratio of the
thermal pressure to the total pressure, PT/ P, in Fig. 2. As the thermal pressure becomes
larger near the surface, it can support much mass there as shown in Fig. 3. Higher T c
causes higher thermal pressure and therefore results in more massive core (see Table I).
To see the Ye,i-dependence of the dynamics we also calculated models of Y e ,i=0.4280
which corresponds to the electron fraction at the center of the presupernova models of 25
Me of Weaver et al.
The assumption of spherical symmetry and adiabaticity except the shock wave is also
adopted in this paper as well as the neglection of the effects of general relativity. The
numerical method and shell division adopted are the same as those in Paper I except the
number of shells which is 94 ~ 121 in this paper. The initiation of the collapse is also the
same as that in Paper I.
§ 3.
Numerical results
General features of the dynamics of the core
The general features of the dynamics of the core are basically the same as those
described in Paper I. In what follows the term core refers to the part which is the Fe-core
at the initial stage. The inner part of the core collapses homologously and then it makes
a bounce when the pressure gradient exceeds the gravity, which generates the strong
3. 1.
528
M. Takahara and K. Sato
reflective shock wave. In the following we call the part interior to the shock wave at the
bounce time the inner core and that exterior to the shock wave the outer core. We also
denote the mass of the core, of the inner core and of the outer core by M, Mcore and Mout
(-=M - Mcore), respectively.
Since the initial thermal pressure is effective only near the surface of the core (Fig. 2), .
it does not affect the collapse of the. inner core. As the collapse proceeds, the cold
pressure Pc grows faster than the thermal pressure PT, which makes the ratio PT/P small
in the inner core as shown in Fig. 2. However, P T contributes to delay the collapse of the
surface layers and keep them in the shallow gravitational potential well. After the
bounce P T becomes dominant in the shocked matter as shown in Fig. 2 because the kinetic
energy of the infalling matter is converted to the thermal energy by the shock dissipation.
In some models surface layers of. the core are blown off by strong shock waves and in
other models shock waves are too weak to eject them. IIi both cases the core left behind
settles down to the hydrostatic equilibrium finally.
3.2. Dependence of dynamics on the trapped lepton fraction (d- or YL-dependence)
In this subsection we discuss only the dependence of dynamics on the trapped lepton
fraction YL or d in the case of Model A. The effects of temperature and/or core mass
will be discussed in the next subsection.
The mass of the homologously collapsing inner part of the core, Mcore , has been
investigated by Goldreich and Weber 25 ) and also by Yahil. 14),15) In the case of the initial
density distribution of a polytrope of n=3, the former found that Mcore is given by the
equation
Mcore=1.0449(x/XO)3/2M,
(5)
where x is the constant of EOS assumed by them,
P=Xp1+1/n.
(6)
The symbol Xo in Eq. (5) denotes the value of x which gives the hydrostatic equilibrium
configuration. As the ratio of x/xocorresponds to the parameter d in our models, we plot
the ratio Mcore/ M versus d 3/2 in Fig. 4.
In Fig. 4 the calculated models are illustrated by various symbols and also the line
given by Eq. (5) for comparison. Since the initial density distribution is almost similar to
a polytrope of n=3 in Models A (and also in Models E), the amount of Mcore/M in these
models lies on this line as pointed out in Paper I. It is to be noticed that Mout (= M
- Mcore), the candidate for the ejecta of the shock wave, decreases to zero as d increases
to unity.
The most interesting quantities, the amount of the ejected mass, Mel, and the kinetic
energy carried out by them, Eel, are plotted versus d in Fig. 5. The dependences of Mel
and Eel on d are determined by the dependences of E core and Mcore/ M (namely Mout / M) on
that parameter, where Ecore is the total energy of the inner core at the bounce time, defined
by
Ecore=
1:
0
.'"(1/2
v2+s- GMr/r )dMr .
(7)
In Eq. (7) the symbols G and Mr denote the gravitational constant and the mass contained
Effects of the Temperature and Mass of Iron Cores
529
I.O.-----------~_r._._._~
~
,," ,
..
Model
A
®
~
""o~
&.
I
I
~ :" 'I
0
2
0.5
,
~
o
~
x
2
0
"
I
: ,\ '..
'.
l
d 3/
0.2
~
" /X"
,.
,
,:,\\ .!\\
, I '.v \...
",
, I I I 'XII "4
I I
E
I
I
.,'
I\'~~
"x" '),~
l4
I I ':
, I I I
(0.8r 2 (1.0)3/2
I
0.0 1--_~J::o.U~
2
Fig. 4. The mass fraction of the inner core versus
the pressure deficit paramet er d. Symbo ls·, 8,
&, G and x denote Models A, B, C, D and E,
respectively. The mass fraction s of each model
stand on almost parallel lines. The relation
found by Goldreich and Weber25 ) (Eq. (5) in the
text) is illustrat ed by the dashed line.
't!J
""Xf"\'
L~I ""
'-" \ ,
: : I I
(0.6)3/2
~
'x ~- '\ ..
Q)
Model
I!J
'
,
~.
0.4
B
C
\
/""'0.
b.O
!....
4:0
Q)
)l
" '.Iji. ••/-
Lt)
a
l . .,fl\F~.
......."
Q)
w
.~
I
2.0
,'1 , JIf' ,¢&
II "
1/
rI.... '.f
~,
t if /~
,i ¢I :~ "'t\
If/I
ff• ~'I II I
•,
within the radius r, and v and c denote the
I, , .
X
I , ., ,
velocity and specific interna l energy at the
I •
O• 0
I
mass coordi nate M r , respectively. As de0.6
0.8
1.0
scribed in Paper I, Ecore has the maxim um at
d
a certain value of d (see Fig. 3(d) in Paper I)
Fig. 5. The ejected mass, Mel, and energy, Eel. versus
and the imler core with larger Ecore bounces
the pressure deficit paramet er, d. Calculated
more strongly.
models are illustrat ed by the same symbols as
In models of small d, small Ecore results
those in Fig. 4. To show the dependence of Mel
and Eel on d clearly, the same symbols are con·
in the weak bounce which genera tes only
nected by the dashed lines.
weak shock waves. Moreover, the shock
wave must propag ate the larger outer core, i.e., larger Mout / M to reach
the surfac e of the
core, which weake ns the shock wave significantly. Both of small Ecore
and large Mout/M
result in no mass ejection; the surfac e veloci ty is too small to make
the core expand when
the shock wave reache s there.
In models of d near the peak of Mej, Ecore is larger, while Mout/ M is
smalle r than those
of the previous models. Theref ore the strong er shock wave arrives
at the surfac e and the
core begins to expand. However, as the shock wave is not strong
enough, the expans ion
velocity is, at the begining, less than the escape velocity. As the surfac
e layers expand,
there occurs the competition betwee n the decele ration by gravity
and decrea se of the
escape velocity by increa sed radius. Finally the latter overcomes
the former, so the
surfac e layers are strippe d layer by layer to result in large Mel
and small E ej. The
expans ion of the outer core is caused by the heatin g due to the dissipa
tion of large E core.
We call this phenomenon the mass ejection due to the therma l expans
ion (see Figs. 6(a)
and 7(a)).
In models of large d, E core is small again, which means the weak bounce
. However,
'---~L.:x~·-A.---I!1---'-_..J
530
M. Takahara and K. Sato
5.0
5.0
(a)
-g
0'\
0
......
S
0.0
0
r-l
:::>
0.0
0'\
0
r-l
-5.0
-5.0
:::>
0.5
Mr
0.5
1.0
1M 0
Mr
-5.0
1.0
1M0
-5.0
0.5
1.0
M
1M0
r
1.5
0.5
1.0
Mr
1.5
2.0
1M0
-Fig. 6. Snap-shots of velocity profiles for several stages. (a), (b), (c) and (d) show ModeIA(d=0.75), B(d
=0.85), C(d=0.85) and D(d=0.85), respectively. (a) and (b) show how shock propagation changes with
the parameter d and (b), (c) and (d) show how it changes with the core mass. (a) shows the case of the
mass ejection due to the thermal expansion and (b) and (c) show the mass ejection by shock waves. (d)
shows the case of no ejection. The numbers in each figure denote the evolutionary sequences.
Mout/ M is extremely small in these models, which causes little weakening of the shock
wave. As the latter effects overcomes the former, the shock wave remains strong during
the propagation to the surface. Therefore the expansion velocity exceeds the escape
velocity before the shock wave reaches the surface of the core, which results in a violent
explosion; the mass ejection by shock waves (see Figs. 6(b) and 7(b».
To summarize, results are classified into three types; i) no ejection, ii) mass ejection
due to the thermal expansion and iii) mass ejection by shock waves. These phenomena
are remarkable in the case 'YT=1.25 (independently of the initial core mass). However,
type ii) disappears as 'YT increases. 26 ) Of course, the boundary values of d among these
three types depend on the initial central temperature and/or core mass;- the boundary
values between cases i) and ii) are obvious in Fig. 5 and those between cases ii) and iii) are
obtained as the value corresponding to EeJ ~ 1.3 X 10 51 ergs. It is to be noticed that only
models of large Eel are responsible to the supernova phenomena.
531
Effects of the Temperature and Mass of Iron Cores
1.24
(b)
8
8
1.24
1.22
S
t.l
p::;
b1
0
S
1.10
7
6
1,150
(ms)
T
p::;
0.44
0
b1
0.89
0.44
6
r-l
5
1,138
501
1,11,1,8
T
(ms)
(d)
1.35
8
......
......
1.97
8
t.l
1.22
7
1.82
7
1.16
p::;
tTl
2.00
S
S
0
1,157
9
1.57
t.l
7
1.08
0.96
0.89
r-l
5
1,100
t.l
0.58
6
r-l
1191,1
T
(ms)
p::;
1. 65
b1
1.33
0.76
0
r-I
6
1259
3505
T
7010
(ms)
Fig. 7. Radial behaviors of the selected mass shells with time. Radius in cm and time in ms. Time is
counted from the beginning of the calculation. (a)~ (d) show the same models as those shown in Fig. 6,
respectively. Numerical values on the right·hand side of each figure are the mass (in solar unit)
contained within the radius shown. The mass shells above the arrows are ejected away finally.
Effects of temperature and/or core mass
As described in the previous subsection, the amount of Mcore/M of Model A lies on the
line given by Eq. (5) (Fig. 4). However, Mcore/M in Models B, C and D are shifted from
this line and form lines almost parallel to it, respectively. As Tc and/or M increases, the
smaller Mcore/M results because the total mass contains not only the mass of the inner part
where P T is ineffective initially (therefore the density distribution is almost similar to a
polytrope of n = 3) but also that of the outer part supported by PT. It is to be noticed that
Mcore/M remains less than unity as d increases to unity. For example, in Model D (i.e.,
the model of the highest T c ), Mcore/M approaches 0.75 as d approaches unity. This is a
remarkable contrast to the Model A and an important point to consider the mass ejection.
3. 3.
532
M. Takahara and K. Sato
The mass ejection is determined by E core and Mout / M as described before. Generally,
for a given d, as Te and/or M increases, Ecore and MoudM both increase as shown in
Fig. 4. The former results in strong bounce, but the latter weakens the shock wave
significantly.
In models of small and moderate value of d the latter overcomes the former to result
in no mass ejection (see Figs. 6(d) and 7(d)). Therefore the critical values of d
distinguishing cases i) and ii), and cases ii) and iii) increase with Te and/or M. This
implies that massive iron cores are difficult to explode.
In models of large d the situation is different. In Model A the increase of d to unity
results in the decrease of MoudM and E core to zero, so there is no candidate mass and
energy to be ejected. However, in the models of large M, Mout/M remains finite even if
d increases to unity. Therefore in these models Ecore and also the mass to be ejected
never vanishes at d ~ 1. In addition to this, the more massive core has much larger radius
at the initial time, hence smaller escape velocity. Accordingly more violent explosion
occurs in models of finite Te when the strong shock wave reaches the surface of the core;
large Mei and Eei result (see Figs. 6(c) and 7(c)). Though we considered until now that
the core means only the initial Fe-core, we find it better to consider that the core partly
contains the layers exterior to Si-burning shell in models of higher Te. Then it is natural
that the large amount of the mass is ejected in these models.
3. 4. Dependence of dynamics on the initial electron fraction (Ye,i-dependence)
The Ye,i-dependence of the hydrodynamics is investigated by changing the value of
Ye,i of the initial cores to Ye,i=0.4280 in the case of Te=OK (Model E). The features of
the dynamics of these models are almost the same as those of Model A (Ye,i=0.4643)
qualitatively. As for the quantitative discussion, Mei and Eei are both scaled roughly by
the Chandrasekhar mass corresponding to each Ye,i.
§ 4.
Summary and discussion
The results obtained in this investigation are summarized as follows. 1) For a fixed
temperature, Te, and/or mass of Fe-cores, M, the final stages of the collapse are classified
into the following three cases according to the trapped lepton fraction, Y L ; in the increasing order of E, i) no ejection, ii) mass ejection due to the thermal expansion and iii) mass
ejection by shock waves. 2) The critical values of Y L distinguishing these cases increase
as Te and/or M increase. 3) Dynamics of cores of different Ye,i is qualitatively similar,
while the ejected mass and energy are both scaled roughly by the Chandrasekhar mass
corresponding to each Ye,i.
Yahil 14),15) obtained similarity solutions of the collapsing cores in use of the polytropic
EOS, P=Kp1, where K is a constant and 'Y is the adiabatic index assumed to be constant
in space and time. Though close comparison of our results to his solution is difficult
because of different EOS, they are in good agreement in the collapsing phase.
He also insists that the critical point to the supernova explosions is the ram pressure
which weakens the shock wave. However, as discussed in this paper, the critical points
are the trapped lepton fraction, Y L , strength of the bounce, i.e., E core and the mass fraction
of the outer core, Mout/ M; iarge Y L , large E core and small Moud M are favorable for violent
explosions.
In the case of small Mout/ M, shock waves are hardly weakened by the spherical
Effects of the Temperature and Mass of Iron Cores
533
damping, which results in violent explosions. This condition, however, can be interpreted
alternatively as follows. If the mass of the homologously collapsing inner core is the
same, the condition of small Moutl M corresponds to the condition of small Mout, hence steep
density gradient in the outer core at the bounce time. In fact, many simulations including
the collapse of Fe-core of small mass have shown almost the same amount of
Mcore. l)-3),5),7),8) Therefore the moderate explosion shown by Hillebrande) can be interpreted also in terms of small Moutl M.
Van Riper 16 ) and Lichtenstadt et al. 17) also carried out many simulations of stellar core
collapses in use of the idealizedEOS (different from ours) on the assumption of adiabatic
collapse. However, they did not take into account the neutrino trapping, which
significantly restricts applicability of their results because the amount of Y L changes the
dynamics of the core greatly as discussed in this paper. Therefore simulations performed
by them are not sufficient to obtain the full understanding of the dynamics of supernovae.
Let us now apply these results to the realistic supernovae. Many of more realistic
simulations show that the plausible value of YL is about 0.34~0.39 at the center of the
core,1l- 3),5),7),8)which corresponds to the pressure deficit parameterd=0.66~0.79 as discussed in Paper I. If we adopt these values,· it is concluded that the Fe-cores more massive
than 1.4 Me cannot explode at all as shown in Fig. 5; only cores of smaller mass can make
violent explosions. This conclusion is in agreement with recent calculations of the
collapse of 10 Me star carried out by Hillebrande) and Wilson lO ) in which Illodenite
explosions occurred.
However, the reliable value of Y L has not been confirmed yet because of the uncertainty of the rate of neutronization of dense matter 27 ),28) as discussed in Paper I. Futhermore, it may be changed much in future by the development of neutrino physics; indeed the
discovery of the neutral currents have already changed the physics of supernovae greatly.29),30) The grand unified theory and the theory of majorons also have the possibility to
change the. neutrino physics. 3l )-33) Therefore it is important to investigate the dynamics
of the stellar core treating YL (i.e., d) as a parameter.
In the case of type II supernova the shock wave is weakened not only by the spherical
damping but also by the photodisintegration of Fe-nuclei. In our models this is takenjnto
account by the parameter YT. The energy loss by neutrinos also weakens the shock wave.
As for the discussion about these factors see Paper I.
Burrows and Lattimer34) obtained the condition for explosion by the comparison of
the shock energy and the energy necessary to dissociate nuclei in the outer core; the
smaller core mass and larger Y L are necessary for explosion. These conditions are
qualitatively in agreement with ours. However, the critical value of Y L for explosion
calculated by them may be changed much by the precise calculation because they calculated that condition keeping the inner core mass fixed. However, it varies with the trapped
lepton fraction itself and also with the mass of the initial Fe-core as is shown in this paper.
Acknowledgements
The authors would like to thank Professor C. Hayashi and Professor D. Sugimoto and
also Dr. K. Nomoto for continual encouragement. This work was supported in parts by
the Grant-in-Aid for Scientific Research (57540197) and Grant-in-Aid for Cooperative
Research (58340023) from the Ministry ·of Education, Science and Culture. Numerical
calculations were carried out in parts with HITAC M-280H of the Computer Center of the
534
M. Takahara and· K. Sato
University of Tokyo and FACOM M-200 of the Nobeyama Radio Observatory (NRO) and
of the Institute of Space and Aeronautical Science (ISAS).
Appendix
- - Construction of the Initial Model-The initial model in hydrostatic equilibrium is constructed as follows. Forgiven T c,
the temperature is calculated from the density at each mass shell by the assumed relation,
Trxp4/30. Then the pressure of degenerate electrons is calculated precisely using the
Fermi-Dirac function. We identify this pressure to the total pressure, P, and calculate the
initial configuration solving the equation of hydrostatic equilibrium.
After the construction of the initial model, the initial value of the cold pressure, Pc,
is calculated by the EOS of completely degenerate electrons of T=OK. 35 ) Then the initial
values of the thermal pressure, PT, and the specific thermal energy, CT, are calculated by
the equation PT=P-PC and Eq. (3), respectively.
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